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 ✓.