Question

Three wheels can complete 25, 21 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?

• A. \(58 \text{ seconds} \)
• B. \(60 \text{ seconds} \)
• C. \(62 \text{ seconds} \)
• D. \(59 \text{ seconds} \)
• E. \(61 \text{ seconds} \)

Hint:

For problems involving rotations: - Convert speeds to time per revolution - Then take LCM to find when events coincide

Answer 1

The Correct Option is B

Concept: Each red spot touches the ground once per revolution. So, the time taken for one full revolution is: \[ \text{Time per revolution} = \frac{60}{\text{revolutions per minute}} \text{ seconds} \] The required time is the LCM of these individual times.
Step 1: Find time for one revolution of each wheel.
\[ \text{Wheel 1: } \frac{60}{25} = \frac{12}{5} \text{ seconds} \] \[ \text{Wheel 2: } \frac{60}{21} = \frac{20}{7} \text{ seconds} \] \[ \text{Wheel 3: } \frac{60}{16} = \frac{15}{4} \text{ seconds} \]

Step 2: Find LCM of the fractions.
LCM of fractions \(=\) \[ \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \] \[ \text{LCM of } (12, 20, 15) = 60 \] \[ \text{HCF of } (5, 7, 4) = 1 \] \[ \text{Required time} = \frac{60}{1} = 60 \text{ seconds} \]