One gear of pulley rotates at a speed of 3 rev/sec another gear roates at 5 rev/sec. if both start together after how many seconds will they be together againa) 3 secb) 5secc) 15 secd) 20 sec

To determine after how many seconds both gears will be together again, we need to calculate the least common multiple (LCM) of the time periods of the two gears. 

1. **Step 1: Find the period of each gear.**
   - Gear 1 rotates at \(3 \text{ rev/sec}\), so its period is \(T_1 = \frac{1}{3} \text{ seconds per revolution}\).
   - Gear 2 rotates at \(5 \text{ rev/sec}\), so its period is \(T_2 = \frac{1}{5} \text{ seconds per revolution}\).

2. **Step 2: Find when they will be together again.**
   The gears will be together again when they both complete an integer number of revolutions, meaning we need to find the LCM of their time periods, which is the smallest time after which both gears align.

   To do this, we can use the fact that the LCM of the periods is found by the LCM of the denominators of their speeds (3 and 5). The LCM of 3 and 5 is:

   \[
   \text{LCM}(3, 5) = 15
   \]

   Therefore, the gears will align again after **15 seconds**.

### Answer: (c) 15 seconds

Would you like more details on any step, or do you have any questions?

Here are 5 related questions for further practice:
1. What is the LCM of 4 and 6, and how does it relate to gear rotations?
2. If one gear rotates at 2 rev/sec and another at 7 rev/sec, after how many seconds will they be together again?
3. How does the concept of greatest common divisor (GCD) relate to rotations of gears?
4. What happens if two gears have the same rotational speed?
5. How would the situation change if the two gears started at different times?

**Tip:** The least common multiple (LCM) is useful when you want to synchronize repeating events or cycles.

One gear of pulley rotates at a speed of 3 rev/sec, another gear rotates at 5 rev/sec. If both start together, after how many seconds will they be together again?
a) 3 sec
b) 5 sec
c) 15 sec
d) 20 sec

Learn how to calculate the least common multiple (LCM) of two gears rotating at 3 and 5 rev/sec. This problem covers key concepts like rotational motion and synchronization, providing a step-by-step solution to find when both gears will be together again. Ideal for students in Grades 6-8.

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