Fraction of a quantity

"$\tfrac{3}{4}$ of $20$" means $\tfrac{3}{4} \times 20 = 15$. The English word "of" is just multiplication in disguise.

Idea

To find a fraction of a quantity, multiply the fraction by the quantity. So: $\tfrac{a}{b} \text{ of } Q = \tfrac{a}{b} \times Q = \tfrac{aQ}{b}.$

For example:

• $\tfrac{1}{2}$ of $20 = 10$.
• $\tfrac{3}{4}$ of $\rupee 240 = \rupee 180$.
• $\tfrac{2}{5}$ of $50$ students $= 20$ students.

To compute fractions of large quantities, divide first by the denominator, then multiply by the numerator. So $\tfrac{3}{4}$ of $\rupee 240$: $240 \div 4 = 60$, then $60 \times 3 = 180$.

Reverse problems. If $\tfrac{2}{3}$ of a number is $24$, what is the number? Let the number be $n$. Then $\tfrac{2}{3} n = 24 \Rightarrow n = 24 \div \tfrac{2}{3} = 24 \times \tfrac{3}{2} = 36$. So the number is $36$.

Percent connection. A percentage is just a fraction over $100$. So $25\%$ of $80$ is $\tfrac{25}{100} \times 80 = 20$. Familiar percent values are easy to remember as fractions: $50\% = \tfrac{1}{2}$, $25\% = \tfrac{1}{4}$, $20\% = \tfrac{1}{5}$, $75\% = \tfrac{3}{4}$.

This is a very practical sub-topic , it appears in discounts, taxes, recipe scaling, and many everyday situations.

Worked examples

Example 1. Find $\tfrac{2}{5}$ of $35$.

$35 \div 5 = 7$, then $7 \times 2 = 14$.

Example 2. Find $\tfrac{3}{8}$ of $\rupee 640$.

$640 \div 8 = 80$, then $80 \times 3 = \rupee 240$.

Example 3. $\tfrac{3}{4}$ of a class is $24$ students. How many students in the class?

Let total $= n$. Then $\tfrac{3}{4} n = 24 \Rightarrow n = 24 \times \tfrac{4}{3} = 32$.

Example 4. A pen costs $\rupee 60$. A shopkeeper offers $\tfrac{1}{5}$ discount. New price?

Discount: $\tfrac{1}{5} \times 60 = \rupee 12$. New price: $60 - 12 = \rupee 48$.

Try it yourself

1. Find $\tfrac{1}{2}$ of $90$.
2. Find $\tfrac{3}{5}$ of $\rupee 200$.
3. Find $\tfrac{4}{7}$ of $42$.
4. $\tfrac{2}{3}$ of a number is $14$. Find the number.
5. Find $\tfrac{5}{6}$ of $\tfrac{12}{25}$.
6. A book has $360$ pages; you've read $\tfrac{2}{9}$. How many pages read?
7. $\tfrac{1}{4}$ off on a $\rupee 800$ jacket , new price?
8. A jar holds $\tfrac{3}{4}$ kg of sugar; if $\tfrac{2}{3}$ is used, how much is left?

Activity

A recipe makes $4$ servings using $\tfrac{1}{2}$ cup oil. Scale it for $6$ servings , how much oil? (Hint: each serving uses $\tfrac{1}{8}$ cup, so $6$ servings use $\tfrac{6}{8} = \tfrac{3}{4}$ cup.)

Take any product you own with a price tag. Calculate the cost after a $\tfrac{1}{5}$ discount.