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# 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,

$$25\%=0.25$$

#### 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.,

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.

#### 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.

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.

#### 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 $$100$$.

#### 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 $$100$$.

#### 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:

$$5x=4×100$$

$$5x=400$$

$$x=\dfrac {400}{5}$$

$$x=80$$

$$\dfrac {4}{5}=\dfrac {80}{100}=80\%$$

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

#### 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.,

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.

$$10x=9×100$$

$$10x=900$$

$$x=\dfrac {900}{10}=90\%$$

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.

$$\Rightarrow4x=3×100$$

$$\Rightarrow4x=300$$

$$\Rightarrow x=\dfrac {300}{4}=75\%$$

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