IPMAT Indore 2023 · Quantitative Ability · Question 40

Step 1 — note that average speed is never the average of the speeds. It is

\text{Average speed}=\frac{\text{total distance}}{\text{total time}}

Step 2 — total distance. Four sides of 100 km each:

4\times 100=400\text{km}

Step 3 — total time. The speeds are 100, 200, 300, 400 kmph, each over 100 km:

\frac{100}{100}+\frac{100}{200}+\frac{100}{300}+\frac{100}{400}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}

=\frac{12+6+4+3}{12}=\frac{25}{12}\text{hours}

Step 4 — divide.

\text{Average speed}=\frac{400}{25/12}=400\times \frac{12}{25}=192\text{kmph}

Answer: Option 4 — 192 kmph

Shortcut. Equal distances means harmonic mean. For n equal stretches at speeds {v}_{1},\ldots ,{v}_{n}:

\text{Average}=\frac{n}{\frac{1}{{v}_{1}}+\cdots +\frac{1}{{v}_{n}}}

Factoring out 100:

\frac{400}{\frac{25}{12}}=192

Option (c) 250 is the arithmetic mean of 100, 200, 300, 400 — the answer to a question nobody asked, and the one most candidates pick. The harmonic mean is always less than the arithmetic mean, which alone rules out 250 and makes 192 the only plausible value near it.

Use 1+1/2+1/3+1/4=25/12 from memory — this little sum shows up constantly in equal-distance problems.