Question

While covering a distance of 30km. Ajeet takes 2 hours more than Amit. If Ajeet doubles his speed, he would take 1 hour less than Amit. Find their speed of walking?

Answer 1

Relation between speed, time and distance is given as $\text{Speed = }\dfrac{\text{Distance}}{\text{Time}}$ . Suppose speed of Ajeet and Amit as two variables. Now, from two equations with the help of the above relation and given conditions in the question. Solve it to get the speed of them.

Complete step-by-step answer:
Let the speed of Ajeet and Amit are ${{V}_{1}}$ km/hr and ${{V}_{2}}$ km/hr respectively. And we know the relation of speed, distance and time can be given by the relation.
$\text{Speed = }\dfrac{\text{Distance}}{\text{Time}}.................\left( i \right)$
So, it is given that Ajeet takes 2 hours more than Amit for covering 30km. Let us calculate the time taken by Ajeet and Amit for covering 30km.
$\text{Time = }\dfrac{\text{Distance}}{\text{Speed}}...........\left( ii \right)$
Now, we can get the value of time with respect to distance and speed from equation (i). Hence, we get time taken by Ajeet to cover 30km can be given as
$\text{Time}{{\left( \text{Ajeet} \right)}_{1}}=\dfrac{30}{{{V}_{1}}}...............\left( iii \right)$
Similarly, time taken by Amit to cover 30km is
$\text{Time}{{\left( \text{Amit} \right)}_{1}}=\dfrac{30}{{{V}_{1}}}...............\left( iv \right)$
Now, we know that Ajeet takes 2 hours more than Amit to cover 30km. Hence, we can write
$\text{Time}{{\left( \text{Ajeet} \right)}_{1}}=\text{Time}{{\left( \text{Amit} \right)}_{1}}+2$
Now, put values of time from the equation (iii) and (iv). Hence, we get
$\begin{aligned} & \dfrac{30}{{{V}_{1}}}=\dfrac{30}{{{V}_{2}}}+2 \\\ & \dfrac{30}{{{V}_{1}}}-\dfrac{30}{{{V}_{2}}}=2 \\\ & \dfrac{1}{{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}=\dfrac{2}{30}=\dfrac{1}{15} \\\ & \dfrac{1}{{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}=\dfrac{1}{15}..............\left( v \right) \\\ \end{aligned}$
Now, the next condition is given that Ajeet will take 1 hour less than Amit to cover 30km if Ajeet will double his speed. Now, the speed of Ajeet is $2{{V}_{1}}$ . So, time taken by Ajeet from equation (ii) can be given as
$\text{Time}{{\left( \text{Ajeet} \right)}_{2}}=\dfrac{30}{2{{V}_{1}}}...............\left( vi \right)$
Similarly, time taken by Amit to cover 30km can be given as
$\text{Time}{{\left( \text{Amit} \right)}_{2}}=\dfrac{30}{{{V}_{2}}}...............\left( vii \right)$
Now, we know that Ajeet will take 1 hour less than Amit if Ajeet will double his speed. So, we get
$\text{Time}{{\left( \text{Ajeet} \right)}_{2}}=\text{Time}{{\left( \text{Amit} \right)}_{2}}-1...........\left( viii \right)$
Now, using the equations (vi), (vii) and (viii), we get
$\begin{aligned} & \dfrac{30}{2{{V}_{1}}}=\dfrac{30}{{{V}_{2}}}-1 \\\ & \dfrac{30}{2{{V}_{1}}}-\dfrac{30}{{{V}_{2}}}=-1 \\\ & \dfrac{1}{2{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}=\dfrac{-1}{30}...............\left( ix \right) \\\ \end{aligned}$
Now, subtract equations (v) and (ix) to get the value of ${{V}_{1}}$ . Hence on subtracting equations (v) and (ix) we get
$\begin{aligned} & \left( \dfrac{1}{{{V}_{1}}}-\dfrac{1}{{{V}_{2}}} \right)-\left( \dfrac{1}{2{{V}_{1}}}-\dfrac{1}{{{V}_{2}}} \right)=\dfrac{1}{15}-\left( \dfrac{-1}{30} \right) \\\ & \dfrac{1}{{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}-\dfrac{1}{2{{V}_{1}}}+\dfrac{1}{{{V}_{2}}}=\dfrac{1}{15}+\dfrac{1}{30} \\\ & \dfrac{1}{{{V}_{1}}}-\dfrac{1}{2{{V}_{1}}}=\dfrac{2+1}{30} \\\ & \left( \dfrac{2-1}{2} \right)\dfrac{1}{{{V}_{1}}}=\dfrac{3}{30} \\\ & \dfrac{1}{2{{V}_{1}}}=\dfrac{1}{10} \\\ & {{V}_{1}}=5\text{km/hr} \\\ \end{aligned}$
Now, put the value of ${{V}_{1}}=5$ in equation (iv) to get the value of ${{V}_{2}}$ . Hence, we get
$\begin{aligned} & \dfrac{1}{5}-\dfrac{1}{{{V}_{2}}}=\dfrac{1}{15} \\\ & \dfrac{1}{5}-\dfrac{1}{15}=\dfrac{1}{{{V}_{2}}} \\\ & \dfrac{3-1}{15}=\dfrac{1}{{{V}_{2}}} \\\ & \dfrac{2}{15}=\dfrac{1}{{{V}_{2}}} \\\ & {{V}_{2}}=\dfrac{15}{2}=7.5 \\\ & {{V}_{2}}=7.5\text{km/hr} \\\ \end{aligned}$
Hence, Speed of Ajeet is 5km/hr and the speed of amit is 7.5km/hr.

Note: Don’t confuse with the formula related to speed, distance and time students get confuse with the formula and they can apply it as $\text{Time}=\text{distance}\times \text{speed}$ or $\text{speed}=\text{distance}\times \text{time}$ which is wrong. It is given as $\text{speed}=\dfrac{\text{distance}}{\text{time}}$ . Hence, be clear with the position of terms with this identity. One may solve the equations
$\dfrac{1}{{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}=\dfrac{1}{15},\dfrac{1}{2{{V}_{1}}}-\dfrac{1}{{{V}_{2}}}=\dfrac{-1}{30}$
By taking $\dfrac{1}{{{V}_{1}}}=x,\dfrac{1}{{{V}_{2}}}=y$ and hence get equation as
$x-y=\dfrac{1}{15},\dfrac{x}{2}-y=\dfrac{-1}{30}.$
The later equations are much more familiar than the ones written in the form of ${{V}_{1}},{{V}_{2}}$ . So, one may solve them by replacing $\dfrac{1}{{{V}_{1}}},\dfrac{1}{{{V}_{2}}}$ by two other variables. Answer will remain the same. Writing the equation in mathematical terms given in the form of words in the problem is the key point of the question. Don’t confuse the symbols 1 and 2 used with time (Ajeet) and time (Amit). They represent the conditions in 1 and 2, as we have two conditions in the problem..