Proportional Reasoning - 2 | IT

Question 6

Context: Puneeth's father went from Lucknow to Kanpur in 3 hours by riding his motorcycle at a speed of 30 km/h. If he takes a car instead and drives at 60 km/h, how long will it take him to reach Kanpur?

Q. Can we represent this problem with the following statement of proportionality — $30 : 60 :: 3 : x$ ? Will the travel time increase or decrease as the speed of the motorcycle increases?

Solution

Understand the Question

For a fixed distance, speed and time are inversely proportional ( $\text{Distance} = \text{Speed} \times \text{Time} = \text{constant}$ ).
This means that as speed increases, the time taken to cover the same distance must decrease.
A direct proportion statement like $s_1 : s_2 :: t_1 : t_2$ incorrectly implies time increases with speed. The correct inverse proportion relation is $s_1 : s_2 :: t_2 : t_1$ .

Step 1 · Find Total Distance

Given speed of motorcycle $= 30 \text{ km/h}$ and time taken $= 3 \text{ hours}$ . Diagram 1

\begin{aligned} \text{Distance} &= \text{Speed} \times \text{Time} \\[0.6em] &= 30 \text{ km/h} \times 3 \text{ hours} \\[0.6em] &= 90 \text{ km} \end{aligned}

Step 2 · Calculate Actual Travel Time by Car

Distance $= 90 \text{ km}$ and car speed $= 60 \text{ km/h}$ .

Let $x$ be the time taken by car:

\begin{aligned} \text{Time} &= \dfrac{\text{Distance}}{\text{Speed}} \\[0.6em] x &= \dfrac{90 \text{ km}}{60 \text{ km/h}} \\[0.6em] &= \dfrac{3}{2} \text{ hours} = 1.5 \text{ hours} \end{aligned}

Step 3 · Evaluate the Proportionality Statement

The statement $30 : 60 :: 3 : x$ represents direct proportion:

\begin{aligned} \dfrac{30}{60} &= \dfrac{3}{x} \\[0.6em] 30 \times x &= 60 \times 3 \\[0.6em] 30x &= 180 \\[0.6em] x &= \dfrac{180}{30} \\[0.6em] x &= 6 \text{ hours} \end{aligned}

Since $6 \text{ hours} \neq 1.5 \text{ hours}$ , the given statement is incorrect.

Because speed and time vary inversely, the correct proportion is: $30 : 60 :: x : 3$

Step 4 · Analyze Effect of Increased Speed

When speed increases from $30 \text{ km/h}$ to $60 \text{ km/h}$ , travel time decreases from $3 \text{ hours}$ to $1.5 \text{ hours}$ .

Therefore, 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 increases.

Common Mistakes

Direct vs. Inverse Proportion: Writing $30 : 60 :: 3 : x$ assumes time increases with speed (direct proportion). For constant distance, the product $\text{speed} \times \text{time}$ remains constant, requiring the inverse proportion $30 : 60 :: x : 3$ .
Intuition Check: Always verify if a higher speed should result in less time. An answer of $6\text{ hours}$ for doubling speed is physically incorrect.

Question 6

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