Three wheels can complete 40, 24 and 16 revolutions per minute respectively. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again?

### Concept & Strategy To find when multiple repeating cyclical events synchronize again, we must calculate the Least Common Multiple (LCM) of their individual time periods. First, we must convert the given speeds (revolutions per minute) into the time taken for a *single* revolution. $$\text{Time per revolution} = \frac{\text{Total Time}}{\text{Number of Revolutions}}$$ For fractions, the LCM is calculated as: $$\text{LCM of fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$ ### Step-by-Step Solution **Given:** * Speed of Wheel 1 = 40 rev/min * Speed of Wheel 2 = 24 rev/min * Speed of Wheel 3 = 16 rev/min **Calculation:** 1. Find the time taken for one complete revolution for each wheel in seconds (since options are primarily in seconds and minutes, seconds is a better base unit): * Wheel 1: $60 \text{ seconds} / 40 \text{ rev} = \frac{60}{40} = \frac{3}{2} \text{ seconds}$ * Wheel 2: $60 \text{ seconds} / 24 \text{ rev} = \frac{60}{24} = \frac{5}{2} \text{ seconds}$ * Wheel 3: $60 \text{ seconds} / 16 \text{ rev} = \frac{60}{16} = \frac{15}{4} \text{ seconds}$ 2. Calculate the LCM of these fractional time periods ($\frac{3}{2}$, $\frac{5}{2}$, $\frac{15}{4}$): * Find the LCM of the numerators (3, 5, 15): The lowest common multiple of 3, 5, and 15 is $15$. * Find the HCF of the denominators (2, 2, 4): The highest common factor of 2, 2, and 4 is $2$. 3. Apply the fraction formula: $$\text{Simultaneous Time} = \frac{15}{2} \text{ seconds}$$ 4. Convert to a mixed fraction to match the options: $$\frac{15}{2} \text{ seconds} = 7.5 \text{ seconds} = 7\frac{1}{2} \text{ sec}$$ ### Exam Strategy & Shortcut Always quickly convert "rate" to "time" before finding the LCM. Taking the LCM of 40, 24, and 16 gives 240, which would incorrectly imply minutes or revolutions. By converting to seconds first ($\frac{3}{2}, \frac{5}{2}, \frac{15}{4}$), you immediately see the numerator must be a multiple of 15, guiding you directly to $15/2$. ### Common Pitfall The most dangerous pitfall is finding the LCM of the given numbers (40, 24, 16) directly. This gives the number of *revolutions* until they align, not the *time*. Always ensure you are finding the LCM of the actual time intervals between the events. ### Final Answer **Therefore, the correct answer is $7\frac{1}{2}$ sec.**