If A:B = 6 : 8 and B:C = 7 : 10, then A:B:C is:

Correct — C. B is 8 in the first ratio and 7 in the second, so scale both until B is the same number.

LCM of 8 and 7 = 56
A : B = 6 : 8, multiply by 7 → 42 : 56
B : C = 7 : 10, multiply by 8 → 56 : 80

A : B : C = 42 : 56 : 80
Divide throughout by 2 → 21 : 28 : 40 → option (c)

Why the others are wrong

• (a)
  14:31:20 — 31 is not a multiple of 7, so no scaling of B : C = 7 : 10 can produce it as the middle term. 14 : 31 does not reduce to 6 : 8 either.
• (b)
  21:23:40 — A and C are right and B is 23 instead of 28. Test the middle term against the first ratio: 21 : 28 reduces to 3 : 4, which is 6 : 8, while 21 : 23 reduces to nothing.
• (d)
  23:32:40 — 23 : 32 does not reduce to 6 : 8 — that would need 24 : 32. A single digit in the first term is enough to break the ratio the question actually gives.

Two ratios can only be joined through the term they share. A : B and B : C describe B with different numbers, so the fix is to rewrite both until B reads the same in each.

The mechanical route is the LCM of B's two values: multiply the first ratio by B's value in the second, and the second ratio by B's value in the first. Here that is ×7 and ×8, giving B = 56 both times.

A ratio is unchanged by multiplying or dividing every term by the same number, which is why 42 : 56 : 80 and 21 : 28 : 40 are the same answer.

Key facts

• To combine A:B and B:C, multiply A:B by B's value in the second ratio and B:C by B's value in the first.
• 6 : 8 reduces to 3 : 4, so 21 : 28 and 42 : 56 are the same ratio.
• Any two terms of a proposed answer can be reduced and checked against the pair given in the question.

Common traps

• Multiplying each ratio by the wrong partner, so the two B values still disagree
• Writing 6 : 8 : 10 by copying the four printed numbers straight down
• Reaching a correct unreduced ratio and then failing to find it among the options

No directly related past PYQ was found.