Question

The speeds of three cars are in the ratio $2:3:4$ . What is the ratio between the times taken by these cars to travel the same distance?
A. $4:3:2$
B. $2:3:4$
C. $4:9:16$
D. $6:4:3$

Answer 1

Hint: Here we have been given the ratio between the speed of three cars and we have to find the ratio between the times taken by the three cars for travelling the same distance. Firstly we will remove the ratio sign by using an unknown variable then we will let our distance and get the time taken by each car using the distance formula. Finally we will form the ratio between the value obtained and get our desired answer.

Complete step-by-step solution:
The speed of three cars ratio is given as follows:
$2:3:4$
Let us take the speed of three cars as,
$2x,3x,4x$…..$\left( 1 \right)$
Now let the distance travelled by the cars be $d$ so
Distance $=d$ ….$\left( 2 \right)$
The distance formula is given as follows:
Speed $=$ Distance $\div $ Time
Time $=$ Distance $\div $ Speed…..$\left( 3 \right)$
So using the value from equation (1) and (2) in equation (3) we get,
Time taken by car $1=\dfrac{d}{2x}$
Time taken by car $2=\dfrac{d}{3x}$
Time taken by car $3=\dfrac{d}{4x}$
So we get the ratio between the time taken by three cars as follows:
$\dfrac{d}{2x}:\dfrac{d}{3x}:\dfrac{d}{4x}$
Next we will make the denominator of all the above values the same by multiplying and dividing a value with each ratio.
So as we can see the denominator has values $2,3,4$ and L.C.M of them is $12$ .
Multiply and divide by $6,4,3$ with the term respectively as follows,
$\dfrac{d}{2x}\times \dfrac{6}{6}:\dfrac{d}{3x}\times \dfrac{4}{4}:\dfrac{d}{4x}\times \dfrac{3}{3}$
$\Rightarrow \dfrac{6d}{12x}:\dfrac{4d}{12x}:\dfrac{3d}{12x}$
Next cancelling out the common value which is $\dfrac{d}{12x}$ from each term we get,
$\Rightarrow 6:4:3$
Hence the correct option is (D).

Note: The most important step is to know the distance formula and how we can eliminate the unknown variables to get answers for such types of questions. A direct method can also be used here as we know that Time is inversely proportional to the speed so we can write the ratio of time as follows:
$\dfrac{1}{2}:\dfrac{1}{3}:\dfrac{1}{4}$
Now we will make the denominator same by multiplying and dividing by $6,4,3$ respectively,
$\dfrac{6}{2\times 6}:\dfrac{4}{3\times 4}:\dfrac{3}{3\times 4}$
$\Rightarrow \dfrac{6}{12}:\dfrac{4}{12}:\dfrac{3}{12}$
We can write them as $6:4:3$ by dividing the above term by $12$ .
We are getting the same answer.