CSAT Solved Papers/ 2024/Q29

2024 CSAT — Q29

Quant Arithmetic & numeracy 2.5 marks Hard

A father said to his son, ”nn years back I was as old as you are now. My present age is four times your age nn years back”. If the sum of the present ages of the father and the son is 130130 years, what is the difference of their ages?

  1. A 30 years Answer
  2. B 32 years
  3. C 34 years
  4. D 36 years

Worked rationale

Let the present ages be FF (father) and SS (son). Translate each clause to an equation.

”nn years back I was as old as you are now”: F−n=SF - n = S, so n=F−Sn = F - S.

“My present age is four times your age nn years back”: F=4(S−n)F = 4(S - n). Substitute n=F−Sn = F - S:

F=4(S−(F−S))=4(2S−F)=8S−4F  ⇒  5F=8S.F = 4\big(S - (F - S)\big) = 4(2S - F) = 8S - 4F \;\Rightarrow\; 5F = 8S.

Use the sum F+S=130F + S = 130. From 5F=8S5F = 8S, F=85SF = \tfrac{8}{5}S, so

85S+S=130⇒135S=130⇒S=50,F=80.\tfrac{8}{5}S + S = 130 \Rightarrow \tfrac{13}{5}S = 130 \Rightarrow S = 50,\quad F = 80.

Difference =80−50=30= 80 - 50 = 30.

Answer: (a) 30 years.

Why the other options miss

  • B
    an arithmetic slip: a slip in solving 135S=130\tfrac{13}{5}S=130 (e.g. S=49S=49), shifting the difference off by a couple of years.
  • C
    mis-translated a clause: reads “four times your age nn years back” as “four times your present age,” giving the wrong ratio F:SF:S.
  • D
    solved a different question: drops the substitution n=F−Sn = F - S and treats nn as an independent unknown, mis-pinning the ages.

Specialist insight

Two phrases must be parsed exactly: nn is not free — the first clause pins n=F−Sn = F - S (the age gap itself). Substituting that into "F=4(S−n)F = 4(S-n)" collapses everything to the single relation 5F=8S5F = 8S, and the sum finishes it. The classic CSAT age-trap is reading “your age nn years back” as “your present age”; here S−n=2S−FS - n = 2S - F, a quantity that depends on both ages. Translate clause by clause, then eliminate nn first — the algebra is then one substitution deep.

The trap, in one line

nn is the age gap itself (n=F−Sn=F-S); substituting it turns the two clauses into 5F=8S5F=8S, giving 8080 and 5050 — difference 3030.

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