Ratios and Their Graphs — Solutions

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1. Simplify each ratio

   1. 12:8 = 3:2
   2. 15:25 = 3:5
   3. 100:75 = 4:3
   4. 18:12 = 3:2
   5. 24:36 = 2:3
   6. 50:80 = 5:8
   7. 9:27 = 1:3
   8. 64:48 = 4:3

2. Find the missing value in equivalent ratios

   1. 3:4 = ?:20: 15
   2. 2:5 = 10:?: 25
   3. 1:3 = ?:18: 6
   4. 5:2 = 25:?: 10
   5. 4:7 = ?:28: 16
   6. 3:8 = 9:?: 24
   7. 6:5 = ?:30: 36
   8. 2:9 = 8:?: 36

3. Ratio to fraction and percentage

   1. 2:3 first quantity: 2/5
   2. 1:4 first quantity: 20%
   3. 3:7 second quantity: 7/10
   4. 3:2 second quantity: 40%
   5. 1:1 each quantity: 50% each
   6. 3:1 first quantity: 3/4
   7. 5:3 first quantity: 5/8
   8. 2:8 first quantity: 20%

4. Real-world ratio problems

   1. Map 1:50000, 3 cm: 3 × 50000 = 150 000 cm = 1.5 km
   2. Map 1:25000, 5 km: 500 000 cm ÷ 25 000 = 20 cm
   3. Paint 3:2, 9 L blue: 6 litres of white
   4. Concrete 1:2:3 for 24 kg: 4 kg cement, 8 kg sand, 12 kg gravel
   5. Flour:sugar 4:1, 120 g flour: 30 g sugar
   6. Lengths 5:3, longer = 40 cm: 24 cm
   7. Students:teachers 20:1, 500 students: 25 teachers
   8. Juice:lemonade 3:2, 500 mL total: 300 mL juice, 200 mL lemonade

5. Extended ratio problems

   1. 20 cupcakes recipe: Scale factor = 20/12 = 5/3; Flour = 300 g, Sugar = 150 g, Butter = 100 g
   2. Scale drawing room: 15 m × 12 m
   3. Revenue:costs 5:3, revenue $40 000: Costs = 3/5 × $40 000 = $24 000
   4. Investment 2:3:5, total $20 000: Person A $4000, Person B $6000, Person C $10 000

6. Ratio bar diagram — Part A (3 parts) to Part B (2 parts)

   1. Ratio A:B: 3:2
   2. Value of each segment ($150 total): 5 total parts; $150 ÷ 5 = $30 per segment
   3. Part A and Part B shares: Part A = 3 × $30 = $90; Part B = 2 × $30 = $60
   4. Fraction of total for Part A: 3/5

7. Complete the ratio table — apples at $3.50/kg

   1. Missing values: 2 kg = $7.00; 3 kg = $10.50; 5 kg = $17.50; 10 kg = $35.00
   2. Equation: C = 3.5m
   3. Graph shape: Straight line through the origin (0, 0) — proportional relationship
   4. Cost of 7.5 kg: C = 3.5 × 7.5 = $26.25

8. Reading proportional relationship graphs

   1. Ratio x:y for Graph A: x:y = 1:2 (for every 1 unit of x, y = 2)
   2. Ratio x:y for Graph B: x:y = 1:3 (for every 1 unit of x, y = 3)
   3. Which graph for “3 cups water per 1 cup cordial”: Graph B (ratio 1:3)
   4. Equations: Graph A: y = 2x; Graph B: y = 3x

9. Mortar mixing — sand:cement = 5:1

   1. 2 kg cement → sand: 5 × 2 = 10 kg sand
   2. Complete table: Sand: 5, 10, 15, 20, 30 kg; Total: 6, 12, 18, 24, 36 kg
   3. Equation S = ?: S = 5C
   4. 42 kg total mortar: Total parts = 6; 1 part = 42÷6 = 7 kg; Cement = 7 kg, Sand = 35 kg
   5. Total cost: Sand: 35 × $0.40 = $14.00; Cement: 7 × $1.20 = $8.40; Total = $22.40

10. Is it proportional?

    1. x: 1,2,3,4 / y: 4,8,12,16: Yes — proportional. y = 4x passes through origin; constant ratio y/x = 4.
    2. x: 0,1,2,3 / y: 3,5,7,9: No — not proportional. When x=0, y=3 (not 0). The line has a y-intercept of 3, so it does not pass through the origin. Rule is y = 2x + 3.
    3. Graph through (0,0) and (5,15): Yes — passes through origin; proportional with ratio 1:3.
    4. Graph crossing y-axis at (0,4): No — does not pass through origin; not a proportional relationship.