Speed Ratio to Time Ratio Calculation

The question asks for the ratio of time taken by three buses to travel the same distance, given the ratio of their speeds.

Inverse Proportion Relationship

For a constant distance ($D$), the relationship between speed ($S$) and time ($T$) is given by $T = \frac{D}{S}$. This means time is inversely proportional to speed ($T \propto \frac{1}{S}$).

If the speeds of the three buses are $S_1, S_2, S_3$, and the times they take to cover the same distance are $T_1, T_2, T_3$, then their time ratio is the inverse of their speed ratio:

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

Time Ratio Calculation Steps

The given speed ratio is:

$S_1 : S_2 : S_3 = 5 : 4 : 6$

To find the time ratio, we take the reciprocal of the speed ratio:

$T_1 : T_2 : T_3 = \frac{1}{5} : \frac{1}{4} : \frac{1}{6}$

To convert this ratio of fractions into a ratio of whole numbers, we find the Least Common Multiple (LCM) of the denominators (5, 4, 6).

• LCM(5, 4, 6) = 60.

Multiply each fraction by the LCM (60):

• $T_1 = \frac{1}{5} \times 60 = 12$
• $T_2 = \frac{1}{4} \times 60 = 15$
• $T_3 = \frac{1}{6} \times 60 = 10$

The resulting time ratio is $12 : 15 : 10$.

Resulting Time Ratio

The ratio between the time taken by the three buses to travel the same distance is $12 : 15 : 10$.