Speed Ratio Problem Analysis

This question asks us to find the ratio of the speed of a jogger to the speed of a cyclist, given their respective distances covered and times taken.

We know the fundamental relationship between speed, distance, and time is:

\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

Cyclist's Speed

Let's denote the distance covered by the cyclist as \(D\) and the time taken by the cyclist as \(T\). The cyclist covers this distance at a constant speed.

The speed of the cyclist, \(S_c\), can be expressed as:

\(S_c = \frac{D}{T}\)

Jogger's Speed

According to the problem, the jogger covers half the distance covered by the cyclist in double the time taken by the cyclist.

• Distance covered by the jogger = \(\frac{D}{2}\)
• Time taken by the jogger = \(2T\)

The speed of the jogger, \(S_j\), can be expressed as:

\(S_j = \frac{\text{Distance covered by jogger}}{\text{Time taken by jogger}}\)

Substituting the values:

\(S_j = \frac{\frac{D}{2}}{2T}\)

Simplifying the expression for \(S_j\):

\(S_j = \frac{D}{2 \times 2T}\)

\(S_j = \frac{D}{4T}\)

Calculating the Ratio of Speeds

We need to find the ratio of the speed of the jogger to that of the cyclist, which is \(S_j : S_c\) or \(\frac{S_j}{S_c}\).

Using the expressions we found for \(S_j\) and \(S_c\):

\(\frac{S_j}{S_c} = \frac{\frac{D}{4T}}{\frac{D}{T}}\)

To simplify this complex fraction, we can multiply the numerator by the reciprocal of the denominator:

\(\frac{S_j}{S_c} = \frac{D}{4T} \times \frac{T}{D}\)

We can cancel out the common terms \(D\) and \(T\) (assuming \(D \neq 0\) and \(T \neq 0\), which must be true for distance and time):

\(\frac{S_j}{S_c} = \frac{\cancel{D}}{4\cancel{T}} \times \frac{\cancel{T}}{\cancel{D}}\)

\(\frac{S_j}{S_c} = \frac{1}{4}\)

So, the ratio of the speed of the jogger to that of the cyclist is \(1:4\).

Comparing this result with the given options:

• Option 1: 1 : 4
• Option 2: 4 : 1
• Option 3: 1 : 2
• Option 4: 2 : 1

The calculated ratio \(1:4\) matches Option 1.