Mathematics • Year 7 • Unit 1 • Lesson 12
Ratios and Rates
Build the basics: simplify a ratio by dividing both sides by their HCF, divide an amount into a ratio using the three-step method (add parts → one part → multiply), and find a missing value in equivalent ratios.
1. I do — fully worked example
Watch a worked "share an amount in a ratio" example. Every step has a short reason on the right.
Problem. Share $72 in the ratio 3:5.
Step 1 — Add the ratio parts to get total parts.
Total parts = 3 + 5 = 8.
Reason: 3:5 means the whole thing is split into 8 equal parts (3 for the first share, 5 for the second).
Step 2 — Divide the amount by total parts to get the value of one part.
One part = $72 ÷ 8 = $9.
Reason: the $72 has to be shared evenly across the 8 parts, so each part is worth $9.
Step 3 — Multiply each ratio number by the value of one part.
First share = 3 × $9 = $27. Second share = 5 × $9 = $45.
Reason: 3 parts of $9 each is $27; 5 parts of $9 each is $45.
Step 4 — Check.
$27 + $45 = $72 ✓, and 27:45 simplifies to 3:5 (÷9) ✓.
Answer: The two shares are $27 and $45.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Share $96 in the ratio 5:7.
Step 1 — Add the ratio parts:
Total parts = 5 + 7 = _______.
Step 2 — Value of one part:
One part = $96 ÷ _______ = $_______.
Step 3 — Multiply each ratio number by the value of one part:
First share = 5 × $_______ = $_______.
Second share = 7 × $_______ = $_______.
Step 4 — Check the two shares add to $96:
$_______ + $_______ = $_______ ✓
3. You do — independent practice
Show working under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Simplify the ratio 18:27. State the HCF you used. 1 mark
3.2 Simplify the ratio 16:24. 1 mark
3.3 Find the missing value: 3:8 = x:24. 1 mark
3.4 $60 is shared in the ratio 2:3. What is the larger share? 1 mark
Standard — combine two ideas
3.5 Share $84 in the ratio 2:5. List both shares and check they add to $84. 2 marks
3.6 Simplify the ratio 45 min : 2 hours. Be careful with units. 2 marks
Extension — push your thinking
3.7 Concrete is mixed in the ratio cement : sand : gravel = 1:2:3. How many kg of each are needed for 120 kg of concrete? Show all three shares and check they sum to 120 kg. 3 marks
3.8 A car travels 315 km in 3.5 hours. What is its average speed in km/h? Show your division. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do ($96 in ratio 5:7)
Step 1: 5 + 7 = 12.
Step 2: $96 ÷ 12 = $8.
Step 3: First = 5 × $8 = $40; second = 7 × $8 = $56.
Step 4: $40 + $56 = $96 ✓.
3.1 — Simplify 18:27
HCF(18, 27) = 9. 18 ÷ 9 = 2, 27 ÷ 9 = 3. Answer: 2:3.
3.2 — Simplify 16:24
HCF(16, 24) = 8. 16 ÷ 8 = 2, 24 ÷ 8 = 3. Answer: 2:3.
3.3 — 3:8 = x:24
Cross-multiply: 3/8 = x/24 → 8x = 3 × 24 = 72 → x = 72 ÷ 8 = 9. Check: 3:8 = 9:24 (× 3 both sides) ✓.
3.4 — $60 in ratio 2:3, larger share
Total parts = 5. One part = $60 ÷ 5 = $12. Larger share = 3 × $12 = $36.
3.5 — $84 in ratio 2:5
Total parts = 7. One part = $84 ÷ 7 = $12. Shares: 2 × $12 = $24 and 5 × $12 = $60. Check: $24 + $60 = $84 ✓.
3.6 — 45 min : 2 hours
Convert to the same unit. 2 hours = 120 min. So 45 : 120. HCF(45, 120) = 15. 45 ÷ 15 = 3, 120 ÷ 15 = 8. Answer: 3:8.
3.7 — Concrete 1:2:3 for 120 kg
Total parts = 1 + 2 + 3 = 6. One part = 120 ÷ 6 = 20 kg.
Cement = 1 × 20 = 20 kg. Sand = 2 × 20 = 40 kg. Gravel = 3 × 20 = 60 kg.
Check: 20 + 40 + 60 = 120 kg ✓.
3.8 — Speed of car
Speed = distance ÷ time = 315 ÷ 3.5. Shift both 1 place: 3150 ÷ 35 = 90 km/h. Check: 90 × 3.5 = 315 ✓.