Question

# Points A and B are 90km apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction they meet in 9 hours and if they go in opposite directions they meet in $\dfrac{9}{7}$hours. Find the speeds of the cars.

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Hint: If the cars move in the same direction, they would take longer to meet each other. If they start from opposite directions, they would meet much faster. Further, in addition to this, we would use the formula below to find the speed of a car-
$speed=\dfrac{dis\tan ce}{speed}$

Complete step-by-step solution:
Let the speed of the car at A be A$\dfrac{km}{hr}$ and the other car at B be B$\dfrac{km}{hr}$. Now, from first condition (for cars going in same direction)-
Let car at A cover distance x km and car at B cover distance y km in duration of 9 hours (time when they meet). We have,
Distance = speed$\times$time
x = A$\times$9
y = B$\times$9
Now, since cars at A and B are 90 km apart and finally after 9 hours, they finally meet, car A would have to cover 90 km more than B since they travel in the same direction. Thus,
x = y+90
Plugging values of x and y,
9A=9B+90
A=B+10 -- (1)
From second condition (for cars going in opposite direction)-
Let a car at A cover distance x km and car at B cover distance y km in duration of $\dfrac{9}{7}$ hours (time when they meet). We have,
Distance = speed$\times$time
x = A$\times$ $\dfrac{9}{7}$
y = B$\times $$\dfrac{9}{7} Now, since cars at A and B are 90 km apart and finally after \dfrac{9}{7} hours they meet, since they are travelling in opposite directions, the total distance covered by cars is 90 km. Thus, x+y=90 A\times \dfrac{9}{7}+ B\times$$\dfrac{9}{7}$=90
$\dfrac{A+B}{7}$=10
A+B=70 -- (2)
2A = 80
A=40$\dfrac{km}{hr}$
Putting this value in (2),
We get, B=70-A
B=30$\dfrac{km}{hr}$
Hence, the car A has a speed of 40kmph and B has a speed of 30kmph
Note: We can solve this problem by using the concept of relative speeds. Thus, if cars move in the same directions, we subtract the speeds. While the cars move in the opposite directions, we add the speeds. In this case, we use the distance as 90 km. Now, for same directions, we have,
Speed of car at A is A$\dfrac{km}{hr}$ and at B is B$\dfrac{km}{hr}$
Distance = speed$\times$time
90 = (A-B)$\times$9 (Since, they meet after 9 hours)
(A-B) = 10
A=B+10 -- (a)
Similarly, for cars travelling in opposite directions, we add the speeds, we would have,
90 = (A+B) $\times$ $\dfrac{9}{7}$
(A+B) = 70 -- (b)
Thus, we get the same equations((a) and (b)) as (1) and (2) in the solution described above. So, we would get the same answers for speeds of car at A and car at B.
Thus, we would get the same answer for A and B as we got in the solution.