D-15MathsWed 26 Aug

Ratio & sharing

Parts vs total — the step children skip.

A ratio compares parts of a whole. To share an amount: add the parts, divide to find one share, then multiply back up. The same one-part method works for two-way splits, three-way splits, “given one person’s share”, and questions where the ratio later changes.

Picture it: share £35 in the ratio 2 : 3

Ava

Ben

£7

5 equal parts. Each part is worth the same — here £7.

Parts

2 + 3 = 5

→

One part

£35 ÷ 5 = £7

→

£14 : £21

Ava’s 2 parts = £14. Ben’s 3 parts = £21. Check: £14 + £21 = £35 ✓.

Ratio 2 : 3 means

2/5 of the whole 3/5 of the whole

Out of 5 equal parts, Ava has 2 and Ben has 3.

It does not mean 2/3

2/3 (wrong picture) left over

2 : 3 is not “Ava gets two-thirds”. Two-thirds would be 2 : 1.

Remember — the method

Step 1

Add the parts

2 : 3 → 5 parts in the whole.

Step 2

Find one part

Total ÷ number of parts.

Step 3

Multiply back

One part × each ratio number.

Always check: the shares add back to the original total. If they don’t, you divided by the wrong number.

Exam trap

Dividing the total by one of the ratio numbers instead of by their sum. For 2 : 3 the whole is 5 parts — not 2, and not 3.

✗ £35 ÷ 2 = £17.50

Only 2 parts drawn — Ben’s 3 parts are missing

✓ £35 ÷ 5 = £7

£7

All 5 parts shown — now multiply by 2 and by 3

Question patterns you might see

Same one-part method every time. Draw the parts first — then the numbers fall out.

1 · Simple — Two-part share
“Share £35 in the ratio 2 : 3.”
£7
£7
£7
£7
£7
1. 1Add the parts: 2 + 3 = 5.
2. 2One part = £35 ÷ 5 = £7.
3. 32 × £7 = £14 and 3 × £7 = £21.
4. 4Check: £14 + £21 = £35.
Shares = £14 and £21
2 · Simple — Three-part share
“Share 60 sweets in the ratio 1 : 2 : 3.”
10
10
10
10
10
10
1 + 2 + 3 = 6 equal parts
1. 1Total parts = 1 + 2 + 3 = 6.
2. 2One part = 60 ÷ 6 = 10 sweets.
3. 3Shares = 10, 20 and 30.
4. 4Check: 10 + 20 + 30 = 60.
Parts
6
→
One part
10
→
Shares
10 : 20 : 30
Shares = 10 : 20 : 30
3 · Medium — Given one share, find the other
“Ravi and Sam share money in the ratio 3 : 5. Ravi gets £24. How much does Sam get?”
Ravi — 3 parts = £24
£8
£8
£8
Sam — 5 parts = ?
£8
£8
£8
£8
£8
1. 1Ravi’s 3 parts = £24, so one part = £24 ÷ 3 = £8.
2. 2Sam has 5 parts: 5 × £8 = £40.
3. 3Total money = 8 parts = £64 (optional check).
4. 4Do not share £24 in the ratio 3 : 5 — £24 is already Ravi’s share, not the total.
3 parts
£24
→
1 part
£8
→
5 parts
£40
Sam gets £40
4 · Medium — Given the difference, find the total
“Money is shared in the ratio 2 : 5. The larger share is £18 more than the smaller. What is the total?”
2
2
gap
gap
gap
The extra 3 parts (5 − 2) are the £18 gap. The full bar is 7 parts.
£6
£6
£6
£6
£6
£6
£6
1. 1The gap is 5 − 2 = 3 parts, and that gap is £18.
2. 2One part = £18 ÷ 3 = £6.
3. 3Total parts = 2 + 5 = 7, so total = 7 × £6 = £42.
4. 4Check: smaller = £12, larger = £30, and £30 − £12 = £18.
Gap (3 parts)
£18
→
1 part
£6
→
Total (7)
£42
Total = £42
5 · Medium — Simplify, then read as a fraction
“A bag has 12 red beads and 18 blue beads. Write the ratio in simplest form. What fraction of the beads are red?”
Before simplifying
12
18
12 red 18 blue
÷ 6 on both sides
2
3
2 red 3 blue
1. 1Divide both numbers by 6: 12 : 18 → 2 : 3.
2. 2Total parts in the simplified ratio = 2 + 3 = 5.
3. 3Red is 2 parts out of 5, so red is 2/5 of the beads.
4. 4Check with the originals: 12 red out of 30 beads is also 12/30 = 2/5.
Simplest ratio 2 : 3 · red = 2/5
6 · Trickier — A ratio that changes
“A class has boys and girls in the ratio 5 : 3, with 20 boys. More girls join so the ratio becomes 5 : 4. How many girls joined?”
Before — 5 : 3
B
B
B
B
B
G
G
G
20 boys → one part = 4 → 12 girls
After — 5 : 4
B
B
B
B
B
G
G
G
G
Boys stay 20, so one part is still 4 → 16 girls
1. 120 boys are 5 parts, so one part = 20 ÷ 5 = 4.
2. 2Girls at the start: 3 × 4 = 12.
3. 3Boys do not change, so one part is still 4. New girls = 4 × 4 = 16.
4. 4Girls who joined = 16 − 12 = 4.
Start girls
12
→
New girls
16
→
Joined
4
4 girls joined
7 · Trickiest — Join two ratios
“A : B = 2 : 3 and B : C = 3 : 4. Write A : B : C.”
A : B = 2 : 3
A
A
B
B
B
B : C = 3 : 4
B
B
B
C
C
C
C
B is 3 parts in both ratios already — they line up. If the B numbers differed, scale one ratio so the B parts match.
A : B : C
A
A
B
B
B
C
C
C
C
1. 1B is the ‘bridge’. Here B is already 3 in both, so no scaling is needed.
2. 2Write A from the first ratio, C from the second, and keep B as 3.
3. 3A : B : C = 2 : 3 : 4.
4. 4If instead B : C were 6 : 8, you would scale 2 : 3 up by 2 first, to 4 : 6, then join.
A : B : C = 2 : 3 : 4

Coach tip: say the total number of parts out loud before dividing — “five parts” — so you never accidentally divide by one of the individual ratio numbers.

Tomorrow (D-14): matrices & two-rule grids.

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