Question

Four people clap after every 20 minutes, 30 minutes, 40 minutes and 50 minutes respectively. All of them clapped together at 10:00 am. Then they will again clap together at ______

• A. 3 pm
• B. 5 pm
• C. 6 pm
• D. 8 pm

Hint:

Find the LCM of the four time gaps and convert it to hours.

Answer 1

The Correct Option is D

To determine the next time all four people will clap together, we need to find the Least Common Multiple (LCM) of the clapping intervals. The intervals given are 20 minutes, 30 minutes, 40 minutes, and 50 minutes. 

The concept of LCM is used to find the first time all periodic events will align again. Here's how to compute the LCM step-by-step:

1. Prime factorize each interval:
   • \(20 = 2^2 \times 5\)
   • \(30 = 2 \times 3 \times 5\)
   • \(40 = 2^3 \times 5\)
   • \(50 = 2 \times 5^2\)
2. Identify the highest power of each prime:
   • \(2^3\) (from 40)
   • \(3\) (from 30)
   • \(5^2\) (from 50)
3. Compute the LCM by multiplying these values: \(\text{LCM} = 2^3 \times 3 \times 5^2 = 8 \times 3 \times 25 = 600\) minutes.

Now, convert 600 minutes into hours:

• \(600 \text{ minutes} = 600 \div 60 = 10 \text{ hours}\)

Since they all clapped together at 10:00 am, adding 10 hours results in the next simultaneous clap occurring at 8:00 pm.

Therefore, the correct answer is: 8 pm.

Answer 2

Step 1: Find the LCM of the four clap intervals.
The four people clap every 20, 30, 40 and 50 minutes. Factorize: \(20 = 2^2 \times 5\), \(30 = 2 \times 3 \times 5\), \(40 = 2^3 \times 5\), \(50 = 2 \times 5^2\). The LCM is \(2^3 \times 3 \times 5^2 = 600\).

Step 2: Convert the LCM to hours.
600 minutes is \(600 / 60 = 10\) hours. So all four will clap together again after exactly 10 hours.

Step 3: Add this to the start time.
They first clapped together at 10:00 am. Adding 10 hours gives 8:00 pm.

Final Answer:
They will clap together again at 8 pm. \[ \boxed{8 \text{ pm}} \]