Ratios and Rates
Comparing quantities with ratios. 2:3 means 2 parts to 3 parts. Simplify, divide, and scale ratios to solve real problems.
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A bag has red and blue marbles in the ratio 3:5. If there are 24 marbles total, how many are red? Try to solve before reading on.
A ratio compares two or more quantities. 2:3 means 2 parts of the first thing for every 3 parts of the second. Ratios have no units — they show relative size. Simplify by dividing both sides by their HCF. To divide an amount in a ratio, add the parts, find the value of one part, then multiply.
A ratio 4:6 simplifies to 2:3 (HCF = 2). This means for every 2 units of A, there are 3 units of B. The total parts = 2 + 3 = 5 parts. If the total amount is 50, each part = 50 ÷ 5 = 10. So A = 2 × 10 = 20, B = 3 × 10 = 30. Check: 20 + 30 = 50, and 20:30 = 2:3. Correct!
Know
- A ratio compares quantities with no units
- How to simplify ratios using HCF
- How to divide an amount in a given ratio
Understand
- Why ratios stay the same when both sides scale
- The connection between ratios and fractions
- How rates differ from ratios
Can Do
- Simplify any ratio to lowest terms
- Divide amounts in a ratio
- Find missing values in equivalent ratios
Wrong: Ratio 2:3 with $50: A gets $20, B gets $30. “I just guessed.” No! Always find total parts first.
Right: Total parts = 2 + 3 = 5. Each part = $50 ÷ 5 = $10. A = 2 × $10 = $20. B = 3 × $10 = $30. Check: $20 + $30 = $50.
Wrong: Simplifying 12:18 by dividing only the first number. 12÷6=2, but 18 stays. 2:18. Wrong!
Fix: Divide BOTH by HCF(12, 18) = 6. 12÷6 = 2, 18÷6 = 3. So 2:3. Check: 12/18 = 2/3. Correct!
Divide both sides by their HCF. If the ratio has different units, convert to the same unit first. If it has decimals or fractions, multiply to make whole numbers first.
Simplify 24:36. HCF(24, 36) = 12. 24÷12 = 2, 36÷12 = 3. So 2:3. Simplify 0.5:1.5. Multiply by 2: 1:3. So 1:3. Simplify 45 min : 2 hours. Convert: 45 min : 120 min. HCF(45, 120) = 15. 45÷15 = 3, 120÷15 = 8. So 3:8.
Add the ratio parts to get total parts. Divide the amount by total parts to get one part. Multiply each ratio number by one part.
Share $72 in the ratio 3:5. Total parts = 3 + 5 = 8. One part = $72 ÷ 8 = $9. First share = 3 × $9 = $27. Second share = 5 × $9 = $45. Check: $27 + $45 = $72, and 27:45 = 3:5 (HCF = 9). Correct!
A rate compares quantities with different units. A unit rate has 1 as the second quantity. Speed = km/h, price = $/kg, heart rate = beats/min.
A car travels 240 km in 3 hours. Speed = 240 km ÷ 3 h = 80 km/h (unit rate). 5 kg of apples cost $12.50. Price per kg = $12.50 ÷ 5 = $2.50/kg. Which is better value: 750 mL for $4.50 or 1 L for $5.80? Find unit price: $4.50/750 = $0.006/mL = $6/L. $5.80/1000 = $0.0058/mL = $5.80/L. The 1 L is better value.
Share $96 in the ratio 5:7.
Total parts = 5 + 7 = 12 parts.
One part = $96 ÷ 12 = $8. First share = 5 × $8 = $40. Second share = 7 × $8 = $56.
Check: $40 + $56 = $96. And 40:56 = 5:7 (HCF = 8). Correct!
Shares: $40 and $56
Concrete is mixed in the ratio cement:sand:gravel = 1:2:3. How much of each is needed for 120 kg of concrete?
Total parts = 1 + 2 + 3 = 6 parts.
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. And 20:40:60 = 1:2:3 (HCF = 20). Correct!
20 kg cement, 40 kg sand, 60 kg gravel
Find $x$ if $3:5 = x:20$.
Method: write as fractions. $\frac{3}{5} = \frac{x}{20}$.
Cross-multiply: $5x = 3 \times 20 = 60$. So $x = 60 \div 5 = 12$.
Check: 3:5 = 12:20. Simplify 12:20 = 3:5 (HCF = 4). Correct!
$x = 12$
Mistake: Ratio 2:3 with $50: first = $20, second = $30. But 20:30 = 2:3 by luck! This only works when total ÷ parts = 10.
Fix: Always use the method: total parts = 5, one part = $50 ÷ 5 = $10. 2×$10=$20, 3×$10=$30. Works for any total.
Mistake: 3:5 means 3/5 of the total. No! 3:5 means 3 parts out of 8 total parts, so 3/8 of the total.
Fix: First share = 3/(3+5) = 3/8. Second share = 5/(3+5) = 5/8. Total parts go in the denominator.
Mistake: Simplifying 15 min : 2 hours as 15:2. Different units! Must convert first.
Fix: 15 min : 120 min = 15:120 = 1:8 (HCF = 15). Always same units first!
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to build your ratio fluency. Work each, then reveal.
-
1 Simplify 18:27.
HCF(18,27) = 9. 18÷9 = 2, 27÷9 = 3.2:3 -
2 Share $84 in the ratio 2:5.
Parts = 7. Each = $84 ÷ 7 = $12. 2×$12 = $24, 5×$12 = $60.$24, $60 -
3 Find $x$: 4:7 = $x$:35.
4/7 = x/35. 7x = 140. x = 20.20 -
4 A car travels 315 km in 3.5 hours. Speed?
315 ÷ 3.5 = 3150 ÷ 35 = 90 km/h.90 km/h
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. (a) Share $150 in the ratio 3:7. Show all working. (b) Three friends invest in a business in the ratio 2:3:5. The total investment is $24,000. How much did each friend invest?
Q7. (a) A train travels 420 km in 3 hours 30 minutes. Find its average speed in km/h. (b) Which is better value: 600 g for $7.20 or 800 g for $9.60? Show working.
Q8. A rectangle has sides in the ratio 3:4. Its perimeter is 84 cm. Find the length and width, and explain why the ratio of sides must be simplified before you can use it.
Quick Check
1. B — 16:24 = 2:3 (HCF = 8).
2. C — $60÷5 = $12. 3×$12 = $36.
3. A — 3/8 = x/24, 8x = 72, x = 9.
4. D — 45:120 = 3:8 (HCF = 15).
5. B — $8.40/3 = $2.80, $11.20/4 = $2.80. Same.
Show Your Working Model Answers
Q6 (4 marks): (a) Parts = 10, each = $150÷10 = $15 [1]. 3×$15 = $45, 7×$15 = $105 [1]. (b) Parts = 10, each = $24,000÷10 = $2,400 [1]. 2×$2,400 = $4,800, 3×$2,400 = $7,200, 5×$2,400 = $12,000 [1].
Q7 (4 marks): (a) 3.5 hours [0.5]. 420÷3.5 = 120 km/h [1.5]. (b) $7.20/600 = $0.012/g = $12/kg [1]. $9.60/800 = $0.012/g = $12/kg [1]. Same value.
Q8 (3 marks): Perimeter = 2(L+W) = 84, so L+W = 42 [0.5]. Ratio 3:4 means 3+4 = 7 parts [0.5]. Each part = 42÷7 = 6 [0.5]. Width = 3×6 = 18 cm, Length = 4×6 = 24 cm [1]. Simplified ratio gives the actual relationship between sides [0.5].
The Golden Ratio
The Golden Ratio is approximately 1.618:1, written as $\phi$ (phi). It appears in nature, art, and architecture. A rectangle with sides in the golden ratio is considered the most pleasing to the eye. If a golden rectangle has a short side of 10 cm, what is the long side? If the long side is 34 cm, what is the short side? Research one place the golden ratio appears in nature or art.
Reveal solution
Long side = 10 × 1.618 = 16.18 cm. Short side = 34 ÷ 1.618 = 21.01 cm (or more precisely: 34/$\phi$ = 21.01 cm). The Golden Ratio appears in: sunflower seed spirals (Fibonacci sequence ratio approaches $\phi$), the Parthenon in Athens, nautilus shell spirals, Leonardo da Vinci’s Vitruvian Man, and even human body proportions.
Ratio
Compares quantities, no units
Simplify
÷ both by HCF
Same units
Convert before simplifying
Divide
Total ÷ parts = one part
Rate
Different units, per 1
Equivalent
Cross-multiply: ad = bc
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