Informative line

Percents Decimals And Fractions

What is Percent?

  • Many times in our daily lives, we hear the word 'Percent'. But what is it? Let's try to understand it.
  • It is a combination of two words - per and cent, where per means "each" and cent means "hundred".
  • It helps us to show things in relative terms which makes the comparison within the choices easier.
  • A percent is one hundredth, and as a fraction it is \(\dfrac {1}{100}\).
  • \(\dfrac {1}{100}\) can be denoted by the symbol '%' or vice versa.
  • Thus, a percent is a part of the whole, where the whole is represented by \(100\).

For example:

\(58\text {%} \) means \(58\) out of \(100\).

For example:

\(58\text {%} \) means \(58\) out of \(100\).

Illustration Questions

What does \(25\text {%} \) mean?

A \(25\) added to \(100\)

B \(25\) out of \(100\)

C \(25\) multiplied by \(100\)

D \(25\) subtracted from \(100\)

×

A percent is a part of the whole where the whole is represented by \(100\).

\(25\text {%} =25\) percent

\(=25\) per \(100\)

\(=25\) out of \(100\)

Hence, option (B) is correct.

What does \(25\text {%} \) mean?

A

\(25\) added to \(100\)

.

B

\(25\) out of \(100\)

C

\(25\) multiplied by \(100\)

D

\(25\) subtracted from \(100\)

Option B is Correct

Fraction as a Percent

  • A fraction can be converted to a percent and vice versa, as both are parts of a whole.
  • To convert a fraction to a percent, consider the following two cases:

Case 1: When the denominator of the fraction is \(100\)

For example:  \(\dfrac {25}{100},\;\dfrac {56}{100},\;\dfrac {16}{100},\;\dfrac {2}{100}\) etc.

  • To convert the fraction where the denominator is \(100\), replace  \(\dfrac {1}{100}\) by the \(\text{%}\) sign, because percent means out of \(100\).

Example: \(\dfrac {25}{100}=25\) out of \(100=25\text{%}\)

Case 2: When the denominator of the fraction is not \(100\)

Consider the following example: Write \(\dfrac {3}{4}\) as a percent.

  • Here, the denominator is not \(100\).
  • But we know that percent means "out of \(100\)".
  • Thus, we have to make the denominator \(100\).

\(\dfrac {3}{4}=\dfrac {?}{100}\)

To get \(100\) in the denominator in place of \(4\), multiply \(4\) with \(25\).

\(4×25=100\)

Thus, we have to multiply the numerator \((3)\) also with \(25\) (to get equivalent fraction).

\(3×25=75\)

Now, \(\dfrac {3}{4}=\dfrac {75}{100}\)

Thus, \(\dfrac {75}{100}=75\) out of \(100=75\text {%}\)

So, we can write \(\dfrac {3}{4}\) as \(75\text {%}\).

Illustration Questions

Which one of the following statements is true?

A \(\dfrac {1}{2}=25\text {%}\)

B \(\dfrac {7}{100}=0.7\text {%}\)

C \(\dfrac {23}{100}=57\text {%}\)

D \(\dfrac {4}{5}=80\text {%}\)

×

Percent means "out of \(100\)" .

Thus, the denominator of the fraction should be \(100\).

Option (A)

Given: \(\dfrac {1}{2}=25\text {%}\)

Let's check it.

Here, we will find the equivalent fraction of \(\dfrac {1}{2}\) having the denominator \(100\).

\(\dfrac {1}{2}=\dfrac {?}{100}\)

To get \(100\) in the denominator, \(1\) and \(2\) should be multiplied with \(50\).

\(\dfrac {1}{2}=\dfrac{1×50}{2×50}=\dfrac {50}{100}=50 \text { out of }100=50\text {%}\)

Hence, option (A) is incorrect.

Option (B)

Given: \(\dfrac {7}{100}=0.7\text {%}\)

Let's check it.

\(\dfrac {7}{100}=7 \text { out of }100=7\text {%}\)

Hence, option (B) is incorrect.

Option (C)

Given: \(\dfrac {23}{100}=57\text {%}\)

Let's check it.

\(\dfrac {23}{100}=23 \text { out of }100=23\text {%}\)

Hence, option (C) is incorrect.

Option (D)

Given: \(\dfrac {4}{5}=80\text {%}\)

Let's check it.

Here, we will find the equivalent fraction of \(\dfrac {4}{5}\) having the denominator \(100\).

\(\dfrac {4}{5}=\dfrac {?}{100}\)

To get \(100\) in the denominator, \(4\) and \(5\) should be multiplied with \(20\).

\(\dfrac {4}{5}=\dfrac {4×20}{5×20}=\dfrac {80}{100}=80 \text { out of }100=80\text {%}\)

Hence, option (D) is correct.

Which one of the following statements is true?

A

\(\dfrac {1}{2}=25\text {%}\)

.

B

\(\dfrac {7}{100}=0.7\text {%}\)

C

\(\dfrac {23}{100}=57\text {%}\)

D

\(\dfrac {4}{5}=80\text {%}\)

Option D is Correct

Decimal as a Percent

  • A decimal can be converted to a percent and vice versa, as they both are parts of a whole.
  • To convert a decimal to a percent, consider the following steps:

For example: Write \(0.23\) as a percent.

Step: 1 Move the decimal point two places to the right.

Step: 2 Add the  \(\text{%}\) (Percent) sign at the end.

\(0.23=23\text{%}\)

Now consider the following cases:

Case-I: If the decimal has zeros

For example: 0.06

  • Here, it has a zero at the tenths place.
  • Simply move the decimal point two places to the right and add the \(\text{%}\) sign at the end.

Case-II: If the decimal does not have two decimal places

For example: \(0.3\)

  • Here, the decimal does not have two decimal places. So add the required zero to the right side.
  • Move the decimal point two places to the right.
  • Add the  \(\text {%}\) sign at the end.

Case-III: If the decimal has more than two decimal places

For example: \(0.125\)

  • Here, it has three decimal places.
  • Move the decimal point two places to the right and add the \(\text {%}\) sign at the end.
  • Now, we have a percent which also has a decimal.

Now, we have a percent which also has a decimal.

Illustration Questions

Which percent correctly represents \(0.007\)?

A \(70\text {%}\)

B \(7\text {%}\)

C \(0.7\text {%}\)

D \(0.07\text {%}\)

×

Given: \(0.007\)

Moving the decimal point two places to the right.

image

Adding the \(\text {%}\) (Percent) sign at the end, 

\(0.7\text {%}\)

Thus,

\(0.007=0.7\text {%}\)

Hence, option (C) is correct.

Which percent correctly represents \(0.007\)?

A

\(70\text {%}\)

.

B

\(7\text {%}\)

C

\(0.7\text {%}\)

D

\(0.07\text {%}\)

Option C is Correct

Percent through Circle Graph-I

  • A circle graph is a way of displaying data.
  • A full circle represents the \(100\text{%}\).
  • It is divided into a number of sections, also known as pie shaped wedges.
  • The items are graphed in each section.
  • Each wedge represents a percent of the whole.
  • First, we will learn to read the circle graph.
  • Consider the following example:
  • The population of a city belonging to different age groups is shown in the circle graph.

From the circle graph, we can say that the city has 

Adults \(=30\text {%}\)

Children \(=25\text {%}\)

Teenagers \(=45\text {%}\)

  • To learn how to find the missing data, consider another example:
  • The circle graph is divided into 6 sections.
  • Each section shows a different color.
  • The data of different colors can be represented as listed below:

Brown \(=19\text {%}\)

Pink \(=18\text {%}\)

Yellow \(=7\text {%}\)

Red \(=27\text {%}\)

Green \(=6\text {%}\)

Thus, the question arises,

"What percent of the circle shows the blue color?"

To find the answer, we should do the following steps:.

The circle graph always represents the  \(100\text {%}\).

\(\therefore\) Percent of blue color 

\(=100\text {%}-\) (Sum of the given data)

The sum of  the given data \(=19\text {%}+18\text {%}+7\text {%}+27\text {%}+6\text {%}\)

\(=77\text {%}\)

\(\therefore\) Percent of blue color \(=100\text {%}-77\text {%}\) \(=23\text {%}\) 

                    

Illustration Questions

The graph represents the percentage of different books in a school library. What is the percentage of novels in the school library?

A \(31\text {%}\)

B \(30\text {%}\)

C \(15\text {%}\)

D \(22\text {%}\)

×

The circle graph always represents the \(100\text{%}\).

image

The percents from the circle graph for:

Comic books \(=10\text{%}\)

Science books \(=25\text{%}\)

Historical books \(=34\text{%}\)

The sum of the given data \(=10\text{%}+25\text{%}+34\text{%}\)

\(=69\text{%}\)

Percentage of Novels \(=100\text{%}\;-\) (Sum of the given data)

\(=100\text{%}-69\text{%}\)

\(=31\text{%}\)

Thus, novels are \(31\text{%}\).

Hence, option (A) is correct.

The graph represents the percentage of different books in a school library. What is the percentage of novels in the school library?

image
A

\(31\text {%}\)

.

B

\(30\text {%}\)

C

\(15\text {%}\)

D

\(22\text {%}\)

Option A is Correct

Conversion of Percent into Fraction

  • A percent can be converted to a fraction and vice-versa, as they both are parts of a whole.
  • To convert a percent to a fraction, replace the \(\text {%}\) symbol by \(\dfrac {1}{100}\) and then simplify it.

For example:

Convert \(25\text {%}\)  to a fraction.

\(25\text {%}=\dfrac {25}{100}\)

  • Since the fraction obtained is not in its simplest form, so we divide it by the greatest common factor. 
  • The greatest common factor of \(25\) and \(100\) is \(25\).
  • Now, we divide numerator and denominator by the greatest common factor.

So,  \(\dfrac {25 \div25}{100\div25}=\dfrac {1}{4}\)

The fraction form of \(25\text {%}\) is \(\dfrac {1}{4}\).

  • Since the fraction obtained is not in its simplest form, so we simplify it by taking the common factor of \(25\) and \(100\).
  • The greatest common factor of \(25\) and \(100\) is \(25\).
  • Now we divide the numerator and denominator by the greatest common factor.

So,  \(\dfrac {25 \div25}{100\div25}=\dfrac {1}{4}\)

The fraction form of \(25\text {%}\) is \(\dfrac {1}{4}\).

Illustration Questions

Which one of the following options represents \(36\text {%}\) as a fraction?

A \(\dfrac {2}{5}\)

B \(\dfrac {3}{4}\)

C \(\dfrac {9}{25}\)

D \(\dfrac {3}{8}\)

×

Given: \(36\text {%}\)

Replacing the  \(\text{%}\) sign with \(\dfrac {1}{100}\),

\(36\text{%}=\dfrac {36}{100}\)

 

Since the fraction obtained is not in its simplest form, so we divide it by the G.C.F.

Fraction = \(\dfrac {36}{100}\)

The greatest common factor (G.C.F.) of \(36\) and \(100=4\)  

Dividing by the greatest common factor,

\(\dfrac {36\div4}{100\div4}=\dfrac {9}{25}\)

 

Thus,

\(36\text{%}=\dfrac {9}{25}\)

 

Hence, option (C) is correct.

Which one of the following options represents \(36\text {%}\) as a fraction?

A

\(\dfrac {2}{5}\)

.

B

\(\dfrac {3}{4}\)

C

\(\dfrac {9}{25}\)

D

\(\dfrac {3}{8}\)

Option C is Correct

Percent as a Decimal

A decimal can be converted to a percent and vice versa, as they both are parts of a whole.

To convert a percent to a decimal, consider the following steps:

For example:  Convert \(26\text {%}\) into a decimal.

Step 1: Remove the \(\text {%}\) sign, so we get \(26\).

Step 2: Put the decimal point two places to the left.

Thus, \(26\text {%}=0.26\)

Consider the following two cases:

(i) To convert a single digit percent

For example: \(6\text {%}\)

Here, the percent doesn't have two digits, thus add a zero to the left of \(6\), i.e.

\(6=06\)

Now, put the decimal point two places to the left.

(ii) To convert a percent having a decimal

For example: 

Here, \(1.5\) already has a decimal point. Thus, shift the decimal point two places to the left.

Illustration Questions

Which one of the following options represents \(12.5\text {%}\) as a decimal?

A \(1.25\)

B \(12.5\)

C \(0.125\)

D \(0.0125\)

×

Given: \(12.5\text {%}\)

After dropping the  \(\text{%}\) sign, we get:

\(12.5\)

 

Moving the decimal point two places to the left,

image

Thus, 

\(12.5\text {%}=0.125\)

Hence, option (C) is correct.

Which one of the following options represents \(12.5\text {%}\) as a decimal?

A

\(1.25\)

.

B

\(12.5\)

C

\(0.125\)

D

\(0.0125\)

Option C is Correct

Ordering Percents, Decimals and Fractions

  • Since percents, decimals and fractions are all parts of the whole, therefore, we can compare them and arrange them either in the order of least to greatest or greatest to least.
  • To understand it clearly, consider an example:
  • Arrange the following in the order of least to greatest:  \(2\dfrac {1}{5},\;4.05\) and \(37.5\text {%}\).

For writing them in the given order, first we need to convert them into the same form, that may be percent, decimal or fraction form.

Here, we are converting them into the percent form.

Step 1:  Convert \(2\dfrac {1}{5}\) into a percent.

  • We can also write \(2\dfrac {1}{5}\) as an improper fraction.

\(2\dfrac {1}{5}=\dfrac {(2×5)+1}{5}=\dfrac {10+1}{5}=\dfrac {11}{5}\)

  • Converting \(\dfrac {11}{5}\) into a percent.

\(\dfrac {11}{5}=\dfrac {11×20}{5×20}=\dfrac {220}{100}=220\text{%}\)

Step 2:  Convert \(4.05\) into a percent.

  • To convert a decimal to a percent, move the decimal point two places to the right.

Here, \(37.5\text {%}\) doesn't need to be converted as it is already in percent form.

Now, we have \(220\text {%},\;405\text {%}\) and \(37.5\text {%}\) to compare.

Step-3: Arrange in the order of least to greatest.

\(37.5\text{%}<220\text{%}<405\text{%}\)

\(\therefore\) The order from least to greatest is  \(37.5\text {%}\)\(220\text {%}\)\(405\text {%}\)

Step-4: Write the original numbers in the required order.

\(37.5\text{%},\;2\dfrac {1}{5},\;4.05\)

Illustration Questions

Which one of the given options represents the following numbers in the order of least to greatest? \(\dfrac {1}{5},\; 5\text{%}\;\,and\;0.5\)

A \(\dfrac {1}{5},\;0.5,\; 5\text{%}\)

B \(5\text{%},\;0.5,\;\dfrac {1}{5}\)

C \(5\text{%},\;\dfrac {1}{5},\;0.5\)

D \(\dfrac {1}{5},\;5\text{%},\;0.5\)

×

Given: \(\dfrac {1}{5},\; 5\text{%},\;0.5\)

To write them in the given order, first we have to convert them into the same form.

Here, we are converting them into percents.

Converting \(\dfrac {1}{5}\) into a percent:

\(\dfrac {1}{5}=\dfrac {?}{100}\)

To get \(100\) in the denominator, multiply both numerator and denominator by \(20\).

\(\dfrac {1×20}{5×20}=\dfrac {20}{100}\)

Hence, \(\dfrac {1}{5}=\dfrac {20}{100}=20\text{%}\)

Now, converting \(0.5\) into a percent.

Moving the decimal point two places to the right.

image

\(5\text{%}\) doesn't need to be converted as it is already in percent form.

Thus, we have \(20\text{%}\)\(5\text{%}\) and \(50\text{%}\) to compare.

So, the order from least to greatest is \(5\text{%},\;20\text{%},\;50\text{%}\).

Writing the original numbers in the required order.

\(5\text{%},\;\dfrac {1}{5},\;0.5\)

Hence, option (C) is correct.

Which one of the given options represents the following numbers in the order of least to greatest? \(\dfrac {1}{5},\; 5\text{%}\;\,and\;0.5\)

A

\(\dfrac {1}{5},\;0.5,\; 5\text{%}\)

.

B

\(5\text{%},\;0.5,\;\dfrac {1}{5}\)

C

\(5\text{%},\;\dfrac {1}{5},\;0.5\)

D

\(\dfrac {1}{5},\;5\text{%},\;0.5\)

Option C is Correct

Representation of a Percent in the form of a Fraction, a Ratio and a Decimal

  • A percent, a ratio, a fraction, and a decimal can be converted into each other.
  • Here, we are considering an example of writing \(36\text {%}\) as a ratio, as a fraction, and as a decimal.

(i) Percent as a Fraction

  • To convert a percent to a fraction, we replace the % symbol by \(\dfrac {1}{100}\), and then simplify the fraction, if necessary.

We know that a percent means out of \(100\).

\(\therefore\) \(36\text {%}=36\) out of \(100\)

\(36\text {%}=\dfrac {36}{100}\)

Since the fraction is not in simplified form, so we divide it by the greatest common factor (G.C.F.).

The G.C.F. of \(36\) and \(100\) is \(4\).

Thus, \(\dfrac {36\div4}{100\div4}=\dfrac {9}{25}\)

(ii) Percent as a Ratio

We have \(36\text {%}=\dfrac {36}{100}\)

[\(a:b\) is written as \(\dfrac {a}{b}\)]

\(\therefore\;\;\dfrac {36}{100}=36:100\)

\(=36\text { out of } 100\)

(iii) Percent as a decimal

To convert a percent to a decimal, we first remove the % sign.

\(36\text {%}=36\)

Now, place the decimal point two places to the left,

\(36\text {%}=.36\)

Hence, we can write all the forms together as:

\(36\text {%}=\dfrac {36}{100}=36:100=.36\)

Illustration Questions

A table is given: Percent Fraction Ratio Decimal 60% - -  - - 0.60 - - - - 3 : 40 - - - -  \(\dfrac {4}{5}\) - - 0.80 25% - -  - - 0.25 Which one of the following options represents the correct table?

A Percent Fraction Ratio Decimal 60% \(\dfrac {2}{5}\)  2 : 5 0.60 35% \(\dfrac {3}{40}\) 3 : 40 3.5 80% \(\dfrac {4}{5}\) 4 : 5 0.80 25% \(\dfrac {3}{4}\) 3 : 4 0.25

B Percent Fraction Ratio Decimal 60% \(\dfrac {3}{5}\) 5 : 3 0.60 40% \(\dfrac {3}{40}\) 3 : 40 0.30 5% \(\dfrac {4}{5}\) 5 : 4 0.80 25% \(\dfrac {5}{2}\) 5 : 2 0.25

C Percent Fraction Ratio Decimal 60% \(\dfrac {3}{5}\) 3 : 5 0.60 7.5% \(\dfrac {3}{40}\) 3 : 40 0.075 80% \(\dfrac {4}{5}\) 4 : 5 0.80 25% \(\dfrac {1}{4}\) 1 : 4 0.25

D Percent Fraction Ratio  Decimal 60% \(\dfrac {3}{2}\) 2 : 3 0.60 30% \(\dfrac {3}{40}\) 3 : 40 0.40 40% \(\dfrac {4}{5}\) 8 : 100 0.80 25% \(\dfrac {1}{5}\) 5 : 1 0.25

×

In the given table, one form of a number is given and we need to write it in other forms.

Completing the first row of the table:

Given: \(60\text {% and }\,0.60\)

Writing \(60\text {%}\) in fraction form:

\(60\text {% }=60\) out of \(100\)

\(\therefore\) \(60\text {% }=\dfrac {60}{100}\)

Simplifying the fraction, 

\(\dfrac {60}{100}=\dfrac {3}{5}\)

\(\therefore\) \(60\text {% }=\dfrac {3}{5}\)

Writing \(\dfrac {3}{5}\) in the form of a ratio: 

\(\dfrac {3}{5}=3:5\)

Hence, we can write

\(60\text{%}=\dfrac {3}{5}=3:5=0.60\)

Completing the second row of the table:

Given: \(3:40\)

Writing \(3:40\)  in fraction form: 

\(3:40=\dfrac {3}{40}\)

To write \(\dfrac {3}{40}\) in percent form, we calculate the equivalent fraction having \(100\) as the denominator.

\(\dfrac {3}{40}=\dfrac {?}{100}\)  (To make the denominator \(100\)\(3\) and \(40\) should be multiplied with \(2.5\))

\(\dfrac {3×2.5}{40×2.5}=\dfrac {7.5}{100}\)

Thus,

\(\dfrac {3}{40}=\dfrac {7.5}{100}\)

\(\dfrac {7.5}{100}=7.5\) out of \(100\)

\(\therefore\;\dfrac {7.5}{100}=7.5\text{%}\) \(\left ( \text {%}=\dfrac {1}{100} \right)\)

\(\dfrac {3}{40}=7.5\text{%}\)

Writing \(7.5\text{%}\)  in decimal form, by putting the decimal point two places to the left.

\(7.5\text{%}=.075\)

\(7.5\text{%}=\dfrac {3}{40}=3:40=0.075\)

Completing the third row of the table 

Given: \(\dfrac {4}{5}\) and \(0.80\)

Writing \(0.80\)  in percent form:

Moving the decimal point two places to the right and putting the %  sign at the end.

\(\Rightarrow 0.80=80\text{%}\)

Writing \(\dfrac {4}{5}\) in the form of a ratio:

\(\dfrac {4}{5}=4:5\)

Hence, we can write

\(80\text{%}=\dfrac {4}{5}=4:5=0.80\)

Completing the fourth row of the table

Given: \(25\text{%}\) and \(0.25\)

Writing \(25\text{%}\) in fraction form:

 \(25\text{%}\) = \(25\) out of \(100\)

\(=\dfrac {25}{100}\)

Simplifying the fraction obtained,

\(\dfrac {25}{100}=\dfrac {1}{4}\)

Thus, 

\(25\text{%}=\dfrac {1}{4}\)

Writing \(\dfrac {1}{4}\) in the form of a ratio:

\(\dfrac {1}{4}=1:4\)

Hence, we can write

\(25\text{%}=\dfrac {1}{4}=1:4=0.25\)

The complete table is:

Percent Fraction Ratio Decimal
60% \(\dfrac {3}{5}\) 3 : 5 0.60
7.5% \(\dfrac {3}{40}\) 3 : 40 0.075
80% \(\dfrac {4}{5}\) 4 : 5 0.80
25% \(\dfrac {1}{4}\) 1 : 4 0.25

Hence, option (C) is correct.

A table is given: Percent Fraction Ratio Decimal 60% - -  - - 0.60 - - - - 3 : 40 - - - -  \(\dfrac {4}{5}\) - - 0.80 25% - -  - - 0.25 Which one of the following options represents the correct table?

A

Percent Fraction Ratio Decimal
60% \(\dfrac {2}{5}\)  2 : 5 0.60
35% \(\dfrac {3}{40}\) 3 : 40 3.5
80% \(\dfrac {4}{5}\) 4 : 5 0.80
25% \(\dfrac {3}{4}\) 3 : 4 0.25
.

B

Percent Fraction Ratio Decimal
60% \(\dfrac {3}{5}\) 5 : 3 0.60
40% \(\dfrac {3}{40}\) 3 : 40 0.30
5% \(\dfrac {4}{5}\) 5 : 4 0.80
25% \(\dfrac {5}{2}\) 5 : 2 0.25

C

Percent Fraction Ratio Decimal
60% \(\dfrac {3}{5}\) 3 : 5 0.60
7.5% \(\dfrac {3}{40}\) 3 : 40 0.075
80% \(\dfrac {4}{5}\) 4 : 5 0.80
25% \(\dfrac {1}{4}\) 1 : 4 0.25

D

Percent Fraction Ratio  Decimal
60% \(\dfrac {3}{2}\) 2 : 3 0.60
30% \(\dfrac {3}{40}\) 3 : 40 0.40
40% \(\dfrac {4}{5}\) 8 : 100 0.80
25% \(\dfrac {1}{5}\) 5 : 1 0.25

Option C is Correct

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