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RATIO AND PROPORTION

If the ratio of two numbers is a : b then the two numbers can be assumed as ax
and bx where x is the H.C.F (Highest Common Factor) of the two numbers.
In the above ratio a and b are known as the terms of the ratio.

Some Formulae:
(i) Two numbers are in the ratio of a : b. If both are increased/decreased by k then
the ratio of the two numbers becomes c : d then the value x (H.C.F) is given by
| c − d | ×k
x=
| ad − bc |
The two numbers will be ax and bx.
(ii) If the ratio of two numbers A and B is N1 : D1 and that of B and C is N 2 : D2 then
the ratio of three numbers i.e. A : B : C = ( N1 × N 2 ) : ( D1 × N 2 ) : ( D1 × D2 ) .
(iii) Three numbers are such that k1 times the first number, k 2 times the second
number and k 3 times the third number are all equal. then the three numbers are in
the ratio of
1 1 1
: : .
k1 k 2 k 3
(iv) If two ratios a : b and c : d have equal values then a, b, c and d are said to be in
proportion i.e. a : b = c : d. Here a and d are called the ends and b and c are called
means of the proportion. In a proportion the product of extremes is equal to the
product of the means.
a× d = b× c
(v) The number to be subtracted from each of a, b, c and d so that they become
proportional
ad − bc
is .
(a + d ) − (b + c)
Similarly the least number to be added to each of a, b, c and d so that they
ad − bc
became proportional is
(b + c) − (a + d )

PROBLEMS
1. The average age of three boys is 25 years and their ages are in the proportion 3 : 5
: 7. The age of the youngest boy is:

1) 21 years 2) 18 years 3) 15 years 4) 9 years 5) None of these

ANSWER: 3
Total age of three boys = 25 × 3 = 75
Their ages are in the ratio of 3 : 5 : 7 the age of youngest boy
⎛ 75 ⎞ 75
=⎜ ⎟×3 = ×3
⎝3+5+7⎠ 15
= 15 years

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2. A box contains 1-rupee, 50-paise and 25-paise coins in the ration 8 : 5 : 3. If the
total amount of money in the box is ` 112.50, the number of 50-paise coins is
1) 80 2) 50 3) 30 4) 42 5) None of
these

ANSWER: 2
The coins are in the ratio of 8 : 5 : 3
So their numbers can be assumed as 8 x , 5 x and 3 x
1 1
Total value of coins = 8 x × 1 + 5 x × + 3 x ×
2 4
5 x 3 x 32 x + 10 x + 3 x 45 x
= 8x + + = =
2 4 4 4
45x
= 112.5
4
112 .5 × 4 450
x= = = 10
45 45
Number of 50-paise coins = 5×10 = 50

3. Two natural numbers are in the ratio 3 : 5 and their product is 2160. The smaller
of the numbers is
1) 36 2) 24 3) 18 4) 12 5) None of these

ANSWER: 1
Since the ratio of two numbers is 3 : 5, the two numbers can be assumed as 3 x
and 5 x
Their product = 3 x × 5 x = 15 x 2
But 15x 2 = 2160
2160
x2 = = 144
15
x = 144 = 12
Smaller number is 3×12 = 36

4. What must be added to each term of the ratio 7 : 11 so as to make it equal to 3 : 4

1) 8 2) 7.5 3) 6.5 4) 5 5) None of these

ANSWER: 4

The terms of the ratio 7 : 11 are 7 and 11 (not 7 x and 11 x )

If ‘k’ be the number added to each term so that ratio becomes 3 : 4
then 7 + k : 11 + k = 3 : 4
4(7 + k) = 3 (11 + k)
28 + 4k = 33 + 3k
k = 33 – 28 = 5

5. If W1 : W2 = 2 : 3 and W1 : W3 = 1 : 2 then W2 : W3 is
1) 3 : 4 2) 4 : 3 3) 2 : 3 4) 4 : 5 5) None of
these

ANSWER: 1

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W1 : W2 = 2 : 3 W2 : W1 = 3 : 2
W2 3 W 1
= also 1 =
W1 2 W3 2
W2 W1 3 1 W2 3
× = × =
W1 W3 2 2 W3 4
W2 : W3 = 3 : 4

2
6. ` 2040 are divided among A, B and C such that A gets of what B gets and B
3
1
gets of what C gets. Then B’s share is:
4
1) ` 180 2) ` 240 3) ` 360 4) ` 480 5) None of
these
ANSWER: 3
A + B + C = 2040
2 1
A = B and B = C
3 4
A 2 B 1
= and =
B 3 C 4
A : B : C = (2 × 1) : (3 × 1) : (3 × 4) = 2 : 3 : 12
⎛ 2040 ⎞ 2040
B’s share = ⎜ ⎟×3 = × 3 = 360
⎝ 2 + 3 + 12 ⎠ 17

7. ` 1870 are divided into three parts in such a way that half of the first part, one-
third of the second part and one-sixth of the third part are equal. The third part is:
1) ` 510 2) ` 680 3) ` 850 4) ` 1020 5) None of
these
ANSWER: 4
x , y and z be the three parts then x + y + z = 1870
1 1 1
Also x = y = z
2 3 6
x:y:z=2:3:6
⎛ 1870 ⎞ 1870
z =⎜ ⎟×6 = × 6 = 1020
⎝2+3+6⎠ 11

8. A sum of ` 1300 is divided among P, Q, R and S such that:

P ' s share Q ' s share R ' s share 2
= = =
Q ' s share R ' s share S ' s share 3
What is P’s share?
1) ` 320 2) ` 240 3) ` 160 4) ` 140 5) None of
these

ANSWER: 3
P + Q + R + S = 1300
P 2 Q 2 R 2
= = =
Q 3 R 3 S 3

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P : Q : R : S = (2 × 2 × 2) : (3 × 2 × 2) : (3 × 3 × 2) : (3 × 3 × 3) = 8 : 12 : 18 :
27
⎛ 1300 ⎞ 1300
P’s share = ⎜ ⎟×8 = × 8 = 160
⎝ 8 + 12 + 18 + 27 ⎠ 65

9. The ages of X and Y are in the ratio of 3 : 1. Fifteen years hence, the ratio will be 2
: 1. Their present ages (in years) are:
1) 30, 10 2) 45, 15 3) 21, 7 4) 60, 20 5) None of
these

ANSWER: 2
The ages of X and Y be 3 x and x
Fifteen years hence their ages will be (3 x + 15) and ( x + 15)
But (3 x + 15) : ( x + 15) = 2 : 1
(3 x + 15) × 1 = ( x + 15) × 2
3 x + 15 = 2 x + 30
x = 15
The present ages are 3 × 15 = 45 and 15 respectively.

10. A and B are two alloys of gold and copper prepared by mixing metals in the ratio
7 : 2 and 7 : 11 respectively. If equal quantities of the alloys are melted to form a third
alloy C, the ratio of gold and copper in C will be:
1) 5 : 9 2) 5 : 7 3) 7 : 5 4) 9 : 5 5) None of
these

ANSWER: 3
Take the quantity of each of A and B as L.C.M of sum of the terms of the ratio.
Quantity of each of A and B = L.C.M of [(7 + 2), (7 + 11)]
= L.C.M of (9, 18) = 18
⎛ 18 ⎞
Gold in alloy A = ⎜ ⎟ × 7 = 14
⎝ 7 + 2 ⎠
⎛ 18 ⎞
Gold in alloy B = ⎜ ⎟×7 = 7
⎝ 7 + 11 ⎠
In alloy C quantity of gold = 14 + 7 = 21
In alloy C quantity of copper = (2×18 – 21) = 15
Gold to copper ratio = 21 : 15 = 7 : 5

11. A mixture contains milk and water in the ratio of 5 : 1. On adding 5 litres of water,
the ratio of milk to water becomes 5 : 2. The quantity of milk in the original mixture
is:
1) 16 liters 2) 25 liters 3) 22.75 liters 4) 32.5 liters 5) None of
these

ANSWER: 2
Milk and water in the mixture be 5 x and x
When 5 litres water is added then milk and water will be 5 x and x + 5
But 5 x : x + 5 = 5 : 2
5 x × 2 : ( x + 5)×5
10 x = 5 x + 25

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x=5
Milk in the original mixture = 5×5 = 25 litres

1 1
12. Two equal glasses are respectively and full of milk. They are then filled with
3 4
water and the contents mixed in a tumbler. The ratio of milk and water in a tumbler
is:
1) 7 : 5 2) 7 : 17 3) 9 : 21 4) 11 : 23 5) None of
these
ANSWER: 2
The glasses be of L.C.M of (3, 4) = 12 lit capacity
1
Then milk in first glass = (12) = 4 lit
3
1
Milk in second glass = (12) = 3 lit
4
Water in first glass = 12 – 4 = 8 lit
Water in second glass = 12 – 3 = 9 lit
Milk in tumbler = 4 + 3 = 7 lit
Water in tumbler = 8 + 9 = 17 lit
Milk : Water = 7 : 17

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