The speeds of three cars are in the ratio 5 : 4 : 6. The ratio between the time taken by them to travel the same distance is

### Concept & Logic Time and speed are inversely proportional when the distance is constant. $$ \text{Time} \propto \frac{1}{\text{Speed}} $$ ### Step-by-Step Solution 1. The given ratio of speeds of the three cars is $5 : 4 : 6$. 2. Because the distance is the same, the ratio of the time taken will be the inverse of their speeds: $$ \text{Time Ratio} = \frac{1}{5} : \frac{1}{4} : \frac{1}{6} $$ 3. To simplify this ratio into whole numbers, find the Least Common Multiple (LCM) of the denominators (5, 4, 6). LCM(5, 4, 6) = 60. 4. Multiply each term of the inverse ratio by 60: $$ \left(\frac{1}{5} \times 60\right) : \left(\frac{1}{4} \times 60\right) : \left(\frac{1}{6} \times 60\right) $$ 5. Simplify to get the final integer ratio: $$ 12 : 15 : 10 $$ ### Exam Strategy & Shortcut To find the inverse ratio of $a : b : c$, simply hide one number and multiply the other two. For the first term (hide 5): $4 \times 6 = 24$ For the second term (hide 4): $5 \times 6 = 30$ For the third term (hide 6): $5 \times 4 = 20$ Ratio: $24 : 30 : 20$. Divide by 2 to get $12 : 15 : 10$. ### Common Pitfall A frequent error is simply reversing the order of the speeds to get the time ratio (e.g., $6 : 4 : 5$). This only works for two variables, not three. ### Final Answer Therefore, the correct answer is **12 : 15 : 10**.