The ratio of present ages (in years) of X to Y is equal to the ratio of present ages (in years) of Y to Z. If the present age of Y is 15 years, then which of the following can be the sum of the ages (in years) of X, Y and Z?
- (a)35
- (b)40
- (c)49
- (d)55
Correct — C, 49. X : Y = Y : Z means Y is the geometric mean of X and Z, so Y² = XZ, that is XZ = 15² = 225. Two constraints then screen the options. The sum of two positive numbers whose product is 225 is at least 2 × 15 = 30, so X + Y + Z is at least 45, which rules out 35 and 40 outright. Testing 49 gives X + Z = 34 with XZ = 225, and the pair 25 and 9 satisfies both — the three ages are 25, 15 and 9, in the ratio 5 : 3 : 9/5, a genuine geometric progression with common ratio 3/5. Testing 55 gives X + Z = 40 with XZ = 225, whose solutions are 20 ± √175, neither of which is a whole number of years.
- (a)35 — 35 needs X + Z = 20 while XZ = 225. No two positive numbers can do that: their product would have to fall to at most 100 for a sum of 20.
- (b)40 — 40 needs X + Z = 25 with XZ = 225, and the maximum product for a sum of 25 is 156.25. The pair does not exist.
- (d)55 — 55 needs X + Z = 40 with XZ = 225, which gives 20 + √175 and 20 − √175, roughly 33.2 and 6.8. Ages in years in this question are whole numbers, and 49 is the option that delivers them.
When X : Y = Y : Z the three quantities are in geometric progression and Y is their geometric mean, so Y² = XZ. Fixing Y therefore fixes the product of the outer terms while leaving their sum free. The inequality between the arithmetic and geometric means then sets a floor on that sum: X + Z ≥ 2√(XZ), with equality only when X = Z.
The word 'can' in the stem is the instruction. The examiner is not asking for the sum of three particular ages but for which offered total is achievable, so the method is to screen the options rather than to solve for X and Z. The floor of 45 kills two options in one line. Between the two survivors, 49 factors cleanly into 25 and 9 while 55 forces a surd, so 49 is the total that corresponds to a real set of ages in completed years.
- X : Y = Y : Z is the same statement as Y² = XZ.
- With Y = 15 the product of the outer ages is fixed at 225.
- For positive X and Z, X + Z ≥ 2√(XZ) = 30, so the three ages total at least 45.
- 225 = 25 × 9 gives X + Z = 34 and a total of 49, with common ratio 3/5.
- Equality in the mean inequality would need X = Z = 15, making all three ages 15 and the total 45.
One inequality removes half the options before any factor pair is tried.
- Treating X : Y = Y : Z as an arithmetic progression and writing 2Y = X + Z.
- Solving for one pair and stopping, without checking whether other options are even possible.
- Overlooking that ages here are whole numbers, which is what separates 49 from 55.
A ratio item whose stem says 'can be', signalling that options are to be tested for possibility rather than a unique answer computed.
Two years ago, the age of A was three times the age of B. If B is currently 9 years old, then after how many years, the age of A will be double of the age of B?
- (a) 2 years
- (b) 3 years
- (c) 4 years
- (d) 5 years
Answer(d) 5 years
An age question built on the same discipline of writing one relation for each sentence of the stem. There the relation is a multiple at a past date; here it is a ratio at the present one.
- practice — not a real PYQ
If a, b, c are in geometric progression with b = 12 and a + c = 30, then ac equals
- (a)100
- (b)121
- (c)144
- (d)169
Answer(c) 144 — for a geometric progression b² = ac, so ac = 12² = 144.
- practice — not a real PYQ
The product of two positive numbers is 100. The smallest possible value of their sum is
- (a)10
- (b)20
- (c)25
- (d)50
Answer(b) 20 — by the mean inequality the sum is at least 2√100 = 20, reached when both numbers are 10.