What is the difference between $\frac{2}{5}$ of 60 and $\frac{2}{5}$ of 30?

$\frac{2}{5}$ of 60 = 24. $\frac{2}{5}$ of 30 = 12. The first is twice the second because 60 is twice 30.

What does "$\frac{2}{5}$ of 60" mean?

It means finding two fifths of 60: divide 60 by 5 to get 12, then multiply by 2 to get 24.

What is the "divide then multiply" rule?

To find $\frac{a}{b}$ of an amount: divide by $b$ (denominator), then multiply by $a$ (numerator).

What does "of" mean in "$\frac{3}{4}$ of 20"?

"Of" means multiply. So $\frac{3}{4}$ of 20 = $\frac{3}{4} \times 20$.

Why do you divide by the denominator first?

Dividing by the denominator splits the amount into that many equal parts. Each part is $\frac{1}{b}$ of the whole.

How do you find $\frac{5}{6}$ of $\frac{8}{3}$?

Multiply the fractions: $\frac{5}{6} \times \frac{8}{3} = \frac{40}{18} = \frac{20}{9}$. This is a fraction of a fraction.

When might you use the decimal method?

When the fraction has a simple decimal form, e.g. $\frac{1}{4} = 0.25$, $\frac{1}{5} = 0.2$, $\frac{9}{10} = 0.9$. Then multiply the decimal by the amount.

Compare finding $\frac{1}{3}$ of 90 with $\frac{2}{3}$ of 90.

$\frac{1}{3}$ of 90 = 90 ÷ 3 = 30. $\frac{2}{3}$ of 90 = 30 × 2 = 60. The second is twice the first.

How does shading $\frac{3}{5}$ of a grid relate to finding $\frac{3}{5}$ of 20?

Both use the same rule: divide by 5, multiply by 3. For a grid of 20 squares, 20 ÷ 5 = 4, 4 × 3 = 12 squares. For an amount, 20 ÷ 5 = 4, 4 × 3 = 12.

What error gives $\frac{2}{5}$ of 60 = 150?

Multiplying first then dividing: 60 × 5 ÷ 2 = 150. The correct order is divide by denominator (5), then multiply by numerator (2): 60 ÷ 5 × 2 = 24.

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Fractions of Amounts

Summary

Finding a fraction of an amount means calculating what part of a quantity the fraction represents. Three methods apply: divide by the denominator then multiply by the numerator; convert the fraction to a decimal and multiply; or write the amount as a fraction and multiply the two fractions together.

1. Definition & Core Concepts

2. Underlying Principles

Finding $\frac{1}{n}$ of an amount means dividing by $n$ (splitting into $n$ equal parts).

Finding $\frac{m}{n}$ means taking $m$ of those parts: divide by $n$ , then multiply by $m$ .

The order "divide by denominator, multiply by numerator" matches the structure of the fraction.

3. Methods & Techniques

Bar model showing 60 split into 3 parts: 2 parts shaded (⅔) equals 40.

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls

7. Connections

Fractions of Amounts

Summary

1. Definition & Core Concepts

A fraction of an amount is the result of taking that fraction of a given quantity.

" $\frac{2}{5}$ of 60" means finding what $\frac{2}{5}$ of 60 equals.

The phrase "of" in this context means multiply: $\frac{2}{5}$ of 60 = $\frac{2}{5} \times 60$ .

All three methods give the same answer; choose the one that suits the numbers.

2. Underlying Principles

Finding $\frac{1}{n}$ of an amount means dividing by $n$ (splitting into $n$ equal parts).

Finding $\frac{m}{n}$ means taking $m$ of those parts: divide by $n$ , then multiply by $m$ .

The order "divide by denominator, multiply by numerator" matches the structure of the fraction.

3. Methods & Techniques

Method 1 (divide then multiply): Divide the amount by the denominator, then multiply by the numerator. For $\frac{2}{5}$ of 60: $60 \div 5 = 12$ , $12 \times 2 = 24$ .

Method 2 (decimal): Convert the fraction to a decimal, then multiply. $\frac{1}{4} = 0.25$ , so $\frac{1}{4}$ of 16 = $0.25 \times 16 = 4$ .

Method 3 (fraction multiplication): Write the amount as $\frac{60}{1}$ , then $\frac{2}{3} \times \frac{60}{1} = \frac{120}{3} = 40$ .

For fractions of fractions, e.g. $\frac{5}{6}$ of $\frac{8}{3}$ , multiply: $\frac{5}{6} \times \frac{8}{3} = \frac{40}{18} = \frac{20}{9}$ .

Bar model showing 60 split into 3 parts: 2 parts shaded (⅔) equals 40.

4. Key Distinctions

Method	When to use	Example
Divide then multiply	Simple fractions, whole numbers	$\frac{2}{5}$ of 60
Decimal	Fractions with easy decimals	$\frac{1}{4}$ of 16
Fraction multiplication	Fractions of fractions	$\frac{5}{6}$ of $\frac{8}{3}$

Method 1 is usually the most efficient for whole number amounts.

5. Exam Strategy & Tips

For whole number amounts, Method 1 (divide by denominator, multiply by numerator) is quick and reliable.

Use Method 2 when the fraction has a simple decimal (e.g. $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{5}$ , $\frac{1}{10}$ ).

Method 3 is essential when finding a fraction of another fraction.

6. Common Pitfalls

Wrong order: $\frac{2}{5}$ of 60 is NOT $60 \times 5 \div 2$ . It is $60 \div 5 \times 2$ .

Confusing numerator and denominator: Divide by the denominator (bottom), multiply by the numerator (top).

Forgetting "of" means multiply: $\frac{3}{4}$ of 20 = $\frac{3}{4} \times 20$ , not $\frac{3}{4} + 20$ .

7. Connections

This topic uses the same "divide by denominator, multiply by numerator" rule as shading fractions on grids.

Fraction multiplication underpins Method 3.

Percentages are fractions of amounts: 25% of 80 = $\frac{25}{100} \times 80$ .