How to Compare Decimal Numbers

by Civil Service Reviewer · February 19, 2016

We have already learned how to compare fractions and in this post, we are going to learn how
to compare decimals. In comparing decimals, it is important to understand place value. In the
number 213.489, the following are their place values. For the whole numbers,

2 – hundred
1 – tens
3 – ones.

For the decimal numbers,

4 – tenths
8 – hundredths
9 – thousandths

In whole numbers, clearly, the larger the number of digits the larger the number. For example,
821 > 92 since 821 has three digits and 92 has only two digits. Since whole numbers are always
greater than decimal numbers in comparing decimal numbers, look at the whole numbers first.
Therefore, we have the following rule.

Rule 1: In comparing decimal numbers, look at the whole number first. The decimal numbers
containing larger whole numbers have larger values.

Example 1: 84.23 > 82.345 since 84 is greater than 82.

Example 2: 12.56 < 15.001 since 12 is less than 15.

Example 3: 141.85 > 123.4 because 141 is greater than 123

Rule 2: If the whole numbers are equal, then compare the numbers by looking at the tenths place
first. The number with the larger digit in tenths place is larger.

Example 1: 18.34 > 18.21 since 3 > 2.

Example 2: 12.95 > 12.15 since 9 > 1.

Example 3: 0. 9 > 0.873 since 9 > 8.

Notice that in Rule 2 Example 3, even if 0.873 has more digits, it is still less than 0.9 since 9 is
greater than 8 and they are in the tenths place.

Rule 3: If the whole numbers and the tenths place are equal, then compare first the hundredths
place. The number with the larger digits in the hundredths place is the larger number.

Rules 2 and 3 can be generalized. That means that you have to compare the digits from the tenths
place first, then hundredths, then thousandths, and so on.

What about negative numbers?

Please take note however the rules of negative numbers.

 Positive numbers are always greater than negative numbers.

 If both numbers are negative, do the following:

1.) Make them positive

2.) Apply the rules above
3.) Reverse your answer.
Example: Compare -82.45 and -82.31

1.) Make them positive: 82.45 and 82.31

2.) Apply the rules above:

From the rules above, the whole numbers are both 82, so we look at the tenths place. 0.4 > 0.3,
so 82.45 > 82.31

Practice Quiz on Converting Decimals to

Fractions
by Civil Service Reviewer · April 3, 2015

We have already learned how to convert decimals to fractions. The idea as we have discussed
in the preceding link is to find the place value of the rightmost significant digit. The decimals
whose place values are tenths, hundredths, thousandths and so on are multiplied by 1/10, 1/100,
1/1000 and so on respectively. After performing multiplication, the fraction must be reduced to
lowest terms.

Practice Quiz: Converting Decimals to Fractions

Convert the following decimals to fractions.

1. ) 0.4

2.) 0.8

3.) 0.18

4.) 0.25

5.) 0.75

6.) 0.35

7.) 0.125

8.) 0.9

9.) 0.05

10.) 0.016

Complete Solutions and Answers

1.) 0.4 is the same as 4 tenths or .

We reduce to lowest terms by dividing both the numerator and denominator by 2. That is,

Answer:

2.) 0.8 is the same as 8 tenths or   .

We reduce to lowest terms by dividing both the numerator and denominator by 2. That is,
 .

Answer:

3.) 0.18 is the same as 18 hundredths or   .

We reduce to lowest terms by dividing both the numerator and denominator by 2. That is,

Answer:

4.) 0.25 is the same as 25 hundredths or   .

We reduce to lowest terms by dividing both the numerator and denominator by 25. That is,

Answer:

5.) 0.75 is the same as 75 hundredths or  

We reduce to lowest terms by dividing both the numerator and denominator by 25. That is,

Answer:

6.) 0.35 is the same as 35 hundredths or  

We reduce to lowest terms by dividing both the numerator and denominator by 5. That is,

Answer:

7.) 0.125 is the same as 125 thousandths or  

We reduce to lowest terms by dividing both the numerator and denominator by 125. That is,

Answer:

8.) 0.9 is the same as 9 tenths or .

Answer:

9.) 0.05 is the same as 5 hundredths or  

We reduce to lowest terms by dividing both the numerator and denominator by 5. That is,

Answer:

10.) 0.016 is the same as 16 thousandths or  

We reduce to lowest terms by dividing both the numerator and denominator by 8. That is,

How to Convert Fractions to Decimals

by Civil Service Reviewer · March 11, 2015

Converting fractions to decimals is one of the basic skills in mathematics that you should learn in
order to pass the Civil Service Examination.  Being able to convert numbers to fractions,
decimals, and percents, will give you an advantage to solve problems better and faster. In this
post, we are going to discuss how to convert fractions to decimals.

Recall that in fractions, the number at the top of the fraction bar is called the numerator and the
number at the bottom of the fraction bar is called the denominator. In converting fractions to
decimals we divide: the numerator becomes the dividend and the denominator becomes the
divisor (don’t switch!).

In converting fractions to decimals, you should divide the numerator by the denominator
manually. Take note of this step because most solvers switch their places.

Example 1: Convert to decimals.

First, 4 divided by 5 cannot be done, so we place 0 in the quotient.

Second, we add the decimal point and place 0 after the decimal point in the dividend. We also
add the decimal point to the quotient aligned with the first decimal point.

Third, ignoring the decimal point, we divide 40 by 5, which gives us 8. We write 8 at the right of
the decimal point and continue our calculation.

So,  in decimals is .

Example 2: Convert  to decimals.

Again, we align the decimals and divide 1 with 8 which cannot be, so we place 0 in the quotient.
Next, we add the decimal point and 0 to the dividend. Now dividing 10 by 8, we get 1 a quotient
as shown below.

After subtraction, we still have a remainder. So, we add another 0 in the dividend as shown.
Performing division, we have the following calculation.

Next, we still have a remainder. Adding 0, we have the following calculation.

Therefore .

Example 3: There are cases that the decimal in non-terminating such as . If you calculate this
fraction, it will give you with never ending 3’s. So, you can just round to 0.33 or
depending on the number of decimal places required.

Example 4: There are cases that the decimals are repeating. For example, if we convert to
fractions, we get 0.142857142857 with 142857 repeating. Again, in examinations, they usually
tell you to round your answers to the nearest place values.

Example 5: For mixed fractions, you can just ignore the whole number, and then convert the
fraction to decimals. After you have calculated the decimal, add the whole number.

For example, how do we convert to decimals.

First, we ignore the whole number. Then, we convert to decimals which is 0.8 in Example 1.
Lastly, we add 9 and 0.8 which is equal to 9.8.
How to Convert Decimal Numbers to Percent
by Civil Service Reviewer · February 14, 2015

Conversions of decimals, fractions, and percent is a very important basic skill in mathematics
and many problems in the Civil Exams require this skill. Being able to convert from one form to
another will help you speed up in calculations.   For example, instead of multiplying a number by
25%, you just have to get its 1/4 or simply divide it by 4.

Percent usually appears in discount and interest problems while fractions and decimals appear in
various types of problems.

How to Convert Decimals to Percent

To convert decimal percent, you just have to multiply the decimal by 100.

Example 1

What is 0.25 in percent?

Solution

0.25 × 100 = 25

So, the answer is 25%.

Example 2

What is 0.08 in percent?

0.08 × 100 = 8

Therefore, the answer is 8%.

Of course, there are cases that the given is more than one such as the next example

Example 3

What is 1.8 in percent?

Solution

1.8 × 100 = 180

Therefore, the answer is 180%.

Example 4

What is 0.009 in percent?

Solution

0.009× 100 = 0.9%

Notice that some percent can also have decimal point such as shown in Example 4. In dealing
with many decimals, if we multiply them with 100, we just move two decimal places to the right.
How to Convert Fraction to Percent Part 1
by Civil Service Reviewer · March 6, 2014

In the previous post, we have learned how to convert percent to fraction. In these series of posts,
we learn the opposite: how to convert fraction to percent. I am going to teach you three methods,
the last one would be used if you forgot the other two methods, or if the first two methods would
not work. Please be reminded though to understand the concept (please do not just memorize).

The first method can be used for fractions whose denominators can be easily related to 100 by
multiplication or division. Recall that from Converting Percent to Fraction, I have mentioned
that when we say percent it means “per hundred.” In effect, n% can be represented by n/100.
Therefore, if you have a fraction and you can turn it into n/100 (by multiplication/division), then
you have turned it into percent.

Example 1: What is the equivalent of 1/5 in percent?

How do we relate the denominator 5 to 100? By multiplying it by 20. Therefore, we also

multiply its numerator by 20:

Now, since we have 100 as denominator, the answer in percent is therefore the numerator.
Therefore, the equivalent of 1/5 in percent is 20%.

Example 2: What is 3/25 in percent?

Again, how do you related 25 to 100? By multiplying it by 4. Therefore,

Therefore, the equivalent of 3/25 in percent is 12%.

Example 3: What is 23/200 in percent?

In this example, we can relate 200 to 100 by dividing it by 2. So, we also divide the numerator
by 2. That is

Therefore, the answer is 11.5%

There are two important things to remember in using the method above.

(1) in changing the form the fractions to n/100, the only operations that you can use are
multiplication and division and

(2) whatever you do to the numerator, you also do to the denominator.

Note that multiplying the denominator (or dividing it) by the same number does not change its
value, it only change its representation (fraction, percent or decimal).

Why It Works

When  you are relating a fraction a/b to n/100, you are actually using ratio and proprotion. For
example, in the first example, you are actually solving the equation
 .

The equation will result to which is equal to 20. Now, this is just the same as multiplying
both the numerator and the denominator by 20.

Note that the method of “relating to 100 by multiplication or division” can only work easily for
denominators that divides 100 or can be divided by 100. Other fractions (try 1/7), you have to
use ratio and proportion and manual division.

You might also like:

How to Convert Fraction to Percent Part 2

by Civil Service Reviewer · March 7, 2014

In the Part 1, we have learned how to convert fraction to percent by relating the denominator to
100 by multiplication or division. In this post, we do its ‘algebraic version.’ This method is a
generalized method to the previous post especially for numbers that do not divide 100 or cannot
be divided by 100 easily. However, to see the relationship between the two methods, let us do the
first example in Part 1 of this series.

Example 1: What is the equivalent of 1/5 in percent.

Recall that in Part 1, we multiplied both the numerator and the denominator by 20, to make the
denominator 100. That is, 

Now, notice how it is related to the new method. In this method, we related 1/5 to n/100. That is,
what is the value of in

To simplify the equation, we multiply both sides of the equation by 100, and we get

Simplifying and switching the position of the expressions, we get the . This means that
.

Of course, Part 1 seems to be easier, but the good thing about putting it into equation is that it
applies to all fractions. For instance, it is quite hard to convert 7/12 using the method in part 1.

Example 2: What is the equivalent of in percent?

We set up the equation with on the left.

To eliminate the fraction, multiply both sides by denominator. This results to

or about 58.33%.

The curly equal sign means approximately equal to since 3 is a non-terminating decimal.

Now, try to examine the expression

because this is where they derived the rule. Recall the rule in converting fraction to percent:
Divide the fraction and then multiply the result to 100. That is exactly it.

So, when you have the fraction, just divide it manually, and then multiply the result to 100.
That is,

Do not forget though that the divisor during division is the denominator (5 in 2/5). as shown
below.

That’s it. I think we don’t have to have the third part, since we already derived the rule here.

How to Convert Percent to Fraction

by Civil Service Reviewer · March 5, 2014

In Civil Service Examinations, as well as other examinations in basic mathematics, knowing how
to convert  percent, fractions, and decimals to each other is very advantageous especially if you
can do it mentally. Let us try with the following example.

A P640 shirt is marked 25% discount. How much will you have to pay for it?

It seems that you need a pencil for this problem, but you can actually do it in your head. Read it
to believe it.

The equivalent of 25% in fraction is 1/4, therefore, you have to take away the fourth of the price.
Now, 1/4 of 640 seems difficult but what if we try to split it to 600 + 40? Now, 1/4 of 600 is 150,
which means that from the 600, you have 450 left. Now, 1/4 of 40 is 10, which means that you
have 30 left. So, 450 + 30 is 480 and that is the discounted price of the t-shirt.

Now, with a little bit of practice, you would be able to do this on your own and you won’t have
to use a pen to perform calculations for problems such as this.

How to Convert Percent to Fraction

There is one important concept to remember when converting percent to fraction. That is, when
you say percent, it means per hundred. The word cent comes from the Latin word centum which
means “hundred”. In effect, when you say, 60%, it means 60 per hundred, 0.4% means 0.4 per
hundred, 125% means 125 per hundred. When you say x per hundred, you can also represent it
by the fraction x/100. This means that the percentages above can be represented as
respectively. Now, all we have left to do is to convert these fractions to lowest terms.

Example 1:

Recall that to convert a fraction to lowest terms, we find the greatest common factor (GCF) of its
numerator and denominator and then divide them both by the GCF.  The GCF of 60 and 100 is
20, so

Therefore, the equivalent of 60% in fraction is .

Example 2:

In this example, we have a decimal point at the numerator and a whole number at the
denominator. We have to “get rid” of the decimal point. To do this, we can multiply both the
numerator and the denominator by 10 (since 0.4 x 10 = 4). Therefore, we have

Now, the greatest common factor of 4 and 1000 is 4, so we divide both the numerator and the
denominator by 4. The final result is .

Therefore, the equivalent fraction of 0.4% is .

Example 3:

The greatest common factor of 125 and 100 is 25, so we divide both the numerator and the
denominator by 25. In doing this, we get .

Therefore, the equivalent fraction of 125% is

Summary

There are three steps to remember in converting percent to fractions.

1. Make a fraction from the given percent with the given as numerator and 100 as
denominator.
2. Eliminate the decimal points (if there are any) by multiplying the numerator and
denominator by the same number which is a power of 10 (10, 100, 1000 and so on).
3. Reduce the resulting fraction to lowest terms.

That’s it. You can now convert any given percent to fraction.