Understanding Speed and Time Relationship

When travelling a fixed or constant distance, the speed of travel and the time taken are inversely proportional to each other. This means that if the speed increases, the time taken decreases, and if the speed decreases, the time taken increases.

The relationship can be expressed as:

\(\text{Speed} \propto \frac{1}{\text{Time}}\)

Or equivalently:

\(\text{Time} \propto \frac{1}{\text{Speed}}\)

Calculating the Time Ratio from the Speed Ratio

We are given that the speeds of the three persons are in the ratio 5 : 4 : 3.

Let the speeds of the three persons be \(S_1\), \(S_2\), and \(S_3\). So, \(S_1 : S_2 : S_3 = 5 : 4 : 3\).

Let the time taken by the three persons to reach B be \(T_1\), \(T_2\), and \(T_3\). Since the distance is constant, the time taken is inversely proportional to the speed.

Therefore, the ratio of the times will be the inverse of the ratio of the speeds:

\(T_1 : T_2 : T_3 = \frac{1}{S_1} : \frac{1}{S_2} : \frac{1}{S_3}\)

\(T_1 : T_2 : T_3 = \frac{1}{5} : \frac{1}{4} : \frac{1}{3}\)

Converting the Fractional Ratio to a Whole Number Ratio

To express the ratio \(\frac{1}{5} : \frac{1}{4} : \frac{1}{3}\) as a ratio of whole numbers, we need to find the Least Common Multiple (LCM) of the denominators (5, 4, and 3).

• The denominators are 5, 4, and 3.
• The LCM of 5, 4, and 3 is 60.

Now, multiply each fraction in the ratio by the LCM (60):

• First person's time component: \(\frac{1}{5} \times 60 = 12\)
• Second person's time component: \(\frac{1}{4} \times 60 = 15\)
• Third person's time component: \(\frac{1}{3} \times 60 = 20\)

So, the ratio of the time taken is 12 : 15 : 20.

Thus, the time ratio to reach B will be 12 : 15 : 20.