Converting Between Fractions, Decimals, and Percentages

Percentage Definition

The term ‘per cent’ means ‘out of a hundred’. In mathematics, percentages are used like fractions and decimals to describe parts of a whole.  When we are using percentages, the whole is considered to be made up of a hundred equal parts.

Symbol of Percentage

The symbol % is used to show that a number is a percentage.

You will see percentages almost everywhere. They appear in discounts in shops, bank interest rates, rates of inflation, in news, even on our devices. Percentages are important for understanding the financial aspects of everyday life. Knowledge of percentages also helps in understanding how to run a business. It helps in analysing and find an optimal solution to what price to charge for the product, how to increase the profits without increasing the price, etc.

Express Percentage as a Fraction

A percentage is a fraction out of a hundred, and any fraction can be converted to a percentage. When converting fractions to percentages, it is easiest to convert the quantity to an equivalent fraction out of a hundred and then write the number as a percentage.

Percentages can be represented using models. Although using 100 squares makes representing and identifying percentages much simpler, percentages can also be represented using squares other than 100. Percentages are fractions or ratios expressed out of 100. 

1) Write down the percentage represented by the shaded section of each grid. 

i. 

'25 out of a hundred' are coloured. We can write $\begin{align*}\frac{25}{100} = 25 \%, \end{align*}$ we say that '25 percent' of the grid is shaded. 

ii. 

'3 out of twenty' are coloured. We can write $\begin{align*}\frac{3}{20} = \frac{3 \times 5}{20 \times 5} = \frac{15}{100} = 15 \%,\end{align*}$ we say that '15 percent' of the grid is shaded. 

CK-12 Interactive: Percentages and Fractions

Use the interactive below to explore how percentages are related to fractions. 

 

 

CK-12 Interactive - Percentage: Your Weight on Other Planets!

In the next interactive, use ratios to convert how much you weigh on Earth to how much you would weigh on different planets. If your weight on Earth represents 100%, determine what percentage is your weight on other planets using the questions that follow.

 

 

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Percentage as a Decimal

Any fraction can be expressed as a decimal (terminating or non-terminating but recurring) and any decimal can be converted to a percentage by multiplying it with 100.

CK-12 Interactive: Converting between Fraction, Decimals and Percents

Use the interactive below to practice converting decimals, fractions and percents. 

 

 

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Percentage of a Number

When dealing with percents, there are always three things to focus on: the part, the whole, and the percent. If you have two of these three values, you can always find the third.

The relationship $\begin{align*}\text{percent} = \frac{\text{part}}{\text{whole}}\end{align*}$ can be used to solve for a percent using the part and the whole. By multiplying the whole to the other side you get the following equation:

$$\begin{align*}\text{percent} \cdot \text{whole}= \text{part}\end{align*}$$

2) Find 36% of 30. 

Part = ? 

Whole = 30. 

Percent = 36%$\begin{align*}\rightarrow \frac{36}{100}.\end{align*}$

$$\begin{align*}\text{Percent} \cdot \text{Whole}& = \text{Part} \\ \frac{36}{100} \cdot 30 & = \text{Part} \\ 10.8 & = \text{Part}\end{align*}$$ ∴ 36% of 30 is 10.8. 

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Percentage Change 

Often we are interested in finding by what percentage a quantity changes, which is a comparison of a change in value to the original value. The new price is expressed as a percentage of the old price. For example, if the new price is 50% of the old price, this means that the new price is half the old price. If the new price is 200% of the old price, this means that the new price is double the old price.

Percentage Change Formula

The percentage change is calculated using the following formula:

$$\begin{align*}\text{Percentage increase/decrease} =\frac{ \text{New amount - Original amount}}{\text{Original amount}}\times 100\end{align*}$$

CK-12 Interactive - Percentage Change: Percent of House Price 

Use the interactive below to explore this idea. 

 

 

CK-12 Interactive - Percentage Change: Car Depreciation

Use the interactive below to explore percentage decreases further. 

 

 

3) The price of tea leaves increased by 20%. If the price of tea leaves was ₹360 per kg, what is the increased price of tea leaves per kg?

Original price of tea leaves= ₹360.  

Let the increased price be $\begin{align*}₹x.\end{align*}$ Then the amount of increase in the price is $\begin{align*}₹x - ₹360. \end{align*}$ 

$$\begin{align*}\therefore \quad \text{Percentage increase} & = \frac{\text{New amount - Original amount}}{\text{Original amount}} \times 100 \\ 20 & = \frac{x - 360}{360} \times 100 \\ \frac{20 \times 360}{100} & = x - 360 \\ 72 & = x - 360 \\ x & = 72 + 360 = 432\end{align*}$$ ∴ The increased price of sugar is ₹432 per kg.  

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Percentage - Examples

Example 1

Convert the following into percentages. 

i. 2 : 5

ii. $\begin{align*}\frac{8}{25} \vphantom{\frac{2}{\frac{2}{2}}}\end{align*}$

iii. 0.006

i. To convert a ratio into a percentage, we first convert the ratio into a fraction and then the fraction is converted into a percentage (by multiplying it with 100)

$$\begin{align*}2: 5 & = \frac{2}{5} \vphantom{\frac{2}{\frac{2}{2}}} \\ & = \frac{2}{5} \times 100 \vphantom{\frac{2}{\frac{2}{2}}}\\ & = 40\%\end{align*}$$ ii. To convert a fraction into a percentage, we multiply the fraction by 100.  $$\begin{align*}\frac{8}{25} & = \frac{8}{25} \times 100 \vphantom{\frac{2}{\frac{2}{2}}} \\ & = 32 \%\end{align*}$$ iii. To convert a decimal into a percentage, we multiply the decimal by 100.  $$\begin{align*}0.006 & = 0.006 \times 100 \\ & = 0.6 \%\end{align*}$$

Example 2

Out of her total salary, Shreya spends 40% on household expenditures, invests 20% salary and pays 18% salary as loan instalment. If her remaining salary is ₹10560, then find her total salary. 

Let Shreya's total salary be $\begin{align*}₹x.\end{align*}$

The percentage of salary spent on household expenses, investment and loan$\begin{align*} = (40+20+18) \% = 78 \%. \end{align*}$

The percentage of salary left$\begin{align*} = (100-78) \% = 22 \%. \end{align*}$

According to the question, 

$$\begin{align*}22 \% \ \text{of} \ x & = ₹10560 \\ \frac{22}{100} \times x & = 10560 \vphantom{\frac{2}{\frac{2}{2}}} \\ x & = \frac{10560 \times 100}{22} \vphantom{\frac{2}{\frac{2}{2}}}\\ x & = 48000\end{align*}$$ ∴ Total salary of Shreya is ₹48000.  

Example 3

Tina and Karan contest an election. Tina gets 47% of the valid votes and is defeated by 1320 votes. Find the total number of valid votes cast in the election. 

Tina gets 47% of the valid votes

⇒Karan gets $\begin{align*}(100 - 47) \% = 53 \%\end{align*}$ of the valid votes. 

∴ Percentage difference between the votes$\begin{align*} = 53 \% - 47 \% = 6 \%. \end{align*}$

According to the question, 

$$\begin{align*}6 \% \text{ of valid votes} & = 1320 \\ \frac{6}{100} \times \text{ valid votes} & = 1320 \\ \text{valid votes} & = \frac{1320 \times 100}{6} \\ & = 22000\end{align*}$$ Hence, the number of valid votes cast in the election were 22000. 

Example 4

A time interval of 6 minutes 30 seconds is wrongly estimated as 6 minutes 40 seconds. What is the percentage error?

Actual time interval = 6 minutes 30 seconds$\begin{align*} = ( 6 \times 60) + 30 = 390\end{align*}$ seconds. 

Estimated time interval = 6 minutes 40 seconds$\begin{align*} = ( 6 \times 60 ) + 40 = 400 \end{align*}$ seconds. 

∴ Error in time interval$\begin{align*} = 400 - 390 = 10 \end{align*}$ seconds. 

∴ Percentage error$\begin{align*} = \frac{10}{390} \times 100 = 2.56 \%. \end{align*}$

Example 5

1% of the people of a village died due to an epidemic. A panic set in during which 15% of the remaining people left the village. If the population is then reduced to 1683, what was it originally?

Let the original population of the village be $\begin{align*}x.\end{align*}$

Since 1% of people died due to an epidemic, the percentage of people who survived$\begin{align*} = (100-1) \% = 99 \%. \end{align*}$

Further, as 15% of the remaining people left the village, people left in the village

$$\begin{align*}& = \left( 1 - \frac{15}{100} \right) \ \text{of} \ 99\% \ \text{of} \ x \\ & = \frac{85}{100} \ \text{of} \ \frac{99}{100} x\end{align*}$$ According to the question,  $$\begin{align*}\frac{85}{100} \times \frac{99}{100} x & = 1683 \\ x & = \frac{1683 \times 100 \times 100}{85 \times 99} = 2000\end{align*}$$ Hence, the original population of the village was 2000.    

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Percentage - Review Questions

1. In a science test, Ridhima got 26 marks out of 40. What percentage is this? 

2. Kajal owed her father ₹1440. She paid back ₹240. What percentage of the original amount did she still owe? 

3. Out of 400 students at a school, 240 came on time, 100 were late and 60 were absent. Write down the percentage of students who 

1.  came on time
2. were late
3. were absent

4. At an RTO last week, 135 people took a driving test. The results are shown below. 

	Number tested	Number passed
Male	75	39
Female	60	42

Find the percentage of 

1. males who passed
2. females who passed
3. males who failed
4. females who failed
5. people who passed
6. people who failed

5. A petrol tanker contains 2580 litres of petrol. If 928.8 litres are drawn off, what percentage of the original amount remains? 

6. In a sports camp, 75% of the athletes are boys. There are 18 girls. How many athletes are there in the group? 

7. The cost of running a car is 38% for petrol, 25% for tax, insurance and repair, and the remaining depreciation. Rohit's costs for a year were ₹18500. By how much did his car depreciate? 

8.  The sides of a rectangle are 24 cm and 18 cm. If each side is increased by 20%, find the percentage increase in the area?

9. Nia's weight is 50 kg and Ria's weight is 42 kg. By what percentage is Ria's weight less than that of Nia?

10. Karan's salary increased by 20% and then decreased by 25%. The effective change in his salary was ₹9250. Find his original salary.