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R6

Express a multiplicative relationship between two quantities as a ratio or a fraction

Ratio as a fraction

Worked answers, methods and verified real exam appearances for R6 on Edexcel GCSE Maths 1MA1.

Explanation

  • A multiplicative relationship compares quantities by division rather than subtraction. If A=kBA=kB, then AB=k\dfrac{A}{B}=k, so A:B=k:1A:B=k:1 and AA is the fraction kk of BB.
  • Convert decimal multipliers to fractions when an integer ratio is required; for example, 1.75=741.75=\dfrac{7}{4} gives A:B=7:4A:B=7:4.
  • For a chain of relationships, substitute one equation into another or match the shared ratio part.
  • Check that a multiplier greater than 11 corresponds to the larger quantity.
  • Examiners expect the direction of the comparison to agree with the wording, because reversing it produces the reciprocal.

Worked example

Quantities PP, QQ and RR satisfy P=53QP=\dfrac{5}{3}Q and Q=34RQ=\dfrac{3}{4}R. Express P:RP:R in simplest form.

  1. 1.Substitute Q=34RQ=\dfrac{3}{4}R into P=53QP=\dfrac{5}{3}Q.
  2. 2.P=53×34R=54RP=\dfrac{5}{3}\times\dfrac{3}{4}R=\dfrac{5}{4}R.
  3. 3.Therefore P:R=54:1=5:4P:R=\dfrac{5}{4}:1=5:4.

Answer: P:R=5:4P:R=5:4.

Common mistakes

  • Don't subtract the quantities and report an additive difference instead of a multiplier.
  • Don't reverse the comparison and give the reciprocal ratio.
  • Don't leave a ratio containing fractions when a simplest integer ratio is required.

Exam tip

Translate “AA is kk times BB” into A=kBA=kB before converting it to a ratio.

Worked practice

Q1
Tier 1 · Easy

1

Quantity AA is 1818 and quantity BB is 3030. Express AA as a fraction of BB and write A:BA:B in simplest form.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
QuestionAnswerMarkMark scheme
1
  • A=35BA=\frac{3}{5}B
  • A:B=3:5A:B=3:5
2A/B=18/30=3/5A/B=18/30=3/5, so A=35BA=\frac{3}{5}B and the corresponding ratio is 3:53:5.
Q2
Tier 2 · Standard

2

The mass of parcel PP is 1.751.75 times the mass of parcel QQ. Express P:QP:Q as an integer ratio and express PP as a fraction of QQ.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
QuestionAnswerMarkMark scheme
2
  • P:Q=7:4P:Q=7:4
  • P=74QP=\frac{7}{4}Q
21.75=741.75=\frac{7}{4}. Therefore P=74QP=\frac{7}{4}Q, which gives P:Q=7:4P:Q=7:4.
Q3
Tier 3 · Hard

3

Quantities PP, QQ and RR satisfy P=53QP=\frac{5}{3}Q and Q=34RQ=\frac{3}{4}R. Express P:RP:R in simplest form and write PP as a fraction of RR.

(3)

(Total for Question 3 is 3 marks)

Mark scheme

Mark scheme for question 3
QuestionAnswerMarkMark scheme
3
  • P:R=5:4P:R=5:4
  • P=54RP=\frac{5}{4}R
3Substitute Q=34RQ=\frac{3}{4}R into the first relationship: P=53×34R=54RP=\frac{5}{3}\times\frac{3}{4}R=\frac{5}{4}R. Therefore P:R=5:4P:R=5:4.
Q4
Tier 1 · Easy

4

Quantity AA is 38\dfrac{3}{8} of quantity BB. Express A:BA:B in its simplest form.

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
QuestionAnswerMarkMark scheme
4
  • A:B=3:8A:B=3:8
1A=38BA=\dfrac{3}{8}B, so the corresponding ratio is A:B=3:8A:B=3:8.
Q5
Tier 2 · Standard

5

Quantity PP is 40%40\% greater than quantity QQ. Express P:QP:Q in its simplest form and write QQ as a fraction of PP.

(2)

(Total for Question 5 is 2 marks)

Mark scheme

Mark scheme for question 5
QuestionAnswerMarkMark scheme
5
  • P:Q=7:5P:Q=7:5
  • Q=57PQ=\dfrac{5}{7}P
2P=1.4Q=75QP=1.4Q=\dfrac{7}{5}Q, so P:Q=7:5P:Q=7:5. Reversing the comparison gives Q=57PQ=\dfrac{5}{7}P.
Q6
Tier 3 · Hard

6

Quantity AA is 35%35\% greater than quantity BB. Express the difference A−BA-B as a fraction of AA, and write (A−B):A(A-B):A in its simplest form.

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
QuestionAnswerMarkMark scheme
6
  • 727\dfrac{7}{27}
  • (A−B):A=7:27(A-B):A=7:27
3Taking BB as 100100 gives A=135A=135, so the difference is 3535. Therefore A−BA=35135=727\dfrac{A-B}{A}=\dfrac{35}{135}=\dfrac{7}{27}, and (A−B):A=7:27(A-B):A=7:27.
Q7
Tier 2 · Standard

7

The ratio M:NM : N is 7:127 : 12. Express N−MN-M as a fraction of MM, and write M:(N−M)M : (N-M) in its simplest form.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
QuestionAnswerMarkMark scheme
7
  • 57\dfrac{5}{7}
  • M:(N−M)=7:5M : (N-M)=7 : 5
3Let M=7kM=7k and N=12kN=12k. Then N−M=5kN-M=5k, so N−MM=5k7k=57\dfrac{N-M}{M}=\dfrac{5k}{7k}=\dfrac{5}{7} and M:(N−M)=7:5M : (N-M)=7 : 5.
Q8
Tier 3 · Hard

8

Quantity AA is 20%20\% less than quantity BB. Quantity CC is 50%50\% greater than quantity AA. Express C:BC : B in its simplest form and write BB as a fraction of CC.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
QuestionAnswerMarkMark scheme
8
  • C:B=6:5C : B=6 : 5
  • B=56CB=\dfrac{5}{6}C
4A=0.8BA=0.8B. Then C=1.5A=1.5×0.8B=1.2B=65BC=1.5A=1.5\times0.8B=1.2B=\dfrac{6}{5}B. Therefore C:B=6:5C : B=6 : 5 and, reversing the comparison, B=56CB=\dfrac{5}{6}C.
Q9
Tier 3 · Hard

9

Quantity A is 1.61.6 times quantity B. The difference between A and B is 4242. Express this difference as a fraction of the total of A and B. Give the fraction in its simplest form.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
QuestionAnswerMarkMark scheme
9
  • 313\dfrac{3}{13}
4Let B be xx, so A is 1.6x1.6x. Their difference is 0.6x=420.6x=42, giving x=70x=70. Therefore B is 7070, A is 112112, and the total is 182182. The required fraction is 42182=313\dfrac{42}{182}=\dfrac{3}{13}.
Q10
Tier 3 · Hard

10

The price of item P is 56\dfrac{5}{6} of the price of item Q. The price of item Q is 30%30\% less than the price of item R. Express the price of P as a fraction of the price of R, and write P:R in its simplest form.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
QuestionAnswerMarkMark scheme
10
  • P is 712\dfrac{7}{12} of R
  • P:R=7:12P : R = 7 : 12
4Q is 0.7=7100.7=\dfrac{7}{10} of R. Therefore P is 56×710=3560=712\dfrac{5}{6}\times\dfrac{7}{10}=\dfrac{35}{60}=\dfrac{7}{12} of R, so P:R=7:12\text{P}:\text{R}=7:12.

Verified exam appearances

SeriesPaperQuestionMarksCalculatorTierLinks
2022-113FQ122AllowedFoundationQPMS
2023-062HQ183AllowedHigherQPMS
2022-111FQ246Non-calculatorFoundationQPMS
2023-113HQ234AllowedHigherQPMS

Other points in R Ratio, proportion and rates of change

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