The traffic lights at three different road crossings change after every 36 sec, 60 sec and 108 sec respectively. If they all change simultaneously at 7 : 20 : 00 hours, then find the time at which they will change simultaneously.
Question
The traffic lights at three different road crossings change after every 36 sec, 60 sec and 108 sec respectively. If they all change simultaneously at 7 : 20 : 00 hours, then find the time at which they will change simultaneously.
Solution
To solve this problem, we need to find the least common multiple (LCM) of the three time intervals (36 sec, 60 sec, and 108 sec). The LCM of these numbers will give us the time in seconds after which all three traffic lights will change simultaneously.
Step 1: Prime factorization of the three numbers
- 36 = 2^2 * 3^2
- 60 = 2^2 * 3 * 5
- 108 = 2^2 * 3^3
Step 2: Find the LCM The LCM is found by multiplying the highest power of all prime numbers obtained from the three numbers. So, LCM = 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540 seconds
Step 3: Convert the LCM from seconds to time format 540 seconds is equal to 9 minutes.
Step 4: Add the LCM to the initial time If the lights all changed simultaneously at 7:20:00, adding 9 minutes to this time gives us 7:29:00.
So, the three traffic lights will all change simultaneously at 7:29:00.
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