Is the quotient smaller?

The idea

When two counting numbers are divided the quotient is always less than the dividend, but when the divisor is a decimal the quotient can be greater than the dividend.

128 ÷ 0.4

When two counting numbers are divided, the quotient is smaller than the dividend: 128 ÷ 4 = 32. Now look at 128 ÷ 0.4. The book says it is 320.

Compared with 128, the quotient 320 is ...

• less than 128
• equal to 128
• more than 128

How many pieces?

A ribbon is cut into pieces of the same length. The number of pieces is the quotient. Tap different piece lengths and see how many pieces fit.

Pieces shorter than 1

A piece longer than 1 needs fewer pieces than units. A piece of length 1 gives one piece for each unit. A piece shorter than 1 gives more pieces than units.

Another way to see it: 128 ÷ 0.4 = 1280 ÷ 4 = 320. Dividing by 0.4 is the same as dividing 10 times as much by 4.

When two counting numbers are divided, the quotient is less than the dividend (the same when the divisor is 1). But when the divisor is a decimal between 0 and 1, the quotient can be greater than the dividend.

• Divisor | Example | Quotient is
• more than 1 | 128 ÷ 4 = 32 | less than 128
• equal to 1 | 128 ÷ 1 = 128 | the same
• between 0 and 1 | 128 ÷ 0.4 = 320 | more than 128

Notes

Counting numbers: the quotient is less than the dividend (equal if the divisor is 1). Divisor between 0 and 1: the quotient is greater than the dividend.

Check yourself

Without dividing: which of these is more than 50?

How many 0.25 L glasses can be filled from a 2 L bottle?

Answer: 8 glasses

2 ÷ 0.25 = 200 ÷ 25 = 8. The quotient 8 is more than the dividend 2.

Rohan says, “Dividing always gives a smaller number.” Which division shows he is wrong?

Which divisor makes 12 ÷ ? greater than 12?

First say whether the answer will be more or less than 36. Then find 36 ÷ 0.9.

Answer: 40

36 ÷ 0.9 = 360 ÷ 9 = 40, which is more than 36.

• 50 ÷ 2.5. The divisor is more than 1, so the quotient is less than 50. (It is 20.)
• 50 ÷ 0.5 — correct. The divisor is between 0 and 1, so the quotient is more than 50. (It is 100.)
• 50 ÷ 5. The divisor is more than 1, so the quotient is less than 50. (It is 10.)
• 9 ÷ 0.3 = 30 — correct. 0.3 is between 0 and 1, so the quotient is more than 9.
• 9 ÷ 3 = 3. This fits Rohan’s claim. We need a quotient that is bigger than 9.
• 9 ÷ 1.5 = 6. This fits his claim too: 1.5 is more than 1.
• 0.6 — correct. It is between 0 and 1. 12 ÷ 0.6 = 20.
• 1.2. 1.2 is more than 1, so 12 ÷ 1.2 = 10 is less than 12.
• 4. 4 is more than 1, so 12 ÷ 4 = 3 is less than 12.