- 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 \(100\).

A \(25\) added to \(100\)

B \(25\) out of \(100\)

C \(25\) multiplied by \(100\)

D \(25\) subtracted from \(100\)

- 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 \(75\text {%}\).

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

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

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

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

- 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.

A \(70\text {%}\)

B \(7\text {%}\)

C \(0.7\text {%}\)

D \(0.07\text {%}\)

- 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 \(=100\text {%}-77\text {%}\) \(=23\text {%}\)

A \(31\text {%}\)

B \(30\text {%}\)

C \(15\text {%}\)

D \(22\text {%}\)

- 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 \(\dfrac {1}{4}\).

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

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

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

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

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.

A \(1.25\)

B \(12.5\)

C \(0.125\)

D \(0.0125\)

- 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.

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

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

- 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.

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

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

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

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

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

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:

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

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