Checkpoint 6 Maths · Number · Ratio and proportion

Ratio and Direct Proportion

Understand ratio and direct proportion: write and simplify equivalent ratios, scale recipes and prices with the unitary method, and share amounts using bar models.

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In the curriculum

Understand what it means when two quantities are in direct proportion — when one is multiplied by a number, the other is multiplied by the same number (6Nf.12)
Use what you know about equivalence to find, simplify and use equivalent ratios, e.g. 2 : 3 = 6 : 9 (6Nf.13)

Ratio and proportion are about comparing amounts and scaling them up or down fairly. You use them when you make a recipe for more people, mix paint or juice, read a map or share sweets. The big idea is the same as for equivalent fractions: multiply or divide both parts by the same number.

Key words

Word / phrase	Meaning / use
ratio	نسبة — compares one part with another part, written with a colon, e.g. 2 : 3 ('for every 2 … there are 3 …')
proportion	تناسب / جزء من الكل — compares a part with the whole, e.g. '2 in every 5' = 2/5
equivalent ratios	نسب متكافئة — ratios that compare the same way, e.g. 2 : 3 = 4 : 6 = 6 : 9
simplest form	أبسط صورة — the ratio cannot be divided any more, e.g. 12 : 18 → 2 : 3
direct proportion	تناسب طردي — two amounts that increase or decrease by the same multiplier
unitary method	طريقة الوحدة — find the value for ONE first, then multiply
scale up / scale down	تكبير / تصغير — multiply / divide all amounts by the same number
share in a ratio	تقسيم بنسبة — split a total into parts that follow a ratio

1. What is a ratio?

A ratio compares part with part. In the pattern below, for every 2 yellow tiles there are 3 blue tiles. We write yellow : blue = 2 : 3 and say 'two to three'. The order matters: blue : yellow is 3 : 2.

The group of 2 yellow and 3 blue tiles repeats 3 times, giving 6 yellow and 9 blue. 6 : 9 and 2 : 3 describe the same pattern, so they are equivalent ratios.

Proportion compares a part with the whole. In each group there are 5 tiles and 2 are yellow, so the proportion of yellow tiles is '2 in every 5', which is the fraction 2/5. Blue tiles are 3 in every 5 = 3/5. To change a ratio into a proportion, add the parts to find the whole: 2 + 3 = 5.

Words	Ratio (part : part)	Proportion (part of the whole)
For every 2 yellow there are 3 blue	yellow : blue = 2 : 3	yellow = 2/5, blue = 3/5
For every 1 teacher there are 8 pupils	teachers : pupils = 1 : 8	teachers = 1/9 of the people
4 in every 10 cars are white	white : not white = 4 : 6 = 2 : 3	white = 4/10 = 2/5

2. Equivalent ratios and simplest form

Equivalent ratios work exactly like equivalent fractions. Multiply or divide both parts by the same number — never add or subtract.

Ratio	× 2	× 3	× 5	× 10
2 : 3	4 : 6	6 : 9	10 : 15	20 : 30
1 : 4	2 : 8	3 : 12	5 : 20	10 : 40
3 : 5	6 : 10	9 : 15	15 : 25	30 : 50

To write a ratio in its simplest form, divide both parts by their highest common factor (HCF). If you do not spot the HCF, divide by any common factor and repeat until there are no more common factors.

12 : 18 → HCF is 6 → 2 : 3.
40 : 60 → ÷ 10 → 4 : 6 → ÷ 2 → 2 : 3.
To find a missing number, find the multiplier first: 3 : 5 = ☐ : 35. 5 × 7 = 35, so ☐ = 3 × 7 = 21.
Check: divide the new ratio by the same number — 21 : 35 ÷ 7 = 3 : 5 ✓

القاعدة الذهبية

النسبة زي الكسر بالظبط: علشان تعمل نسبة مكافئة اضرب أو اقسم الجزأين في نفس الرقم. ممنوع تجمع أو تطرح! مثلاً 2 : 3 = 4 : 6 = 6 : 9. وانتبه للترتيب: لو السؤال قال "أصفر إلى أزرق" يبقى الأصفر الأول. والنسبة (جزء : جزء) غير التناسب أو الكسر (جزء من الكل): 2 : 3 يعني الأصفر 2/5 من الكل.

3. Sharing in a ratio

Sometimes you know the total and the ratio. Draw a bar model: one box for each part of the ratio. Then find the value of one box.

Share 35 sweets in the ratio 3 : 4. There are 3 + 4 = 7 boxes, so each box is 35 ÷ 7 = 5. Sami gets 3 × 5 = 15 and Nada gets 4 × 5 = 20.

Add the parts of the ratio to find the number of equal boxes.
Divide the total by the number of boxes to find one part.
Multiply to find each person's share.
Check that the shares add up to the total.

4. Direct proportion

Two quantities are in direct proportion when they grow or shrink together by the same multiplier: if you buy 2 times as many pens, you pay 2 times as much; one third of the pens cost one third of the money. Zero pens cost $0. The ratio between the two quantities always stays the same.

3 pens cost $4.50. Divide both lines by 3: 1 pen costs $1.50. Multiply both by 7: 7 pens cost $10.50.

The safest method is the unitary method: find the value for one, then multiply. When the numbers are friendly you can also scale in one step (e.g. 4 people → 12 people is × 3).

Cakes	1	2	3	4	5
Flour (g)	200	400	600	800	1000

In this table every amount of flour is the number of cakes × 200. That is what direct proportion looks like in a table: the same multiplier links every pair. If you plot the pairs on a graph, the points lie on a straight line through 0.

Flour and number of cakes are in direct proportion: the points lie on a straight line that starts at (0, 0).

Is it direct proportion?	Yes or no	Why
Cost of apples at $3 per kg and their mass	Yes	2 kg cost 2 times as much as 1 kg; 0 kg costs $0
Number of legs and number of chairs (4 legs each)	Yes	legs = chairs × 4 every time
A child's age and height	No	a child of 10 is not twice as tall as a child of 5
Taxi fare with a $5 starting charge plus $2 per km	No	0 km already costs $5; 2 km ($9) does not cost twice 1 km ($7)

5. Worked examples

Worked example 1 — Write 12 : 18 in its simplest form. [1]

Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. HCF = 6. 12 ÷ 6 = 2, 18 ÷ 6 = 3. Answer: 2 : 3. Common mistake: dividing only once by 2 and stopping at 6 : 9.

Worked example 2 — Write the missing number: 3 : 5 = ☐ : 35 [1]

5 → 35 is × 7. So 3 × 7 = 21. Common mistake: 3 + 30 = 33 (adding the same amount instead of multiplying).

Worked example 3 — For every 2 yellow tiles there are 3 blue tiles. (a) There are 12 yellow tiles. How many blue tiles are there? (b) What fraction of all the tiles are yellow? [2]

(a) 2 → 12 is × 6, so 3 × 6 = 18 blue tiles. (2 : 3 = 12 : 18.)
(b) Each group has 2 + 3 = 5 tiles, and 2 are yellow: 2/5. (Check: 12 out of 30 = 2/5 ✓.) Common mistake: answering 2/3, which compares yellow with blue, not with all the tiles.

Worked example 4 — A recipe uses 300 g of rice for 4 people. How much rice is needed for 10 people? [2]

Unitary method: 1 person needs 300 ÷ 4 = 75 g. 10 people need 75 × 10 = 750 g. Another way: 2 people need 150 g, and 10 = 5 × 2, so 150 × 5 = 750 g. Mark scheme style: M1 for 75 (or 150 for 2 people) seen, A1 for 750.

Worked example 5 — Tick the table that shows direct proportion. Explain your choice. [1]

Table	Values	Multiplier
Table A: litres of juice → cost	2 → $6, 5 → $15, 8 → $24	× 3 every time
Table B: tickets → total cost	1 → $5, 2 → $10, 4 → $18	× 5, × 5, then × 4.5

Table A ✓. Every cost is the number of litres × 3. In Table B, 4 tickets should cost 4 × $5 = $20 if it were direct proportion, but they cost $18.

Worked example 6 — Sami and Nada share 35 sweets in the ratio 3 : 4. How many sweets does each get? [2]

3 + 4 = 7 parts. 35 ÷ 7 = 5 sweets per part. Sami: 3 × 5 = 15. Nada: 4 × 5 = 20. Check: 15 + 20 = 35 ✓ (see the bar model above). Common mistake: dividing 35 by 3 and by 4.

Worked example 7 — 3 pens cost $4.50. How much do 7 pens cost? [2]

1 pen: 4.50 ÷ 3 = $1.50. 7 pens: 1.50 × 7 = $10.50 (see the double number line). Estimate: 7 is a bit more than twice 3, so a bit more than $9 ✓.

Common mistakes

Adding instead of multiplying: 2 : 5 → 8 : 11 ✗. Correct: 2 : 5 = 8 : 20.
Writing the ratio in the wrong order. 'Boys to girls' means boys first.
Mixing up ratio and fraction: in 3 : 7, the fraction of the first part is 3/10, not 3/7.
Sharing in a ratio by dividing the total by each number of the ratio instead of by the total number of parts.
Not simplifying fully: 6 : 9 still has a common factor 3 → 2 : 3.
Assuming every relationship is direct proportion — check that the same multiplier works for every pair (and that 0 matches 0).

How Checkpoint asks this

Patterns of tiles, beads or counters: 'For every 2 red there are 3 blue …' — find the missing number or write the ratio.
Missing-number equivalent ratios, e.g. 4 : ☐ = 12 : 27, and 'Write in its simplest form'.
Recipes and prices ('6 pancakes need 150 ml of milk …') — write the value for one, or a friendly step, in your working for the method mark.
Complete a table of values that are in direct proportion, or choose which table/graph shows direct proportion.
'Is … correct? Explain' — say yes or no and give the mathematical reason (multiply both parts by the same number).
Ratio and proportion can appear on either paper; with a calculator on Paper 2 the numbers may be less friendly, but the method is the same.

Practice

Remember

A ratio compares part with part (2 : 3); a proportion compares part with the whole (2 in every 5 = 2/5).
Equivalent ratios: multiply or divide BOTH parts by the same number — never add.
Simplest form: divide both parts by the HCF (12 : 18 = 2 : 3).
Sharing a total in a ratio: add the parts, find one part, then multiply.
Direct proportion: the same multiplier links every pair; use the unitary method (find one first).

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Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

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