If a/b = 2, b/c = 3, c/d = 4, d/e = 3, e/f = 2 and f/g = 1.5, what is the value of (a+b+c)/(e-f+g)?

Correct — A, 144. Work backwards from the last letter, because every ratio is given as one letter over the next. Put g = 2, chosen to keep f whole. Then f = 1.5g = 3, e = 2f = 6, d = 3e = 18, c = 4d = 72, b = 3c = 216 and a = 2b = 432. The numerator a + b + c is 432 + 216 + 72 = 720. The denominator e − f + g is 6 − 3 + 2 = 5. The quotient is 720 ÷ 5 = 144. The starting value does not matter — the whole expression is homogeneous of degree zero, so g = 1 gives 360 over 2.5, which is again 144.

• (b)216 — The value of b in the chain when g is taken as 2. It is one of the intermediate quantities, offered to catch a candidate who writes out the letters and picks the wrong line.
• (c)225 — Does not appear anywhere in the chain. It would require the denominator to be 3.2 when the numerator is 720, and e − f + g works out at 5.
• (d)120 — Comes from mis-reading the denominator as e + f − g or e − f − g. With 6 + 3 − 2 = 7 or 6 − 3 − 2 = 1 the result is not 120 either; the printed expression is e − f + g = 5.

A chain of ratios a/b, b/c, c/d and so on fixes every letter once one of them is chosen, because each ratio ties a neighbour to its successor. The resulting expression is homogeneous — multiply every letter by the same factor and a quotient of sums is unchanged — so any convenient starting value works. Choosing the last letter and multiplying upwards avoids fractions and division.

Two things go wrong in the hall. The first is direction: a/b = 2 means a is twice b, so moving from a towards g the numbers shrink, and it is easier to start at g and multiply up. The second is the denominator's minus sign. The expression is e − f + g, which for g = 2 is 6 − 3 + 2 = 5, a small number, and a small denominator makes the answer large; a candidate who reads it as e + f + g gets 11 and lands nowhere near any option. Pick g = 2 rather than 1 and every letter stays a whole number, which removes the last chance of a slip.

• In a ratio chain each equation fixes one letter in terms of the next, so one free choice determines all of them.
• Taking g = 2 gives f = 3, e = 6, d = 18, c = 72, b = 216 and a = 432.
• a + b + c = 720 and e − f + g = 5, so the quotient is 144.
• The expression is homogeneous of degree zero, so the answer is independent of the value chosen for g.
• Choosing g = 2 rather than 1 keeps every term a whole number, since f = 1.5g.

• Building the chain downwards from a and drowning in fractions.
• Reading the denominator as e + f + g and losing the minus sign.
• Picking an intermediate value such as b as the final answer.

A ratio-chain item where the arithmetic is easy and the marks go to the candidate who reads the signs in the denominator correctly.

No directly related past PYQ was found.

• practice — not a real PYQ

  If p/q = 3 and q/r = 2, what is the value of (p + q)/(q + r)?

  • (a)2
  • (b)8/3
  • (c)3
  • (d)4

  Answer(b) 8/3 — take r = 1, then q = 2 and p = 6, so the expression is (6 + 2)/(2 + 1) = 8/3.

• practice — not a real PYQ

  If x/y = 4 and y/z = 3, then x : y : z is

  • (a)12 : 3 : 1
  • (b)4 : 3 : 1
  • (c)12 : 4 : 1
  • (d)12 : 3 : 2

  Answer(a) 12 : 3 : 1 — take z = 1, then y = 3 and x = 12.