Three wheels can completed respectively 60, 36, 24 revolutions per minute . 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 ?

Options

A. 5/2 seconds
B. 5/3 seconds
C. 5 seconds
D. 7.5 seconds

Explanation

To solve the problem of when all three red spots on the wheels will touch the ground again, we need to determine the time it takes for each wheel to complete one full revolution and then find the least common multiple (LCM) of these times.

### Step-by-Step Solution:

1. **Identify the Revolutions per Minute (RPM)**:
- Wheel A: 60 revolutions per minute (RPM)
- Wheel B: 36 revolutions per minute (RPM)
- Wheel C: 24 revolutions per minute (RPM)

2. **Convert RPM to Revolutions per Second**:
- For Wheel A:
\[ \text{Revolutions per second} = \frac{60 \text{ RPM}}{60} = 1 \text{ revolution/second} \]
- For Wheel B:
\[ \text{Revolutions per second} = \frac{36 \text{ RPM}}{60} = 0.6 \text{ revolutions/second} \]
- For Wheel C:
\[ \text{Revolutions per second} = \frac{24 \text{ RPM}}{60} = 0.4 \text{ revolutions/second} \]

3. **Calculate Time for One Revolution**:
- For Wheel A:
\[ \text{Time for one revolution} = \frac{1}{1} = 1 \text{ second} \]
- For Wheel B:
\[ \text{Time for one revolution} = \frac{1}{0.6} = \frac{5}{3} \text{ seconds} \]
- For Wheel C:
\[ \text{Time for one revolution} = \frac{1}{0.4} = \frac{5}{2} \text{ seconds} \]

4. **Find the Least Common Multiple (LCM)**:
- We need to find the LCM of the times for one revolution:
- \(1\) second
- \(\frac{5}{3}\) seconds
- \(\frac{5}{2}\) seconds

- To find the LCM, we first convert all times to a common fraction:
- \(1 = \frac{6}{6}\)
- \(\frac{5}{3} = \frac{10}{6}\)
- \(\frac{5}{2} = \frac{15}{6}\)

- The LCM of the numerators (6, 10, 15) can be calculated:
- The prime factorization is:
- \(6 = 2 \times 3\)
- \(10 = 2 \times 5\)
- \(15 = 3 \times 5\)
- The LCM is \(2^1 \times 3^1 \times 5^1 = 30\).

- Now, since we have the LCM of the numerators, we divide by the common denominator (6):
\[ \text{LCM} = \frac{30}{6} = 5 \text{ seconds} \]

5. **Conclusion**:
- Therefore, all three red spots will simultaneously touch the ground again after **5 seconds**.