If (x+y):(x−y) =5:2, find (x³ + y³) : (x³ − y³)

Correct — A.
Write the ratio with a multiplier: x + y = 5k, x − y = 2k
Add the two: 2x = 7k, so x = 3.5k
Subtract them: 2y = 3k, so y = 1.5k

Reduce: x : y = 3.5 : 1.5 = 7 : 3
Cube each: x³ = 343, y³ = 27

Sum: 343 + 27 = 370
Difference: 343 − 27 = 316
Halve both: 370 : 316 = 185 : 158 → option (a)

Why the others are wrong

• (b)
  158 : 185 — 158 : 185 is the right pair upside down, which is (x³ − y³) : (x³ + y³). With x = 7 and y = 3 the sum of cubes, 370, beats the difference, 316, so the larger number comes first.
• (c)
  17 : 23 — 17 : 23 is less than 1, but (x³ + y³) ⁄ (x³ − y³) = 370⁄316 is more than 1. A ratio with the smaller term first cannot be the answer.
• (d)
  23 : 17 — 23 : 17 ≈ 1.35, while the true ratio 185 : 158 ≈ 1.17. And 185 : 158 is already in lowest terms, so it does not reduce to 23 : 17.

Componendo–dividendo: if (x + y) : (x − y) = a : b, then x : y = (a + b) : (a − b). Adding and subtracting the two parts of the given ratio separates x from y.

Here 5 : 2 gives x : y = 7 : 3.

Both x³ + y³ and x³ − y³ are made of cubes, so a common multiplier cancels out of the ratio. Put x = 7 and y = 3 straight in, compute, then reduce.

Key facts

• If (x + y) : (x − y) = a : b, then x : y = (a + b) : (a − b)
• 7³ = 343 and 3³ = 27
• x³ + y³ = (x + y)(x² − xy + y²) and x³ − y³ = (x − y)(x² + xy + y²)

Common traps

• Cubing 5 and 2 as if they were x and y, which gives 133 : 117 — 5 : 2 compares x + y with x − y
• Stopping at 370 : 316 and not recognising it among the options until it is halved
• Writing the ratio upside down as (x³ − y³) : (x³ + y³)

No directly related past PYQ was found.