Question 13

When the distance traveled is fixed, speed and time are inversely proportional, meaning if one increases, the other decreases.

Step 1 — Understand the relationship

Let us consider how speed and time are connected when the distance between two places remains the same. If Puneeth's father drives his motorcycle at a higher speed, he will take less time to cover the same distance. Conversely, if he drives at a lower speed, it will take him more time. This shows that speed and time have an inverse relationship.

This means their product is constant, which is the distance. So, Speed $\times$ Time = Distance. We cannot use a direct proportion like $50 : 2 :: 75 : \_\_$ because that would imply that as speed increases, time also increases, which is incorrect for a fixed distance.

Step 2 — Calculate the total distance

First, we need to find the distance between Lucknow and Kanpur. We are given the initial speed and the time taken.

Let $S_1$ be the initial speed, which is $50 \text{ km/h}$. Let $T_1$ be the initial time taken, which is $2 \text{ hours}$. Let $D$ be the distance between Lucknow and Kanpur.

$D = S_1 \times T_1$

$D = 50 \text{ km/h} \times 2 \text{ hours}$

$\boxed{D = 100 \text{ km}}$

Step 3 — Calculate the new time

Now, Puneeth's father drives at a new speed, and we need to find how long it will take him. The distance remains the same.

Let $S_2$ be the new speed, which is $75 \text{ km/h}$. Let $T_2$ be the new time taken.

Since the distance is constant, we can write: $S_1 \times T_1 = S_2 \times T_2$. We know that $S_1 \times T_1$ is the distance, which is $100 \text{ km}$.

$100 \text{ km} = 75 \text{ km/h} \times T_2$

To find $T_2$, we divide the distance by the new speed.

$T_2 = \frac{100}{75} \text{ hours}$

$T_2 = \frac{4}{3} \text{ hours}$

To express this in hours and minutes, we convert the fraction of an hour to minutes. There are $60 \text{ minutes}$ in $1 \text{ hour}$.

$T_2 = 1 \text{ hour} + \frac{1}{3} \times 60 \text{ minutes}$

$T_2 = 1 \text{ hour} + 20 \text{ minutes}$

$\boxed{T_2 = 1 \text{ hour } 20 \text{ minutes}}$

Since the speed increased from $50 \text{ km/h}$ to $75 \text{ km/h}$, it will take less time to reach Kanpur. The initial time was $2 \text{ hours}$, and the new time is $1 \text{ hour } 20 \text{ minutes}$, which is indeed less.

Answer

(i) It will take Puneeth’s father less time to reach Kanpur. (ii) The time taken to reach Kanpur at $75 \text{ km/h}$ is 1 hour 20 minutes. (iii) We cannot form this problem as a direct proportion like $50 : 2 :: 75 : \_\_$ because speed and time are inversely proportional when the distance is constant.