The circumferences of the front and rear wheels of a bicycle are 3.5 m and 3 m respectively. If the vehicle is moving at a speed of 15 m/sec, the shortest time in which both the wheels will make a whole number of turns is

### Concept & Formula Both wheels will complete a whole number of turns when the total distance covered is a common multiple of their circumferences. The shortest distance is their Least Common Multiple (LCM). $$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$$ ### Step-by-Step Solution * **Given:** * Circumference of front wheel ($C_1$) = $3.5\text{ m} = \frac{7}{2}\text{ m}$ * Circumference of rear wheel ($C_2$) = $3\text{ m}$ * Speed ($S$) = $15\text{ m/sec}$ * **Find the LCM of the circumferences:** * To find the LCM of a fraction and an integer: $\text{LCM}\left(\frac{7}{2}, \frac{3}{1}\right) = \frac{\text{LCM of numerators (7, 3)}}{\text{HCF of denominators (2, 1)}}$ * $\text{LCM}(7, 3) = 21$ * $\text{HCF}(2, 1) = 1$ * $\text{Shortest distance} = \frac{21}{1} = 21\text{ m}$ * **Calculate the time taken:** $$\text{Time} = \frac{21\text{ m}}{15\text{ m/sec}} = \frac{7}{5}\text{ seconds}$$ $$\frac{7}{5} = 1.4\text{ seconds}$$ ### Exam Strategy & Shortcut Multiply by 10 to work with integers: LCM of 35 and 30 is 210. Divide by 10 to revert to meters: 21m. Time $= \frac{21}{15} = 1.4\text{ sec}$. This avoids fraction rules entirely. ### Common Pitfall Miscalculating the LCM of a decimal number. Always convert decimals to fractions or scale them up to integers before taking the LCM. ### Final Answer Therefore, the correct answer is **1.4 seconds**.