Question

The speeds of three cars are in the ratio 5: 4: 6. The ratio between times taken by them to travel the same distance is
$
  (a){\text{ 5:4:6}} \\
  (b){\text{ 6:4:5}} \\
  (c){\text{ 10:12:15}} \\
  (d){\text{ 12:15:10}} \\
$

Answer 1

Hint: The distance travelled by each car is given as the same. So directly use the basic formula that resembles the relation between distance travelled, speed and the time taken to get the ratio between times taken by the car.

Complete Step-by-Step solution:
The given ratio of the speed of cars is 5 : 4 : 6.
 We have to find out the ratio between the times taken by them to travel the same distance.
As we know speed, distance and time is related as
${\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{speed}}}}$
So as we see time and speed has an inverse ratio.
So the ratio of time taken by them to travel the same distance is,
Ratio of time = $\dfrac{1}{5}:\dfrac{1}{4}:\dfrac{1}{6}$
Now take the L.C.M of 5, 4 and 6.
So first factorize the number we have,
Factors of 5 are
$ \Rightarrow 5 = 1 \times 5$
Factors of 4 are
$ \Rightarrow 4 = 1 \times 2 \times 2$
Factors of 6 are
$ \Rightarrow 6 = 1 \times 2 \times 3$
Therefore L.C.M of numbers is
$ \Rightarrow L.C.M = 1 \times 2 \times 2 \times 3 \times 5 = 60$
So, multiply by 60 in the ratio of time we have,
Therefore ratio of time = $\dfrac{{60}}{5}:\dfrac{{60}}{4}:\dfrac{{60}}{6}$
Now simplify them we have,
Therefore ratio of time = $12:15:10$
So this is the required ratio of time taken by them to travel the same distance.
Hence option (D) is correct.

Note: Whenever we face such types of questions the key concept is simply to have the gist of direct relation between the speed, distance and time. Note point of this question is that the distance which is travelled by the three cars is given the same thus we can easily find out the ratios of time which the cars would have taken.