Grade by Grade
B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Dividing Decimals by Decimals

B2.9 A1About 45 minutes
Learning goals
  • I can explain dividing by a decimal as "how many of this fit into that?"
  • I can change a decimal divisor into a whole number by multiplying both numbers by 10, 100 or 1 000.
  • I can divide decimals by decimals and check with multiplication.
  • I can find unit prices and compare better buys.

Words to know

dividend
The number being divided. In 4.8 ÷ 0.6, the dividend is 4.8.
divisor
The number you divide by. In 4.8 ÷ 0.6, the divisor is 0.6.
quotient
The answer to a division.
equivalent division
A division with the same quotient, made by multiplying the dividend and divisor by the same number: 4.8 ÷ 0.6 = 48 ÷ 6.

Remember from Grade 6

In Grade 6 you divided whole numbers by tenths (240 ÷ 0.6 = 2 400 ÷ 6 = 400) and decimals by whole numbers (7.5 ÷ 3 = 2.5). Now you will divide a decimal by a decimal. The good news: you can turn it into a division you already know.

Let's start with a shelf

A board is 1.2 m long. You cut it into pieces that are 0.3 m long. How many pieces do you get?

You are asking how many 0.3s fit into 1.2: 1.2 ÷ 0.3. The picture shows 4 pieces. Notice that 12 tenths ÷ 3 tenths is the same as 12 ÷ 3 = 4.

0.3 m0.3 m0.3 m0.3 m01.2 m1.2 ÷ 0.3 = 4 pieces
A 1.2 m board cut into 0.3 m pieces. Four pieces fit, so 1.2 ÷ 0.3 = 4.

The method: make the divisor a whole number

If you multiply the dividend and the divisor by the same number, the quotient does not change. (Think of sharing: 12 cookies among 3 people gives the same amount each as 120 cookies among 30 people.)

  1. Multiply both numbers by 10, 100 or 1 000 so the divisor becomes a whole number.
  2. Divide as you did in Grade 6.
  3. Check by multiplying the quotient by the original divisor.
  • 4.8 ÷ 0.6 → (× 10) → 48 ÷ 6 = 8
  • 7.5 ÷ 0.25 → (× 100) → 750 ÷ 25 = 30
  • 2.88 ÷ 1.2 → (× 10) → 28.8 ÷ 12 = 2.4
Worked example

Example 1: Hundredths divisor

How many 0.25 L glasses can you fill from 7.5 L of lemonade?

  1. 7.5 ÷ 0.25. The divisor has hundredths, so multiply both by 100.
  2. 750 ÷ 25 = 30. (25 × 30 = 750.)
  3. Check: 30 × 0.25 = 7.5. ✓

30 glasses

$125¢25¢25¢25¢$125¢25¢25¢25¢$125¢25¢25¢25¢$125¢25¢25¢25¢$125¢25¢25¢25¢$125¢25¢25¢25¢$125¢25¢25¢25¢25¢25¢50¢$7.50 ÷ $0.25 = ?7 dollars × 4 quarters = 2850¢ = 2 more quarters28 + 2 = 30 quarters
7.5 ÷ 0.25 asks "how many quarters make $7.50?" Four quarters in each dollar gives 28, plus 2 more: 30, the same as 750 ÷ 25.
Worked example

Example 2: Decimal answer

Find 2.88 ÷ 1.2.

  1. Multiply both by 10: 28.8 ÷ 12.
  2. Estimate: 24 ÷ 12 = 2 and 36 ÷ 12 = 3, so the answer is between 2 and 3.
  3. 28.8 ÷ 12: 12 × 2 = 24, leaving 4.8. 4.8 ÷ 12 = 0.4. Total 2.4.
  4. Check: 2.4 × 1.2 = 2.88. ✓

2.88 ÷ 1.2 = 2.4

Worked example

Example 3: Unit price

A 1.2 kg bag of carrots costs $4.68. What is the price per kilogram?

  1. Price per kg = 4.68 ÷ 1.2.
  2. Multiply both by 10: 46.8 ÷ 12.
  3. 12 × 3 = 36, leaving 10.8. 10.8 ÷ 12 = 0.9. Total 3.9.
  4. Check: 3.90 × 1.2 = 4.68. ✓

$3.90 per kilogram

Think like a mathematician

Dividing by a number less than 1 gives an answer bigger than the dividend, because many small pieces fit. 4.8 ÷ 0.6 = 8, which is bigger than 4.8. If your answer to a division by 0.6 came out smaller than the dividend, check your decimal point.

Hands-on activity

Coin division

You need: a handful of coins (quarters, dimes, nickels), or paper circles labelled $0.25, $0.10 and $0.05.

  1. Make $3.75. How many quarters is that? Write it as a division: 3.75 ÷ 0.25 = ?
  2. Make $1.20 using dimes. How many dimes? 1.20 ÷ 0.10 = ?
  3. Make $0.85 using nickels. How many nickels? 0.85 ÷ 0.05 = ?
  4. Solve each one on paper by multiplying both numbers by 100, and check that it matches your coins.

What to look for: 3.75 ÷ 0.25 = 375 ÷ 25 = 15. 1.20 ÷ 0.10 = 120 ÷ 10 = 12. 0.85 ÷ 0.05 = 85 ÷ 5 = 17.

Math talk: "Why do we multiply by 100 and not by 10 here?"

Just for fun

How many dimes fit into a toonie? 2 ÷ 0.1 = 20. Dividing decimals is basically counting coins with style.

Your turn

1

1.2 ÷ 0.3 = ?

2

4.5 ÷ 0.5 = ?

3

0.56 ÷ 0.08 = ?

4

6.3 ÷ 0.9 = ?

5

3.6 ÷ 0.12 = ?

6

9.6 ÷ 1.2 = ?

7

2.25 ÷ 1.5 = ?

More practice: warm-up

1

0.8 ÷ 0.2 = ?

2

1 ÷ 0.1 = ?

3

2.4 ÷ 0.4 = ?

4

0.9 ÷ 0.3 = ?

5

5 ÷ 0.5 = ?

More practice: on your own

1

7.2 ÷ 0.6 = ?

2

1.44 ÷ 0.12 = ?

3

13.5 ÷ 2.7 = ?

4

A 4.2 m roll of ribbon is cut into 0.35 m pieces. How many pieces?

5

A 2.5 kg bag of apples costs $6.25. What is the price per kilogram?

Challenge

1

Which is the better buy: 1.5 L of juice for $3.30 or 2 L for $4.20?

2

8.1 ÷ 0.45 = ?

3

Mila snowshoes 7.8 km in 1.5 hours at a steady pace. What is her speed in km per hour? How far would she go in 2.5 hours at that speed?

Check your understanding

Quiz

1.

What is 3.6 ÷ 0.4?

2.

Which division has the same answer as 0.75 ÷ 0.05?

3.

What is 1.8 ÷ 0.06?

4.

A 0.5 kg bag of nuts costs $4.50. What is the price per kilogram?

5.

Without calculating, which is true about 6.4 ÷ 0.8?

6.

How would you check 5.04 ÷ 1.2 = 4.2?

For parents and teachers

Big idea: The Ontario curriculum (B2.9) asks Grade 7 students to multiply and divide decimal numbers by decimal numbers, in various contexts. The key idea is that multiplying the dividend and the divisor by the same power of 10 keeps the quotient the same, which turns the problem into one students already know.

Common difficulties:

  • Multiplying only the divisor by 10 or 100 (and not the dividend).
  • Expecting a smaller answer when dividing. Dividing by a decimal less than 1 gives a larger answer.
  • Forgetting to check with multiplication.

How to help: Compare unit prices on the shelf while shopping ("Which is cheaper per litre?"). Use coins to show how many quarters or dimes make an amount. This lesson builds A1 skills: reasoning about equivalent forms, estimating, and making connections to money decisions.