Whole number times a fraction

The idea

Multiplying a fraction by a whole number works just as with whole numbers, as repeated addition: the fraction is added as many times as the whole number says.

A tortoise on a walk

Aaron walks 3 km in one hour. In 5 hours he walks 5 × 3 = 3 + 3 + 3 + 3 + 3 = 15 km.

His pet tortoise is slow. It walks only 1/4 km in one hour.

How far does the tortoise walk in 3 hours?

• 3/4 km
• 3/12 km
• 1/12 km

Take the hops

Each hop is the distance covered in one hour. Take the hops and watch the pieces under the line.

Add the fraction again and again

Aaron’s 5 × 3 is 3 added 5 times. The tortoise’s distance is 1/4 added 3 times. The multiplication works the same way, even though the distance in one hour is a fraction.

Three hops of one quarter make 3 quarters. The pieces are still quarters, so the denominator 4 does not change.

Multiplying a fraction by a whole number is repeated addition: the fraction is added as many times as the whole number says. 3 × 1/4 = 1/4 + 1/4 + 1/4 = 3/4.

• Repeated addition | Product
• 1/4 + 1/4 + 1/4 | 3 × 1/4 = 3/4
• 2/3 added 5 times | 5 × 2/3 = 10/3

Notes

Multiplying a fraction by a whole number is repeated addition: the fraction is added as many times as the whole number says. 3 × 1/4 = 1/4 + 1/4 + 1/4 = 3/4.

Check yourself

A snail crawls 1/8 metre in one minute. How far does it crawl in 5 minutes? Type a fraction, like 2/9.

Answer: 5/8

5 × 1/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 5/8 metre.

Which repeated addition is the same as 4 × 3/7?

One bottle holds 3/10 litre of juice. How many litres are there in 4 bottles? Type a fraction or a mixed number, like 2 1/10.

Answer: 1 1/5

4 × 3/10 = 12 tenths = 12/10 = 1 1/5 litres.

Ria writes 3 × 1/4 = 3/12, because 3 × 1 = 3 and 3 × 4 = 12. What is wrong?

• 3/7 + 3/7 + 3/7 + 3/7 — correct. Yes! The fraction 3/7 is added 4 times.
• 4/7 + 4/7 + 4/7. That is 3 × 4/7: the fraction 4/7 added 3 times. In 4 × 3/7 the fraction is 3/7 and it is added 4 times.
• 3/7 + 4. The whole number says how many times to add the fraction. It is not added itself.
• Nothing. 3/12 is right.. 3/12 is the same as 1/4: that is only one hour of walking, not three.
• Three hops of a quarter make 3 quarters. The pieces are still quarters, so it is 3/4. — correct. Yes! Adding quarters gives quarters. Only the number of pieces grows.
• The denominator should be 3 + 4 = 7.. The size of the piece does not change when we add more of the same piece.