10 questions that get harder as you go. No calculator. Write your working on paper, pick A–E, then open the solution. Q1–3 are warm-ups, Q4–6 a step up, Q7–10 ESAT standard.
Q1 · Warm-up
£60 is shared in the ratio 2 : 3. What is the larger share?
A £12 · B £20 · C £24 · D £36 · E £40
Show solution
D. 2 + 3 = 5 parts, so one part = £60 ÷ 5 = £12. The larger share is 3 parts: 3 × £12 = £36.
Q2 · Warm-up
What is 15% of 80?
A 68 · B 12 · C 15 · D 5.3 · E 1.2
Show solution
B. 10% of 80 = 8 and 5% = 4. So 15% = 8 + 4 = 12.
Q3 · Warm-up
A £40 price goes up by 25%. What is the new price?
A £41 · B £45 · C £50 · D £65 · E £100
Show solution
C. 25% of 40 is 10, so the new price is £50. Or use a multiplier: 40 × 1.25 = 50.
Q4 · Step up
y is directly proportional to x. When x = 3, y = 12. What is y when x = 5?
A 60 · B 36 · C 15 · D 4 · E 20
Show solution
E. y = k x. From 12 = 3k, k = 4. So when x = 5, y = 4 × 5 = 20.
Q5 · Step up
y is inversely proportional to x. When x = 4, y = 6. What is x when y = 8?
A 3 · B 5⅓ · C 6 · D 12 · E 32
Show solution
A. Inverse proportion means x × y stays the same: 4 × 6 = 24. So x × 8 = 24 and x = 3.
Q6 · Step up
In a 20% off sale, a coat costs £64. What was its original price?
A £51.20 · B £76.80 · C £84 · D £80 · E £320
Show solution
D. £64 is 80% of the original. So 1% = 64 ÷ 80 = 0.80, and 100% = £80.
💡 B is the trap: adding 20% of 64. Percentages are always of the original.
Q7 · ESAT standard
y is proportional to x². If x increases by 50%, by what percentage does y increase?
A 50% · B 100% · C 125% · D 150% · E 225%
Show solution
C. A 50% increase multiplies x by 1.5. Then y is multiplied by 1.5² = 2.25. That’s an increase of 1.25, or 125%. (E is the trap: 2.25 is the new value as a percentage of the old, not the increase.)
Q8 · ESAT standard
a : b = 3 : 4 and b : c = 6 : 5. What is a : c?
A 3 : 5 · B 9 : 10 · C 10 : 9 · D 18 : 25 · E 1 : 2
Show solution
B. Make b the same in both. The LCM of 4 and 6 is 12. a : b = 9 : 12 and b : c = 12 : 10. So a : b : c = 9 : 12 : 10, and a : c = 9 : 10.
Q9 · ESAT standard
A price is increased by 10% and then the new price is decreased by 10%. What is the overall change?
A 1% decrease · B No change · C 1% increase · D 10% decrease · E 2% decrease
Show solution
A. Multipliers: 1.1 × 0.9 = 0.99. That’s 99% of the original, so a 1% decrease. The 10% off is taken from a bigger number than the 10% added.
Q10 · ESAT standard
A force F is directly proportional to a mass m and inversely proportional to the square of a distance r. If m is doubled and r is tripled, F is multiplied by:
A 2/3 · B 4/9 · C 6 · D 9/2 · E 2/9
Show solution
E. F ∝ m / r². Doubling m multiplies F by 2. Tripling r multiplies r² by 9, so F is divided by 9. Overall: 2 ÷ 9 = 2/9.
💡 This is exactly how gravity and electric force work, so the same skill turns up in the physics module.
🔥 Challenge: full ESAT level
Five harder questions with up to 8 options, answers in symbols and “which statements are correct?” questions, just like the real test. Aim for 90 seconds each.
C1
y is inversely proportional to x². If x increases by 25%, what happens to y?
A 25% decrease · B 36% decrease · C 20% decrease · D 64% decrease · E 56.25% decrease · F 44% decrease · G 50% decrease · H 6.25% decrease
Show solution
B. x becomes 1.25x = (5/4)x, so x² becomes (25/16)x². y is multiplied by 16/25 = 0.64, which is a 36% decrease.
💡 D is the trap: 64% is what is left, not what is lost.
C2
y = 3x + 2. Which of these statements are correct?
I y is directly proportional to x.
II If x doubles, y doubles.
III A graph of y against x is a straight line.
A none · B I only · C II only · D III only · E I and II only · F I and III only · G II and III only · H I, II and III
Show solution
D. Direct proportion needs a straight line through the origin. Here y = 2 when x = 0, so I is false. Check II: x = 1 gives y = 5, x = 2 gives y = 8, not 10, so II is false. III is true: it is a straight line with gradient 3.
💡 Straight line ≠ proportional. Proportional means straight line through (0, 0).
C3
Pink paint is made by mixing red and white paint in the ratio 2 : 3. You have 6 litres of red and 6 litres of white. What is the greatest volume of pink paint you can make?
A 6 L · B 8 L · C 10 L · D 9 L · E 12 L · F 15 L · G 7.5 L · H 11 L
Show solution
C. White runs out first. All 6 L of white is 3 parts, so 1 part = 2 L and red needed = 4 L (you have 6, fine). Total = 4 + 6 = 10 L.
💡 E (12 L) ignores the ratio; F (15 L) uses all the red, which would need 9 L of white.
C4
A cyclist rides the first half of a route at 40 km/h and the second half (same distance) at 60 km/h. What is the average speed for the whole route?
A 45 km/h · B 48 km/h · C 50 km/h · D 52 km/h · E 55 km/h · F 42 km/h · G 24 km/h · H 100 km/h
Show solution
B. Pick an easy distance: 120 km each half. Time = 120/40 + 120/60 = 3 + 2 = 5 h. Average = 240 ÷ 5 = 48 km/h.
💡 C (50) is the trap: more time is spent at the slower speed, so the average is below 50.
C5
The value of a rare coin doubles every 3 years. What is the percentage increase in its value over 12 years?
A 400% · B 800% · C 1600% · D 1200% · E 1500% · F 300% · G 1400% · H 3200%
Show solution
E. 12 years = 4 doublings, so the value is multiplied by 2⁴ = 16. New value = 1600% of the old, so the increase is 1600 − 100 = 1500%.
💡 C is the trap: 1600% is the new value as a percentage, not the increase.
Next: M4 Algebra · All ESAT topics