Question

A man travelled a distance of 22km in one hour, partly in a car at a speed of \[30{\rm{km/hr}}\] and partly on a motorcycle at \[18{\rm{km/hr}}\] . Find the distance travelled by him in the car.

Answer 1

Hint:
Here we will first assume the distance travelled by a man in a car to be any variable. Then we will apply the time distance formula to calculate the time taken by man with a car and motorcycle. Then we will add them to find the total time taken by hi, and equate it to the given distance. Then we will solve the equation further to get the required distance.

Formula used:
\[{\rm{time}} = \dfrac{{{\rm{distance}}}}{{{\rm{speed}}}}\]

Complete step by step solution:
Here we need to find the distance travelled by a man in the car.
It is given that the total distance travelled by a man \[ = 22{\rm{km/hr}}\]
Total time taken to cover that distance \[ = 1hr\]
Speed while traveling in a car \[ = 30{\rm{km/hr}}\]
Speed while traveling on a motorcycle \[ = 18{\rm{km/hr}}\]
Let the distance travelled by the man in a car to be \[x\].
So the distance travelled by the man on a motorcycle will be equal to \[\left( {22 - x} \right){\rm{km/hr}}\].
The time distance formula is given by \[{\rm{time}} = \dfrac{{{\rm{distance}}}}{{{\rm{speed}}}}\]
Therefore,
Time taken by man in a car to travel a distance \[ = \dfrac{x}{{30}}hr\] ……….. \[\left( 1 \right)\]
Time taken by the man on motorcycle to travel a distance \[ = \dfrac{{22 - x}}{{18}}hr\] ……….. \[\left( 2 \right)\]
Total time taken by man to cover total distance will be equal to the sum of the time taken to cover distance by a car and time taken to cover distance by a motorcycle.
So we can write it as
\[ \Rightarrow 1 = \dfrac{x}{{30}} + \dfrac{{22 - x}}{{18}}\]
On adding the terms on right had side, we get
\[ \Rightarrow 1 = \dfrac{{3x + 5\left( {22 - x} \right)}}{{90}}\]
On further simplifying the terms, we get
\[\begin{array}{l} \Rightarrow 1 = \dfrac{{3x + 5 \times 22 - 5x}}{{90}}\\ \Rightarrow 1 = \dfrac{{110 - 2x}}{{90}}\end{array}\]
On cross multiplying the terms, we get
\[ \Rightarrow 90 = 110 - 2x\]
On adding and subtracting the terms, we get
\[\begin{array}{l} \Rightarrow 90 - 110 = 110 - 2x - 110\\ \Rightarrow - 20 = - 2x\end{array}\]
On dividing both sides by -2, we get
\[\begin{array}{l} \Rightarrow \dfrac{{ - 20}}{{ - 2}} = \dfrac{{ - 2x}}{{ - 2}}\\ \Rightarrow 10 = x\end{array}\]

Therefore, the distance travelled by the man in the car is equal to 10 km.

Note:
Here we have obtained the distance travel by the man in the car. As we wanted total time by man to cover the distance, so we added the time taken by him with the car and the motorcycle. WE can make a mistake by subtracting the time taken by him. We know that time and speed are inversely proportional to each other but are directly proportional to distance. This means that if the distance is constant, and speed increases then the time taken to cover the distance will be less and vice versa.