Checkpoint 6 Maths · Number · Fractions

Multiplying and Dividing Fractions by Whole Numbers

Learn how to estimate, multiply and divide proper fractions by whole numbers, with diagrams that show why the rules work, worked Checkpoint-style examples and a 14-question practice quiz.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum

Estimate the answer, then multiply a proper fraction by a whole number, e.g. 3/4 × 5 (6Nf.06)
Estimate the answer, then divide a proper fraction by a whole number, e.g. 3/4 ÷ 2 (6Nf.06)
Write answers in simplest form and as mixed numbers when the answer is bigger than 1 — skills from Simplest Form (6Nf.03) that Checkpoint expects you to use here

In this lesson you learn two short rules — one for multiplying and one for dividing — and, more importantly, why they work. A proper fraction is a fraction smaller than 1, like 3/4 or 2/7. Here we only multiply or divide it by a whole number (2, 3, 12 …). Checkpoint tests this on Paper 1 (no calculator), so you must be able to do it by hand quickly and check your answer with an estimate.

Key words

Word / phrase	Meaning / use
numerator	البسط — the top number; it counts how many parts you have
denominator	المقام — the bottom number; it tells you the size of each part (how many equal parts make 1 whole)
proper fraction	كسر حقيقي — numerator smaller than denominator, e.g. 5/8 (less than 1)
improper fraction	كسر غير حقيقي — numerator equal to or bigger than denominator, e.g. 15/4 (1 or more)
mixed number	عدد كسري — a whole number and a fraction together, e.g. 3 3/4
simplest form	أبسط صورة — numerator and denominator have no common factor except 1, e.g. 6/8 → 3/4
product	حاصل الضرب — the answer to a multiplication
quotient	خارج القسمة — the answer to a division
estimate	تقدير — a quick, rounded answer to check that the real answer is sensible

1. Multiplying a fraction by a whole number

Multiplication is repeated addition. 2/7 × 3 means 'three lots of 2/7': 2/7 + 2/7 + 2/7. The parts are all sevenths, so you just count the sevenths: 2 + 2 + 2 = 6 sevenths.

2/7 × 3 = 2/7 + 2/7 + 2/7 = 6/7. The size of the parts (sevenths) does not change — only the number of parts does.

So the rule is: multiply the numerator by the whole number and keep the denominator the same. The denominator only names the size of the pieces, and multiplying does not change the size of the pieces.

Estimate first: is the fraction near 0, near 1/2 or near 1? (see section 3)
Multiply the numerator by the whole number. Keep the denominator.
If the answer is an improper fraction, change it to a mixed number: divide the numerator by the denominator; the remainder is the new numerator.
Write the fraction part in its simplest form.
Compare with your estimate.

Here is a bigger example on a number line: 3/4 × 5 is five jumps of 3/4.

Five jumps of 3/4 land on 15/4. 15 ÷ 4 = 3 remainder 3, so 15/4 = 3 3/4.

Calculation	Multiply the numerator	Change / simplify	Answer
2/9 × 4	8/9	already proper and simplest	8/9
3/4 × 5	15/4	15 ÷ 4 = 3 r 3	3 3/4
3/8 × 6	18/8	18/8 = 9/4; 9 ÷ 4 = 2 r 1	2 1/4
2/3 × 12	24/3	24 ÷ 3 = 8 exactly	8
5/6 × 4	20/6	20/6 = 10/3; 10 ÷ 3 = 3 r 1	3 1/3

Order does not matter

3 × 2/5 and 2/5 × 3 give the same answer (6/5), just like 3 × 4 = 4 × 3. Also, 2/3 × 12 is the same as '2/3 of 12' — you can find 1/3 of 12 (= 4) and double it (= 8). Use whichever way is easier.

2. Dividing a fraction by a whole number

Division is sharing into equal groups. 4/5 ÷ 2 means 'share 4 fifths between 2'. Each group gets 2 fifths.

4/5 ÷ 2 = 2/5: four fifths shared equally into 2 groups gives 2 fifths in each group.

Case A — the numerator divides exactly: divide the numerator and keep the denominator. 4/5 ÷ 2 = 2/5, 9/10 ÷ 3 = 3/10, 8/9 ÷ 4 = 2/9.

Case B — the numerator does not divide exactly (for example 3/4 ÷ 2: you cannot share 3 quarters equally into 2 groups of whole quarters). Cut every part into 2 smaller equal parts. Quarters become eighths, and each group gets 3 eighths.

Cutting each quarter in half makes 8 equal parts (eighths). Half of the shaded 3/4 is 3 of those eighths: 3/4 ÷ 2 = 3/8.

So the general rule is: keep the numerator and multiply the denominator by the whole number. This rule always works — even in Case A. For example 4/5 ÷ 2 = 4/10, and 4/10 simplifies to 2/5, the same answer as before.

Estimate: the answer must be smaller than the fraction you started with.
If the numerator divides exactly by the whole number, divide it and keep the denominator.
Otherwise, keep the numerator and multiply the denominator by the whole number.
Simplify if you can.
Check by multiplying back: answer × whole number should give the starting fraction.

Calculation	Method	Answer	Check (multiply back)
9/10 ÷ 3	9 ÷ 3 = 3, keep 10	3/10	3/10 × 3 = 9/10 ✓
3/4 ÷ 2	keep 3, 4 × 2 = 8	3/8	3/8 × 2 = 6/8 = 3/4 ✓
2/3 ÷ 5	keep 2, 3 × 5 = 15	2/15	2/15 × 5 = 10/15 = 2/3 ✓
4/7 ÷ 6	keep 4, 7 × 6 = 42 → 4/42	2/21	2/21 × 6 = 12/21 = 4/7 ✓

Dividing by 3 is the same as finding one third

3/4 ÷ 3 is the same as 1/3 of 3/4. Dividing by 2 is halving, dividing by 4 is finding a quarter. Word problems often say 'half of', 'a third of' or 'shared equally between' — all of these mean divide.

3. Estimating before you calculate

The curriculum asks you to estimate as well as calculate. Compare the fraction with the 'friendly' fractions 0, 1/2 and 1. A fraction is near 1/2 when the numerator is about half of the denominator (4/9, 5/9), and near 1 when the numerator is almost as big as the denominator (7/8, 9/10).

Calculation	Think	Estimate	Exact answer
7/8 × 11	7/8 is nearly 1, so a little less than 11	about 10	77/8 = 9 5/8
5/9 × 8	5/9 is just over 1/2, so just over 4	about 4 or 5	40/9 = 4 4/9
1/10 × 30	1/10 is small: one tenth of 30	3	30/10 = 3
4/5 ÷ 4	the answer must be less than 4/5, about a quarter of it	about 1/5	1/5

proper fraction × whole number → the answer is smaller than the whole number (you take only part of it each time).
proper fraction ÷ whole number → the answer is smaller than the fraction you started with.
If your answer breaks one of these facts, check your working again.

القاعدة اللي لازم تحفظها

في الضرب: اضرب البسط بس في العدد الصحيح والمقام زي ما هو (لأن حجم الجزء ماتغيرش، عدد الأجزاء بس اللي زاد). مثال: 2/7 × 3 = 6/7. في القسمة: البسط زي ما هو واضرب المقام في العدد الصحيح (لأنك بتقطع كل جزء لقطع أصغر). مثال: 3/4 ÷ 2 = 3/8. ولو البسط بيقبل القسمة، ممكن تقسمه على طول: 4/5 ÷ 2 = 2/5. وفي الآخر بسّط الكسر وحوّله لعدد كسري لو أكبر من 1.

4. Worked examples

Worked example 1 — Work out 3/8 × 6. Give your answer as a mixed number in its simplest form. [2]

Estimate: 3/8 is a bit less than 1/2, so the answer is a bit less than 3.
Multiply the numerator: 3 × 6 = 18, so 3/8 × 6 = 18/8.
Change to a mixed number: 18 ÷ 8 = 2 remainder 2, so 18/8 = 2 2/8.
Simplify: 2/8 = 1/4. Answer: 2 1/4 — a bit less than 3 ✓

Common mistake: stopping at 2 2/8. The question says simplest form, so 2 2/8 would lose the second mark.

Worked example 2 — Work out 6/7 ÷ 3. [1]

6 divides exactly by 3: 6 ÷ 3 = 2. Keep the sevenths. Answer: 2/7. Check: 2/7 × 3 = 6/7 ✓. (Using the general rule, 6/21 also simplifies to 2/7.)

Worked example 3 — Work out 2/3 ÷ 5. [1]

2 does not divide by 5, so keep the numerator and multiply the denominator: 3 × 5 = 15. Answer: 2/15. Check: 2/15 × 5 = 10/15 = 2/3 ✓. Common mistake: writing 10/3 (multiplying the numerator) — that makes the answer bigger, which is impossible when you divide.

Worked example 4 — Ring the best estimate for 7/8 × 11. 5 8 10 13 [1]

7/8 is very close to 1, so 7/8 × 11 is a little less than 11. Ring 10. (13 is too big — more than 11 is impossible; 5 would be the estimate for 1/2 × 11.) The exact answer is 77/8 = 9 5/8.

Worked example 5 — A jug holds 3/4 litre of juice. Amira pours all of it equally into 6 glasses. What fraction of a litre is in each glass? [2]

'Equally into 6 glasses' means divide: 3/4 ÷ 6.
3 does not divide by 6, so multiply the denominator: 3/24.
Simplify (÷ 3): 3/24 = 1/8. Answer: 1/8 litre.
Check: 1/8 × 6 = 6/8 = 3/4 ✓ (M1 for 3/4 ÷ 6 or 3/24; A1 for 1/8)

Worked example 6 — Omar says 4/5 ÷ 2 = 4/10. Sara says 4/5 ÷ 2 = 2/5. Who is correct? Explain your answer. [1]

Both are correct. 4/10 and 2/5 are equivalent fractions (divide the numerator and denominator of 4/10 by 2 to get 2/5). Omar used 'multiply the denominator', Sara used 'divide the numerator'. Sara's answer is in its simplest form. This is the kind of convincing (TWM) answer Checkpoint likes: say who is right and give the mathematical reason.

Worked example 7 — Write the missing number: 2/5 × ☐ = 2 2/5 [1]

Change the mixed number to an improper fraction: 2 2/5 = 10/5 + 2/5 = 12/5. The denominator stays 5, so 2 × ☐ = 12, ☐ = 6. Check: 2/5 × 6 = 12/5 = 2 2/5 ✓

Common mistakes

Multiplying both numbers: 2/7 × 3 = 6/21 ✗. That is just an equivalent fraction of 2/7 — nothing was multiplied! Correct: 6/7.
Dividing the denominator: 3/4 ÷ 2 = 3/2 ✗. Dividing must make the fraction smaller. Correct: 3/8.
Leaving an improper fraction or an unsimplified answer when the question asks for a mixed number in simplest form (18/8 instead of 2 1/4).
Wrong remainder when changing to a mixed number: 15/4 = 3 3/4, not 3 1/4. Check: 3 × 4 + 3 = 15 ✓.
Not reading the word problem: 'shared equally', 'each', 'a third of' → divide; 'how much altogether in 12 bottles' → multiply.
Not checking with an estimate: an answer like 3/8 × 6 = 18 should look wrong at once, because 3/8 × 6 must be less than 6.

How Checkpoint asks this

Short calculations with an answer box: 'Work out 5/6 × 4. Give your answer as a mixed number.' Read the last sentence — it tells you the form of the answer.
Missing-number boxes: 3/8 × ☐ = 15/8 or 4/9 ÷ ☐ = 4/27. Use the inverse: what was the numerator or denominator multiplied by?
Ring or tick the best estimate — compare the fraction with 0, 1/2 and 1.
Word problems (ribbons, juice, recipes, distances) — decide multiply or divide first and write the calculation down: it can earn a method mark even if the arithmetic slips.
'Explain' or 'Who is correct?' questions — give a reason using maths words: 'dividing makes it smaller', 'the denominator shows the size of the parts', 'these fractions are equivalent'.
These questions usually appear on Paper 1 (no calculator), so practise doing them by hand. Show your working in the space given.

إزاي تتأكد من إجابتك

اسأل نفسك سؤالين: هل الإجابة منطقية؟ كسر أقل من 1 مضروب في عدد صحيح لازم يدي أقل من العدد ده. وكسر مقسوم على عدد صحيح لازم يصغر. وبعدين اعمل العملية العكسية: لو قسمت، اضرب الإجابة في العدد الصحيح ولازم يرجع لك الكسر الأصلي.

Practice

Remember

Fraction × whole number: multiply the numerator, keep the denominator (2/7 × 3 = 6/7).
Fraction ÷ whole number: keep the numerator, multiply the denominator (3/4 ÷ 2 = 3/8) — or divide the numerator if it divides exactly (4/5 ÷ 2 = 2/5).
Change improper answers to mixed numbers and always simplify.
Estimate with 0, 1/2 and 1: proper fraction × 11 is less than 11; dividing always makes a proper fraction smaller.
Check division by multiplying back.

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