B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Dividing Whole Numbers by Decimal Tenths

B2.8 B2.1 A1About 45 minutes

Learning goals

I can explain what dividing by a decimal like 0.5 means.
I can divide a three-digit whole number by a decimal in tenths by changing it to an equal division with whole numbers.
I can decide what to do with a remainder in a real problem.

Words to know

dividend
The number being divided. In 240 ÷ 0.6, the dividend is 240.

divisor
The number you divide by. In 240 ÷ 0.6, the divisor is 0.6.

quotient
The answer when you divide.

remainder
The amount left over when a number can't be divided into equal groups exactly.

equivalent division
A different division with the same answer, like 240 ÷ 0.6 = 2 400 ÷ 6.

Remember from Grade 5

In Grade 5 you learned that 3 ÷ 1/4 asks how many fourths fit in 3? (12). You also divided by 2-digit numbers and decided what to do with remainders. Dividing by a decimal uses both ideas.

Let's start with a question

How many 0.5 m pieces of ribbon can you get from 3 m?

Each metre holds 2 half-metres. 3 metres hold 3 × 2 = 6. So 3 ÷ 0.5 = 6.

The answer, 6, is bigger than 3. When you divide by a number less than 1, you are counting small pieces, so you get lots of them.

There are 6 jumps of 0.5 from 0 to 3, so 3 ÷ 0.5 = 6.

The strategy: make it tenths

Think of both numbers in tenths.

240 = 2 400 tenths.
0.6 = 6 tenths.
How many groups of 6 tenths fit in 2 400 tenths? 2 400 ÷ 6 = 400.

So 240 ÷ 0.6 = 2 400 ÷ 6 = 400.

The shortcut: multiply both numbers by 10. The quotient stays the same, and now you are dividing by a whole number. It's like sharing: 10 times as much stuff in groups that are 10 times as big gives the same number of groups.

Change dollars to dimes (tenths): 2 ÷ 0.5 becomes 20 ÷ 5 = 4. That is the "multiply both by 10" shortcut, using coins.

Worked example

Example 1: Divide by a decimal less than 1

240 ÷ 0.6 = ?

Multiply both numbers by 10: 240 ÷ 0.6 = 2 400 ÷ 6.
2 400 ÷ 6 = 400 (because 24 ÷ 6 = 4).
Check: 400 × 0.6 = 240. ✓

240 ÷ 0.6 = 400

Worked example

Example 2: Divide by a decimal greater than 1

378 ÷ 1.4 = ?

Multiply both by 10: 3 780 ÷ 14.
Partial quotients: 14 × 200 = 2 800, leaving 980. 14 × 70 = 980, leaving 0.
200 + 70 = 270.
Check: 270 × 1.4 = 270 + 108 = 378. ✓

378 ÷ 1.4 = 270

Worked example

Example 3: What about the remainder?

A roll has 125 m of ribbon. Each prize ribbon needs 0.4 m. How many prize ribbons can be made? How much ribbon is left?

Think in tenths: 125 m = 1 250 tenths of a metre. 0.4 m = 4 tenths.
1 250 ÷ 4 = 312 R2.
The remainder is 2 tenths of a metre, which is 0.2 m.
Check: 312 × 0.4 = 124.8. 124.8 + 0.2 = 125. ✓

312 prize ribbons, with 0.2 m left over. (A remainder of 2 here means 2 tenths, because we were counting tenths.)

Remainders: think about the situation

Read the question to decide what the answer should look like.

How many whole pieces? Give the whole number and say what is left over.
How many buses or boxes do we need? Round up so nobody is left out.
An exact amount (like money)? Keep dividing to get a decimal: 1 250 ÷ 4 = 312.5, so 125 ÷ 0.4 = 312.5.

Just for fun

Dividing by 0.5 is the only time you can say "I divided my cookies and ended up with MORE pieces!" Sadly, there's still the same amount of cookie.

Hands-on activity

Dimes and dollars

You need: 20 dimes or 20 small paper squares labelled 0.1 (one tenth of a dollar), paper and a pencil.

Two dollars is 20 dimes. Put them into groups worth $0.20 (2 dimes each). How many groups? (10) So 2 ÷ 0.2 = 10.
Put the 20 dimes into groups worth $0.50 (5 dimes each). How many? (4) So 2 ÷ 0.5 = 4.
Now use the pattern: how many $0.50 groups would there be in $100? (1 000 dimes ÷ 5 = 200.) So 100 ÷ 0.5 = 200.
Make up your own: how many $0.40 groups are in $120?

Math talk: "Why does changing dollars into dimes make the division easier?"

Your turn

100 ÷ 0.5 = ?

120 ÷ 0.3 = ?

360 ÷ 0.9 = ?

256 ÷ 0.8 = ?

135 ÷ 1.5 = ?

450 ÷ 2.5 = ?

101 ÷ 0.2 = ?

More practice: warm-up

10 ÷ 0.1 = ?

5 ÷ 0.5 = ?

2 ÷ 0.2 = ?

3 ÷ 0.1 = ?

4 ÷ 0.5 = ?

More practice: on your own

432 ÷ 1.2 = ?

510 ÷ 0.6 = ?

288 ÷ 3.2 = ?

700 ÷ 0.4 = ?

A 252 km cycling trail is marked into sections that are each 0.7 km long. How many sections are there?

Challenge

200 L of apple cider is poured into bottles that hold 0.3 L. How many bottles can be filled? How much cider is left over?

154 m of rope is cut into skipping ropes 2.5 m long. How many skipping ropes? How much rope is left?

Explain why 300 ÷ 0.5 is greater than 300.

Check your understanding

Quiz

What is 160 ÷ 0.4?

Which division has the same answer as 300 ÷ 0.6?

What is 455 ÷ 0.5?

What is 175 ÷ 3.5?

100 m of ribbon is cut into 0.3 m pieces. What is the answer?

Which of these is greater than 240?

For parents and teachers

Big idea: B2.8 asks Grade 6 students to represent and solve problems involving the division of three-digit whole numbers by decimal tenths, using appropriate tools, strategies and algorithms, and expressing remainders as appropriate. The main strategy is an equivalent division: multiplying both numbers by 10 (thinking in tenths) turns 240 ÷ 0.6 into 2 400 ÷ 6.

Common difficulties:

Multiplying only one number by 10. 300 ÷ 0.6 is not 300 ÷ 6.
Reading a remainder in the wrong unit. In 1 250 ÷ 4 = 312 R2 (from 125 ÷ 0.4), the 2 means 2 tenths, so 0.2 m is left.
Expecting division to always make numbers smaller.

How to help: Use money: how many 25-cent stickers can you buy with $10? (10 ÷ 0.25 = 40.) Ask your child to check every division with multiplication. Thinking about what a remainder means in a real situation builds the A1 skill of reflecting on whether an answer makes sense.