Informative line

Conversion Of Percents

Percent as a Decimal

Decimals can be converted into percents and vice versa as they both are parts of a whole.

To convert a percent into 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: \(1.5\text {%}\)

Here, \(1.5\) already has a decimal point.

Thus, shift the decimal point two places to the left.

Example: Aron paid \(25\%\) of his total income to the charity. Show the amount paid by Aron in decimal form.

First of all, remove the \(\%\) sign.

\(25\%=25\)

Insert the decimal point two places to the left. So,

\(25\%=0.25\)

Illustration Questions

Convert the given percent into a decimal:  ​​ \(2.5\text {%}\)

A \(0.025\)

B \(0.25\)

C \(0.0025\)

D \(0.00025\)

×

Given: \(2.5\text {%}\)

Removing the \(\text{%}\) sign.

\(2.5\%=2.5\)

 

Here, \(2.5\) already has a decimal point, so we shift the decimal point two places to the left i.e.,

image

Thus, the decimal form of \(2.5\text {%}\) is \(0.025\)

 

Hence, option (A) is correct.

Convert the given percent into a decimal:  ​​ \(2.5\text {%}\)

A

\(0.025\)

.

B

\(0.25\)

C

\(0.0025\)

D

\(0.00025\)

Option A is Correct

Conversion of Decimals into Percents

  • Decimals can be converted into percents and vice versa as they both are parts of a whole.
  • To convert the decimal into 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 a \(\text{%}\) (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 zero at the tenths place.

Simply move the decimal point two places to the right and add a \(\text{%}\) 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 a \(\text {%}\) 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 a \(\text {%}\) sign at the end.

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

Example :

Sam takes \(0.25\) liter milk in his breakfast. Show the amount of milk he takes in percent.

First, move the decimal point two places to the right.

Add a percent \((\%)\) sign at the end.

\(0.25=25\%\)

Thus, Sam takes \(25\%\)  of \(1\) liter milk.

Illustration Questions

Which one of the following shows the correct representation of \(0.0015\) in percent?

A \(0.015 \%\)

B \(0.15\%\)

C \(1.5\%\)

D \(15\%\)

×

Given: \(0.0015\)

Moving the decimal point two places to the right.

image

Adding a \(\text {%}\) (Percent) sign at the end, i.e. \(0.15\text {%}\)

Thus, \(0.0015=0.15\%\)

Hence, option (B) is correct.

Which one of the following shows the correct representation of \(0.0015\) in percent?

A

\(0.015 \%\)

.

B

\(0.15\%\)

C

\(1.5\%\)

D

\(15\%\)

Option B is Correct

Percent as Fraction

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

For example:

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

\(=25\text {%}\)

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

  • Since, the fraction obtained is not in its simplest form, so we simplify it by taking out the greatest 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 \(\dfrac {1}{4}\).

Example :

Mr. Watson withdrew \(30\%\) from his savings bank account. Show the amount withdrawn from the account in fraction form.

First of all, replace the \(\%\) sign with \(\dfrac {1}{100}\).

\(30\%=\dfrac {30}{100}\)

Since the fraction obtained is not in its simplest form, so simplify it by taking out the greatest common factor (G.C.F.).

Fraction = \(\dfrac {30}{100}\)

The greatest common factor (G.C.F.) of \(30\) and \(100\) is \(10\) .

So, we divide the numerator and denominator by the greatest common factor.

\(\dfrac {30\div10}{100\div10}=\dfrac {3}{10}\)

Thus, Mr. Watson withdrew \(\dfrac {3}{10}\) part of his savings from his bank account.

Illustration Questions

Which one of the following options represents \(45\text {%}\) as a fraction?

A \(\dfrac {9}{5}\)

B \(\dfrac {9}{10}\)

C \(\dfrac {9}{15}\)

D \(\dfrac {9}{20}\)

×

Given: \(45\%\)

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

\(45\text{%}=\dfrac {45}{100}\)

Since, the fraction obtained is not in its simplest form, so simplify it by taking out the greatest common factor (G.C.F.).

Fraction = \(\dfrac {45}{100}\)

The greatest common factor (G.C.F.) of \(45\) and \(100=5\)  

Dividing numerator and denominator by the greatest common factor,

\(\dfrac {45\div5}{100\div5}=\dfrac {9}{20}\)

Thus, \(45\%=\dfrac {9}{20}\)

Hence, option (D) is correct.

Which one of the following options represents \(45\text {%}\) as a fraction?

A

\(\dfrac {9}{5}\)

.

B

\(\dfrac {9}{10}\)

C

\(\dfrac {9}{15}\)

D

\(\dfrac {9}{20}\)

Option D is Correct

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

Conversion of Fractions into Percents (Using Proportions)

  • Both fractions and percents are parts of whole.
  • A fraction can be converted into a percent by using a proportion.
  • To solve, we first need to make a new equivalent fraction with the numerator \(x\) and the denominator \(100\).

i.e., \(\dfrac {x}{100}\)

  • Solve the proportion by using cross multiplication method.

Eg.  Sarah has finished \(\dfrac {2}{5}\) part of her work. What percent of work has Sarah finished?

First, we make a new equivalent fraction with the numerator \(x\) and the denominator \(100\),i.e. \(\dfrac {x}{100}\)

So, the equivalent fraction of \(\dfrac {2}{5}\) is:

\(\dfrac {2}{5}=\dfrac {x}{100}\)

 

Now, solve the proportion by using cross multiplication method:

\(\Rightarrow5x=2×100\)

\(\Rightarrow5x=200\)

\(\Rightarrow x=\dfrac {200}{5}\)

\(\Rightarrow x=40\)

\(\dfrac {2}{5}=\dfrac {40}{100}=40\%\)

NOTE:  This method is used when the fraction does not have \(100\) as the denominator. When fraction has \(100\) as the denominator, we can convert it directly into a percent by removing \(100\).

\(\Rightarrow5x=2×100\)

\(\Rightarrow5x=200\)

\(\Rightarrow x=\dfrac {200}{5}\)

\(\Rightarrow x=40\)

\(\dfrac {2}{5}=\dfrac {40}{100}=40\%\)

NOTE:  This method is used when the fraction does not have \(100\) as the denominator. When fraction has \(100\) as the denominator, we can convert it directly into a percent by removing \(100\).

Illustration Questions

Alex has eaten \(\dfrac {4}{5}\) of the total cookies he had. What percent of the cookies has Alex eaten?

A \(50\%\)

B \(60\%\)

C \(75\%\)

D \(80\%\)

×

Given: \(\dfrac {4}{5}\)

Make a new equivalent fraction with the numerator \(x\) and the denominator \(100\), i.e. \(\dfrac {x}{100}\)

Thus, the equivalent fraction of \(\dfrac {4}{5}\) is:

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

Now, solve the proportion using cross multiplication method:

image

\(5x=4×100\)

\(5x=400\)

\(x=\dfrac {400}{5}\)

\(x=80\)

\(\dfrac {4}{5}=\dfrac {80}{100}=80\%\)

Thus, Alex has eaten \(80\%\) of his cookies.

Hence, option (D) is correct.

Alex has eaten \(\dfrac {4}{5}\) of the total cookies he had. What percent of the cookies has Alex eaten?

A

\(50\%\)

.

B

\(60\%\)

C

\(75\%\)

D

\(80\%\)

Option D is Correct

Ordering of Fractions, Decimals and Percents

  • We can easily compare percents, decimals and fractions with each other as they are all parts of a whole.
  • They all can be arranged in the order of least to greatest or greatest to least.
  • For ordering, we first make all the numbers in the same form.
  • After conversion, we compare all the numbers with each other.
  • Lastly, we arrange them in the order of least to greatest or greatest to least.

Example : 

Aron, Carl and Tim, all are having \($100\) in their pockets. They go to buy \(10\) candies each from different markets. Aron buys the candies for \(4\%\) of the total amount. Carl buys for \(\dfrac {3}{100}\) of the total amount and Tim buys the candies for \(0.05\) of the total amount. Now, we want to arrange their spending amount from least to greatest.

First, we make all the numbers in the same form.

Aron buys the candies for \(4\%\) of the total amount. Carl buys the candies for \(\dfrac {3}{100}\) of the total amount. Tim buys the candies for \(0.05\) of the total amount.

To convert \(\dfrac {3}{100}\) into a percent, remove the \(100\) and add a \(\%\) sign.

\(\therefore\;\dfrac {3}{100}=3\%\)

So, Carl buys the candies for \(3\%\) fo the total amount.

To convert \(0.05\) into a percent,

we move the decimal point two places to the right.

i.e., 5%

So, Tim buys the candies for \(5\%\) of the total amount.

Hence,Tim buys for \(5\%\),Carl buys for \(3\%\) and Aron buys for \(4\%\) of the total amount.

Therefore, \(3\%<4\%<5\%\)

Carl, Aron, Tim

Illustration Questions

Keith, Kevin, Marc and Alex are practicing for the upcoming running competition in the school. In an hour, Keith runs \(85\%\) of the track, Kevin covers \(1.25\) of the track, Marc runs \(\dfrac {9}{10}\) of the track and Alex covers \(\dfrac {3}{4}\)  of the track. Which one of the following options shows the correct sequence of names, considering the distance covered by them, in the order of greatest to least?

A Alex, Marc, Kevin, Keith

B Alex, Keith, Marc, Kevin

C Kevin, Marc, Keith, Alex

D Keith, Kevin, Marc, Alex

×

First, make the numbers in the same form.

Keith runs \(85\%\) of the track.

Kevin covers \(1.25\) of the track.

To convert \(1.25\) into a percent, move the decimal point two places to the right, i.e.,

image

So, Kevin covers \(125\%\) of the track.

Marc runs \(\dfrac {9}{10}\) of the track.

To convert \(\dfrac {9}{10}\) into a percent, we cross multiply.

image

\(10x=9×100\)

\(10x=900\)

\(x=\dfrac {900}{10}=90\%\)

Marc runs \(90\%\) of the track.

Alex covers \(\dfrac {3}{4}\) of the track.

To convert \(\dfrac {3}{4}\) into a percent, we cross multiply.

image

\(\Rightarrow4x=3×100\)

\(\Rightarrow4x=300\)

\(\Rightarrow x=\dfrac {300}{4}=75\%\)

Alex covers \(75\%\) of the track.

Arranging the numbers from greatest to least.

\(125\%,\;90\%,\;85\%,\;75\%\)

or \(1.25\%,\;\dfrac {9}{10},\;85\%,\;\dfrac {3}{4}\)

Thus, Kevin covers the maximum distance than others.

Marc covers more distance than Keith.

Alex covers the least.

Hence, option (C) is correct.

Keith, Kevin, Marc and Alex are practicing for the upcoming running competition in the school. In an hour, Keith runs \(85\%\) of the track, Kevin covers \(1.25\) of the track, Marc runs \(\dfrac {9}{10}\) of the track and Alex covers \(\dfrac {3}{4}\)  of the track. Which one of the following options shows the correct sequence of names, considering the distance covered by them, in the order of greatest to least?

A

Alex, Marc, Kevin, Keith

.

B

Alex, Keith, Marc, Kevin

C

Kevin, Marc, Keith, Alex

D

Keith, Kevin, Marc, Alex

Option C is Correct

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