Places A and B are 20 km apart on the highway. A car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in one hour. If they travelled towards each other, they met in 12 minutes. What are the speeds (in km/h) of two cars?
- (a)51, 49
- (b)60, 40
- (c)55, 45
- (d)52, 48
Answer
Why
Correct — B. Let the faster car run at u km/h and the slower at v km/h. The gap between A and B is 20 km.
Same direction — the faster closes the gap at the difference of the speeds, and takes 1 hour:
u − v = 20 ⁄ 1 = 20
Towards each other — they close at the sum, and take 12 minutes = 1⁄5 hour:
u + v = 20 ÷ 1⁄5 = 20 × 5 = 100
Add the two equations:
2u = 120 → u = 60
Subtract them:
2v = 80 → v = 40
The speeds are 60 and 40 km/h → option (b).
Why the others are wrong
- (a)51, 49 — 51 and 49 differ by 2, so in the same direction the faster car would need 20 ⁄ 2 = 10 hours to close the gap, not 1 hour. The sum of 100 is right, which is what makes the option look safe.
- (c)55, 45 — 55 and 45 differ by 10, giving 20 ⁄ 10 = 2 hours in the same direction. The head-on condition passes and the overtaking condition fails.
- (d)52, 48 — 52 and 48 differ by 4, so the same-direction meeting would take 20 ⁄ 4 = 5 hours. Only a difference of exactly 20 delivers the required 1 hour.
Concept
Two bodies, two conditions, one idea: relative speed.
Travelling the same way, the gap between them closes at the difference of the speeds. Travelling towards each other, it closes at the sum. Both are just distance ÷ relative speed = time applied to the same 20 km.
That gives a pair of linear equations in u and v which add and subtract straight into the answer — no substitution needed. The only unit care required is turning 12 minutes into 1⁄5 of an hour before dividing.
Every option here sums to 100, so all four satisfy the head-on condition of 12 minutes.
Only the same-direction condition separates them, and only 60 and 40 differ by the required 20. Test the difference first and the item takes seconds.
Key facts
- Moving in the same direction, the relative speed is the difference of the two speeds.
- Moving towards each other, the relative speed is the sum of the two speeds.
- 12 minutes is 1⁄5 hour, so covering 20 km in that time needs a relative speed of 100 km/h.
Study next
Common traps
- Dividing 20 by 12 instead of by 1⁄5, which returns 1.67 and no usable equation.
- Swapping the conditions, so the difference is set to 100 and the sum to 20.
- Assuming the cars must meet somewhere between A and B, when in the same-direction case they meet 60 km from A, well past B.
The pair of conditions — one same-direction, one head-on — is the standard shape for this topic, and it is the difference that discriminates.
The same relative-speed idea runs a chase in this shift at Quant Q.4, where a policeman starts 500 metres behind a thief and catches him in 15 minutes.
Related PYQs
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