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Daily Maths

Daily Maths

Ratio and proportion

Ratios Core + Extended 12 min read

On this page
  1. In this note
  2. Formula bank
  3. What a ratio means
  4. Simplifying a ratio
  5. Sharing in a ratio
  6. Ratios that change; increasing in a ratio
  7. Proportion: recipes and best value
  8. Ways they test it
  9. Mistakes to avoid
  10. Summary
  11. Practice
  12. Can I do this?
  13. Quiz

In this note

Syllabus 1.11: ratio and proportion
  • 1.11Write a ratio in its simplest form (units, decimals, fractions, three parts) and in the form 1 : n.
  • 1.11Share an amount in a ratio when you know the total, one share, or the difference.
  • 1.11Follow a ratio that changes; increase or decrease an amount in a ratio.
  • 1.11Scale a recipe and find the best value.

Before you start: fractions and the HCF. Map scales: maps and scales.

Formula bank

One partLearn it\(\text{one part}=\dfrac{\text{amount}}{\text{number of parts it is}}\)total ÷ sum of parts; a share ÷ its parts; a difference ÷ difference in parts.Use it when every sharing question.
Ratio → fractionLearn itin \(a:b\), the first share is \(\dfrac a{a+b}\) of the wholeboys : girls = 3 : 5 → boys are \(\tfrac38\).Use it when "what fraction is ...".
Increase / decrease in a ratioLearn itold : new \(=a:b\Rightarrow\text{new}=\text{old}\times\dfrac ba\)Use it when "increase 40 in the ratio 5 : 8".
Unitary methodLearn itamount for 1 = amount ÷ number; then × the number you wantUse it when recipes and proportion.
Best valueLearn itprice per unit (per 100 g, per item)lowest price per unit = best value.Use it when comparing packs.

What a ratio means

A ratio compares parts. "Squash : water = 1 : 4" means 1 cup of squash for every 4 cups of water, so 5 equal parts in all. In \(a:b\) the first share is \(\dfrac a{a+b}\) of the whole: squash is \(\tfrac15\) of the drink, not \(\tfrac14\).

Ratio bar 1:4Squash : water = 1 : 4 (5 equal parts)squashwaterwaterwaterwater
One bar, five equal boxes: each box is one part.

Try it yourself

1
Red : white = 2 : 7. What fraction is red?
Show answer

Answer \(\tfrac29\)

2
\(\tfrac4{11}\) of a bag are mints, the rest toffees. Mints : toffees?
Show answer

Answer 4 : 7

Simplifying a ratio

General methodHow to simplify any ratio

  1. Same units first (change to the smaller unit): 1.5 litres : 450 ml → 1500 : 450.
  2. Clear decimals (× 10 or 100) or fractions (× the common denominator).
  3. Divide every part by the HCF.
  4. Form 1 : n: divide both parts by the first part (n may be a decimal).

Example Simplify 250 ml : 1.5 litres.

Same units, then divide by the HCF.

  1. 1.5 litres = 1500 ml
  2. 250 : 1500, ÷ 250

Answer 1 : 6

Try it yourself

1
Simplify 24 : 36 : 60.
Hint
HCF 12.
Show answer

Answer 2 : 3 : 5

2
Simplify 2.4 kg : 600 g.
Show answer

Answer 4 : 1

3
Write 4 : 9 in the form 1 : n. n =
Show answer

Answer 2.25

Sharing in a ratio

Every sharing question is really "find ONE part". Then each share is its number of parts × one part.

Sharing $360 in 4:5Share $360 in the ratio 4 : 54 parts5 parts9 parts = $360, so 1 part = $40$160$200
Total ÷ number of parts = one part.

General methodHow to answer any sharing question

  1. Find what you are given: the TOTAL, ONE SHARE, or the DIFFERENCE between two shares.
  2. Divide it by the matching number of parts: total ÷ (sum of parts); share ÷ its parts; difference ÷ (difference in parts).
  3. That is one part. Multiply by each person's parts.
  4. Check: the shares add up / differ by the right amount.

Example P : Q = 7 : 4. P gets 18 more than Q. Find both shares.

The difference 18 is 7 − 4 = 3 parts.

  1. 3 parts = 18, so 1 part = 6
  2. P \(=7\times6=42\), Q \(=4\times6=24\)

Answer 42 and 24

Try it yourself

1
Share $252 in the ratio 2 : 3 : 4. The largest share is $
Show answer

Answer $112

2
A : B = 9 : 2 and A gets 63. Find B.
Show answer

Answer 14

3
C : D = 5 : 2. C gets 21 more than D. Find C.
Show answer

Answer 35

General methodHow to join two ratios with a shared letter

  1. Find the letter in both (\(b\) in \(a:b\) and \(b:c\)).
  2. Make its number the same in both (use the LCM).
  3. Write \(a:b:c\).

Example \(a:b=2:3\) and \(b:c=4:5\). Find \(a:b:c\).

Make b equal to 12 in both.

  1. \(2:3=8:12\)
  2. \(4:5=12:15\)

Answer 8 : 12 : 15

Ratios that change; increasing in a ratio

General methodHow to handle a ratio that changes

  1. Find the REAL amounts first (never add to the ratio numbers - they are parts).
  2. Make the change to the real amounts.
  3. Write the new ratio and simplify.
  4. If no amounts are known, call one part \(k\) and make an equation.

Example A class of 30 has boys : girls = 3 : 2. Four more girls join. New ratio?

Real numbers first, then add the girls.

  1. 5 parts = 30, so 18 boys and 12 girls
  2. now 18 boys, 16 girls
  3. 18 : 16 = 9 : 8

Answer 9 : 8

Try it yourself

1
Increase 40 in the ratio 5 : 8.
Hint
× \(\tfrac85\).
Show answer

Answer 64

2
Decrease 72 in the ratio 9 : 4.
Show answer

Answer 32

Proportion: recipes and best value

General methodHow to scale any recipe (unitary method)

  1. Find the amount for ONE (one person, one item).
  2. Multiply by the number you want.

Example A recipe for 6 uses 450 g of flour. How much for 10?

Find the amount for 1 person, then × 10.

  1. \(450\div6=75\) g
  2. \(75\times10=750\) g

Answer 750 g

General methodHow to find the best value

  1. Choose one unit: price per 100 g, per item, or amount per dollar.
  2. Work it out for every pack.
  3. Lowest price per unit = best value. Write a sentence quoting the figures.

Example Rice: 500 g for $2.40 or 750 g for $3.45. Which is better value?

Price per 100 g for each.

  1. \(2.40\div5=\$0.48\) per 100 g
  2. \(3.45\div7.5=\$0.46\) per 100 g

Answer the 750 g bag ($0.46 < $0.48)

Try it yourself

1
A recipe for 4 uses 300 ml of milk. How much for 7?
Show answer

Answer 525 ml

2
6 eggs for $1.74 or 15 eggs for $4.20. Which is better value?
Show answer

Answer 15 eggs ($0.28 each against $0.29)

Ways they test it

Type 1 Simplify with units

"Write 1.5 litres : 450 ml in its simplest form." (2 marks)

Spot it two quantities in different units.

  1. Same units.
  2. ÷ HCF; keep the order.

For example 1.5 litres : 450 ml

  1. 1500 : 450
  2. ÷ 150

Answer 10 : 3

Type 2 Share a total

"Share $360 in the ratio 4 : 5."

Spot it a total and a ratio.

  1. Total ÷ sum of parts.
  2. × each part.

Type 3 One share or the difference given

"Sara : Tom = 11 : 6. Sara gets $350 more than Tom." (3 marks)

Spot it "more than", or one person's share.

  1. Difference ÷ difference in parts (or share ÷ its parts).
  2. × each part.

For example 11 : 6, difference $350

  1. \(350\div5=70\)
  2. \(11\times70=770\), \(6\times70=420\)

Answer $770 and $420

Type 4 Ratio that changes

"Red : blue = 4 : 7, 28 blue. 6 more red are added. New ratio?"

Spot it something added or removed.

  1. Real numbers first.
  2. Change, then simplify.

For example 4 : 7 with 28 blue; +6 red

  1. one part = 4; red = 16
  2. 22 : 28

Answer 11 : 14

Type 5 Unknown part: k

"Two numbers are 3 : 5. Add 8 to each and it becomes 5 : 7. Find them." (4 marks)

Spot it no amounts given, a change described.

  1. Numbers \(3k\), \(5k\).
  2. \(\dfrac{3k+8}{5k+8}=\dfrac57\), cross-multiply, solve.

For example \(3k:5k\), +8 → 5 : 7

  1. \(7(3k+8)=5(5k+8)\)
  2. \(4k=16\Rightarrow k=4\)

Answer 12 and 20

Type 6 Recipe / best value

"Which pack is better value? Show your working."

Spot it packs or recipes.

  1. Unit price for each.
  2. Conclusion with figures.

Mistakes to avoid

  • ✗ 1.5 : 450✓ 1500 : 450 = 10 : 3Units must match.
  • ✗ Boys are \(\tfrac35\) of a 3 : 5 class✓ \(\tfrac38\)Add the parts to get the whole.
  • ✗ Difference given: ÷ total parts✓ ÷ difference in partsThe extra amount is the extra parts.
  • ✗ Adding new items to the ratio numbers✓ add them to the real amountsRatio numbers are parts.
  • ✗ "B is better value" with no figures✓ quote both unit pricesThe comparison earns the marks.

Summary

  • Find ONE part, then multiply.
  • Total → ÷ sum of parts; one share → ÷ its parts; difference → ÷ difference in parts.
  • Same units before simplifying; clear decimals and fractions.
  • A changing ratio: real amounts first. Unknown part: k.
  • Best value: same unit for every pack, then a sentence.

If you see …

  • "more than" in a sharing question÷ the difference in parts.
  • different unitsconvert first.
  • "increase in the ratio a : b"× \(\tfrac ba\).

Practice

Practice: easy, medium, hard

Easy

1
Simplify 45 : 60.
Show working
  1. ÷ 15

Answer 3 : 4

2
Share 56 in the ratio 3 : 4. The larger share is
Show working
  1. \(56\div7=8\)
  2. \(4\times8=32\)

Answer 32

3
A : B = 4 : 9 and A = 20. Find B.
Show working
  1. one part = 5
  2. \(9\times5=45\)

Answer 45

Medium

1
Write 750 g : 2 kg in its simplest form.
Hint
2 kg = 2000 g.
Guide me

What is it asking? simplest form

What do we know? different units

Plan convert, then ÷ HCF 250

Show working
  1. 750 : 2000
  2. ÷ 250: 3 : 8

Answer 3 : 8

2
X : Y = 7 : 4 and X gets 21 more than Y. Find Y.
Guide me

What is it asking? Y's share

What do we know? difference 21 = 3 parts

Plan one part = 7

Show working
  1. \(21\div3=7\)
  2. \(4\times7=28\)

Answer 28

3
A club has adults : children = 5 : 3 with 54 children. 6 adults leave. New ratio adults : children?
Guide me

What is it asking? the new ratio

What do we know? 54 children = 3 parts

Plan real numbers, change, simplify

Show working
  1. one part = 18; adults 90
  2. 84 : 54 = 14 : 9

Answer 14 : 9

Hard

1
Green paint is blue : yellow = 2 : 5. Leo has 1.6 litres of blue and plenty of yellow. Greatest amount of green (litres)?
Hint
The blue runs out first.
Guide me

What is it asking? most green paint

What do we know? blue = 2 parts = 1.6 litres

Plan one part, then 7 parts

Show working
  1. \(1.6\div2=0.8\)
  2. \(0.8\times7=5.6\)

Answer 5.6 litres

PartAnswer / workingMarksGuidance
\(1.6\div2\)M1
\(0.8\times7\)M1
5.6A1
Total: 3 marks
2
Two numbers are in the ratio 3 : 5. When 8 is added to each, the ratio becomes 5 : 7. Find the two numbers.
Guide me

What is it asking? the two numbers

What do we know? \(3k\) and \(5k\)

Plan equation from the new ratio

Show working
  1. \(7(3k+8)=5(5k+8)\)
  2. \(21k+56=25k+40\)
  3. \(k=4\)

Answer 12 and 20

PartAnswer / workingMarksGuidance
\(7(3k+8)=5(5k+8)\)M1
\(k=4\)A1
12 and 20A1
Total: 3 marks
3
Amir and Bina share $189 in the ratio 2 : 7. How much more does Bina get?
Guide me

What is it asking? the difference

What do we know? 9 parts = 189

Plan one part, then 5 parts

Show working
  1. \(189\div9=21\)
  2. \(5\times21=105\)

Answer $105

PartAnswer / workingMarksGuidance
\(189\div9\)M1
105A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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