If A, B, and C are three amounts of money such that B is the simple interest on A, and C is the simple interest on B, all for the same time and at the same rate of interest, then which of the following is true?

Correct — B.
Let r = RT⁄100, the same for both loans

SI on A: B = A × r
SI on B: C = B × r = A × r²

Divide: B⁄A = r and C⁄B = r, so B⁄A = C⁄B
Cross-multiply: B² = AC → option (b)

Why the others are wrong

• (a)
  A² = BC — A² = BC means A² = A²r³, true only when r = 1, i.e. interest equal to principal. Try A = 100, B = 20, C = 4: A² = 10,000 but BC = 80.
• (c)
  C² = AB — C² = AB means A²r⁴ = A²r, again true only when r = 1. With A = 100, B = 20, C = 4: C² = 16 but AB = 2,000.
• (d)
  A + B = C — A + B = C needs A + Ar = Ar², i.e. r² = r + 1, which fixes r to one special value. With A = 100, B = 20, C = 4: 120 ≠ 4.

Once the rate and time are fixed, simple interest is a fixed fraction of the principal: SI = P × RT⁄100 = P × r.

So each amount here is r times the one before: A, Ar, Ar². Three quantities in that pattern are in continued proportion (A : B = B : C), and the middle one is the mean proportional: B² = AC, or B = √(AC).

Key facts

• SI ⁄ P = RT⁄100, the same fraction for any principal at a fixed rate and time
• a, b, c are in continued proportion when a : b = b : c
• The mean proportional of a and c is √(ac)

Common traps

• Taking C as the interest on A instead of the interest on B
• Reading C as the amount A + B instead of the interest on B

No directly related past PYQ was found.