N11: Fractions in Ratio Problems
Work with fractions in ratio problems
📋 Key Concepts
Ratio: Compares two or more quantities in order. Written as a:b or a:b:c
Fractions and Ratios: A ratio can be expressed as fractions of the total. In ratio a:b, the first part is a/(a+b) of the total.
Key Relationships:
For ratio a : b (total = a + b):
• First part = a/(a+b) of the total
• Second part = b/(a+b) of the total
• First part = a/b of the second part
For ratio a : b (total = a + b):
• First part = a/(a+b) of the total
• Second part = b/(a+b) of the total
• First part = a/b of the second part
📝 Expressing Ratio as Fractions
Example 1
A bag contains red and blue counters in the ratio 3:2. What fraction of the counters are red?
Solution:
Total parts = 3 + 2 = 5
Red fraction = 3/5
Blue fraction = 2/5
Example 2
A mixture contains apple juice and water in ratio 4:1. What fraction is apple juice?
Solution:
Total parts = 4 + 1 = 5
Apple juice fraction = 4/5
📝 Finding One Quantity as a Fraction of Another
Example 3
In a class, the ratio of boys to girls is 2:3. What fraction of the class are boys?
Solution:
Total = 2 + 3 = 5 parts
Boys = 2/5 of the class
Example 4
Alex has 12 sweets and Sam has 18 sweets. Express Alex's sweets as a fraction of the total.
Solution:
Total = 12 + 18 = 30
Alex's fraction = 12/30 = 2/5
📝 Using Fractions to Solve Ratio Problems
Example 5
A prize of £500 is shared between Tom and Jerry in the ratio 2:3. How much does each receive?
Solution:
Total parts = 2 + 3 = 5
Tom gets: 2/5 × £500 = £200
Jerry gets: 3/5 × £500 = £300
Example 6
In a recipe, flour and sugar are mixed in ratio 5:2. How much flour is needed for 350g total mixture?
Solution:
Total parts = 5 + 2 = 7
Flour = 5/7 × 350g = 250g
📝 Comparing Quantities Using Fractions
Example 7
In school A, 3/8 of students study music. In school B, 2/5 study music. Which school has a higher proportion?
Solution:
Convert to common denominator:
3/8 = 15/40
2/5 = 16/40
School B has a higher proportion (16/40 > 15/40)
📝 Combining Ratios and Fractions
Example 8
A box contains red, blue and green balls in ratio 3:4:5. What fraction are NOT green?
Solution:
Total parts = 3 + 4 + 5 = 12
Green = 5/12
Not green = 1 - 5/12 = 7/12
Example 9
A sum of money is shared between A, B and C in ratio 2:3:5. If C gets £150, find the total amount.
Solution:
C's fraction = 5/(2+3+5) = 5/10 = 1/2
If 1/2 of total = £150
Total = £150 × 2 = £300
🧮 Worked Examples — Exam Style
Ratio and fraction are two views of the same division: parts ÷ total parts. The total is ALL parts added together.
A necklace has gold and silver beads in ratio 3 : 7. What fraction are silver, and what percentage?
Silver = 7/10 of the beads = 70%
M1 for 7/10, A1 for 70%. Fractions-to-percentages inside ratio questions is the standard double-mark.
2/3 of a class are girls. Write the ratio girls : boys.
Girls 2 parts, boys 1 part → 2 : 1
M1 for recognising boys = 1/3, A1 for 2 : 1. The reverse direction: fraction → ratio by taking complements.
Exam wording to expect: "What fraction of the … are …?", "Write as a ratio …".
❓ Practice Questions
Q1: A ratio is 5:3. What fraction does the first part represent?
Q2: A drink is made from juice and water in ratio 1:4. What fraction is juice?
Q3: £120 is shared in ratio 3:1. How much does the smaller share receive?
Q4: In a car park, the ratio of cars to vans is 7:3. What fraction are vans?
Q5: A cake is shared in ratio 2:3:4. What fraction is the largest share?
Q6: A rope is cut in ratio 2 : 5. What fraction of the whole is the LONGER piece?
Q7: In a class, 0.4 of students walk. Write the ratio walkers : non-walkers.
✅ Answers
- 5/(5+3) = 5/8
- 1/(1+4) = 1/5
- 1/4 × £120 = £30
- 3/(7+3) = 3/10
- 4/(2+3+4) = 4/9
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 5/(5 + 3) = 5/8
Q2. 1/(1 + 4) = 1/5
Q3. 4 parts → £30 each → smaller share (1 part) = £30
Q4. 3/(7 + 3) = 3/10
Q5. 4/(2 + 3 + 4) = 4/9
Q6. Longer = 5 parts of 7 → 5/7
Q7. 0.4 = 2/5 walk → non-walkers 3/5 → ratio 2 : 3
📌 Key Facts to Memorise
- Part a of ratio a : b → fraction a/(a + b) — the DENOMINATOR is the total of all parts
- Fraction p/q → ratio p : (q − p) (when p + q = whole)
- Decimals convert: 0.35 = 35/100 = 7/20 → ratio 7 : 13
- Simplify fractions before converting to ratios
- Three-part ratios: each part over the grand total (2 : 3 : 4 → 2/9, 3/9, 4/9)
🗺️ In the Exam
Fractions-with-ratio: 2–3 marks, Papers 1/2. It fuses two heavy-hitter topics — the errors here are usually the denominator (forgetting to add ALL parts).
Links to: simplifying ratio (R4), fractions of amounts (R3), percentages (R9).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always find the total number of parts first
- Check that your fractions add up to 1
- To find a quantity: multiply the fraction by the total
- To find a total: divide the quantity by its fraction
- Read carefully - "fraction of" and "fraction more than" are different
🧠 Problem-Solving Strategies
Problem-Solving
For ratio problems involving fractions, always find the total number of parts first, then express each share as a fraction of the total. When comparing ratios, convert to fractions with the same denominator. For "what fraction remains" questions, subtract the given fraction from 1.
Multi-Step Problem
A prize is shared between A, B and C in the ratio 3:5:7. If C receives £84 more than A, find the total prize.
Solution: Total parts = 3 + 5 + 7 = 15. C − A = (7 − 3)/15 of total = 4/15 of total. 4/15 of total = £84. Total = £84 × 15/4 = £315.
⚠️ Common Errors
Watch Out!
1. Wrong: In ratio 2:3, the first part is 2/3 of the total Correct: Total parts = 5, so the first part is 2/5 of the total
2. Wrong: In ratio 1:4, the second part is 4 times the total Correct: The second part is 4/5 of the total, not 4 times the total
3. Wrong: To find the total from one share in ratio 3:5, just multiply by 5 Correct: If one share is 3/8 of total = £30, then total = £30 × 8/3 = £80
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A business divides its profit between three departments in the ratio 2:3:5. (a) What fraction of the profit does the largest department receive? (b) The smallest department receives £24,000. Find the total profit. (c) The following year, the total profit increases by 20%. The largest department's fraction increases to 6/15. Has the largest department received more money than the previous year? Show working.
(a) Total parts = 10. Largest = 5/10 = 1/2 of the profit.
(b) Smallest = 2/10 = 1/5 of total = £24,000. Total = £24,000 × 5 = £120,000.
(c) Previous year: largest = 1/2 × £120,000 = £60,000. New total = £120,000 × 1.2 = £144,000. New largest = 6/15 × £144,000 = 2/5 × £144,000 = £57,600. No — the largest department received less (£57,600 < £60,000) despite the total increasing, because their share fraction decreased.
Mark scheme: 1 mark for (a), 2 marks for (b), 3 marks for (c) including comparison and conclusion
📊 AO3: Reason & Interpret
Reasoning and Interpretation
School X has 180 students and the ratio of boys to girls is 5:4. School Y has 240 students and the ratio of boys to girls is 7:5.
(a) Which school has a higher proportion of boys?
(b) How many more boys than girls are there in School X?
(c) A student says "School Y has more boys than School X because it has more students." Is this necessarily true? Calculate to check.
Answers: (a) School X: boys = 5/9 ≈ 55.6%. School Y: boys = 7/12 ≈ 58.3%. School Y has a higher proportion. (b) Boys = 5/9 × 180 = 100, Girls = 4/9 × 180 = 80. Difference = 20 more boys. (c) School Y boys = 7/12 × 240 = 140. School X boys = 100. Yes, School Y does have more boys (140 vs 100), but this is because of BOTH the proportion and the total size, not just the total size.
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🎬 Video Resources
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