Worked Examples: Ratio and Proportion Problems

This final lesson presents complete UCAT-style data sets focusing on ratios, percentages, and proportions. Each set includes 4 questions with full worked solutions, timing guidance, and strategy notes. These examples bring together all the skills covered in this course.

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Data Set 1: Business Partnership

Data: Three business partners — Ahmed, Beth, and Carlos — invest in a company in the ratio 3:5:2. The total investment is £150,000.

In the first year, the company makes a profit of £42,000, which is distributed in the same ratio as the investment.

Question 1: How much did Beth invest?

• Total ratio parts: 3 + 5 + 2 = 10
• One part: £150,000 ÷ 10 = £15,000
• Beth (5 parts): 5 × £15,000 = £75,000

Time: ~5 seconds.

Question 2: How much profit does Ahmed receive?

• Ahmed's share of profit: 3/10 × £42,000
• 42,000 ÷ 10 = 4,200
• 3 × 4,200 = £12,600

Time: ~5 seconds.

Question 3: What is Ahmed's profit as a percentage of his investment?

• Ahmed invested: 3 × £15,000 = £45,000
• Profit: £12,600
• Percentage: (12,600 ÷ 45,000) × 100 = 28%

Quick check: 12,600/45,000 = 126/450 = 14/50 = 28/100 = 28% ✓

Time: ~8 seconds.

Question 4: If the profit were £56,000 instead, how much more would Carlos receive compared to the actual profit?

• Carlos's actual profit: 2/10 × £42,000 = £8,400
• Carlos's new profit: 2/10 × £56,000 = £11,200
• Difference: £11,200 − £8,400 = £2,800

Time: ~10 seconds.

Strategy note: Reuse the "one part" value. For actual profit: one part = £4,200. For new profit: one part = £5,600. Carlos has 2 parts. Difference: 2 × (5,600 − 4,200) = 2 × 1,400 = £2,800.

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Data Set 2: Nutritional Information

Data:

Nutrient	Per 100 g	% Daily Value (per 100 g)
Calories	250 kcal	12.5%
Protein	8 g	16%
Carbohydrates	35 g	13.5%
Fat	10 g	15.4%
Fibre	3 g	12%
Sugar	12 g	13.3%

A serving size is 40 g.

Question 1: How many calories are in one serving?

• Per 100 g: 250 kcal
• Per 40 g: 250 × 40/100 = 250 × 0.4 = 100 kcal

Time: ~3 seconds.

Question 2: If someone eats 3 servings, how much protein do they consume?

• Protein per serving: 8 × 0.4 = 3.2 g
• 3 servings: 3.2 × 3 = 9.6 g

Time: ~5 seconds.

Question 3: What percentage of the daily recommended fat intake does one serving provide?

• Fat per serving: 10 × 0.4 = 4 g
• Daily value per 100 g: 15.4%. This means 10 g = 15.4% of daily value.
• So daily value = 10 ÷ 0.154 = ~64.9 g (but we don't need this)
• Simpler: per serving = 40% of per-100g values. So % daily value per serving = 15.4% × 0.4 = 6.16%

Or directly: if 100 g provides 15.4% of daily value, then 40 g provides 15.4% × 40/100 = 6.16%

Time: ~8 seconds.

Question 4: What is the ratio of protein to fat to fibre per serving?

• Per serving: Protein = 3.2 g, Fat = 4 g, Fibre = 1.2 g
• Ratio: 3.2 : 4 : 1.2
• Multiply by 10: 32 : 40 : 12
• Simplify (÷4): 8 : 10 : 3

Time: ~8 seconds.

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Data Set 3: School Trip Budget

Data: A school is planning a trip for 120 students. The costs are:

Item	Cost
Coach hire	£1,800 (total, regardless of numbers)
Entrance fee	£12.50 per student
Lunch	£8 per student
Insurance	£3 per student

The school covers 40% of the total cost. The remaining 60% is split equally among the students.

Question 1: What is the total cost of the trip?

• Coach: £1,800
• Entrance: 120 × £12.50 = £1,500
• Lunch: 120 × £8 = £960
• Insurance: 120 × £3 = £360
• Total: £4,620

Time: ~12 seconds.

Question 2: How much does each student pay?

• Students' share: 60% of £4,620 = 0.6 × 4,620 = £2,772
• Per student: £2,772 ÷ 120 = £23.10

Time: ~10 seconds.

Question 3: If 10 students drop out (now 110 students), what is the new cost per student? (Assume the school still covers 40% of the new total.)