Understanding the Relationship Between Speed, Time, and Distance

The fundamental relationship is given by the formula: Distance = Speed × Time.

For a constant distance, speed and time are inversely proportional. This means if the speed increases, the time taken decreases, and vice versa. Mathematically, if Distance ($D$) is constant, then $D = S \times T$, which implies $T = D/S$. Therefore, the ratio of times is the reciprocal of the ratio of speeds.

Speed Ratio Calculation

Let the speeds of the three cars be $S_1, S_2, S_3$. The given ratio of speeds is:

$S_1 : S_2 : S_3 = 2 : 3 : 4$

Time Ratio Derivation

Since the distance traveled by all cars is the same, the ratio of the time taken ($T_1 : T_2 : T_3$) is the reciprocal of the ratio of their speeds.

Therefore, the ratio of times is:

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

$T_1 : T_2 : T_3 = \frac{1}{2} : \frac{1}{3} : \frac{1}{4}$

Simplifying the Time Ratio

To simplify this ratio, we find the least common multiple (LCM) of the denominators (2, 3, and 4). The LCM is 12.

We multiply each term in the ratio by the LCM:

• $T_1 = \frac{1}{2} \times 12 = 6$
• $T_2 = \frac{1}{3} \times 12 = 4$
• $T_3 = \frac{1}{4} \times 12 = 3$

So, the ratio of the time taken is $6 : 4 : 3$.