In covering a distance of 130 km, Amit takes 4 hours more than Vinay. If Amit doubles his speed, then he would take 9 hour less than Vinay. Amit's original speed is:

The correct answer is 5 km/h.

Let us solve this step-by-step to find Amit's original speed.

Step 1: Define the variables and equations based on the given problem statement.
Let the total distance be $D=130\text{ km}$.
Let Amit's original speed be $v_A$ km/h and Vinay's speed be $v_V$ km/h.
Let the time taken by Vinay to cover 130 km be $t_V$ hours.

According to the first condition, Amit takes 4 hours more than Vinay to cover 130 km:
Time taken by Amit at original speed = $t_V+4$ hours.

Using the formula for speed, $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$, we have:

$v_A=\frac{130}{t_V+4}$    --- (Equation 1)

According to the second condition, if Amit doubles his speed to $2v_A$, he takes 9 hours less than Vinay:
Time taken by Amit at doubled speed = $t_V-9$ hours.

Therefore, at double speed:

$2v_A=\frac{130}{t_V-9}$    --- (Equation 2)

Step 2: Relate the two equations to solve for $t_V$.
Divide Equation 2 by Equation 1:

$\frac{2v_A}{v_A}=\frac{130/(t_V-9)}{130/(t_V+4)}$

$2=\frac{t_V+4}{t_V-9}$

Cross-multiplying gives:

$2(t_V-9)=t_V+4$

$2t_V-18=t_V+4$

$2t_V-t_V=4+18$

$t_V=22\text{ hours}$

Step 3: Calculate Amit's original speed $v_A$.
Substitute the value of $t_V=22$ back into Equation 1:

$v_A=\frac{130}{22+4}$

$v_A=\frac{130}{26}=5\text{ km/h}$

Thus, Amit's original speed is 5 km/h.