IPMAT Indore 2023 · Quantitative Ability · Question 11

Step 1 — translate the two relations. Let the four speeds be {v}_{1},{v}_{2},{v}_{3},{v}_{4}.

“{v}_{4} is five times {v}_{1}”: {v}_{4}=5{v}_{1}

“Twice the time in 2nd as in 3rd, for the same distance”: time is inversely proportional to speed, so {v}_{3}=2{v}_{2}

Step 2 — work with reciprocals. Time = distance \div speed, so put

X=\frac{1}{{v}_{1}}, Y=\frac{1}{{v}_{2}}

Then \frac{1}{{v}_{4}}=\frac{X}{5} and \frac{1}{{v}_{3}}=\frac{Y}{2}.

Step 3 — the first journey (25 km in each gear).

25X+25Y+25\cdot \frac{Y}{2}+25\cdot \frac{X}{5}=585

30X+\frac{75}{2}Y=585

Step 4 — the second journey (4, 4, 32, 60 km).

4X+4Y+32\cdot \frac{Y}{2}+60\cdot \frac{X}{5}=4X+12X+4Y+16Y=16X+20Y

Step 5 — notice that (2) is a multiple of (1). Compare the coefficients:

\frac{16}{30}=\frac{8}{15}, \frac{20}{75/2}=\frac{40}{75}=\frac{8}{15}

They match, so \left( 2 \right)=\frac{8}{15}\times \left( 1 \right) — the two unknowns never need to be found separately.

\text{Time}=\frac{8}{15}\times 585=8\times 39=312

Answer: 312 minutes

Shortcut. You cannot find the four speeds — and you do not need to. Two relations and one equation leave a free parameter, which is the clue that the answer must be proportional to the given 585.

Write both journeys in terms of X=1/{v}_{1} and Y=1/{v}_{2}:

\text{first}=30X+37.5Y, \text{second}=16X+20Y

Check the ratio once: 16/30=20/37.5=8/15. Hence

\text{answer}=8/15\left( 585 \right)=312

The transferable insight: when a problem is under-determined but still has a unique answer, the target must be a fixed multiple of what you are given. Test that ratio first instead of hunting for the individual values.