Less common Papers 1 & 2

Ratio and Proportion: Scale Factors, Direct/Inverse Variation, Percentages and Compound Growth

Ratio and proportion turn "how quantities compare" into exact arithmetic: splitting amounts, scaling shapes, and modelling growth. On the TMUA the payoff is speed: converting every percentage into a multiplier and every relationship into y = k × xn lets you reason to the answer by hand and eliminate options fast.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • A ratio a:b has no units and simplifies like a fraction; to share an amount in ratio a:b, cut it into (a+b) equal parts and take a and b of them (share = total × a/(a+b)).
  • Direct proportion means the ratio y/x stays constant, so y = k × x and the graph is a straight line through the origin.
  • Inverse proportion means the product x*y stays constant, so y = k/x; doubling x halves y.
  • For a power law y = k × xn, multiplying x by a factor m multiplies y by mn (an x2 law with x tripled sends y to 9 times its value).
  • Every percentage change is a multiplier: +r% means × (1 + r/100) and -r% means × (1 - r/100); successive changes multiply and never simply add.
  • Compound interest and exponential growth use the same rule: value after n periods = P × (1 + r/100)n.
  • Exponential decay and depreciation multiply by a fixed factor below 1 each period: value = A × (1 - r/100)n.
  • A length scale factor k gives an area scale factor k2 and a volume scale factor k3, so doubling every length multiplies area by 4 and volume by 8.

Formulae

y = k × x

Direct proportion: find k from one known (x, y) pair, then predict any other value.

y = k / x

Inverse proportion: use when the product x*y is constant, e.g. speed and time over a fixed distance.

y = k × xn

General variation with a power; the ratio of two outputs equals (ratio of inputs)n.

A = P × (1 + r/100)n

Compound interest or exponential growth over n equal periods at rate r% per period.

A = P × (1 - r/100)n

Exponential decay or depreciation: value falls by r% per period for n periods.

area SF = k2, volume SF = k3

Convert a length scale factor k into area and volume scale factors for similar shapes.

Definitions

Ratio
A comparison of two or more quantities measured in the same unit, written a:b; it carries no units and can be scaled or simplified like a fraction.
Direct proportion
A relationship in which y = k × x for a fixed constant k, so y/x is constant and each quantity is a fixed multiple of the other.
Inverse proportion
A relationship in which y = k/x, so the product x*y is constant; as one quantity grows the other shrinks in the same ratio.
Compound interest
Interest paid on the running total (principal plus interest already earned), so the balance is multiplied by the same factor (1 + r/100) every period.

Worked examples

1

The time T taken to drain a tank varies inversely with the square of the pipe radius r. When r = 2 cm the tank drains in 45 minutes. Find T when r = 3 cm.

  1. 1

    Model the link:

    T = k/r2
  2. 2
    Substitute r=2:
    45 = k/22
  3. 3

    Solve for k:

    k = 180
  4. 4
    Now use r=3:
    T = 180/32
  5. 5

    Simplify:

    T = 20

Answer: T = 20 minutes.

2

A shop raises the price of a jacket by 25%, then in a sale reduces the new price by 20%. Express the final price as a percentage of the original and state the overall percentage change.

  1. 1

    A 25% rise:

    f1 = 5/4
  2. 2

    A 20% fall:

    f2 = 4/5
  3. 3

    Multiply factors:

    f1 × f2 = 1
  4. 4

    Final equals start:

    final = 100%
  5. 5

    Net movement:

    change = 0%

Answer: The final price is 100% of the original, an overall change of 0%.

Common mistakes

  • ×Adding successive percentage changes: a 25% rise then a 20% fall is not a 5% rise; the multipliers 5/4 and 4/5 give exactly 1, i.e. no net change.
  • ×Mixing up direct and inverse proportion: in direct proportion y/x is constant, in inverse proportion x*y is constant, so 'doubling x doubles y' only holds for direct.
  • ×Dropping the power in a power law: if y = k × x2 and x triples, y grows by a factor 32 = 9, not 3.
  • ×Scaling area or volume by the length factor: enlarge lengths by k and area grows by k2, volume by k3, never just by k.
  • ×Reversing a percentage from the wrong base: adding back the same percent you took off does not restore the original, because the second percentage is of a smaller amount.

No-calculator tips

  • ✓Convert every percentage to an exact fraction or decimal multiplier first (+30% → 13/10, -40% → 3/5), then multiply; fractions keep the answer exact.
  • ✓For proportion questions, work with the multiplier between the two x-values and raise it to the power n; you rarely need to compute k itself.
  • ✓To share in a ratio, add the parts to get the number of shares, divide the total by that, then scale each part and check they sum back to the total.
  • ✓Expand small compound-interest powers by hand: 1.12 = 1.21, then × 1.1 = 1.331; keep exact rather than rounding early.
  • ✓Sanity-check the options by size before computing exactly: a 15% rise on 400 must land between 400 and 500, which usually kills two or three answers.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.y is inversely proportional to the square of x. When x = 1, y = 12. What is the value of y when x = 2?

  • A. 6
  • B. 24
  • C. 3
  • D. 48
Show answer

Answer: C — 3

y = k/x2 with k = 12*12 = 12, so at x = 2, y = 12/4 = 3. The trap 6 comes from using y proportional to 1/x (forgetting to square), and 48 from treating it as direct proportion y = 12x2.

Q2.An investment of 2000 pounds earns compound interest at 10% per year. What is its value, in pounds, after 2 years?

  • A. 2200
  • B. 2420
  • C. 2400
  • D. 2662
Show answer

Answer: B — 2420

Compound value = 2000 × (1.1)2 = 2000 × 1.21 = 2420. The distractor 2400 is the simple-interest answer (200 per year), 2200 stops after one year, and 2662 uses one power too many (3 years).

Q3.In a box the numbers of red and blue counters are in the ratio 3 : 5. After 4 more red counters are added, the ratio of red to blue becomes 5 : 7. How many blue counters are in the box?

  • A. 35
  • B. 21
  • C. 10
  • D. 56
Show answer

Answer: A — 35

Write red = 3k, blue = 5k; then (3k + 4) : 5k = 5 : 7 gives 7(3k + 4) = 25k, so 4k = 28, k = 7 and blue = 5*7 = 35. The trap 21 is the red count, and 10 comes from wrongly adding 4 to the blue counters as well.

Read this topic in the official UAT-UK TMUA content specification →

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