Maths Grade 7 PDF
Maths Grade 7 PDF
Maths Grade 7 PDF
Grade 7 Mathematics
1 WHOLE NUMBERS
Revision
Exercise 1
1. Calculate the least common multiple (L.C.M) of:
a) 12 b) 18 c) 27 d) 42
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Mathematics Grade 7
9. Copy and complete:
Number Number in words
Seven million two hundred thousand five hundred and forty five
8 000 006
Ninety nine million eight thousand two hundred and sixty five
59 403 671
10.Copy and complete the table. the first line has been done for you:
21 792 418
12 951 900
2 373 286
99 999 999
a) 27 834 913 27 834 914 b) 178 676 114 178 676 113
a) 17 b) 21 c) 12 d) 15
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Grade 7 Mathematics
M illions
The chart shows the place value of numbers
Millions Thousands Ones
1 000 000 1 0 0 0 0 0 0
10 000 000 1 0 0 0 0 0 0 0
9 046 125 9 0 4 6 1 2 5
12 709 451 1 2 7 0 9 4 5 1
Examples
(i) 572 816 134 = (5 x 100 000 000) + (7 x 10 000 000) + (2 x 1 000 000)
+ (8 x 100 000) + (1 x 10 000) + (6 x 1 000) + (1 x 100)
+ (3 x 10) + (4 x1 )
(ii)950 832 704 = (9 x 100 000 000) + (5 x 10 000 000) + (0 x 1 000 000)
+ (8 x 100 000) + (3 x 10 000) + (2 x 1000) + (7 x 100)
+ (0 x 10) + (4 x1)
In these example there is 0 in the millions column and the tens column.
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Mathematics Grade 7
Exercise 2
1. Write in digit form:
c) Fifteen million seven hundred and five thousand and twenty five
d) Eight hundred and forty million, two hundred and four thousand, nine hundred and thirty
(a)30 787 003 b) 85 373 000 c) 101 101 101 d) 3 912 853
4. Give the total value of the underlined digit in each of the following.
a) 375 500 021 b) 21 808 309 c) 255 260 800 d) 123 456 789
5. Arrange each of the following groups of numbers from the largest to smallest.
a) 47 286 798 523 051 622 85 116 041 126 990 258
b) 4 315 623 18 351 462 9 760 264 11 879 649
c) 94 678 850 901 413 226 99 854 277 94 786 985
d) 702 410 526 193 738 565 320 100 201 98 675 092
e) 7 240 195 21 062 514 962 570 150 628 070
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Grade 7 Mathematics
Squa re s
The square of a number is the product of multiplying a number by itself.
The square of a number n is n x n or n 2.
Examples
(i) Calculate the square of 5
We can obtain the square of a number by multiplying the number by itself.
5 2 = 5 x 5 = 25 therefore the square of 5 = 5 2 = 25
(ii) Calculate the square of 7
7 2 = 7 x 7 = 49
(iii) Calculate the square of 223 223
x223
669
4460
44600
therefore 223 2 = 49729 49729
Exercise 3
1. Calculate the squares:
a) 21 b) 35 c) 75 d) 93
i) 1000 j) 110 k) 42 l) 50
3. Calculate
a) 9 2 b) 15 2 c) 1022 d) 45 2
e) 220 2 f) 10 2 g) 12 h) 13 2
i) 211 2 j) 85 2
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Mathematics Grade 7
Squa re Root s
Some numbers can be arranged as a square array of dots. They are called square
numbers or perfect squares.
Examples
(i)
9 16 25 36
(ii) The length of one side of the square of 25 dots is 5. 5 is the square root of 25.
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Grade 7 Mathematics
Exercise 4
1) Calculate the square root:
a) 121 b) 361 c) 400 d) 196 e) 169
2. Calculate the value:
a) 36 b) 64 c) 81 d) 49 e) 361
3. The area of a square plot of land is 529 m2. Calculate the length of one side.
4. The area of a garden is 800 m2. 120 m of wire is provided to make a fence.
Will the wire be long enough?
Cube s
2 x 2 x 2 = 8 and 2 x 2 x 2 = 23 23 is read as two cubed
The cube of any number is product of that number multiplied by itself three times.
Examples
(i) Calculate the cubes:
a) 5 b) 1 c) 0 d) 7
a) 53 = 5 x 5 x 5 b) 13 = 1 x 1 x 1
= 25 x 5 =1
= 125
3
(c) 0 = 0 x 0 x 0 d) 73 = 7 x 7 x 7
=0 = 49 x 7
= 343
(ii) Find the difference between 93 and 83
93 = 9 x 9 x 9 = 729
83 = 8 x 8 x 8 = 512
Therefore 93 - 83 = 729 - 512 = 217
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Mathematics Grade 7
The diagram shows a cube of side 3 cm
V = LxLxL
= L3
= 33 = 3 x 3 x 3
3 cm = 27 cm3
L = 3 cm
3 cm
Note: 27 cubes measuring l cm x l cm x l cm will fit into 27 cm3
(iii) The volume of the cube shown is 64 cm3 3 64
V = LxLxL
= L3
so L = 3 64 = 4 cm 4 cm
4 cm
4 cm
(iv) Calculate the cube root: 5 125
a) 125 b) 216 c) 5 25
a) 125 = 5 x 5 x 5 5 5
3 125 = 5 1
b) 216 = 2 x 2 x 2 x 3 x 3 x 3 = 23 x 33 2 216
3 216 = 2 x 3 2 108
=6 2 54
3 27
3 9
3 3
1
Exercise 5
1. Calculate the cubes:
a) 5 b) 9 c) 10 d) 6 e) 8
6. A cubic water tank has the volume of 3375 cm3. What are the dimensions of the tank?
7. The volume of a cubic room is 2744 m3. What are the measurements of the room?
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Grade 7 Mathematics
N um be r Ba se s
Base 10
36 can be grouped in tens and ones.
Base 5
Other bases can be used to count. When we count and work in another base, we indicate the
base by writing it as a sub-script.
For example: 34 5. This numeral is read as three - four base five. Working in base five means
counting in five - fives, fives and ones.
Examples
(i) These dots are grouped in fives.
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Mathematics Grade 7
Base 10 1 2 3 4 5 6 7 8 9 10
Base 5 10
Base 10 11 12 13 14 15 16 17 18 19 20
Base 5 34
In base 10 we use these digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
One group of ten is made up of two digits 1 and 0
Base 10 1 2 3 4 5 6 7 8 9 10
Base 5 10 20
Base 10 11 12 13 14 15 16 17 18 19 20
Base 5 30
Base 10 21 22 23 24 25 26 27 28 29 30
Base 5 100
Note that 24 10 = 445
2510 is five fives, 0 fives and 0 >
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Grade 7 Mathematics
(iv) Convert 49 to base 5
9
Here there are 9 groups of fives and 4 ones.
5 49
45 How many groups of five fives and how many
4 Ones groups of fives are in the 9 groups of fives.
1 five-fives
There is 1 group of five fives and 4 groups of fives. 5 9
5
4 fives
Therefore 49 = 1 group of five fives, 4 groups of fives and 4 ones.
Therefore 4910 = 1445.
Exercise 6
1. Convert to base 5;
a) 15 b) 10 c) 18 d) 20
e) 36 f) 47 g) 30 h) 45
Base 10 1 2 3 4 5 6 7 8 9 10
Base 5
Base 10 11 12 13 14 15 16 17 18 19 20
Base 5
Base 10 21 22 23 24 25 26 27 28 29 30
Base 5
Base 10 31 32 33 34 35 36 37 38 39 40
Base 5
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Mathematics Grade 7
2 SETS
Wha t is a se t ?
When we describe groups of objects, people, or animals, we use words such as bundle, swarm,
flock, bunch, team, herd, heap, brigade etc.
All these words describe a collection of things. In mathematics we use the word set to
represent a collection of objects.
A set is a collection of things. The things in a set can be called members or elements.
This picture shows members of Jama's family. Together they form a set. Each member of this
set is called an element. The elements of this set are Jama, Seinab, Adam and Faduma.
Students in your class are a set. Each student in your class is an element or member of the set.
The days of the week are a set and its elements or members are Saturday, Sunday,
Monday,Tuesday, Wednesday, Thursday and Friday.
The elements in the word school are s, c, h, o, o, l.
The individual objects that collectively make up a given set are called its elements or
members.
The symbol for element is (∈). (∉) means not an element.
Examples
(i) Friday ∈ of the set of the days of the week.
(ii) Sunday ∉ of the set of the months of the year.
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Grade 7 Mathematics
Exercise 1
1. Write the elements of each of these sets.
a) Set of whole numbers less than ten.
b) Set of odd numbers between 2 and 10.
c) Set of the letters in the word Hussein.
d) Set of the months of the year.
e) Set of the countries in East Africa.
f) Set of the days of the week.
g) Set of the seasons of the year.
h) Set of the countries of the Arab League.
i) Set of the Prayers of the day.
j) Set of countries in the Horn of Africa.
Se t not a t ion
There are two ways to specify a set.
A. The roster method
All elements of the set are listed and separated by commas.
The elements are enclosed in braces or brackets, { }.
Note: • The order of the listed elements is not important.
• An element of a set cannot be repeated when listing the elements of the set.
• We use capital letters to name sets.
Examples
(i) Write the following sets using the roster method and name each set.
a) Rivers in Somalia b) Vowels in the Somali alphabet
A = {Shabbelle, Juba}
B = {a, e, i, o, u, aa, ee, ii, oo, uu}
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Mathematics Grade 7
b) R = {1, 3, 5, 7, 9}
R = {odd numbers less than 10}
c) M = {a, e, i, o, u, aa, ee, ii, oo, uu}
M = {vowels in the Somali alphabet}
Exercise 2
1. Write the following sets using the roster method and name each set.
a) The set of letters in the word “science”.
b) The set of prime numbers less than 19.
c) The set of multiples of 2 less than 18.
d) The set of East African countries.
e) The set of digits in 3003.
f) The set of the digits in 671239.
g) The set of the names for your fingers.
h) The set of districts in Bay region.
i) The set of whole numbers between 5 and 12.
j) The set of letters in the word “Somalia”.
T he e m pt y se t (t he null se t )
A physical education teacher came to a Grade 7 classroom and took all the pupils to the
school playground. After a while he asked a pupil to check if there were any left in the
classroom. The pupil came back and told the teacher that there was no one in the classroom.
The set of pupils in the classroom at that moment had no elements. Also the set of pupils
whose height are more than 3 metres, has no elements or members.
These two sets are an empty set.
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Grade 7 Mathematics
Exercise 3
1. Which of the following sets are empty sets? Write the answer in the form of
X = { } or Y ≠ { }.
a) A = {triangles with four sides}
b) R = {odd numbers less than 1}
c) C = {pupils with 6 legs}
d) M = {even numbers less than 10}
e) F = {whole numbers less than 0}
f) H = {mango trees on the moon}
g) N = {pupils whose height is 0.2 cm}
h) G = {0}
i) Q = {1, 2, 0}
j) D = { whole numbers between 7 and 8}
k) E = { , , , }
l) K = {hills in Somalia}
Finit e a nd I nfinit e se t s
Set E = {Prime numbers less than 10} can be listed.
Set E = {2,3,5,7}
The number of members or elements in E is limited.
A finite set has a limited number of members
Can you write all the elements of the set of whole numbers by listing all the elements?
It is not possible to list all the elements of this set. To represent such sets we use dots… to
show that the elements of the set are continuing. This type of set is known as infinite set.
Examples
(i) Write the set of natural numbers by listing its elements.
N = {1, 2, 3, 4, 5, ……}
(ii)Write the set of multiples of 5 using the roster method.
F = {0, 5, 10, 15, 20, …….}
This way of writing infinite sets can also be used for finite sets with large numbers of elements.
After writing a few elements of the set and a few dots, we write the last elements of the set.
Examples
(i) Write the set of whole numbers less than 1000.
W = {0, 1, 2, 3, 4, ……..999}
(ii)List the set of even numbers less than 100.
E = {2, 4, 6, 8, ….., 98}
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Mathematics Grade 7
Exercise 4
1. Identify the finite and infinite sets.
a) B = {1, 2, 3, 4, 5}
b) X = set of letters in the word Somalia
c) W = set of prime factors of the number 210.
d) M = set of the pupils in your school.
e) T = set of the population in the Horn of Africa.
f) N = set of cars in Arab countries.
g) H = set of prime numbers between one and one million.
h) Y = set of banana trees in Somalia.
i) A = set of whole numbers less than 1000.
j) C = set of multiples of 5.
k) E = set of whole numbers.
l) F = set of fractions.
2. Which of the following sets are finite and which are infinite?
a) The set of capital cities in East Africa.
b) The set of people in the Horn of Africa.
c) The set of all Muslims in the world.
d) The set of multiplies of 2.
e) The set of even numbers between 2 and 300.
f) The set of people in Somalia
g) The set of all integers.
h) The set of pupils in your school.
i) The set of 1 , 1 , 1 , ........ .
2 3 4
j) The set of Somali alphabet symbols.
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Grade 7 Mathematics
Equa l se t s
Set A is the set of letters in the name "Ahmed" and set B is the set of letters in the name
"Hamed".
We can write the elements of A and B using the roster method as follows:
A = {a, h, m, e, d} B = {h, a, m, e, d}
What do you notice about the members of these sets.
A = {a, h, m, e, d} B = {h, a, m, e, d}
Writing the numbers of elements in a set A and set B
n(A) = 5 and n(B) = 5
The number of elements in A and B are equal and the elements are identical.
Every element in A is also an element in B and vice-versa.
Examples
(i) If X = set of digits in the number 75 227 and Y = set of digits in the number 572 are
X and Y equal?
X = {7, 5, 2}
Y = {5, 7, 2} Note: We cannot repeat numbers in a set
n(X) = n(Y) = 3
Every element of X is also in Y and vice versa then X = Y.
(ii) If M = {1, 2, 3} and W = {3, 2}
Are M and W equal?
n (M) = 3 n (W) = 2
1 ∈ M but 1 ∉ W
Therefore M and W have different elements. Thus M ≠ W
Exercise 5
1. If N = {1, 3, 2, 5, 4} and H = {1, 2, 3, 4, 5}
Are N and H equal? Give your reasons.
2. If L = set of digits in the number 375 822 and
F = set of digits in the number 753 832
a) Write L and F using the roster method
b) Are L and F equal? Give your reasons.
3. Copy and complete the statement by writing = or ≠
i) {2, 3} {3, 2}
iii){a, b, c} {a, b, c, d}
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Mathematics Grade 7
4. Copy and complete the missing elements to write a true statement.
a) {3, 6, 9, 12} = {6, 9, ….., 12}
b) {7, 11, 5, 9} = { 11, ….., 7, 9}
c) {4, …, 1} = {4, 1, a}
5. If {1, 2, m, 5} = {2, 4, 1, 5} What is the value on m?
Equiva le nt se t s
If A = {a, b, c} then the number of elements in A ie. n(A) = 3 and if B = {1, 2, 3} then the
number of elements in B is 3 because n(B) = 3.
Therefore n(A) = n(B) = 3
Also if M = {1, 3, 5, 7} then the number of elements in set M is written n(M) = 4 and if L =
{x, y, w, } then the number of elements in set L is written n(L) = 3
Hence n(M) ≠ n (L)
Therefore we say:
Set Ais equivalent to set B and is written as A B and M is not equivalent to Land is written as M L.
If two sets have equal numbers of elements we say that the two sets are equivalent.
We use the symbol for ‘equivalent’and the symbol for ‘not equivalent’
Examples
(i) If X = {2, 3, 5, 8} and Y is a set of digits in the number 835. Are X and Y equivalent?
X = {2, 3, 5, 8} and Y = {8, 3, 5}
n(X) = 4 n(Y) = 3
Therefore n(X) ≠ n (Y) Therefore X ≠ Y
(ii)Let W = {1, 3, 5, 7, 9, 8) and H be the set of digits in the number 38 1957.
Are W and H equivalent?
n(W) = 6 n(H) = 6
Set W and set H have the same number of elements.
Therefore W = H and W H
Note: Equal sets are equivalent. Equivalent set are not necessarily equal.
Exercise 6
1. a) Write the members of these sets:
(i) All the factors of 4 (set F)
(ii) Prime numbers equal to or less than 7 (set P)
(iii) All the days of the week (set W)
(iv) Even counting numbers less than 10 (set C)
(v) The sides of ABC (set T)
b) Which sets are equivalent?
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Grade 7 Mathematics
2. a) Write the members of these sets
(i) Set A = {the factors of 10 }
(ii) Set B = {the factors of 8}
(iii) Set C = {the first four prime numbers}
(iv) Set D = {the digits in 8214 }
b) Which sets are equivalent?
c) Which sets are equal?
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Mathematics Grade 7
3 INTEGERS
Revision
Exercise 1
1. Read these thermometers and write each temperature in two ways.
a) 8
b) 8
c) 8
d) 8
7 7 7 7
6 6 6 6
5 5 5 5
4 4 4 4
3 3 3 3
2 2 2 2
1 1 1 1
0 0 0 0
-1 -1 -1 -1
-2 -2 -2 -2
-3 -3 -3 -3
-4 -4 -4 -4
-5 -5 -5 -5
-6 -6 -6 -6
-7 -7 -7 -7
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Grade 7 Mathematics
Weather forecasts tell us about the likely atmospheric conditions. One measure is the
temperature. For example, the temperature of some places can be 5° below zero,
10° below zero or 14° below zero.
How can we express the following information in mathematical expressions?
5°above zero can be expressed as +5 and read as positive 5.
10° above zero can be expressed as +10 and read as positive 10.
5° below zero can be expressed as -5 and read as negative 5.
10°below zero can be expressed as -10 and read as negative 10.
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7 This thermometer has a vertical scale.
Positive integers are above 0o
Negative integers are below 0o
T he num be rline
Placing the thermometer horizontally, you can see that all the positive integers
are on the right of ‘0’ and all negative integers are on the left of ‘0’.
-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
Positive integers can be represented by points A, B, C, D….They are on the right side of
0 - 0A, AB, BC, CD are equal segments.
Negative integers can be represented by points K, L, M, N…. They are on the left side of 0.
KL, LM, MN are equal segments.
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Mathematics Grade 7
P N M L K 0 A B C D E
-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
Positive integers Negative integers
Note: You can write positive numbers with or without a + sign.
Examples
(i) Write the integers that represent the position of. A, B, C, D, M, K.
C B A D M K
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
A represents -1 D represents +2
B represents -5 M represents +5
C represents -7 K represents +7
(ii) Draw a number line and plot the following integers using the letters.
-5 ; A 0;B -1 ; K
+4 ; X -2 ; H +1 ; L
+8 ; Y +3 ; C -6 ; P
P A H K B L C X Y
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
(iii) Graph these integers on a number line 6, -8, +7, -1, 0, +3.
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
x x x x x x
Exercise 2
1. Write the integer that represents the position of each point.
i) M ii) N iii) P iv) Q
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
Q N O M P
2. Draw number lines to represent the following integers and number them:
a) +6, -7, -3, +2, 0, -1, +4 b) 6, -9, 7, -1, 0, 3,
c) –5, +1, 0, -8, -2, +3, +9, -7
3. Write down the set of integers between
a) +9, +11 b) -7, -9 c) -2, 0
d) +1, -1 e) +2, +4 f) -10, -8
g) -4, + 6 h) +2, +7 i) -7, -3
4. Write the integer that is represented by x in each case
x 0 1 2 -6 -5 -4 x
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Grade 7 Mathematics
x 0 10 20 x 5 10 15
Opposit e s
Every positive integer has an opposite negative integer. Two numbers that are the same
distance from 0 on the number line, but in different directions are opposites. Such
numbers are referred to as additive inverses.
Examples
3 units 3 units
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
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Mathematics Grade 7
Exercise 3
1. Write the opposite negative integer:
a) +3 b) +19 c) +4 d) +10
2. Write the opposite positive integer:
a) -4 b) -17 c) -12 d) -2
3. Write the opposite integer:
a) -7 b) +8 c) +50 d) -13 e) +17
f) -9 g) -18 h) +31
Examples
(i) Compare +7 and -3
+7 lies to the right of -3 on the number line. ∴ +7 > -3, -3 < +7
(iii) Write the following integers in descending order. -18, -14, +15, +14, -8
In descending order: +15, +14, -8, -14, -18.
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Grade 7 Mathematics
Exercise 4
1. Write these integers in ascending order:
a) +1, +4, -9, +7, -8, +9, -1
b) -14, -19, +15, +27, -30, +36, -15
c) +38, -43, +29, -54, +60, +12, -45
2. Write these integers in descending order:
a) -4, +3, -8, +6, -11, +12, -9
b) +34, -36, +71, -48, -50, +12, +20
c) -1, +3, -2, -9, +4, -7, +10
3. Draw a number line from -8 to +8.
a) Mark -7 with A and +7 with B.
b) How many units are there from A to B?
c) Mark -4 with C and +6 with D.
d) How many units are there from C to D?
Adding int e ge rs
Integers can be added on a number line.
The addition of integers can be viewed as a series of moves along a number line. A positive
integer is represented by a move to the right. A negative integer is represented by a move to
the left. Arrows are used to show these moves.
Examples
2 units 5 units
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
So + 2 + (+ 5) = 7
(ii)Add -2 and -5
Start at 0 and move 2 positions to the left, then move 5 positions further.
5 units 2 units
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
So, -2 + (-5) = -7
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Mathematics Grade 7
The addition of two integers with the same sign is their sum with the sign.
e.g. -8 + (-2) = -10 (+8) + (+2) = +10
(ii) Add 2 and –5
Start at 0, move 2 positions to the right, then move 5 units to the left.
5 units
2 units
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
So, 2 + (-5) = -3
(iii) Add -2 and 5
Start at 0 move 2 units to the left , then move 5 positions to the right
5 units
2 units
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
So, -2 + 5 = 3
The addition of two numbers with different signs is their difference with the sign of the
greater number e.g.
12 + ( -8) = +4 -12 + 8 = -4
16 + (-11) = +5 -16 + 11 = -5
Exercise 5
1. Write the addition statements represented by the moves on the number lines.
a)
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
b)
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
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Grade 7 Mathematics
c)
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
d)
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8
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Mathematics Grade 7
Subt ra c t ing I nt e ge rs
Examples
Evaluate:
(i) +5 – (+3) (ii) +5 – (+3) (iii) +5 – (+3)
+5
-3
(i) number line
0 1 2 3 4 5 6 7
So +5 – (+3) = 5 – 3 = 2
Also (+5) – (+3) = (+5) + (-3) = 2
This shows that subtracting a positive number is the same as adding a negative
number. -(+4)
-2
So -2 – (+4) = -6
Also -2 – (+4) = (-2) + (-4)
= -(2 + 4)
= -6
Suppose 5 – (-2) = a
then a + (-2) = 5
Adding a negative number is the same as subtracting a positive number.
So a = 7
The number of steps between 5 and -2 is 7.
5- (-2) = 7 steps
-3 -2 -1 0 1 2 3 4 5
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Grade 7 Mathematics
Subtracting a negative number is the same as adding a positive number.
Generally:
a) a – (+b) = a – b
b) a – (-b) = a + b
When a is negative, (-5) for example:
-5 – (-7) = -5 + 7 = 2
then –a – (-b) = -a + b = b - a
In summary:
a) a – (+b) = a – b
b) a – (-b) = a + b
c) -a – (-b) = b – a
Exercise 6
1. Evaluate:
a) 5 – 8 b) 9 – 5 c) -2 – 6
d) -3 – 9 e) 9 – (-4) f) -2 – (-3)
g) -5 – (-6) h) 0 – 4 i) 0 – (-9)
M ult ipying I nt e ge rs
Multiplication is a way of interpreting continued addition.
In continued addition:
2 + 2 + 2 + 2 + 2 + 2 = 12 As multiplication: 6 x 2 = 12
(-3) + (-3) + (-3) + (-3) = 12 As multiplication: 4 x (-3) = -12
Here we have:
• the multiplication of a positive number by a positive number;
• the multiplication of a positive number by a negative number.
Generally:
• (+a) x (+b) = + product
• (+a) x (-b) = - product
• (-a) x (+b) = - product
The product of a negative number and a negative number can be shown by looking at a
pattern of multiplicands and multipliers alongside their products:
5 x – 3 = -15
4 x – 3 = -12
3 x – 3 = -9
2 x – 3 = -6
1 x – 3 = -3
0x–3= 0
-1 x – 3 = +3
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Mathematics Grade 7
-2 x – 3 = +6
-3 x – 3 = +9
-4 x – 3 = +12
-5 x – 3 = +15
Generally:
• (-a) x (-b) = + product;
In summary:
•+ x += +
•+ x - = -
•- x += -
•- x -= +
Exercise 7
1. Evaluate:
a) 5 x –2 b) -5 x –2 c) 4 x 4
d) -2 x –2 e) -3 x –4 f) -9 x –3
g) 9 x 3 h) -4 x –4 i) -10 x +3
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Grade 7 Mathematics
Dividing I nt e ge rs
Multiplication and division are linked. The rules for dividing integers are the same, in terms
of quotient sign, as product signs.
In summary:
•+ ∏ += +
•+ ∏- = -
•- ∏ += -
•- ∏ -= +
Exercise 8
1. Evaluate:
a) 2 π 2 b) 8 π 2 c) 8 π (–2)
d) 6 π (–3) e) -9 π 3 f) (-10) π (–2)
g) (-12) π (–4) h) (-18) π (–9) i) -22 π 10
j) (-2) π (–2) k) 12 π (–4) l) (-15) π (–3)
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Mathematics Grade 7
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Grade 7 Mathematics
8. Calculate:
a) (225 ÷ 15) – (9 ÷ 3) b) (95 – 21) – (36 ÷ 4)
c) (125 ÷ 5) + (12 + 3) – (40 – 30) d) 18 ÷ (3 x 4) + 6 - 12
Addit ion
When adding
• Place each digit in the correct place value position.
• Add the ones first and write the carried digits clearly.
• Pay particular attention to the carried digits.
Examples
(i) Arrange and calculate: 23 898 172 + 27 014 653
23898172
+27014653
50912825
(ii) Calculate the sum of 342 898 and 2654.
342898
+ 2654
345552
(iii) Work out 6 930 821 + 1 269 230 + 400 028
6930821
1269230
+ 400028
7600079
Exercise 1
1. Calculate:
a) 4 2 9 7 1 8 0 4 b) 5621729 c) 84210729
+27348123 +87124583 +6354320
2. Find the sum of 193 248 617 and 215 682 904.
3. What is 365 421 385 plus 98 724 347?
4. Find the total of 2 543 789, 385 278 and 16 234 568.
5. Add 21 678 423 to 298 784 308.
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Mathematics Grade 7
6. In the year 2001 there were 521 526 children in Grade One, 126 369 in Grade Two,
562 078 in Grade Three, 220 994 in Grade Four in Somalia. How many children were there
altogether in the lower primary classes in the year 2001?
7. The number of animals exported to four countries in the year 2000 was as follows: Country
A 42 132 341; Country B 18 127 423; Country C 15 118 082 and Country D 29 108 189.
What was the total number of animals exported in the four countries?
8. In 1999 the total number of children born in Jawhar was 152 503. This number increased
by 17 340 in year 1, and then in year 2, the number increased by 12 904 over the first year.
Find the total number of children born in the second year.
9. The table shows the populations of African countries to the nearest thousand.
Country Population
Libya 13 410 000
Egypt 89 615 000
Somalia 8 765 000
Sudan 20 819 000
Kenya 35 810 000
Ghana 13 529 000
Nigeria 96 897 000
Uganda 19 142 000
a) What was the total population of:
(i) Somalia, Uganda, Libya
(ii) Kenya, Nigeria, Egypt
(iii) Sudan, Ghana
b) What was the total population of the eight countries?
10. Dhusa mareeb district got four million, sixty three thousand and eighty seven shillings for
development projects in the first half of the year 2001. In the second half, another nine
hundred and thirty seven thousand, six hundred and one shillings was allocated to the
district. How much money did Dhusamareeb get for development projects in that year?
Subt ra c t ion
In subtraction:
• Write the numbers clearly with no gaps.
• Make sure each digit is in the correct place value column.
• Start with the ones
• Remember when you borrowed.
There are many ways of saying subtract, for example;
take away, find the difference, how many more is, how many are left, minus and less
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Grade 7 Mathematics
Examples
(i) What is 54 071 982 - 12 346 385?
54071982
-12346385
41725597
(ii) Find the difference between 68 924 and 946 384.
Remember to put the larger number at the top.
946384
-68924
877460
Exercise 2
1. Calculate:
a) 5358228 b)4 0 5 2 6 1 9 8 4 c) 5854812
-3702539 - 854327 -2847983
d) 5853624 e) 2 7 1 3 5 2 0 7 f) 48239427
-6000275 - 165008 -1503625
2. Find the difference between the sum of 375 298 and 8 742 935, and the sum of 7 082 164
and 1 494 360.
3. How many more is 245 610 723 than the sum of 48 906 237 and 78 910 632?
4. The sum of two numbers is 58 965 673. One of the numbers is 2 176 458. What is the other
number?
5. The population of a country is 34 815 496. Of these 26 918 180 live in rural areas while
the rest live in urban areas. How many people live in urban areas?
6. The table below shows the number of livestock exported in the last five years.
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Mathematics Grade 7
7. In the first quarter of the year 9 520 000 people travelled by air.A similar number travelled
in the second quarter. Of the total 12 102 500 were women.
How many men travelled by air in the two quarters?
8. Of the 6 728 906 people living in a city 3 421 318 have electricity in their homes.
How many do not have electricity in their homes in this city?
9. A company has a total amount of sh. 65 917 400 before paying zakat. After paying the
zakat the total amount became sh. 64 269 465.
Find how much the company paid for zakat.
10.In a year there were six hundred and fifty thousand, two hundred and twenty men and nine
hundred thousand women who performed the haj.
Of these there were five hundred thousand married couples while the rest were single
people. How many single people were there?
Examples
i) What is 35 728 x 964?
35728
x 964
142912 (x 4)
2143680 (x 60) put down a 0 then multiply by 6
32155200 (x 900) put down two 0s then multiply by 9
34441792
(ii) Find the product of 32 746 and 1 324
32746
x 1324
130984 (x 4)
654920 (x 20) put down a 0 then multiply by 2
9823800 (x 300) put down two 0s then multiply by 3
32746000 (x 1000) put down three 0s then multiply by 1
43355704
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Grade 7 Mathematics
Exercise 3
1. Calculate the products
a) 1386 x 652 b) 1526 x 983 c) 2705 x 503
d) 5812 x 624 e) 6156 x 3251 f) 3482 x 8126
g) 7569 x 9435 h) 3782 x 5380 i) 32 561 x 2145
j) 59723 k) 93452 l) 175432
x 1507 x 3643 x 4615
Division
When dividing :
• There should be no gaps between the digits in the arrangement.
• Start from the left.
• Use a work column to guess and check multiplications.
• Write a table of divisor products
• Multiply and subtract to find the remainders
• Bring down the dividend digits and attach them to the remainders.
The other ways of saying divide are: find the quotient, share and go into.
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Mathematics Grade 7
Examples
(i) What is 9 034 780 ÷ 6340
9 034 780 is the dividend, 6340 is the divisor.
1 4 2 5 rem. 280
6340 9 0 3 4 7 8 0
6 3 4 0
2 6 9 4 7
2 5 3 6 0 6340 x
1 5 8 7 8 6340 1
1 2 6 8 0 12680 2
3 1 9 8 0 19020 3
3 1 7 0 0 25360 4
2 8 0 31700 5
1425 is the quotient and the remainder is 280.
(ii) Divide 975 432 841 by 2678
1 4 2 5 3 9 rem. 799
2678 9 7 5 4 3 2 8 4 1
8 0 3 4
1 7 2 0 3
2678 x
1 6 0 6 8
2678 1
1 1 3 5 2
5256 2
1 0 7 1 2
8034 3
6 4 0 8
5 3 5 6 10712 4
1 0 5 2 4 13390 5
8 0 3 4 16068 6
2 4 9 0 1 18746 7
2 4 1 0 2 21424 8
7 9 9 24102 9
Exercise 4
1. Calculate
a) 43 916 826 ÷ 1 234 b) 34 441 792 ÷ 964 c) 37 084 380 ÷ 3 004
d) 256 768 ÷ 2 560 e) 13 871 514 ÷ 2 122 f) 11 477 816 ÷ 5 432
g) 132 500 ÷ 2 121 h) 153 750 ÷ 1 625 i) 21 546 228 ÷ 3 124
j) 3 567 891 ÷ 7 008
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Grade 7 Mathematics
2. Divide 7 543 218 by 639
3. Find the quotient when 327 565 is divided by 245
4. What is the remainder when Sh 756 215 is shared among 560 people?
5. Find how many times does 4321 go into 24 379 082.
6. A bookseller collected Sh 724 955 after selling books. One book costs Sh 245.
How many books did he sell?
7. A factory produced 724 955 kg of sugar in one day. This sugar was packed in sacks each
holding 25 kg. How many sacks were used?
8. At a party, 511 875 people were served with a soda each. How many cartons
of soda were bought if each carton holds 175 sodas?
9. A total of Sh 16 338 976 was distributed equally amongst 356 schools.
How much did each school get?
10. A plot of land measures 35 901 720 m2. What is the area of one plot?
Exercise 5
1. 46 - 11 + 9 x 3 234 ÷ 3 + 29 2 1
2. 215 + 11 - 9 x 6 750 ÷ (10 x 15) + 7 x 8 - 60 ÷ 1043
3. 108 + 9 - (5 x 6) ÷ 2 16 x 7 ÷ 4 + 21 x 565
4. 31 x 7 + 21 x 3 - 13 x 8 90 - 50 ÷ (25 ÷ 5) 87
5. (48 ÷ 4) + (13 x 6) - (45 ÷ 3) (18 x 4) ÷ (3 + 3) 109
6. 30 + (66 ÷ 6) + (10 x 8) (50 + 80) ÷ (25 x 6) 1211
7. (195 - 55) ÷ (210 ÷ 30) (67 - 4) ÷ 9
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Mathematics Grade 7
d) 7 12 - 5 78 e) 8 311 - 4 56
d) 6 23 x 2 14 e) 3 23 x 4 16
d) 9 13 ÷3 7
10 e) 6 23 ÷2 1
5 ÷2 1
5
4. Evaluate.
1
of (2 15 ÷ 45 ) + 3 14 - 1 78 x 1 12 1
of (3 14 + 13 ) x 1
a) 2
b) 5 + 4 2
÷ 1 14 - 2 12
3
c) 4 of 78 ÷ 12 x 3
5
+ 2
3
- 7
2
d) 7
8 of 3
4
x 2 12 ÷ 1 14 - 3
5
+ 4 12
2 1
7. Mohamed spends 5 of his salary on food and 4 on house rent.
What percentage of his income is spent on other things?
3
8. Warsame had 400 hectares of land. He gave his children 4 of the land.
If each of his children received 50 hectares. How many children did he have?
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Grade 7 Mathematics
Examples
(i)
(ii)
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Mathematics Grade 7
(iii) (1 16 - 3
4) +5- 3
8
Work the brackets first:
(1 12 - 2
3) +5- 3
8 = 1 212
-9
+5- 3
8
5 3
= 12 +5- 8
(add first, then subtract)
5 3
=5 12 - 8
10 - 9
=5 24
1
=5 24
(iv) (2 23 - 1
3) -( 3
4 - 1
2 )
2-1 3-2
= (2 ) - ( )
3 4
1 1
= (2 3
) - ( 4
)
4-3 1
= (2 12
) = 2 12
Exercise 2
Evaluate
3 3 4 2 1 5 5 5 5
1) 8
+ 4
- 5
2) 5
- 2
+ 6
3) 12
- 11
+ 11
3 7 7 2 1 3 2 3 1
4) 8
+ 12
- 10
5) 1 3
+ 3 5
- 2 4
6) 1 3
- 1 4
+ 3 5
4 7 23 7 8
7) 5 5
+ 3 12
- 4 60
8) 1 15
- 8 10
+ 11
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Grade 7 Mathematics
1 3 5 3 1 5
9) 2
+ 4
- 1 10) 2 - 8
+ 5
11) 6 3
- 2 - 6
5 3 1 2 4 1
12) 10 + 7 12
- 20
13) ( 2
+ 3
) - ( 5
+ 6
)
5 1 3 3
14) ( 7
- 5
) - ( 4
- 8
)
1 3 7 1
15) ( 3 3
- 5
) - ( 1 10
- 12
)
5 7 3 5
16) ( 4 6
- 1 12
) - ( 24
+ 12
)
4 5 5 5 1 7
17) 1 5
+ ( 4 11
+ 11
) 18) 7 8
- ( 2 4
+ 2 16
)
3 2 1 1 3 3 1 5
19) 5
x 7
÷ 3
20) 5 x 2
÷ 10
21) 1 8
x 2 4
÷ 1 6
1 1 1 1 3 5 5
22) 4 2
÷ 1 5
x 3 3
23) ( 2
x 4
) + ( 1 6
÷ 2 6
)
1 1 1 5 1
24) 10
of ( 2 2
÷ 4 25) 2
of ( 7
÷ 4 3
1 3 3 4 3 2
26) 5 3
- ( 10
of 1 4
) 27) 1 5
x 1 10
÷ 15
1 3 7 5 3 1 1
28) 7 7
x 21 ÷ 1 11
29) 2 10
x 6
÷ 3 30) 5
÷ 2
- 4
Examples
(i) Find the square of 12
The square of 12 = 122 = 12 x 12 = 144
3
(ii) Find the square of 7
3 2 3 3 9
( 7
) = 7
x 7
= 49
2
(iii) Find the square of 7 3
2 7x3+2 21 + 2 23
First change the fraction into improper fraction 7 3
= 3
= 3
= 3
2 2 23 2 23 23 529 7
then square the result ( 7 3
) = ( 3
) = 3
x 3
= 39
= 58 9
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Mathematics Grade 7
Exercise 3
1. Calculate the square of these fractions and mixed numbers:
1 2 3 1 3 7
a) 4
b) 5
c) 8
d) 5
e) 5
f) 9
1 2 11 17 7 8
g) 8 h) 15
i) 20
j) 25
k) 19
l) 31
1 2 1 2 3 1
m) 1 4
n) 2 5
o) 3 3
p) 4 5
q) 5 7
r) 7 3
3 1 1 7 3 4
s) 3 15
t) 6 4
u) 3 10
v) 15 12
w) 18 20
x) 9 21
2. Evaluate
2 2 3 2 4 2
a) (1 ) b) (2 4
) c) (3 5
)
5
1 2 5 2 1 2
d) (2 8
) e) (1 6
) f) (3 7
)
3 2 1 2 5 2
g) (3 8
) h) (3 4
) i) (2 8
)
1 2 4 2 4 2
j) (1 10
) k) (2 7
) l) (5 11
)
4
3. The length of a side of a square room is 2 7
m. Find its area.
1 1
4. A square piece of land is 13 m by 13 m. What is its area in metres square?
5 5
2 7
5. A surveyor subdivided a piece of land into two square plots of sides km and km
5 12
what is the area of each plot?
Examples
4
(i) Calculate the square root of 25
4 4 2
25
= 25
= 5
1
(ii)Evaluate 3 16
1 16 x 3 +1 48 + 1 49
3 16
= 16
= 16
= 16
1 49 7 3
So 3 16
= 16
= 4
= 1 4
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Grade 7 Mathematics
1
(iii) What is the length of a side of a square whose area is 14 16 m2?
1 16 x 14 +1 224 +1 225
14 16
= 16
= 16
= 16
225
To get the length of the side we take the square root of 16
3
The length of the side = 3 4
Exercise 4
1. Calculate the square roots of
9 16 4 1 9
a) 4
b) 25
c) 25
d) 25
e) 10
25 36 40 81 169
g) 36 h) 49
i) 100
j) 576
k) 625
2. Evaluate
225 16 144 121
a) 16
b) 25
c) 169
d) 625
225 4 1 14
i) 1 576
j) 13 9
k) 11 9
j) 2 25
14
3. The area of a square room is 2 25
m2. Find the length of one side and the perimeter of
the room.
29
4. The area of a square carpet is 4 m2. Find the length of one side and its perimeter.
49
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Mathematics Grade 7
Examples
Exercise 5
Evaluate
1) 7.52 ÷ 0.2 – 11.8 2) 11.7 – 0.5 x 0.5 2 1
3) 16.5 ÷ 0.5 + 96.84 4) 5.27 + 0.36 ÷ 0.3 4 3
5) 48.7 x 92 + 36.4 6) 36.5 – 18 x 0.7 6 5
7) 24.54 – (2.84 x 1.03) 8) 142.3 x 2.5 – 26.75 8 7
9) (2.31 ÷ 0.77) x (11.5 ÷ 2.94) 10) 16.04 + 79.03 – 12.94
Squa re of a De c im a l
The square of a decimal is calculated by multiplying the decimal by itself
Remember: Count the number of decimal places in the number you are multiplying; these will
be the number of decimal places in the answer.
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Grade 7 Mathematics
Examples
(i) Find the square of 1.4 1 4
Square of 1.4 = (1.4)2 x 1 4
= 1.4 x 1.4
5 6
Multiply as whole numbers
1 4 0
1 9 6
Count 2 decimal places from the right and insert the decimal point.
So,the square of 1.4 = (1.4)2 = 1.96
Also:
This diagram shows an area of 1.4 x 1.4
1 0.4
1 x 1 =1
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Mathematics Grade 7
Squa re root s of a de c im a l
To find the square root of a decimal, express the decimal as a fraction with a denominator of
100, 10000, 1000000, ….. depending on the number of decimal places.
Note: There must be an even number of 0s. Then find the square root of the fraction.
Examples
(i) Find the square root of 0.04
square root of 0.04 = 0.04
4 4 2
= = 100
= = 0.2
100 10
∴ 0.04 = 0.2
(ii) Find the square root of 6.76
square root of 0.04 = 6.76
676 676 2 x 2 x 13 x 13 26
= = 100
= 10 x 10
=
100 10
∴ 6.76 = 2.6
(iii) Find the square root of 0.000144
square root of 0.04 = 0.000144
144 2x2x2x2x3x3
= = 10 x 10 x 10 x 10 x 10 x 10
1000000
2 x 2 x 13 12
= =
10 x 10 x 10 1000
∴ 0.000144 = 0.012
Exercise 6
1. Calculate the square root
a) 0.81 b) 0.25 c) 0.49 d) 0.16
e) 0.01 f) 0.04 g) 0.09 h) 0.01
i) 1.96 j) 2.89 k) 5.29 l) 7.29
2. Evaluate
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Grade 7 Mathematics
Pe rc e nt a ge s
The symbol, %, means 'percent' or 'out of hundred', so 60% is read as sixty percent and it
60
means 60 out of 100 or . Percentages can be written as fractions, decimals and ratios.
100
Examples
3
(i) Write as a percentage
5
3
To express as a percentage we write as a fraction with denominator 100
5
We multiply denominator 5 by 20 to make it 100. We do the same to the numerator?
3 3 20
5
= 5
x 20
3
∴ as percentage is 60%
5
1
(ii) Write 6
as a percentage
2
1 1 100 100 100 16 3 2
6
= 6
x 100
= 600
= 6
= = 16 3
%
600
100
6
4
(iii) Write 7
as a percentage
1
4 4 100 100 400 57 7 1
7
= 7
x 100
= 700
= 7
= = 57 7
%
700
100
7
Percentages as a fraction
(iv) Express these percentages as fractions
a) 20% b) 45% c) 95%
20 2 1 1
a) 20% = 100
= 10
= 5
∴ 20% as a fraction is 5
45 9 9
b) 45% = 100
= 20
∴ 45% as a fraction is 20
95 19 19
c) 95% = 100
= 20
∴ 95% as a fraction is 20
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Mathematics Grade 7
Exercise 7
1. Express as percentages
4 19 11 3 17
a) b) c) d) e)
5 20 20 8 40
13 23 33 17 43
f) g) h) i) j)
25 24 50 4 80
Pe rc e nt a ge s a nd de c im a ls
When changing a decimal to a percentage:
• Write the decimal as a fraction
• Find an equivalent fraction with a denominator of100.
Examples
(i) Write the decimals as percentages.
a) 0.25 b) 0.7 c) 0.55 d) 0.0005
25 7 70
a) 0.25 = = 25% b) 0.7 = = = 70%
100 10 100
55 5 5 0.05
c) 0.55 = = = 5% d) 0.0005 = = = 0.05%
100 100 10000 100
Note: Write the percentage out of a hundred (a fraction with denominator 100)
(ii) Express these percentages as decimals
a) 45% b) 35% c) 4% d) 9.6%
45 35
a) 45% = = 0.45 b) 35% = = 0.35
100 100
4 9.6
c) 4% = = 0.04 d) 9.6% = = 0.096
100 100
Exercise 8
1. Change to percentages
a) 0.5 b) 0.1 c) 0.75 d) 0.35 e) 0.2 f) 0.05
g) 0.4 h) 0.16 i) 0.004 j) 0.15 k) 0.56 m)0.003
2. Express as decimals
a) 24% b) 54% c) 67% d) 6% e) 74% f) 17%
g) 112% h) 9% i) 20% j) 25% k) 1% l) 80%
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Grade 7 Mathematics
Exercise 9
1. Write in order of size, the smallest first.
3 4 1
a) 50%, 0.6, 8 b) 75%, 5 , 0.7 c) 30%, 3 , 0.3,
1 4
d) 0.25, 5
, 26% e) 15
, 28%, 0.35
3. Zahar's mark in Science is 70%, in History she scored 44 out of 60, and in mathematics she
scored 20 out of 30. Which was:
a) Her highest mark?
b) Her lowest mark?
Ca lc ula t ing pe rc e nt a ge s
Examples
(ii)Find 20% of 72 kg
20
20% of 72 kg = 100
x 72 kg = 14.4 kg
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Mathematics Grade 7
Exercise 10
1. Calculate:
a) 10% of Sh 800 b) 15% of Sh 720 c) 75% of 80 marks
1
d) 12 2 % of 80 days e) 5% of 120 km
2. Hassan spent 30% of his year’s salary on developing his farm. His salary was Sh 788 000
per month.
a) How much money did he spend on the farm?
b) How much money was he left with?
3. Ashopkeeper bought 60 kg rice and sold 15% of it in two days. How much rice did he sell?
4. A goat costing Sh 40 000 is sold at 20% profit. Calculate the farmers profit.
= 66.7%
(ii)A football team played 25 games and won 17 of them. What percentage of the games
played did the team win?
17
proportion of games won = 25
17
percentage of games won = x 100
100
25
17 x 4
= 100
= 68%
Exercise 11
1. In a basket of 80 eggs 12 are broken. What percentage is broken?
2. Aziza had 68 marks out of 85 in an English test.
Write her marks as a percentage.
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3. In a football competition Marine Stars played 20 games and lost 8.
What percentage of games did they lose?
What percentage of games did they win?
4. There were 72 000 litres of fresh water in a tank.
Some water was used and there were 45 000 litres left.
a) How many litres have been used?
b) What percentage of the whole tank is used?
c) What percentage of the water is left?
5. There are 40 pupils in a class. If 25 pupils are absent:
a) What percentage of the whole class is absent?
b) What percentage of the whole class is present?
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Mathematics Grade 7
6 GEOMETRY
Circle
To draw a circle, radius 4 cm:
• open the compasses and measure 4 cm on a rule as shown.
• with any centre draw the circle
4 cm
radius radius
diameter diameter
The distance across the circle, through the centre, is the diameter. The diameter is twice the
radius. The radius is half of the diameter.
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Exercise 1
1. Draw a circle with radius:
a) 6 cm b) 4.5 cm c) 3 cm d) 3.5 cm e) 5 cm
2. What is the diameter of a circle with radius:
a) 9 cm b) 21 cm c) 12 cm d) 6 cm e) 8 cm
3. What is the radius of a circle with diameter:
a) 16 cm b) 20 cm c) 32 cm d) 9 cm e) 14 cm
4. Draw a pattern with different sized circles. Copy one of these or make your own.
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Mathematics Grade 7
5. Draw a circle of radius 4 cm and mark a point P on the circumference. Centre P, draw
an arc, cutting the circumference at two other points. Repeat and continue until the pattern
is complete.
Intersecting lines
Transversals:
A line drawn across two or more parallel lines is called a transversal.
Q J O
L M F F L
N
M
D
N O E R
R L
K R P
G I
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Grade 7 Mathematics
Exercise 2
1. Name the transversals in figures (i) to (vi)
(i) (ii) (iii) X
P
E N
A B Q V
M
C D S
F Q
L O U R
K D
S J
V H
G
J
G I U X E
2. Which of the statements about figures (i) to (iv) are true and which are false?
(i) E (ii)
L
C
N O
G H
P Q
F
D L
N
W
O
Y V Q S
(iii) UV is a transversal (iv) RS is a transversal
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Mathematics Grade 7
2 1
A 3 4 B 1 and 5 are corresponding angles.
2 and 6 are corresponding angles.
3 and 7 are corresponding angles.
6 5 4 and 8 are corresponding angles.
C 7 8 D
1 5
2 6
3 7
4 8
Name the corresponding angles in the table.
What do you notice about the value of corresponding angles.
corresponding angles are equal.
Examples
Find the sizes of angles L, Y and r in the figure
alongside
E
L
A 120o B
y
C r D
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Exercise 3
1. Identify and name the corresponding angles by completing the statements:
a) b) c)
o
l
g h m
n w
f e z
f
g y x
v a
b h i
i j u
x d
k l w c
72o
110o v
s f
a
b
c) d)
120o
c
l f
m n g 35o
d
e)
i j
k l
b d o
c 25
b d
c e
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Mathematics Grade 7
Draw a large diagram like the above. Copy and complete the table
Use a protractor to measure the angles.
3 7
4 8
Examples
Find the value of (i) d (ii) e (iii) f in the figure.
Exercise 4
1. Write the alternate angles number which is equal to the letters:
a) b)
2 3 1
1 4 4 3 2
m s
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c) d)
4 3
1 2
n x 4 1
3 2
2. Write the value of the angles indicated by letters in the figures below:
a) b)
f
80o
O 154o
c) d)
i x
48o
e) f)
d s
r
123o 145o
g) h)
L m
n
c g
50o f
q k
w 45o
i) j)
c m
g
x 140o g r t
35o
w s u
t
b
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Mathematics Grade 7
I nt e rior a ngle s
Interior angles are formed by parallel lines and a transversal.
d and h are also co-interior angles.
c a Make a large diagram and measure the angles.
d b
Copy and complete the table.
b e
h e d h
g f Name the interior angles.
What is the value of b + e?
What is the value of d + h ?
Co- interior angles add up 1800.
Co-interior angles are supplementary .
∠s and e are interior angles.
∠d and h are interior angles.
∠a = 80. Find the value of the other angles. c a = 80
d b
∠b = 1800 - 800 = 1000
∠c = 1000 (vertically opposite to b)
∠d = 800 (vertically opposite to 800)
h e
∠e = 800 (Alternate to d) g f
∠h = 100 (Alternate to b)
Co-interior angles b and e add up to 180
Co-interior angles are supplementary;
they add up to 180o.
Pa ra lle logra m s
z y
w x
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Grade 7 Mathematics
Constructing parallelograms
Examples
Construct parallelogram ABCD with AB
parallel to DC.
AB = 3 cm, BC = 4 cm
A D
and ∠ ABC = 55°.
Draw line BC = 10 cm.
3 cm
Using a protractor, draw ∠CBE = 55°
Centre B and radius 8 cm draw an arc to cut
BE at A. Centre A and radius 4 cm draw AB. 55o
4 cm
Centre C and radius 8 cm draw an arc to cut B C
the previous arc at D. Join CD and AD to
complete parallelogram ABCD.
Measure the value of ∠ADE. Which angle has the same value?
∠ABC and ∠ADC are opposite angles in ABCD.
∠s BAD and BCD are opposite angles in ABCD.
Measure ∠BAD and ∠BCD.
Complete:
The opposite angles of a parallelogram are -------------- .
The opposite sides of a parallelogram are ------------- and parallel.
Exercise 5
1. Find the sizes of the angles marked with letters in the parallelograms:
a) b)
a
b a 125o
48o c c
b
c) d)
h n m
i g
130o
k L
j
35o
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Mathematics Grade 7
2. Construct a parallelogram PQRS with PQ parallel to SR; PQ = 4 cm, QR = 5.5 cm.
Use a protractor to make ∠PQR = 45°.
3. Using ruler and a pair of compasses only construct:
a) ABCD with AB parallel to DC; AB = 5 cm, BC = 6 cm and ∠ABC = 60°.
b) MNPQ with line MN parallel to QR; line MN = 3 cm, NP = 5 cm and ∠NPQ = 105°.
The Pythagorean relationship
Measure the sides of the right-angled triangles below. Which is the longest side in every
triangle?
a) b) c) d)
B E Q T
A F
C
X Y
D
R S
The longest side in a right-angled triangle is called the hypotenuse.
The hypotenuse is always opposite to the right angle.
Exercise 6
1. Name the hypotenuse in each triangle.
a) b) c) d)
C E H I J
F
D G K L
A B
2. Construct a right-angled triangle ABC with AB = 4 cm, BC = 6 cm and
∠ABC = 90°.
a) Name the hypotenuse.
b) What is the length of the hypotenuse?
3. Construct a right angled triangle XYZ with YZ = 4 cm, ZX = 7 cm and angle XYZ = 90°.
a) What is the length of XY?
b) Which is the hypotenuse in ∆XYZ?
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Triangles with sides in the ratio 3 : 4 : 5 and 5 : 12 : 13 are right angled.
∆ABC has sides of 3 cm, 4 cm and 5 cm.
∆ABC is a right-angled triangle.
A 25 cm2
P 9 cm2
4 cm
B C
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Mathematics Grade 7
Examples
(i) Calculate the length of the hypotenuse in
the triangle shown below.
c2 = a 2 + b2
= 92 + 122
= 81 + 144
= 225 12
c = 225
c = 15 cm.
9
The length of the hypotenuse is 15 cm
Exercise 7
1. Find the length of the side marked with a letter.
a) b) 9 cm
c
c
8 cm
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c) d) 21 cm
10 cm
35 cm
e) f)
c 7.5 cm
16 cm b
3. The diagonals of a rectangle are 35 cm each. If one of the sides is 21 cm, what is the length
of the other side.
4. To repair the roof of a 15 m high building, a 16 m ladder was used. How far is the foot of
the ladder from the wall to the nearest centimetre?
Sym m e t r y
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Mathematics Grade 7
A line of symmetry divides a shape into two parts of the same area and shape. If a shape is
folded about a line of symmetry, one part will fit exactly on the other.
Does the rectangle have more than one line of symmetry ?
Do the circle and hexagon have other lines of symmetry?
Write these letters and draw their lines of symmetry.
AV O U T E N R
Which letters do not have a line of symmetry?
Exercise 8
1. Identify the symmetric figures
a) b) c)
d) e)
2. Copy these figures. Draw the lines of symmetry for each shape.
a) b) c) d)
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b) c) d)
e) f) g)
h) i) j)
Examples
(i) A(1,1), B(5,1) and C(3,7)
Plot the points using the x-y co-ordinates.
Join the points and draw the dotted line
CD.
a) What shape is produced?
b) If you fold the shape at the dotted line,
what do you notice?
c) CD is a line of symmetry for ∆ABC.
x-axis
When we join the points, we get a triangle. If you fold the shape on the line, you will
see that point B will go on point A. The dotted line will divide the triangular region
into two congruent triangular regions.
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Mathematics Grade 7
x-axis
You will notice that the dotted line divides the shape into two congruent shapes.
The line that divides a shape into two congruent shapes is called line of symmetry.
When this happens the shape is said to be symmetrical about the line of symmetry.
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In figure a) x-y is the line of symmetry. If you fold the shape on x-y, point A will
fall on A and point B will fall on point B on the other side of x-y.
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Mathematics Grade 7
The point A and A are equidistant from x-y, also B and B are equidistant from x-y.
Now, complete shape a), which is ABY B A.
Also identify the point C, D and E and then complete the shape, following the same
the steps as for shapes (a).
Examples
Assume the dotted line is a line of symmetry copy and complete the polygon.
a) b) c)
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7 cm 14 mm 2.5 m 14 m
5m
7m
3. Calculate the area of these triangles.
a)
b)
4 cm
12 cm
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Mathematics Grade 7
4. Calculate the area of triangles with the following measurements.
a) Base 5 cm, b) Base 16 cm,
perpendicular height 8 cm perpendicular height 12 cm
c) Base 17 cm, d) Base 24 cm,
perpendicular height 14 cm perpendicular height 18 cm
2m
The metre (m) is equal to 100 cm.
Metres and centimetres can be used to
measure heights.
Hussein is 1 m 72 cm tall.
The largest unit of length is the kilometre (km).
1m
The kilometre is equal to 1000 m.
This table shows the relationship between
the units of the International System.
It can also be used to convert units.
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km m cm mm
10 mm = 1 cm 1 0
100 cm = 1 m 1 0 0
1000 m = 1 km 1 0 0 0
Examples
(ii) convert to cm
a) 4 5 mm b) 68 mm c) 93 mm
km m cm mm
4 5 a) 45 mm = 4.5 cm
b) 68 mm = 6.8 cm
6 8 c) 93 mm = 9.3 cm
9 3
(iii) convert to m:
a) 238 cm b) 465 cm c) 609 cm
km m cm mm
2 3 8 a) 238 cm = 2.38 m
b) 465 cm = 4.65 m
4 6 5 c) 609 cm = 6.09 m
6 0 9
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Mathematics Grade 7
(v) convert to m:
a) 6 km b) 8.436 km c) 15.904 km
km m cm mm
6 0 0 0 a) 6 km = 6000 m
b) 8.436 km = 8436 m
8 4 3 6 c) 14504 km = 14.504 m
14 5 0 4
Note: Before the introduction of the International System of metric measure, the units
of length table looked like this:
km Hm Dm m dm cm mm
The Hm, Dm, and dm are no
5 0 longer used but the table
3 0 remains in the same form.
Their spaces are still there.
8 0
Exercise 1
Draw a metric units table to convert the units in these questions.
1. Convert to cm
a) 84 mm b) 37 mm c) 50 mm
2. Convert to mm
a) 6 cm b) 4 cm c) 9 cm
3. Convert to km
a) 4382 m b) 6084 m c) 5608 m
4. Convert to m
a) 265 cm b) 4632 mm c) 3806 cm
d) 7 km e) 7.365 km f) 38.6 5 km
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= 33 + 42
= 75 m
(ii) Find the perimeter.
22
(Take π as )
7
Label the figure ABCD
The Perimeter of the figure
= rectangle BADC + semi circle on BC
= (AB + DC +AD) + 1 π x 14
2
1 22 1
= (28 +28 + 14) + x x 14 Circumference of semicircle = πxd
2 7 2
= 70 + 22 = 92 cm
(iii) Find the perimeter of the given figure.
Perimeter = Semi circles on AD + AO + OD
1 1 1
=( π x 42) + ( π x 21) +( π x 21)
2 2 2
1
= (42 + 21 + 21)
2
1 22
= x x 84
2 7
= 11 x 12
= 132 cm
22
(iv) Calculate the perimeter: (Use π = )
7
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Mathematics Grade 7
1
The 2 semi-circular lengths = πd x 2
2
1 22
= x x 105 x 2
2 7
= 330 m
The length of the sides = 235 x 2
= 470 m
The perimeter of the figure is = 330 m + 470 m
= 800 m
Exercise 2
22
1. Calculate the perimeter of the figures. (Take π = )
7
a) b)
c) d)
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22
1. Calculate the perimeter of the figures. (Take π = )
7
a) b)
c) d)
e) f)
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Mathematics Grade 7
g) h)
a) b)
4. This figure is made up of two semicircles of radius 3.5 cm and an isosceles triangle
whose equal sides are 14 cm.
Calculate its perimeter.
5 Su’ad and Shariif set out to roll their tyres along a 100 m part of a straight road. Su’ad ’s
tyre has a diameter of 42 cm, while Shariif’s has a radius of 35 cm. How many revolutions
will each tyre make along the road.
Give the answers to the nearest whole number.
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6. This figure shows a grazing area. The
diameter of the semi-circle is 35m.
What is the perimeter of the grazing area
in metres?
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Mathematics Grade 7
Exercise 3
1. Calculate the areas of the shaded regions:
a) b)
c) d)
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e) f)
g) h)
g)
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Mathematics Grade 7
3. The curved part of the figure is a
semi-circle of diameter 20 m.
Calculate the:
a) Length of y.
b) Area of the figure.
c) Perimeter of the figure.
Surfa c e Are a
Surface area is the sum of the areas of all of the surfaces or faces.
Cubes
A closed cube has six faces. The edges are equal.
The surface area of the cube = Area of one face x Number of faces
= L2 x 6 = 6L 2
Surface Area of a closed cube = 6L2 , where L is the length of one its edges.
An open cube has 5 outer surfaces or faces.
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Examples
(i) Calculate the outer surface area of an open cube of side 5 cm.
The outer surface area of an open cube = 5L2
= 5 (5) 2
= 5 x 25
= 125 cm 2
(ii)Calculate the side of a closed cube whose surface area is 253.5 m
Surface area of a closed cube = 6L 2
Surface area of a closed cube
L2 = 6
L2 = 253.5 ÷ 6
L2 = 42.25
L = √42.25
L = 6.5 m
Exercise 4
1. Find the surface area of these closed cubes of sides:
a) 7 cm b) 2.4 cm c) 5 cm d) 6.7 cm
e) 9.6 cm f) 8 cm g) 3.8 cm h) 4 cm
2. Find the outer surface area of these open cubes of side:
a) 4.5 cm b) 6 cm c) 7.9 cm d) 1.5 cm
e) 9 cm f) 2.9 cm g) 15 cm h) 3 cm
3. Find the side of these closed cubes whose surface areas are.
a) 384 cm b) 121.5 cm c) 201.84 cm d) 54 cm
e) 150 cm f) 216 cm g) 6 cm h) 24 cm
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Mathematics Grade 7
Cuboids
This is a closed cuboid and its net.
Surface area = Lw + Lw + Lh + Lh + wh + wh
= 2Lw + 2Lh + 2wh
= 2(Lw + Lh + wh)
This is an open cuboid and its net. The top or bottom could be open.
Surface area = Lw + Lh + Lh + wh + wh
= Lw + 2Lh + 2wh
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Cuboid with inward or outward end open.
Examples
(ii) Calculate the surface area of an open (top) cuboid of length 8 cm, width 4 cm, height
7 cm.
A = Lw + 2Lh + 2wh
= (8 x 4) + (2 x 8 x 7) + (2 x 4 x 7)
= 32 + 112 + 56
= 200 cm
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Mathematics Grade 7
(iii) Calculate the width of a closed cuboid with length 9 cm, height 3 cm and surface area
105 cm2.
A = 2Lw + 2Lh + 2wh
150 = (2 x 9 x w) + (2 x 9 x 3) + (2 x w x 3)
150 = 18w + 54 + 6w
24 w + 54 = 150
24 w = 150 – 54
24 w = 96
96
w=
24
w = 4 cm
The width of the cuboid is 4 cm.
Exercise 5
1. Calculate the surface area of a closed cuboid with:
a) length 5 cm, width 3 cm and 2 cm
b) length 6.8 cm, width 4 cm and 5 cm
c) length 9 cm, width 6 cm and 4 cm
d) length 1.1 m, width 1 m and 0.8 m
e) length 0.17 m, width 0.12 m and 0.08 m
f) length 0.4 m, width 0.3 m and 0.2 m
2. Copy and complete the given table
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3. Calculate the height of a closed cuboid with length 5 cm, width 6 cm and surface area
214 cm 2.
4. Calculate the width of a closed cuboid with length 10 cm, height 4 cm and surface area
136 cm2.
5. Calculate the surface area of the following closed cuboids.
Cylinders
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Mathematics Grade 7
Area of A = π r2 Area of C = π r2
B is the curved surface area of the cylinder.
It becomes rectangular in shape when laid flat.
The area of the rectangle = L x h (h being the height of the cylinder)
Here L = circumference of the cylinder = 2πr
So Area of B = 2πrh
Surface area of the cylinder = Area of A + Area of B + Area of C
= πr2+ 2πr h +πr2
= 2πr2 + 2 πrh
= 2πr(r + h)
The surface area of a cylinder = 2πr(r + h)
Examples
(i) Calculate the surface area of a cylinder of diameter 18 cm and height 12 cm.
(use π = 3.14)
h = 12 cm
d 18
r = = = 9 cm
2 2
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Grade 7 Mathematics
Exercise 6
1. Complete the tables. Take π as 22
or 3.14 where appropriate.
7
Note that an open cylinder has only one circular end.
Table A
Table B
2. Calculate the surface area of a pipe, open at both ends, of length 3 m and a radius of
28 cm. Give your answer in m2.
3. Calculate the length in metres, of a closed pipe of radius 5 cm and a surface area of 2.2 m2.
4. Calculate the radius of an open tube of length 10 cm and surface area of 88 cm2.
5. Calculate the length of a closed cylinder with radius 12 cm and surface area 1733 cm2.
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Mathematics Grade 7
a) b)
c) d)
c) d)
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a) b)
1 cm
5 cm
2 cm
7 cm
1 cm
5 cm
9 cm
8 cm
2 cm
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Mathematics Grade 7
4. The length of a rectangular water tank is 6.5 m and its breadth is 5 m.
If its height is 4 m, calculate 1 m3 = 1000 l:
a) The volume of the water tank.
b) The capacity of the tank in litres.
5. A boarding school has an open cylindrical container with radius 50 cm and a height of 1 m
for the storage of the water.
What is the volume of the container in a) m3? b) cm3?
What is the volume of the container’s material if its uniform thickness is 0.5 cm?
6. Convert to litres (l):
a) 200 ml b) 2674 ml c) 5070 ml d) 12 465 ml
7. Convert to millilitres (ml):
a) 2.5 l b) 4.5 l c) 9.248 l
d) 4.375 l e) 24.608 l
8. a) l ml b) l ml c) l ml
4 3 22 6 49 8
+ 3 9 + 12 5 + 16 4
d) l ml e) l ml f) l ml
18 426 15 345 43 537
- 10 872 - 12 793 - 40 863
g) l ml h) l ml i) l ml
2 424 15 612 20 32
x 6 x 2 x 7
9. Fadhia made lemon juice and sold it in 100 ml containers. Each 100 ml container sold at
Sh 1500. If altogether she sold 280 containers, calculate:
a) how much money she made?
b) the quantity of juice sold in litres.
10. Express in tonnes and kilograms. (reminder 1000 g = 1 kg 1000 kg = 1 t )
a) 8624 kg b) 97160 kg c) 3.25 t d) 4.75 t
3 1 1 2
e) 2 t f) 4 t g) 5 t h) 7 t
4 2 8 5
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11.a) t kg g b) t kg g c) t kg g
4 260 430 12 140 360 47 925 375
3 934 167 9 593 485 - 38 865 792
5 678 475 8 349 268
+1 345 198 + 1 293 867
d) t kg g e) t kg g f) t kg g
198 472 141 274 630 497 216 402 319
- 159 869 759 - 186 438 968 102 328 545
198 595 392
+ 86 398 193
12.a) t kg g b) t kg g c) t kg g
18 425 312 15 195 248 143 324 516
x 7 x 15 x 24
13.A truck has 168 sacks of rice each weighing 90kg. How many tonnes and kilograms is the
total weight of the rice?
Volum e of a c ube
Examples
(i) Calculate the volume of a closed cube with sides 7 cm
Volume of a closed cuboid.
Volume of a cube =LxLxL
= L3
= (7)3
= 7 x 7 x 7 cm3
= 343 cm 3
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(ii)Calculate the volume of a cuboid with a length of 8 cm, width of 6 cm and a height of
4 cm.
(iii) Calculate the width of a cuboid with a volume of 60 m3, height of 3 m and length
10 m.
Volume of a cuboid = L x w x h
60 = 10 x w x 3
60
3 x 10
=w
w = 2m
Volum e of a c ylinde r
Examples
(i) Calculate the volume of a closed cylinder of radius 14 cm and height 20 cm.
Volume of a cylinder = π r2 h
V = x 14 x 14 x 20
= 22 x 2 x 14 x 20
= 12 320 cm 3
(ii)Calculate the radius of a cylinder with volume 616 cm3 and height 4 cm
Volume of a cylinder = π r2 h
616 = π r2 h
22
616 = x r2 x 4
7
616 x 7 = 88 r 2
4312 88
88
= 88 r2
r2 = 49
r 49 = 7
The radius of the cylinder is 7 cm.
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Exercise 1
1. Calculate the volume of the following closed figures:
a) b)
c) d)
e) f) g)
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2. Calculate the volume of a closed:
a) cuboid with length 8 cm, width 7 cm and height 5 cm.
b) cube with sides 13 cm.
c) cuboid with length 14.7 cm, width 11 cm and height 5 cm.
d) cylinder with radius 21 cm and height 3.5 cm (use π22
= ).
7
e) cylinder with radius 2 m and height 1.5 m (use π = 3.14).
22
f) cylinder with radius 9.8 cm and height 3 cm (use π = )
7
3. A cube has a volume 64 cm3. Calculate the side of the cube
4. What is the volume of a cube with side 0.11m?
5. A cuboid with length 12 cm and height 12 cm has a volume of 480 cm3.
What is the width of the cuboid?
6. A cuboid with width 6 cm and height 4 cm has a volume of 216 cm3.
What is the length of the cuboid?
7. A cuboid with length 9 cm and width 5 cm has a volume of 315 cm3.
What is the height of the cuboid?
8. Calculate the height of a cylinder with:
a) radius 8 cm and volume 1205.76 cm3 (use π = 3.14)
b) radius 0.7 cm and volume 1.386 cm3 (use π = 3.14)
22
c) radius 14mm and volume 2464 m3 (use π = )
7
9. Calculate the radius of a closed cylinder with:
22
a) height 5 cm and volume 0.077 m3 (use π = )
7
22
b) height 1.6 cm and volume 22.176 cm3 ( use π = )
7
c) height 0.8 m and volume 3.04 m3 (use π = 3.14 )
10.The radius of a closed cylindrical tank is 14 cm and its height is 12 cm. Calculate its
volume.
11.The radius of a closed cylindrical oil storage tank is 2.8m and its height is 9.2 m.
Calculate its volume in
a) cm3 b) m3
12.A bridge is constructed using cylindrical bars each of diameter 2.1 cm and length 500 cm.
What is the volume of each bar?
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A lit re c ube
A cube measuring 10 cm by 10 cm by
10 cm has a volume:10 x 10 x 10 = 1000
cm3
1 m3 = 1000 litres
1000 l is known as 1 kilolitre (kl)
1 m = 100 cm
Examples
(i) How many litres are there in a container measuring 24 cm by 17 cm by 20 cm
V = 24 x 17 x 20 = 8160 cm3
1000 cm 3= 1 l
1
1 cm3 = l
1000
1
8160 cm 3= 8160 x 1000
l
8160 cm 3= 816 l
100
= 8.16 l
(ii)A bucket has a volume of 968 cm3. What is its capacity in litres?
We know that
1
1 cm3 = l
1000
968
So 968 cm 3 = l
1000
= 0.968 l
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Exercise 2
2. Convert to cm 3.
a) 0.54 l b) 5 000 l c) 24.6 l d) 3 952 ml
3. Convert to m3.
a) 745 000 l b) 620 l c) 24.6 l d) 250 l
5. A water pump delivers water into a tank at the rate of 150 litres per minute.
How long will it take to fill a tank whose volume is 12 m3 ?
6. A tank has a volume of 80 000 cm 3. How much water, in litres, would this tank hold when
full?
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M illilit re (m l)
1000 cm3 = 1 l
1000 cm3 = 1000 ml
So 1 cm 3 = 1 ml
A container measuring 1 cm by 1 cm by 1 cm
has a volume of 1 cm3.
This container will hold 1 ml of liquid.
Examples
(i) How many ml are there in:
a) 5 cm3 b) 8.9 cm3 c) 45.7 cm3
a) 5 cm 3 = 5 x 1 ml = 5 ml
b) 8.9 cm 3 = 8.9 x 1 ml = 8.9 ml
c) 45.7 cm 3 = 45.7 x 1 ml = 45.7 ml
(ii) How many cm 3 are there in:
a) 120 ml b) 1.98 ml c) 260 ml
a) 120 ml = 120 x 1 cm3 = 120 cm 3
b) 1.98 ml = 1.98 x 1 cm3 = 1.98 cm 3
c) 260 ml = 260 x 1 cm3 = 260 cm 3
Exercise 3
1. How many millilitres are in:
a) 98.2 cm 3 b) 1 cm3 c) 2658 cm3 d) 198.57 cm3
2. How many cm 3 are in:
a) 7.5 ml b) 120 ml c) 519.6 ml d) 690 ml
3. A rectangular tank holds 120 l of liquid when full.
a) How many cm 3 of water does it hold?
b) What is the height of the tank with width 50 cm and length 60 cm.
4. A closed cylindrical tank has a radius of 14 cm3 and 25 cm 3 height. Calculate the:
a) volume of the tank in m3 and cm3
b) capacity of the tank in litres and millilitres.
5. A cylindrical tank has 4600 litres of water.
The radius of the tank is 10 m, what is the height of the water in the tank?
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6. Calculate the capacity of these containers in litres.
a) b)
c) d)
c) d)
The tonne
1 tonne (t) = 1000 kg
1 kg = 1000 g
1 t = 1000 kg = 1000 x 1000 g = 1 000 000 g
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Examples
(i) Convert into kilograms
a) 0.842 tonnes b) 763 000 g
a) We know that 1 t = 1000 kg
therefore 0.842 t = 0.842 x 1 000
= 842 kg
b) We know that 1 kg= 1000 g
763 000
So 763 000 g =
1000
= 763 kg
(ii) Convert to grams
a) 45 kg b) 0.98 tonnes c) 63.2 kg
a) 45 kg = 45 x 1000
= 45 000 g
b) 0.98 t = 0.98 x 1 000 000
= 980 000.00
= 980 000 g
c) 63.2 kg = 63.2 x 1000
= 63 200.00
= 63 200 g
Exercise 2
1. Convert to kg:
a) 17 t b) 7.96 t c) 48.961 t
d) 12.49 t e) 623 400 g f) 54 912 g
2. Convert to g:
a) 6.31 kg b) 2 t c c) 16 kg d
d) 0.002563 t e) 1.83 t f) 5 kg g
3. Convert to tonnes:
a) 6 000 000 g b) 47.89 kg c) 70 500 g
d) 62 500 g e) 81 479 kg f) 3 500 000 g
5. After selling 25 kg of meat at Sh 450 000 a butcher made a profit of 50%.
How much had he paid per kilogram?
6. A lorry was loaded with 200 cartons each containing 20 tins of fat. Each empty carton
weighs 500 g and each tin of fat weighs 2 kg.
What was the total weight carried by the lorry in tonnes.
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7. Tahliil weighs 62.3 kg. He wears shoes which weigh 750 g and carries a school bag full of
books weighing 10.5 kg. If he steps on the weighing scale in his shoes and carrying the
school bag full of books, what will be the reading on the scale.
8. Warsame harvested a total of 2.69 tonnes of maize and sold it in 50 kg bags.
How many bags of maize did he sell?
9. A box containing 24 books weighs 9.6 kg. If the box weighs 1.2 kg when empty, what is
the weight of each book in grams, if all the books are of equal weight?
1 1
10. Miss Kaltuun packed equal number of 2
kg flour and 4 kg flour.
1 1
If she had 180 kg of flour, how many packets of and kg did she pack?
2 4
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5. A train left station A towards station B at 0645 h. It reached station B at 2015 h. How long
did the train take to travel from station A to station B?
6. Express the following in the 24 hours system:
a) 2:16 a.m b) 5:45 a.m c) 11:24 a.m d) 12:00 noon
e) 3:48 p.m f) 6:20 p.m g) 7:30 p.m h) 10:00 p.m
7. Express the following in a.m and p.m
a) 0118 b) 1245 c) 1306 d) 2108 e) 2348
f) 0259 g) 0615 h) 0930 i) 1416 j) 1800
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8. It took 3 hours for a car to travel from Hargeisa to Berbera a distance of 175 km. At what
speed was the car travelling?
9. A bus was travelling at 75 km/h for a distance of 300 km.
How long did the bus takes on the journey?
T im e
The 24 hour clock
Reminder
Examples
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Grade 7 Mathematics
Exercise 1
1. Write these times using the 12-hour clock
a) 0700 hours b) 1710 hours c) 1300 hours d) 2115 hours
e) 2300 hours f) 2030 hours g) 1610 hours h) 1930 hours
2. Write the following time using 24-hour clock.
a) 10:05 am b) 4:08 pm c) 9:25 pm d) 11:30 am
e) 5:08 pm f) 3:34 am g) Half past ten in the morning
h) Midnight i) Quarter to two in the morning j) Noon
3. Calculate the hours and minutes between these times.
a) 0413 h and 1609 h b) 1427 h and 2203 h
c) 0110 h and 1114 h d) 0750 h and 2330 h
e) 0314 h and 1104 h f) 0916 h and 1604 h
4. Use the timetable to answer the questions:
a) What time does the bus for Marka leave Mogadishu on Mondays?
b) What time does the bus for Wanlawey leave Mogadishu on Saturdays?
c) What time does the bus for Marka leave Mogadishu on Thursdays?
d) What time does the bus for Wanlawey leave Mogadishu on Tuesdays?
e) Omar takes 45 minutes to walk to the bus station, if he wants to catch the bus to
Wanlawey on Sunday, when must he leave home?
f) Sadio takes 15 minutes to walk to the bus station, If she wants to catch the bus to Marka
on Thursday, when must she leave home?
5. The table shows the departure and arrival times from Jidda to Mogadishu via D’jabuti and
Hargeisa.
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How long is the flight between:
a) Jidda and D’jabuti
b) D’jabuti and Hargeysa
c) Hargeysa and Mogadishu
6. Ahmed, who lives in Marka, plans to get the Mogadishu plane to Jidda. It takes 2 hours
and 15 minutes to travel by car from Marka to Mogadishu.
What is the latest time that Ahmed should leave Marka to board the plane?
Spe e d
Distance
Speed =
Time
Also, Distance = Speed x Time
Distance
And Time =
Speed
Using S for speed, T for time and D for distance the formulae can be shortened:
D D
So, S = T
D=SxT T= S
Examples
(i) Nur drove from Mogadishu to Baidoa, a distance of 250 km.
1
If the journey took 2 hours, what was his average speed?
2
1
D = 250 km T = 2 h S=?
2
250
D
S= = 2
1
T 2
2
S = 250 x
5
S = 100 km/h
(ii)Ali took 5 hours to complete a journey. If his average speed was 75 km/h, how long
was his journey?
S = 75 km/h T=5h D=?
D=SxT
D = 75 x 5 = 375 km
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iii)A rally car covered a distance of 3600 km at an average speed of 180 km/h.
How many hours did it take to cover this distance.
D = 3600 km S = 180 km/h T=?
D
T=
S
3600
T=
180
T = 20h
Exercise 2
1. Asha took 6 hours to travel a distance of 360 km. At what speed was she driving in km/h?
2. Mahad was driving at 65 km/h. He drove for 9 hours. How long was the journey?
3. A bird flew a distance of 56km at a speed of 7 km/h. How many hours did it fly?
1
4. A bullet fired at a speed of 250 m/s took 2 seconds to hit the target.
2
Average Speed
Different sections of a journey may take a longer or shorter time than other parts.
The overall speed for the journey is called the average speed.
Average speed = Total distance travelled
Total time taken
Total time taken includes time when: driving fast or slow, stopping for fuel, changing a tyre,
eating lunch, etc.
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Examples
(iii)A car driver left town A for town B at 8.30 a.m. He travelled at a speed of 60 km/h for
1
4 hours. He stopped for lunch which took 30 minutes. He then drove at an speed of
2
1
120 km/h and reached town B after 2 2 hours.
a) What is the distance between town A and B?
b) How long did he take to travel from town A to B?
c) What was his average speed for the whole journey?
d) At what time did he reach town B?
a) Distance travelled before lunch = S x T = 60 x 4 1 = 270 km
2
b) Distance travelled while having lunch = 0 km
1
c) Distance travelled after lunch = S x T = 100 x 2 2 = 250 km
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Exercise 3
1
1. Nur took 2 hours to travel the first 40 km of his journey. He then stopped for a hour
2
break. He travelled the remaining 30 km of his journey in 1 hour.
What was the average speed for the whole journey?
1 1
2. Omar travelled 125 km by lorry in 1 hours . He stopped to load cement for 1 2
hours.
2
1
He then travelled 120 km in 1 2 hours to reach his destination.
What was his average speed for whole journey?
1 1
3. Mahad travelled 150 km in 2 2
hours and a further 270 km in 3 2
hours.
What was his average speed?
4. Oday swam across a 100 m wide river in 12 minutes and swam back in 13 minutes.
What was his average swimming speed?
5. Luul travelled 150 km in 2 hours and then a further 180 km in 3 hours.
a) What was the total distance travelled?
b) What was the total time taken
c) What was her average speed?
1
6. A bus took 2 2
hours to travel 240 km. What was its average speed?
7. Ruqiyo travelled at 90 km/h for 4 hours and then stopped for a 1 h lunch break. She then
travelled for 2 hours at 65 km/h.
What was her average speed for the whole journey?
8. Sahra travelled 120 km in 1 1 hours and then 240 km at 100 km/h.
2
a) How long did the whole journey take?
b) What was her average speed for the whole journey?
9. A motorist travelling at 60 km/h completed a journey in 6 hours.
How long would it take to do the same journey at 80 km/h?
10.A train travelled 45 km in 30 minutes. How long will it take to travel 75 km?
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Mathematics Grade 7
11. Adan drove at 75 km/h for 4 hours. He then stopped for 30 minutes to attend a meeting.
He drove back the same distance at 60 km/h.
a) Calculate the distance to the meeting place?
b) What was his average speed for the whole journey?
12. A woman drove from town X to town Y at 6.00 a.m. She travelled at 100 km/h for
1
1 2 hours and then stopped for breakfast which took 30 minutes. She then drove at 120
km/h and reached town Y after travelling for 3 hours.
a) What is the distance between towns X and Y?
b) How long did she take to travel from town X to Y?
c) What was her average speed?
d) At what time did she reach town Y?
1
13. Sudi left town A for town B at 7.30 a.m. She travelled at 90 km/h for 5 2
hours. She
1
stopped for lunch which lasted for 1 hours. She then drove at 105 km/h and reached town
2
B after 8 hours.
a) What is the distance between towns A and B?
b) How long did she take to travel from town A to town B?
c) What was her average speed?
d) At what time did she reach town B?
Te m pe rat ure
Temperature means how hot or cold an object is.
An object temperature is measured by a thermometer. The unit of measuring temperature is
the degree Centigrade (or degrees Celsius) written as °C.
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Activity
Exercise 4
1. Record the temperature shown by each thermometer:
a) b) c) d)
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5. At 9:00 a.m. the temperature is 18oC. It rises by 0.5oC every minute.
What is the temperature at 9:08 am?
6. The minimum temperature is below 12oC during July, August and September. How many
days is the minimum temperature is below 12oC.
7. The temperature falls by 8.2oC to 16 oC.
What was the temperature before the fall?
8. The temperature rises from 4 oC to 22 oC. How many degrees is this?
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Grade 7 Mathematics
10 MONEY
Profit And Loss
Revision
1. A co-operative society had 372 members. If each member contributed Sh 5000 per month,
how much money in total will be collected in one month?
2. Faisal bought these items for his shop.
12 packets of rice @ Sh 2500 a packet.
8 loaves of bread @ Sh 550 a loaf.
4 bundles of wheat flour @ Sh 5800 per bundle.
10 kgs of sugar @ Shs 800 per kg.
9 litres of cooking oil @ Sh 3500 a litre.
5 packets of tea leaves @ Sh 2500 a packet.
Pe rc e nt a ge profit a nd pe rc e nt a ge loss:
Profit = selling price - buying price Loss = buying price – selling price
Buying price = selling price - profit Buying price = loss + selling price
Selling price = buying price + profit Selling price = buying price - loss
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Mathematics Grade 7
Profit
Percentage profit = x 100%
Buying price
Loss
Percentage loss = x 100%
Buying price
Examples
(i) Ahmed bought a goat for Sh 200 000 and sold it Sh 230 000.
Calculate his percentage profit.
Buying Price (BP) = Sh 200 000 Selling Price (SP) = Sh 230 000
Profit (P) = 230 000 - 200 000
= Sh 3000
Profit
Percentage profit (%P)= x 100%
Buying price
30000
= x 100% = 15%
200000
(ii)Anab bought 5 cartons of pasta at Sh 500 000 and sold at Sh 450 000.
Calculate her percentage loss.
BP = Sh 500 000 SP = Sh 450 000
Loss = BP - SP
= 500 00 - 450 000
= Sh 50 000
Loss
% loss = x 100%
Buying price
50000
= x 100%
500000
= 10%
(iii) A trader bought a pair of trousers at Sh 220 000 and sold them at a profit of 50%.
Calculate his selling price.
BP = Sh 220 000 %P = 50%
Profit
%P = x 100%
Buying price
Profit
50% = x 100%
220 000
Profit x 100
50 =
220 000
P = 50 22 000
P = Sh 110 000
SP = BP + P
SP = 220 000 + 110 000
∴SP = Sh 330 000
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Grade 7 Mathematics
Examples
(i) The market price of a book is Sh 45 000. The discount is Sh 5000. Calculate:
a) the selling price b) percentage discount.
Market price = Sh 45 000
Discount = Sh 5 000
Selling price = market price - discount
= 45 000 - 5 000
= 40 000
Discount
% discount = Market price
x 100%
Discount
= x 100%
Market price
= 11.11%
= 11% (to nearest whole number)
(ii)Amina purchased a table whose market price was Sh 50 000 and got a 20% discount.
a) How much was the discount? b) Calculate the selling price.
Market price = Sh 50 000
Discount = 20%
∴Discount = 20% market price
= 20 x 50 000
50
= Sh 10 000
selling price = market price - discount
= 50 000 - 10 000
= 40 000
Z a k at
Zakat is one of the five pillars of Islam.
The conditions for performing Zakat are:
1. A person’s goods or money must be equal to the NISAB.
2. One complete Hijra year must pass without decrease in the value of goods or money from
NISAB.
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Mathematics Grade 7
The nisab of silver is equal to 200 Dirham (about 624 g )
200 1
The nisab of gold is equal to 7 Dirham (about 89 7 g ) or the value of that.
If payment of the zakat on someone’s goods is obligatory, it has to be calculated at the fixed
measure ( 1 = 2.5%)
40
Examples
i) A woman has 1 kg of gold. Calculate the Zakat to be paid after a complete Hijra year?
The amount of zakat = 2.5% of her gold.
25
= x 100 = 2.5 gram of gold
1000
Exercise 2
1. Hussein bought a radio for Sh 120 000 and sold it at a loss of 30%.
Calculate his selling price.
2. By selling a piece of land at Sh 860 000, a farmer makes a profit of 40%.
How much did he pay for the land?
3. By selling a cow at Sh 80 000, a butcher makes a loss of 20%.
What was the buying price for the cow?
4. Abshir has Sh 80 000 for a complete Hijra year.
Calculate the obligatory zakat that Abshir should pay from this amount.
5. A shopkeeper wants to pay the zakat from his goods and money. After one complete year
he found that he has : 1 kg of gold 4 kg of silver Sh 200 000
2
Calculate the zakat he has to pay?
6. Husni started a business with Sh 150 000. After one complete year of Hijra, his capital was
Sh 200 000.
Calculate the zakat Husni paid at the end of the year.
7. A man wanted to pay the zakat on his goods and money. After calculating all his goods and
money, he found: 2 1 kg of gold 5 kg of silver Sh 250 000
2
Calculate the obligatory zakat payment.
8. A person’s wealth is Sh 1 600 000. Calculate the zakat that should be paid after a complete
year of Hijra.
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Grade 7 Mathematics
The ratio of the cost of Husni’s pens to the cost of Halima’s pens is
Cost of Husnis pens 3 x 4000 12000 12
Cost of Halimas pens
= 4 x 4000
= 16000
= 16
Examples
2 20 3 4 5 10
a) 3
, 30
b) 15
, 20
c) 6
, 18
20 2 3 1 10 5
Taking = (÷ 10) Taking = (÷ 3) Taking = (÷ 2)
30 3 15 5 18 9
2 20 4 1 5 10
∴ = Taking = (÷ 4) ∴ ≠
3 30 20 5 6 18
3 4
∴ =
15 20
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Mathematics Grade 7
Examples
2 3
Calculate ‘n’in = =
5 n
2n2xn = 5 x 3 (equate cross products)
2n 15
= (divide by 2)
2 2
1
n =7 2
Exercise 1
1. Write the number of elements in the fourth set to make a correct proportion statement.
number of elements in set 1 number of elements in set 3
=
number of elements in set 2 number of elements in set 4
6
4. The ratio of Ali to Nur ’s height is . If Nur’s height is 175cm, what is Ali’s height?
7
a) 175 cm b) 185 cm c) 90 cm d) 150 cm
5. A trader bought goods at Sh 825 000. He sold the goods at a profit.
2
If the ratio between the profit and the buying price is , find the profit.
11
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Grade 7 Mathematics
Examples
(i) If the cost of 7 books is Sh 7000, what is the cost of 13 books of the same type?
When the number of books increases, the cost will also increase:
number of ebooks in the first case
= cost of ebooks in the first case
number of ebooks in the second case cost of ebooks in the second case
7 7000
13
= x
13 x 7000 = 7 x x (using cross products)
13 x 7 000 7xx
7
= 7
(dividing both sides by 7)
x = Sh 13 000 13 books cost Sh 13 000.
(ii)5 shirts cost Sh 420 000. How much will 12 such shirt cost?
Find the cost of 1 shirt then find the cost of 12 shirts.
420000
1 shirt will cost 5
= Sh 84 000
∴12 shirts will cost 84 000 x 12 = Sh 1 008 000
(iii)The cost of 8 T–shirts is Sh 6000. If Abdi paid Sh 1500 how many T-shirts did he
buy?
8 6000
By proportion ∴ a
= 1500
a x 6000 = 8 x 1500
a = 12 000 ÷ 6000
a = 2 T-shirts
Abdi bought 2 T-shirts
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Exercise 2
1. 20 sacks of lemons cost Sh 300 000. How much do 175 sacks cost?
2. 6 books cost Sh 2100. What is the cost of 15 books?
3. 4 cartons of apples cost Sh 140 000. How much do 25 cartons of apples cost?
4. The weight of 7 cartons of soap is 315 kg. What is the weight of 16 cartons?
5. Ahmed’s taxi used 6 litres of petrol to travel 60 km.
a) How much petrol will it use to travel:
i) 105 km? ii) 75 km iii)225 km?
iv)90 km? v) 205 km?
b) How far will it go if it is filled with:
i) 2 litres? ii) 13 litres? iii)225 litres?
iv)20 litres? v) 303 litres?
6. 12 shirts cost Sh 35 000. How much will 40 shirts of the same kind cost?
7. For unloading work Hassan earned Sh 52 500 for 3 hour’s work.
How much will he earn in:
1 1
a) 2 2
h b) 18 h c) 6 h d) 9 h e) 8 2
h
Examples
(i) 12 workers can complete a piece of work in 5 days. But 60 workers can finish the same
work in 1 day.
1 worker can do the work in 60 days.
2 workers can do the work in less time, that is 30 days.
3 workers can do the work in even less time, 20 days.
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(ii) 5 men take 8 hours to build a wall. How long would 4 men take, working at the
same rate, to build the same wall?
5 men take 8 hours
1 man would take (8 x 5) hours = 40 hours
40
4 men would take 4
hours = 10 hours
(iii) 3 workers can complete a piece of work in 12 days. How many days will 9
workers take to complete the same task?
3 workers take 12 days
1 worker takes 3 x 12 days
9 workers take (3 x 12) ÷ 9
= 4 days
Exercise 3
1. Six men can weed a piece of land in 13 days.
How long would it take 4 men to clear the same piece of land?
2. At an average speed of 100 km/h Ali takes 6 hours to travel from Eldeer to Mogadishu.
How long would he take when driving at an average speed of 60 km/h?
3. When the price of sugar was Sh 7000 per kg a household sugar budget would buy 10 kg a
month. Sugar costs increased to Sh 18 000 per kg,
How much sugar can the same household budget buy?
4. Five builders complete the floor of a house in 12 hours. How long will it take 18 builders
to complete the same task?
5. 14 cows eat an amount of grass in 30 days. How long will the grass last if the cows are
reduced to 6?
6. A man travelled 25 km in 5 hours, how many hours will he take to travel a distance of
45 km at the same rate?
7. A dairy produces 4 kg of butter from 46 litres of milk, how many litres of milk does the
factory need to produce 200 kg of butter?
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Mathematics Grade 7
12 SCALE DRAWING
Line a r sc a le
Scale drawing is used to enlarge or reduce a plan or image.
A plan of a rectangular house, which is 20 m long and 15 m wide, cannot be drawn in an
exercise book or other similar sized book. However the plan can be, represented by a scale
drawing.
Consider these scales for the house plan:
(i) 1 cm represents 2m;
Using a scale of 1 cm to represent 2 m:
20 15
length 2
= 10 cm width 2
= 7.5 cm
(ii) 1 cm represent 4 m;
Using a scale 1 cm represents 4 m:
20 15
length 4
= 5cm width 4
= 3.75cm
(iii) 1 cm represents 5 m;
Using a scale 1 cm represents 5 m:
20 15
length 5
= 4cm width 5
= 3cm
(iv) 1 cm represents 10 m.
Using a scale 1 cm represents 10 m:
20 15
length 10
= 2cm width 10
= 1.5cm
5 cm
2 cm 2 cm
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Grade 7 Mathematics
To calculate the real perimeter of the plot: Scaled Real
Add the scale drawing lengths and multiply by 10 m: AB 1.5 cm
AB + BC + CD + DE + EF + FG + HG + AH BC
2 + 2 + 1 + 2 + 2 + 4 + 5 + 4 = 22cm CD
1710 = 170 cm DE
Real perimeter = 22 x 10 = 220m. EF
FG
HG 5 cm 5 x 10 = 50 m
AH 4 cm 4 x 10 = 40 m
Examples
A B
D C
What is:
a) the real length of AB? b) the real length of BC?
a) scaled AB = 11 cm b) scaled = 4 cm
∴ real AB = 11 x 2 = 22 m ∴ real BC = 4 x 2 = 8 m
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Exercise 1
1. The diagram represents a scale drawing of a roundabout road section. The scale is
1 cm: 10 m.
a) What is the real distance? b) What is the real width of the:
(i) between A and B? (i) North - South road?
(ii) P and Q? (ii)East - South road?
c) What is the real diameter of the roundabout (XY)?
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Examples
(i) What is the meaning of these RF scales in centimetres?
a) 1 : 200 b) 1 : 100 000 c) 1: 25 000
a) 1 : 200
1 cm, drawn length 200 cm or 2 m real length.
b) 1 : 100 000
1 cm, drawn length 100 000 cm or 1000 m or 1 km actual length.
c) 1: 250000
1 cm, drawn length 250 000 cm or 2500 m or 2.5 km real length.
(ii)Write 1 cm 8 m as a RF
8 m = 800 cm.
In ratio form - 1 : 800
RF is 1 : 800
Exercise 2
1. Write these representative fractions (RF) as ‘1 cm a real measurement’ in an appropriate
unit.
a) 1 : 400 b) 1 : 3000 c) 1 : 450 000 d) 1 : 600 000
e) 1 : 1000 000 f) 1 : 5500 g) 1 : 750 000 h) 1 : 400 000
2. Write these as RF:
a) 1 cm 10 m b) 1 cm 15 m c) 1 cm 25 m d) 1 cm 150 m
3 1
e) 1 cm 4
km f) 1 cm 12 km g) 1 cm 4.5 km h) 1 cm 3.5 km
1
i) 1 cm 10 km j) 1 cm 3 4
km k) 1 cm 12.5 km l) 1 cm 0.6 km
Dra w ing t o sc a le
Examples
(i) Using the scale 1 : 200 make a scale drawing of a rectangular plot of land measuring 8 m by 4 m.
Calculate the length and width of the scale diagram to be drawn.
1 : 200 means that
1 cm 200 cm or 2 m 4m
8 4
length is = 4 cm; width is = 2 cm 8m
2 2
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(ii) Using the scale 1 : 5000 make a scale drawing of a rectangular plot of land measuring
450 m by 50 m.
Calculate the length and width of the scale diagram to be drawn.
1 : 5000 means 1 cm 5000 cm or 50 m.
450 4
length is 50
= 9 cm; width is 2
= 1 cm.
1 cm
9 cm
Exercise 3
1. Using 1 cm 10m, make a scale drawing of these rectangles;
a) 50 m by 45 m. b) 65 m by 50 m
c) 85 m by 55 m d) 100 m by 80 m
e) 90 m by 70 m f) 75 m by 60 m
g) 45 m by 90 m h) 150 m by 50 m
2. Using a scale of 1 : 100 make scale drawing of:
a) rectangles measuring:
(i) 5 m by 3 m ii) 6.5 m by 5.5 m
(iii) 4.8 m by 4.2 m (iv) 8.4 m by 7.3 m
b) equilateral triangles with sides of:
(i) 6 m (ii) 8 m
(iii) 5.5 m (iv) 10 m
3. Using 1:200 make scale drawing of the these rectangles:
a) 12 m by 8 m b) 9 m by 7 m
c) 10 m by 10 m d) 14 m by 12 m
e) 16 m by 14 m f) 13 m by 13 m
g) 8.6 m by 7.4 m h) 9.2 m by 6.8 m
4. Using 1 : 5000 make scale drawings of the these rectangles:
a) 200 m by 150 m b) 350 m by 30 m
c) 420 m by 360 m d) 450 m by 400 m
e) 280 m by 200 m f) 380 m by 320 m
5. Using 1 : 100 000 make scale drawings of the these rectangles:
a) 2.5 km by 2 km b) 7.5 km by 6.4 km
c) 8 km by 7.2 km d) 9 km by 8 km
e) 5.2 km by 4.5 km f) 4.7 km 4.1 km
g) 7 km by 7 km h) 6 km by 6 km
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13 GRAPHS
Coordina t e pla ne
y-axis, or vertical
Origin, where number line
axes intersect
x-axis, horizontal
number line
A coordinate plane
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Examples
(i) Plot point A with coordinates (3, - 2)
(0, 5)
(-3, 1)
(1, -2)
(-4, -4)
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Points can be referred to as being in one of the four sections or quadrants of the coordinate
plane.
(0, 5) is in the 1st quadrant
(-3, 1) is in the 2nd quadrant Quadrant Quadrant
rd
(-4, -4) is in the 3 quadrant II I
(1, -2) is in the 4th quadrant
Quadrant Quadrant
III IV
Exercise 1
1. Identify the coordinates of:
A, B, C, D, E, F, G and H.
y- axis
x- axis
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Answer questions 4 to 6 using this coordinate plane.
C N M
D L
F J K
G H
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9. Identify the quadrant in which each can be located:
a) (3, 2) b) (-17, 2) c) (-6, -40) d) (9, -11) e) (-1, 100)
f) (3, 38) g) (0, 2) h) (-4, 1) i) (-1, -3) j) (2, -2)
10.Which of these points is in the fourth quadrant?
a) (-2, 2) b) (2, -2) c) (-2, -2) d) (2, 2)
11.Three corners of a rectangle have coordinates (4, 2), (4, 7) and (-3, 2). Find the coordinates
of the fourth corner.
Ba r gra phs
A bar graph can represents information or data. It should pictorially aid the comparison of
quantities.
Drawing a bar graph involves:
• using suitable horizontal and vertical scales for the data;
• using suitable scales for the size of the page;
• using scales which could be in 1, 2, 5, 10, 20, units per cm;
• drawing bars of equal width;
• labelling the axes;
• providing a title for the graph.
Examples
(i) The table represents class attendance for one week
Draw a bar graph to represent the data.
Day Saturday Sunday Monday Tuesday Wednesday Thursday
Attendance 35 38 40 35 37 32
To draw a bar graph:
Choose a suitable scale: 1 cm 10 pupils on the vertical axis
This means that the highest class attendance of 40 pupils can be represented by 4 cm.
Class attendance per week
Notice that
the bar graph has a title Days
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(ii)Draw a bar graph representing the information which shows the amount of rainfall
recorded over a period of six month in mm.
mm
(iii) The table below gives the number of pupils attending eight primary schools. Draw a
bar graph representing this information:
School A B C D E F G H
Pupils 500 350 100 250 200 300 150 400
Enrolment at eight primary schools
Schools
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Exercise 2
1. The bar graph shows the daily rainfall, over a week in Balad.
a) How much rain fell in the week?
b) Which was the wettest day?
c) What was the difference between highest and the lowest rainfall?
mm
a) On which day did the largest number of people visit the restaurant?
b) On which day did the least number of people visit the restaurant?
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c) How many more people went to the restaurant on Tuesday than on Thursday?
d) Does the graph show which day the restaurant earned most money?
4. Draw a bar graph to show this information for 1 month’s egg production:
Hen A B C D E
Eggs 26 17 24 20 16
5. This bar graph shows primary 7 enrolment at Darwish primary school (1996 - 2001).
Year
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7. This bar graph shows the score of a grade 7 pupil in subject tests.
Test scores
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Examples
(i) The table shows the votes given to 4 candidates in an election.
Draw a pie chart to illustrate this information:
64o Ahmed
96o
Osman
80o Hussein
120o Jama
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36o
Banana
72o Livestock
180o
Vegetable
Other products
72o
(iii) A farmer has 20 goats, 5 cows, 15 camels, 10 sheep. Draw a pie chart to show this
information:
Sum of the animals = 20 + 5 +15 +10 = 50
20 5
sector ∠ for goats = 50
x 360 = 144o sector ∠ for cows = 50
x 360 = 36o
15 10
sector ∠ for camels = 50 x 360 = 108o sector ∠ for sheep = 50
x 360 = 72o
72o Goats
144o Cows
Camels
108o Sheep
36o
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Exercise 3
1. Ahmed records the amount of time he spends on various activities during a day.
Grey
Green
Blue
Red
Yellow
Visit friends
Cultivating
Reading
Resting
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14 AVERAGE
Mean
A survey of 10 Grade 7 pupils showed how many oranges they eat every 3 days.
The survey showed that 9 pupils eat
Pupils Oranges 2 oranges each:
9 2 9 x 2 = 18 oranges
1 12 1 pupil eats 12 oranges
1 x 12 = 12 oranges
The total oranges eaten in 3 days by the 10 pupils is:
18 + 12 = 30
Total eaten
Mean number of oranges eaten = Number of pupil
30
Mean = 10
= 3 oranges
Total of item
∴Mean = Number of item
Examples
(i) Mohamed scored 65, 94, 50, 86, 66 and 70 in his Grade 6 examination. Find
Mohamed’s mean score.
Total scored
Mean = Number of exams
65 + 94 + 50 + 81 + 66 + 70
= 6
426
= 6
Mean = 71 marks
(ii)The mean height for 5 pupils is 1.61 m. The heights for 4 of them are 1.55 m, 1.65 m,
1.71 m, and 1.56 m. What is the height of 5th pupil?
Total height of the 5 pupils = mean height number of pupils
= 1.61 x 5 = 8.05 m
Total height of 4 pupils = 1.55 + 1.65 + 1.71 + 1.56
= 6.47 m
∴ The height of the 5th pupils = 8.05 - 6.47
= 1.58 m
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Exercise 1
1. What is the mean of:
a)10 m, 8 m, 14 m, 15 m, 16 m, 11 m, 9 m, 13 m,?
b) 0.5, 0.3, 0.9, 0.7, 0.4, 0.8?
c) 3.5 m, 3.2 m, 2.4 m, 2.6 m, 2.8 m?
2. Mohamed sat for three Science tests. He scored 55 marks, 68 marks and 72 marks. What
was his mean mark for the three tests?
3. The mean age of three girls is 9 years. Two of the girls are 11 years and 7 years
respectively. Find the age of the third girl.
4. The mean temperature for the first two days of a week was 38oC and the mean
temperature for the next two was 34oC. If the mean temperature for the fifth day of the
week is 32oC, what is the mean temperature for the last two days of the week?
5. The mean weight of 8 boys is 44 kg. The weights of 7 of the boys are:
43 kg, 45 kg, 43 kg, 48 kg, 45 kg, 46 kg, 45 kg.
What is the weight of the eighth boy?
6. This table shows the number of shoes sold by a trader in a week
M ode
Look at these numbers 2, 3, 4, 6, 6, 9.
The mode is 6, because 6 happens most often.
Mode is the number, event or item which appears more than others or which is the most
frequent.
Examples
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Exercise 2
1. Find the mode of each of the following numbers
a) 7, 2, 6, 8, 2, 5, 7, 2 b) 13, 15, 17, 15, 16, 12, 15, 16
c) 30, 70, 30, 40, 30, 80 d) 740, 730, 780, 740, 780, 740
e) 22, 23, 25, 20, 24, 23, 20, 23 f) 81, 82, 78, 74, 83, 85, 78, 80, 81
2. A fisherman caught 10 fish. Their weights were: Score Pupils
2.2 kg, 0.7 kg, 1.5 kg, 2.1 kg, 1.9 kg, 2.1 kg, 50 6
2.2 kg, 1.8 kg, 0.6 kg, 1.9 kg 65 8
a) What was the mean weight of the fish? 70 9
b) What is the modal weight of fish? 75 6
3. The table shows examination scores and the number 80 5
of pupils who attained each score. 85 4
a) How many pupils sat for the examination? 90 2
b) What is the modal score? 95 1
c) What was the mean score? 100 1
4. The number of people who attended eight inter-school football matches were:
1200, 1000, 1200, 1100, 1300, 1200, 1200, 1000
a) What was the mean attendance?
b) What is the modal attendance?
5. Identify the modes:
a) 2.5, 2.7, 2.9, 2.5, 2.8 b) 12, 27, 13, 12, 18, 13, 14, 12, 16, 13
c) 45, 49, 47, 48, 47, 46, 45
M e dia n
Median is the midway (halfway) mark between the highest and the lowest value. To identify
the median of: 7, 20, 18, 11, 15, 17, 6 and 12?
Cancel the numbers from both ends until a middle number is reached:
6, 7, 11, 12, 15, 18, 20
or
20, 18, 15, 12, 11, 7, 6
∴12 is the median
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Mathematics Grade 7
Sometimes we can get two numbers.
Add the two numbers and divide the sum by 2. The result is the median.
Examples
(i) Identify the median of 1, 8, 5, 2, 3, 6, 6, 7, 10
In ascending order: 1, 2, 3, 6, 6, 7, 8, 10
∴The median = 6
(ii)Identify the median of 22, 40, 83, 20, 16, 48, 52, 11
In ascending order: 11, 16, 20, 22, 40, 48, 52, 83
22 + 40 62
Median = 2
= 2
∴Median = 31
Exercise 3
1. Order the data and find the median of each group:
a) 70, 90, 76, 102, 79, 80
b) 3, 18, 20, 17, 22, 20, 19
c) 10, 5, 7, 13, 14, 12, 12, 10
d) 111, 105, 100, 101, 92, 87, 96, 92, 95
2. The daily temperatures in oC at noon for one week were:
26, 24, 30, 25, 27, 29 and 23
a) Order the data values
b) Calculate the median temperature.
c) Calculate the mean temperature.
3. This table shows the tally of students’ test scores in a test.
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5. Halima kept a record of the number of eggs laid each day by her hens:
9, 10, 13, 12, 10, 11, 12, 9, 9, 9, 9, 11, 12, 11, 12, 13, 11, 10, 10, 11
a) Calculate the mean numbers of eggs per day.
b) What is the median number of eggs per day?
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15 ALGEBRA
U sing indic e s
The number 9 can be written as:
9 = 3 x 3 or 3 2
For 3 2 we say:
• 3 to the 2 nd power or
• 3 to the power of 2 or
• 3 squared.
The small 2 is known as an index. In 32, 2 is the index of 3.
Also 2 is the power to which 3 is raised.
The numbers 27 and 8 can be thought of as numbers raised to the power of 3.
27 can be written as: 3 x 3 x 3 or 33
This is read ‘three to the third power’ or ‘three cubed’.
8 can be written as: 2 x 2 x 2 or 23
This is read as ‘two to the third power’ or three cubed.
34 = 3 x 3 x 3 x 3 = 81 Here the power or index 4 shows how many times 3
4 factors occurs as a factor.
2 x 2 x 2 x 2 x 2 = 25 = 32 Here the index 5 shows how many times 2
5 factors occurs as a factor
therefore
x x x x x x x = x4 x x x x x x x x x x x = x6
4 factors 6 factors
xn = x x x x ……. x
n factors or n times
The index shows the number of times the base occurs as a factor.
Indices is the plural of index.
index
xn = the nth power of x
base
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Examples
I nde x la w s
Expressions involving indices can often be simplified using the index laws.
Multiplication
34 x 32 = 3 x 3 x 3 x 3 x 3 x 3 x 3 = 36 x4 x x2 = x x x x x x x x x x x = x6
So 34 x 3 2 = 34 + 2 = 36 So x4 x x2 = x4 + 2 = x6
Division
2x2x2x2x2 xxxxxxxxx
25 ÷ 2 3 = 2x2
= 22 x5 ÷ x3 = xxx
= x2
So, 2 5 ÷ 2 = 25 - 3 = 22
3 So, x5 ÷ x = x = x2
3 5-3
Powers of Powers
(52)3 = 52 x 52 x 52 = 52 + 2 + 2 = 56 (x2)3 = x2 x x2 x x2 = x2+2+2 = x6
(52)3 = 52 x 3 = 56 (x2)3 = x2 x 3 = x6
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Mathematics Grade 7
Powers of a products
(53 x 74)2 = (53 x 74) x (5 3 x 74) (x3 x y4)2 = (x3 x y 4) (x3 x y4)
= 53 x 53 x 74 x 74 = x3 x x3 x y4 x y4
= 56 x 78 = x6 x y8
(53 x 74)2 = 53 x 2 x 74 x 2 = 56 x 7 8 (x3 x y4)2 = x3 x 2 x y4 x 2 = x6 x y8
The index law for the power of a product:
(xayb)n = xan x ybn
Power of a quotient
75 75 75 x5 x5 x5
( )2 = ( ) ( ) ( )2 = ( ) ( )
53 53 53 y3 y3 y3
75 x 75 710 x5 x x5 x10
= = = =
53 x5 3 56 y3 xy 3 y6
75 75 x 2 710 x5 x5 x 2 x10
( )2 = ( )= ( )2 = ( )=
53 53 x 2 56 y3 y3 x 2 y6
Examples
(i) simplify:
a) (2x) (3x2)
b) (x 2y) (x4y3)
14x 5
c)
7x 3
a) (2x) (3x2) = 2 x 3 x x x x2 = 6x3
b) (x2y) (x4y3) = x2 x x4 x y x y 3 = 6y4
14x 5 14 x5 14
c) = 7 x = 7 x5-3 = 2x2
7x3 x3
(ii) Simplify:
(6a2b3) (2a3b5) 6 x 2a2+3 b3+5 12
= = 4 a5-2b8-2 = 3a 3b6
(2ab)2 22a2b2
(iii) Simplify
a) (2x2y)(3x3y4)
18a 4b5
b) 12a2b
a) (2x2y)(3x3y4) = 2 x 3 x x2 x x3 x y x y 4 = 6x5y5
18a 4b5 18 a4 b5 18 3
b) = 12 x x = 12 x a4-2b5-1 = 2 a2b4
12a2b a2 b
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Grade 7 Mathematics
Exercise 1
1. Simplify:
a) (x3) (x5) b) (a2) (a10) c) (b) (b) (b)
d) (m2) (m3) (m) e) (a2) (a3) (b) (b 4) f) (a) (b) (a2) (b3)
2. Simplify:
a) a7 ÷ a3 b) b3 ÷ b2 c) n12 ÷ n3
d) x8 ÷ x5 e) a5 ÷ a f) x10 ÷ x9
3. Simplify:
a) (x4)2 b) (a3)3 c) (a2b)3
d) (xy3)5 e) (abc)5 f) (b8)3
g) (2x3)3 h) (a5b2)3 i) (3a5)2
4. Simplify:
x a2 x2 a
a) ( y
)5 b) ( )2 c) ( )3 d) ( )4
3 y b5
3x 2x2 3 2 3a
e) ( y
)2 f) ( ) g) ( 5a3 )2 h) ( )2
w 2b b3
5. Simplify:
a) (5a3)(3a6) b) (4x2)(-2x3) c) (3a2b3)(2ab2)
d) (2x)(5x3) e) (5mn)(3m) f) (-4x3)(-3x2)
g) (3y)(5y)(2y2) h) (a)(2a2)(-3a5)
6. Simplify:
(81b 2)(3a2b) (3x2y5)3 (12m2n5)(-5mn3)
a) b) c)
12a3 9xy2 15m3n2
7. Simplify:
a) 12a5 ÷ 3a 3 b) 21x2y5 ÷ 7xy c) 8a2 ÷ 8a
The number part of a term is called the numerical coefficient. A term may also have a letter or
variable part. Expressions are made up of terms. Like terms can be added or subtracted like
you would in arithmetic.
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Exercise 2
1. Add the like terms as you would in arithmetic.
a) b) c) (d)
3x + 4y 4x2 + 3x 7xy + 5w 5x - 6y
2y + y 7x2 + 10x 20xy - 2w 2x + 5y
Examples
To add expressions, collect the like terms
(i) Add
(-1) + 4 + 3 = 6. Write 6 in the coefficients column.
2x2 + 3x - 1
3x + 2x + (-x) = 4x. Write 4x in the x column.
x2 + 2x + 4
2x2 + x2 + 3x2 = 6x2 Write 6x2 in the x2 column.
2x2 - x + 3
6x2 + 4x + 6
(ii)Simplify: (x2 - 3xy + 2y 2) + (2x2 - xy - y 2)
(x2 - 3xy +2y2) + (2x2 - xy - y2)
= x2 - 3xy + 2y 2 + 2x2 - xy - y2
= 3x2 - 4xy + y 2
To subtract expressions write the opposite subtrahend and add.
(iii) Subtract.
4x2 - 3x + 1
x + 2x -3 is the subtrahend - the value to be subtracted.
x2 + 2x - 3
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Exercise 3
1. Subtract the first expression from the second.
a) 2x + 3y; 5x + 6y
b) 3w + 2x + 5y; 7w + 11x + 9y
2
c) 6x – 5x + 8; 4 – 3x + 2x2
2. Simplify.
a) (x3 + 6) + (x + 3)
b) (x2 + 2x ) + ( - 7x + 2)
c) (y3 + 2y2 + 3) + (4y 2 - 3y – 1)
3. Simplify.
a) (6x – 3) – (7x + y)
b) (7y2 - xy) – (8x2 + xy)
c) (13x3 - 3x2 - xy) – (13y 3 - 3y2 - xy)
4. Simplify.
a) (3x2 - 2y) – (2y + 3x2) + (x2 -2y)
b) (-3xy – 2y 2) – (x2 - y2) + (3x2 - xy)
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The product (x + 5)(x + 3) can be shown in a diagram as the area of a rectangle with sides
(x + 3) and (x + 5).
Area of rectangle = (x + 5)(x + 3)
Area of rectangle = A1 + A2 + A3 + A4
= x2 + 3x + 5x + 15
= x2 + 8x + 15
(x + 3)(x +5) = x2 + 8x + 15
The distributive property is also used to multiply two expressions.
(a + b) (c + d) = a(c + d) + b(c + d)
= ac + ad + bc + bd
Similarly: (x + 5) (x + 3) = x (x + 3) + 5(x + 3)
= x2 + 3x + 5x + 15
= x2 + 8x + 15
or
(x + 5) (x + 3) = (x + 5)( x + 3)
= (x + 5)x + (x + 5)3
= x2 + 5x + 3x + 15
= x2 + 8x + 15
Examples
(i) Expand and simplify. (2x – 3)(5x + 7)
(2x – 3)(5x + 7) = 10x2 + 14 x – 15 x – 21
= 10x2 - x - 21
(ii)Expand and simplify. 2(2x + 1)(x – 3) – 4(x + 5)
2(2x + 1)( x – 3) – 4(x + 5) = 2(2x2- 6 x + x – 3) – 4(x + 5)
= 2(2 x2 - 5x – 3) – 4(x + 5)
= 4 x - 10 x – 6 – 4 x – 20
= 4 x - 14 x – 26
Exercise 4
1. Determine the area of each of the smaller rectangles, then add them together to expand
(x + 3) (x + 2). x+3
x
+
2
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2. Draw rectangles to help you expand and simplify these:
a) (P + 3) (p + 1) b) (r + 2) (r + 5) c) (m + 4)
3. Expand.
a) (x + 1)(x + 2) b) (x + 2)(x + 3) c) (x + 4)(x + 2)
d) (x – 1)(x – 3) e) (x – 3)(x + 5) f) (x + 2)(x – 1)
g) (y – 5)(y – 4) h) (t + 7)(t + 8) i) (m – 4)(m + 9)
j) (n – 2)(n – 9) k) (x + 8)(x + 6) l) (y + 1)(y – 7)
m) (x + 7)(x + 6) n) (t – 5)(t – 9) o) (m – 6)(m – 11)
4. Expand and simplify.
a) (2x + 3) (x + 5) b) (3x + 4)(2x + 7) c) (7y – 2)(2y + 5
d) (2m – 5)(3m – 1) e) (2x – 5)(2x + 5) f) (3x + 5)
g) (5m + 2n)(4m – n) h) (4t + 7)(2t + 3) i) (3x – 2y)(4x –3y)
j) (4m – 3) k) (5t – 6) l) (7t + 4)
5. Expand and simplify
a) (3x + 5y) b) (3x + 7y)(4y – x) c) (7 – 8t)(7 + 8t)
d) (4x – 9y)(2y + 7x) e) (9x + 10y)(8x + 3y) f) (2m – 5n)
6. Expand and simplify.
a) 2(x + 3)(x + 4)+ 3(2x + 3) b) 3(x + 1)(x + 2) + 2 (x + 4)(x + 5)
c) 5(t – 3)(t + 4) – 5(t – 6) (t – 5) d) 2(m + 3) + 3(m – 1) - 2(m – 4)
Spe c ia l produc t s
These examples suggest a pattern for squaring.
Add the square of the first term to the square of the last, with twice the product of both terms.
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Examples
(i) Using the pattern for special products simplify: (2x + 3y).
(a + b) 2 = a 2 + 2ab + b 2
Exercise 5
1. Expand
a) (x + 3) 2 b) (x – 2)2 c) (x + 5) 2
d) (x – 4)(x + 4) e) (y + 2) 2 f) (m – 7)2
g) (t + 5)(t – 5) h) (x + 6)(x – 6) i) (y + 1)2
j) (x – 9) 2 k) (x + 10) 2 l) (x – 6)2
m) (x + 12)2 n) (x – 6)2 o) (y – 1)(y + 1)
2. Expand and simplify.
a) (3x – 5)(3x + 5) b) (2x + 7) 2 c) (4x + 5) 2
d) (2x – 3y) 2 e) (5x – y)(5x + y) f) (a + 2b)2
g) (ab + 2)(ab – 2) h) (x2 - 3m)(4 + 3m) i) (4 – 3m)(4 + 3m)
3. Evaluate using one of the special products (a + b)2 or (a – b) 2 :
a) 522 b) (40 – 5)(40 + 5) c) 95 2
d) 712 e) 101 f) (30 + 3)(30 – 3)
g) 832 h) (4 + 50)(4 - 50)
4. Expand and simplify.
a) 2x + (x + 4)2 b) x2 - (2x – 1) 2 c) 3ab + (2a – 7b)2
d) –4x2 + (3x – y)(3x + y) e) (2a – 3b) 2 - (a 2 + 4ab)
f) (ab – 2c)(ab + 2c) – (ab)2
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5. Expand and simplify.
a) (x + 2) 2 + (x – 5) 2 b) 2(x – 4)(x + 4) - x2
c) (x + 5)(x – 3) + (x + 4)2 d) 7 – (x – 6)2
e) 3 + 2x)(3 – 2x) – (3 + 2x)2 f) 4(5x – 1) - 2(3x + 1)(3x – 1)
6. Evaluate using the special product ( a + b)(a – b):
a) 87 x 93 = (90 – 3)(90 + 3) b) 105 x 95 = (100 + 5)(100 -5)
c) 48 x 52 d) 29 x 51
Examples
(i) Solve the equation 2(6x + 2) = 2(2x + 10)
2(6x + 2) = 12x + 4
12x + 4 = 4x + 20
12x – 4x + 4 = 4x – 4x + 20 First work the brackets on both sides of
8x + 4 = 20 the equal sign.
8x + 4 – 4 = 20 – 4 Subtract 4x from both sides.
8x = 16 Subtract 4 from both sides.
8x
= 8
16 Divide both sides by 8.
8
x=2
Note: A term such as 4x is regarded as having a + sign in front of it.
(ii)Solve 6n = 2n - 56
6n – 2n = 2n - 56 – 2n
4n = -56 Subtract 2n from both sides.
4n -56 Divide both sides by 4.
4
= 4
n = -14
Subst it ut ion
a(abc + d 2)
(i) Evaluate a when a = 1; b = 3; c = 5 and d =7
b2- a 2
1(3 x 5 + 7 2)
Substitute numbers for the letters: 32 – 12
1(15 + 49) 1 x 15 + 1 x 49
9–1 = 8 remove the brackets
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Mathematics Grade 7
15 + 49
= 8
64
= 8
=8
2(a2 + b2)
(i) Evaluate when a = 3; b = 5; c = 4 and d = 2
3c – d 2
substitute numbers for the letters:
2(32 + 5 2) 2(9 + 25)
3 x 4 – 22
= 12 – 4
2 x 9 + 2 x 25
= 12 – 4
18 + 50
= 8
68
= 8
17
= 2
1
= 82
Exercise 6
1. Solve:
a) 3a + 2 = 7 b) s + 9 = - 11 c) 3(4 + p) – 2(2 – p = 20
d) 2(m + 3) = 12 e) 2(v + 4) = 3(v – 5) f) 7d – 2(d + 3) = 9
g) 5(c + 5) = 4(2c + 5) h) 4(q + 1) – 3(5q – 7) = 16
2. Evaluate the following given that r = 4; s = 3; t = 2 and u = 1.
a) 2t + u b) 5(r + t) c) t + r 2 + u2
2(r – t)
d) s2 + t2 e) r2 – t2 f) 5(s + u)
r2 + u 2 3s 2 – r2
g) h)
s2 r2 – 2u2
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4. If w = 3 x = 2 and y = 4 evaluate:
a) 2xy b) 3wxy c) 3x + 4w + 5y d) 3x-24
2 2
e) w + x + y 2 f) w - x - y g) 2x2 x 2 – 3x – 4h) wx - xy - 2wy
2 2
i) 3x – 2xy – y j) 4(3x - 2y) k) 2w2 x -3xy2 l) w3 - x4 -y2
5. If w = -1, x=-z and y = 3, evaluate
a) 3w + 2 x 4y b) 5x - 34 c) 3wxy d) – 4wxy
e) w - x - y f) w2 + x2 + y2 g) x2 - w 2 - y2 h) 3wx - xy + y
i) 3w2x2 - 4xy2 j) w5 – 2x3 – 7 k) –2(w3 - 3w2) l) w2 - 2x3 -7
I ne qua lit y
Reminders:
{1, 2, 3, 4, 5, 6, 7…..} is a set of counting numbers
{0, 1, 2, 3, 4, 5, ……} is a set of whole numbers
{….., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ………} is a set of integers
This line segment shows whole number between 1 and 6 but not including 1 and 6.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Examples
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Exercise 7
1. Write inequalities of each of the following:
a) p is greater than 4 b) q is less than 12 c) r is greater than 15
d) k is less than –3 e) m is greater than –8 f) n is less than –17
2. Write inequalities for the integer i that satisfy the following sets:
{7, 8} {5, 4, 3} {-3, -2}
{-8, -7, -6, -5} {-2, -3, -4} {-2, -1, 0, 1, 2, 3}
{-3, -2, -1} {1, 2} {-1}
3. What inequalities are represented in each number line below?
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