Number · Fractions, Decimals and Percentages · Lesson 7

Dividing Fractions, Decimals and Percentages

How many quarter-loaves can you cut from 3 loaves of bread? How much does each of 3 friends get if they share a quarter of a pizza? Both questions are divisions with fractions. In this lesson you will see what dividing by a fraction really means, learn why we "multiply by the reciprocal", and divide common fractions, decimals and percentages with confidence.

NaCCA B7.1.3.3.3 ⏱ about 50 min

By the end you will be able to

✔Explain what a whole number ÷ a fraction means ("how many of the fraction fit in?").
✔Find the reciprocal of a whole number, a fraction or a mixed number.
✔Divide a whole number or a fraction by a fraction by multiplying by the reciprocal of the divisor.
✔Divide a common fraction, a decimal (up to thousandths) or a percentage by a 1-digit whole number.

Key words

Dividend: The number being divided. In 3 ÷ $\frac{1}{4}$ , the dividend is 3.
Divisor: The number you divide by. In 3 ÷ $\frac{1}{4}$ , the divisor is $\frac{1}{4}$ .
Quotient: The answer to a division. 3 ÷ $\frac{1}{4}$ = 12, so the quotient is 12.
Reciprocal: The number you multiply by to get 1. Turn a fraction upside down: the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ , and of 3 is $\frac{1}{3}$ .

Curriculum reference

Content standard: B7.1.3.3 Demonstrate an understanding of the process of multiplying and dividing positive fractions and apply this in solving problems.

Indicator: Explain the process of dividing a fraction (i.e. common, percent and decimal fractions up to thousandths) by a 1-digit whole number and by a fraction.

Starter

Auntie Esi's bread kiosk

Auntie Esi runs a small kiosk at the Kaneshie lorry station. Every morning she buys 3 big loaves of tea bread and cuts each loaf into quarters to sell to drivers and passengers. How many quarter-pieces can she sell? Her nephew Yaw says "3 divided by a quarter must be less than 3, because division makes things smaller." Is Yaw right? Count the pieces in your head... you will soon see why dividing by a fraction can make a number much BIGGER.

1. What does dividing by a fraction mean?

Think about 12 ÷ 3. It asks: how many 3s are there in 12? The answer is 4. Dividing by a fraction asks exactly the same kind of question. So $3 \div \frac{1}{4}$ asks: how many quarters are there in 3 wholes?

Each whole loaf gives 4 quarters, so 3 loaves give 3 × 4 = 12 quarters. Yaw was wrong: because a quarter is small, many quarters fit into 3, and the quotient is bigger than 3. Another way to see it is as a missing-number multiplication: $3 = \frac{1}{4} \times ?$ The missing number is 12, because $\frac{1}{4} \times 12 = 3$ .

▶ Count the quarters: every whole bar holds 4 of them, so 3 wholes hold 12. That is why 3 ÷ $\frac{1}{4}$ = 12.

2. The reciprocal method

Drawing bars works for easy numbers, but we need a method that always works. It uses the reciprocal. Two numbers are reciprocals if they multiply to give 1: $\frac{1}{4} \times 4 = 1$ , so 4 is the reciprocal of $\frac{1}{4}$ . To find a reciprocal, write the number as a fraction and turn it upside down.

Reciprocals
Number	As a fraction	Reciprocal	Check
$\frac{1}{4}$	$\frac{1}{4}$	$\frac{4}{1} = 4$	$\frac{1}{4} \times 4 = 1$
$\frac{2}{5}$	$\frac{2}{5}$	$\frac{5}{2}$	$\frac{2}{5} \times \frac{5}{2} = 1$
3	$\frac{3}{1}$	$\frac{1}{3}$	$3 \times \frac{1}{3} = 1$
$1 \frac{2}{3}$	$\frac{5}{3}$	$\frac{3}{5}$	$\frac{5}{3} \times \frac{3}{5} = 1$

Why it works. Multiplying the dividend and the divisor by the same number does not change a division (12 ÷ 3 = 24 ÷ 6 = 4). So multiply both by the reciprocal of the divisor: $3 \div \frac{1}{4} = (3 \times 4) \div (\frac{1}{4} \times 4) = 12 \div 1 = 12$ . The divisor always becomes 1, so the answer is simply dividend × reciprocal of the divisor.

1Change whole numbers and mixed numbers into common fractions.
2Keep the first fraction (the dividend) as it is.
3Change ÷ into × and turn the divisor upside down (use its reciprocal).
4Multiply numerators and denominators, cancelling where you can, then simplify.

▶ Notice that exactly 4 jumps of $\frac{3}{4}$ reach 3. The reciprocal method agrees: $3 \times \frac{4}{3} = 4$ .

3. Dividing a fraction by a whole number

Now share a fraction among a few people. $\frac{1}{4} \div 3$ means a quarter of a loaf shared equally by 3 people. Each person's piece is small: cutting each quarter into 3 makes 12 equal pieces in the whole loaf, so each person gets $\frac{1}{12}$ of the loaf.

The reciprocal method gives the same answer: the reciprocal of 3 is $\frac{1}{3}$ , so $\frac{1}{4} \div 3 = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$ . Dividing by 3 is the same as finding $\frac{1}{3}$ of the fraction. When the numerator divides exactly, there is a shortcut: $\frac{6}{7} \div 3 = \frac{6 \div 3}{7} = \frac{2}{7}$ .

▶ Notice that one quarter shared by 3 becomes one twelfth of the whole bar.

4. Dividing decimals and percentages

To divide a decimal by a 1-digit whole number, use short division and keep the decimal point in the same column: 0.375 ÷ 3 = 0.125 (3 into 3 is 1, 3 into 7 is 2 remainder 1, 3 into 15 is 5). Check: 0.125 × 3 = 0.375 ✓. A percentage divided by a whole number works the same way: 45% ÷ 5 = 9%, because 45 hundredths shared into 5 is 9 hundredths.

To divide by a decimal, multiply both numbers by 10, 100 or 1,000 so that the divisor becomes a whole number — the same "multiply both" idea you used with reciprocals. For example 0.6 ÷ 0.2 = 6 ÷ 2 = 3, and 1.25 ÷ 0.05 = 125 ÷ 5 = 25.

Decimal and percentage divisions
Problem	Working	Answer
0.9 ÷ 3	9 tenths ÷ 3 = 3 tenths	0.3
0.375 ÷ 3	short division, point kept in place	0.125
45% ÷ 5	45 ÷ 5 = 9	9%
0.6 ÷ 0.2	× 10 both: 6 ÷ 2	3
1.25 ÷ 0.05	× 100 both: 125 ÷ 5	25

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Whole number ÷ unit fraction

Find

3 \div \frac{1}{4}

Show the solution

Step 1Meaning: how many quarters are in 3?
Step 2The reciprocal of $\frac{1}{4}$ is 4.
Step 3 $3 \div \frac{1}{4} = 3 \times 4 = 12$ .
Step 4Check: $12 \times \frac{1}{4} = 3$ ✓

Answer: 12

Example 2

Fraction ÷ whole number

Find

\frac{1}{4} \div 3

Show the solution

Step 1The reciprocal of 3 is $\frac{1}{3}$ .
Step 2 $\frac{1}{4} \div 3 = \frac{1}{4} \times \frac{1}{3}$ .
Step 3= $\frac{1}{12}$ .

Answer: $\frac{1}{12}$

Example 3

Dividing by a mixed number

Divide:

5 \div 1 \frac{2}{3}

Show the solution

Step 1Change to fractions: $5 = \frac{5}{1}$ and $1 \frac{2}{3} = \frac{5}{3}$ .
Step 2Multiply by the reciprocal of the divisor: $\frac{5}{1} \times \frac{3}{5}$ .
Step 3Cancel the 5s: $\frac{1}{1} \times \frac{3}{1} = 3$ .
Step 4Check: $3 \times 1 \frac{2}{3} = 3 \times \frac{5}{3} = 5$ ✓

Answer: 3

Example 4

Fraction ÷ fraction

Divide:

\frac{5}{8} \div \frac{1}{2}

Show the solution

Step 1Keep $\frac{5}{8}$ , change ÷ to ×, flip $\frac{1}{2}$ to $\frac{2}{1}$ .
Step 2 $\frac{5}{8} \times \frac{2}{1} = \frac{10}{8}$ .
Step 3Simplify: $\frac{10}{8} = \frac{5}{4} = 1 \frac{1}{4}$ .

Answer: $1 \frac{1}{4}$

Example 5

Mixed number ÷ fraction

Simplify

2 \frac{1}{4} \div \frac{3}{8}

Show the solution

Step 1 $2 \frac{1}{4} = \frac{9}{4}$ .
Step 2 $\frac{9}{4} \div \frac{3}{8} = \frac{9}{4} \times \frac{8}{3}$ .
Step 3Cancel 9 and 3 by 3, and 8 and 4 by 4: $\frac{3}{1} \times \frac{2}{1} = 6$ .

Answer: 6

Summary

Whole number ÷ fraction asks "how many of this fraction fit into the number?" — the answer can be bigger than the number.
The reciprocal of a number is the number that multiplies with it to give 1; for a fraction, turn it upside down.
To divide by a fraction, multiply by the reciprocal of the divisor (keep, change, flip). Flip only the divisor.
Change mixed numbers to improper fractions before dividing.
Divide a decimal or percentage by a whole number with short division; to divide by a decimal, first multiply both numbers by 10, 100 or 1,000.

Auntie Esi's bread kiosk

1. What does dividing by a fraction mean?

2. The reciprocal method

3. Dividing a fraction by a whole number

4. Dividing decimals and percentages

Worked examples

Whole number ÷ unit fraction

Fraction ÷ whole number

Dividing by a mixed number

Fraction ÷ fraction

Mixed number ÷ fraction

Summary

Practice: 15 questions