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We use percentages to describe different quantities like scores, discounts, rate of interest, and so on. Here, we will learn how we can use percentages to find unknown quantities. We will look at some solved examples where we calculate the value of missing values using the percent equation. ...Read MoreRead Less

A percent proportion is a relation between two ratios such that the percent expressed as a fraction with the denominator 100 is equal to the ratio of the part to its whole.

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \)

This formula can be used to calculate the percentage of a given ratio as well as the missing value of a part or a whole.

**Numerical Applications:**

Two out of five is 40%.

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \)

\( \frac{2}{5}=\frac{40}{100} \)

**Example: **Three out of five is 60%.

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \)

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

The model to represent the above example:

In this model, the whole is 100% and is divided into 10 parts. So each part is 10%. Hence, to represent 60%, we shade 6 parts.

The same model as above is divided into 5 parts. To express three out of five or \( \frac{3}{5} \), we need to shade 3 parts.

A percent equation is another way to express the percent proportion.

Let’s understand this with the help of an example,

Use the percent equation to represent that “x is p percent of y.”

\( x=p\%\times y \)

This is the percent equation, where x is a part of the whole, p is the percent, and y is the whole.

\( \text{part}=\text{percent %}\times \text{whole} \)

\( \text{part}=\frac{\text{percent}}{100}\times \text{whole} \)

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \)

Hence, you can see that the percent equation is the same as the percent proportion but both are written differently.

**Example: **

For instance, if someone wanted to know what 20 percent of 80 is, we can use the percent equation to find the answer.

\( \text{part}=\text{percent %}\times \text{whole} \)

\( \text{part}=0.2\times 80=16 \)

Therefore, 20 percent of 80 is 16.

**Example 1: **Find what percent of 20 is 12?

**Solution:**

\(\frac{\text{Part}}{\text{Whole}}=\frac{\text{Percent}}{100}\) Write the percent proportion

\(\frac{12}{20}=\frac{\text{Percent}}{100}\) Substitute 12 for part and 20 for whole

\( 100\times \frac{12}{20}=100\times \frac{\text{percent}}{100} \) The multiplication property of equality

\( \text{percent}=60 \) Simplify

Hence, 60% of 20 is 12.

**Check:** We can use a model to check the answer.

\( \frac{\text{part}}{\text{whole}}=\frac{12+4}{20+4} \)

On simplifying further,

\( \frac{\text{part}}{\text{whole}}=\frac{3}{5} \)

The model below represents 20, the whole. The whole is divided into 5 parts. Therefore, to represent \( \frac{3}{5} \), we need to shade 3 parts. Also, when considering the whole to be 100%, each part is 20%.

Hence, to represent 60%, we need to shade 3 parts. In both cases, we need to shade 3 parts, and therefore our answer is correct.

**Example 2: **Find what part of a number is 2% of 50?

**Solution:**

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \) Write the percent proportion

\( \frac{\text{part}}{50}=\frac{2}{100} \) Substitute 2 for part and 50 for whole

\( 50\times \frac{\text{part}}{50}=50\times \frac{2}{100} \) Multiply each side by 50

\( \text{part}=1 \) Simplify

Hence, 1 is 2% of 50.

**Example 3: **25% of a number is 50. Find the number.

**Solution:**

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \) Write the percent proportion

\( \frac{50}{\text{whole}}=\frac{25}{100} \) Substitute 25 for part and 50 for whole

\( 5000=25\times \text{whole} \) The cross products property

\( \text{whole}=\frac{5000}{25}=200 \) Simplify

Hence, 25% of 200 is 50.

**Example 4: **According to a survey in a city, 60% of students aged 16–24 watch movies at least once a day. The total number of students in the city is 1200. Find the number of students who watch movies daily.

**Solution:**

\( \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100} \) Write the percent proportion

\( =\frac{\text{part}}{1200}=\frac{60}{100} \) Substitute 60 for part and 1200 for whole

\( =1200\times \frac{\text{part}}{1200}=1200\times \frac{60}{100} \) Multiply each side by 1200

\( \text{part}=720\) Simplify

Hence, 720 students watch movies once a day.

**Example 5: **Find what number is 20% of 50?

**Solution:**

Let x be the part, p be the percent, and y be the whole.

\( x=p\%\times y \) Write the percent proportion

\( x=\frac{20}{100}\times 50 \) Substitute \( \frac{20}{100} \) for p% and 50 for y

\( x=10\) Simplify

Hence, 20% of 50 is 10.

**Example 6: **What percent of 30 is 10?

**Solution:**

\( x=p\%\times y \) Write the percent proportion

\( 10=p\%\times 30 \) Substitute 10 for x and 30 for y

\( \frac{1}{30}=\frac{p\%\times 30}{30} \) The division property of equality

\( p\%=0.333\) Simplify

\( \frac{p}{100}=0.333 \)

\( \frac{p}{100}\times 100=0.333\times 100 \) The multiplication property of equality

\( p=33.3 \)

Hence, 33.3% of 30 is 10.

**Example 7: **70 is 25% of what number?

**Solution:**

\( x=p\%\times y \) Write the percent proportion

\( 70=25\%\times y \) Substitute 70 for x and 25 for p%

\( \frac{70}{25}=\frac{25\%\times y}{100\times 25} \) The division property of equality

\( y=\frac{70\times 100}{25}=280\) The cross products property

Hence, 25% of 280 is 70.

**Example 8: **According to a survey, 80% of people aged 18-24 play the popular war game at least once a day. How many people were surveyed if a total of 4416 people played the game?

**Solution:**

Let y be the number of people who play the game.

\( x=p\%\times y \) Write the percent proportion

\( 4416=80\%\times y \) Substitute 70 for x and 25 for p%

\( 4416=\frac{80}{100}\times y \)

\( \frac{4416}{80}=\frac{80\times y}{100\times 80} \) The division property of equality

\( y=\frac{4416}{80}\times 100\) The cross products property

\( y=5520 \) Simplify

5520 people play the game.

Frequently Asked Questions

In mathematics, a percentage is a number or ratio that represents a fraction of 100. Percentage is usually represented by the symbol ‘%’. It is also written simply as ‘percent’. For example, the decimal 0.55, or the fraction \( \frac{55}{100} \), is equivalent to 55% or 55 percent.

Proportion is the equality between two ratios. That is, when one ratio is equal to another, they are said to be in proportion.

Let us take an example to understand this concept. Suppose you have 10 apples. 50% of these apples is 5. So here 5 acts as a part of the whole, which is 10 apples, and 50% is the percent representing 5 apples to 10 apples.