Question 6

When distance is fixed, speed and time are inversely proportional.

Step 1 — Find the total distance

Let us first find the distance between Lucknow and Kanpur. Distance is calculated by multiplying speed and time.

$\text{Distance} = \text{Speed} \times \text{Time}$

$= 30 \text{ km/h} \times 3 \text{ hours}$

$\boxed{90 \text{ km}}$

Step 2 — Calculate the new travel time

Puneeth's father now uses a car. Its speed is 60 km/h. The distance remains the same, which is 90 km. Let $x$ be the time taken to travel by car.

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

$x = \frac{90 \text{ km}}{60 \text{ km/h}}$

$x = \frac{3}{2} \text{ hours}$

$\boxed{1.5 \text{ hours}}$

Step 3 — Evaluate the proportionality statement

The statement $30 : 60 :: 3 : x$ shows a direct proportion. It means the speed ratio equals the time ratio. Let us calculate $x$ using this statement.

$\frac{30}{60} = \frac{3}{x}$

$30 \times x = 60 \times 3$

$30x = 180$

$x = \frac{180}{30}$

$\boxed{6 \text{ hours}}$

This result (6 hours) differs from the actual time. Speed and time are inversely proportional for a fixed distance. This means as speed increases, time decreases. For inverse proportion, the statement should be $30 : 60 :: x : 3$. So, the given statement does not represent this problem.

Step 4 — Analyze the effect of increased speed on time

Speed increased from 30 km/h to 60 km/h. The time taken decreased from 3 hours to 1.5 hours. So, travel time decreases as speed increases.

Answer

(i) No, the problem cannot be represented by $30 : 60 :: 3 : x$. (ii) The travel time will decrease as the speed of the motorcycle increases.