Aptitude · Quant

Arithmetic progressions & train problems

Aptitude rounds are timed, so speed matters. Two formulas cover most AP questions, and one idea, "what distance is actually covered?", covers most train questions.

Your win todaySolve a standard AP or train question in under 60 seconds using the right formula on the first try.

Part A · Arithmetic progressions

An AP adds the same common difference d each step: 3, 7, 11, 15 … (a = 3, d = 4).^{[Khan Academy]}

nth term

aₙ = a + (n − 1)d

You take (n − 1) jumps from the first term, not n.

Sum of first n terms

Sₙ = n/2 · (first + last)

= n/2 · [2a + (n − 1)d]. Pair the first with the last: they all add to the same total.

The sum formula is "number of terms × average term". The average of an AP is (first + last) ÷ 2.

Speed facts: sum of 1 to n = n(n+1)/2 (1 to 100 = 5050). Number of terms from a to l in steps of d = (l − a)/d + 1. For example, the two-digit multiples of 7 are 14 to 98 → (98 − 14)/7 + 1 = 13.

Part B · Speed, time, distance, and trains

Distance = Speed × Time

km/h → m/s: multiply by 5/18. m/s → km/h: multiply by 18/5. (1 km/h = 1000 m / 3600 s.)

The train idea: what length gets covered?

The train has to clear itself plus whatever it passes, and a pole has length 0. With two moving objects, use the relative speed: add the speeds if they approach each other, subtract if one chases the other. These are the standard relations used in IndiaBIX train problems.

Worked examples

1 · A 150 m train at 54 km/h passes a pole. Time?

54 km/h × 5/18 = 15 m/s. Distance = 150 m (its own length). Time = 150 / 15 = 10 s.

2 · The same train crosses a 250 m platform. Time?

Distance = 150 + 250 = 400 m. Time = 400 / 15 = 26.67 s (26⅔ s).

3 · Trains of 120 m and 80 m run towards each other at 50 and 40 km/h. Time to cross?

Relative speed = 90 km/h = 25 m/s. Distance = 120 + 80 = 200 m. Time = 8 s.

4 · Same trains, same direction (50 chasing 40). Time to cross?

Relative speed = 10 km/h = 25/9 m/s. Time = 200 ÷ (25/9) = 72 s.

5 · Average speed: 60 km/h out and 40 km/h back over the same road?

It's not 50. For equal distances, the average speed = 2xy/(x+y) = 2·60·40/100 = 48 km/h.

Two traps(1) Forgetting to convert km/h to m/s when lengths are in metres. (2) Forgetting the platform length, or adding the pole's.

Retrieval check

What is the 20th term of 5, 8, 11, …?

What is the sum of the first 10 terms of 2, 4, 6, …?

What is 72 km/h in m/s?

Crossing a platform, the distance covered is …

Type it (time yourself: 60 s each)

A 300 m train at 90 km/h passes a signal post. Seconds?

A 100 m train at 36 km/h crosses a platform in 30 s. Platform length (m)?

How many two-digit numbers are divisible by 7?

Your turn

Write the 2 AP formulas and the 4 train rules on an index card from memory.
Cover each worked example's answer and solve it yourself before revealing.
Do 10 questions from IndiaBIX Problems on Trains with a 60-second timer each. Log your score.
Do 5 AP questions from Khan Academy: Sequences.
Memorise: ×5/18 and ×18/5, 1+…+n = n(n+1)/2, average speed = 2xy/(x+y).

Go deeper

Primary source: IndiaBIX: Problems on Trains. It's the standard question bank for Indian aptitude rounds, with formulas and solved examples. For AP concepts: Khan Academy: Sequences.

💬 Ask your teacher: "give me a timed set of 10 mixed train questions" or "boats and streams?"