If M : N = 4 : 5, N : O = 6 : 7, and O : P = 9 : 11, then find N : P.

Correct — A. Join N : O and O : P through their shared term, O.

N : O = 6 : 7 and O : P = 9 : 11
Make O equal: LCM of 7 and 9 = 63

Scale N : O by 9: 54 : 63
Scale O : P by 7: 63 : 77
Combine: N : O : P = 54 : 63 : 77

Read N against P: N : P = 54 : 77 → option (a)

Why the others are wrong

• (b)
  45 : 77 — 45 : 77 swaps the digits of 54. N : P = (6 × 9) : (7 × 11) = 54 : 77. A 45 would need N = 5, the N of M : N, which sits on a different scale.
• (c)
  42 : 55 — 42 : 55 is a different ratio from 54 : 77. Cross-multiplying, 42 × 77 = 3,234 while 55 × 54 = 2,970, so the two are not equal.
• (d)
  36 : 63 — 63 is O's scaled value, not P's. Once O is 63 on both sides, P becomes 77. And 36 : 63 reduces to 4 : 7, which is not 54 : 77.

When two ratios share a term, make that term equal before comparing the outer ones. Here O is 7 in N : O and 9 in O : P, so scale both to 63, their LCM. N becomes 54 and P becomes 77.

A faster route multiplies the links: N⁄P = (N⁄O) × (O⁄P) = 6⁄7 × 9⁄11 = 54⁄77. Only the ratios that connect N to P matter.

Key facts

• If a : b = p : q and b : c = r : s, then a : c = pr : qs.
• To join two ratios through a shared term, scale both so that term becomes the LCM of its two values.
• 54 : 77 is in lowest terms, since 54 = 2 × 3³ and 77 = 7 × 11 share no factor.

Common traps

• Taking N = 5 from M : N instead of N = 6 from N : O: each ratio has its own scale until the shared term is made equal.
• Working M : N = 4 : 5 into the answer, when N : P needs only N : O and O : P.

No directly related past PYQ was found.