तीन ट्रेफ़िक सिग्नल हैं । प्रत्येक सिग्नल का रंग हरे से लाल और फिर लाल से हरा बदलता है । हरे से लाल रंग बदलने में पहले सिग्नल को 25 सेकंड, दूसरे सिग्नल को 39 सेकंड और तीसरे सिग्नल को 60 सेकंड लगते हैं । हरे व लाल रंगों की अवधियाँ समान हैं । 2:00 बजे अपराह्न को, वे एक साथ हरे हो जाते हैं । अगली बार किस समय पर वे एक साथ हरे होंगे ?
सही उत्तर (B) 4:10 p.m. है।
व्याख्या:
We are given three traffic signals with durations for green and red being equal. - First signal changes colour from green to red in 25 seconds. Since green and red durations are equal, a complete cycle (green + red) takes 25 + 25 = 50 seconds. (Wait, let's re-read: 'The first signal takes 25 seconds... to change the colour from green to red. The durations for green and red colours are same.' This means green lasts 25 seconds, and red lasts 25 seconds. So a full cycle of green-to-red and red-to-green takes 25 + 25 = 50 seconds? Wait! Let's read carefully: 'The first signal takes 25 seconds... to change the colour from green to red.' Does it mean green duration is 25 seconds? Yes, green takes 25 seconds, then red takes 25 seconds, total cycle = 50 seconds. Similarly, second signal takes 39 seconds for green, so total cycle = 39 + 39 = 78 seconds. Third signal takes 60 seconds for green, so total cycle = 60 + 60 = 120 seconds.) Wait, when do they turn green simultaneously? They turn green together at the start of their green cycles. The time interval between successive simultaneous green occurrences is the Least Common Multiple (LCM) of the durations of their green phases (or cycle lengths). Let's check the green phase durations: 25 seconds, 39 seconds, 60 seconds. Let us find the LCM of 25, 39, and 60: - 25 = 5^2 - 39 = 3 × 13 - 60 = 2^2 × 3 × 5 LCM(25, 39, 60) = 2^2 × 3^1 × 5^2 × 13^1 = 4 × 3 × 25 × 13 = 300 × 13 = 3900 seconds. So all three signals will turn green together again after 3900 seconds. Now, let us convert 3900 seconds into hours and minutes: 3900 / 60 = 65 minutes. 65 minutes = 1 hour and 5 minutes? Wait! Let us re-verify the cycle or LCM: If they turn green together at 2:00 p.m., and the interval is 65 minutes (1 hour 5 minutes), then the next time would be 3:05 p.m. But 3:05 p.m. is not in the options! The options are: (a) 4:00 p.m. (b) 4:10 p.m. (c) 4:20 p.m. (d) 4:30 p.m. Let us re-calculate the LCM or cycle: Does 'takes 25 seconds to change colour from green to red' mean the green phase is 25s, red is 25s, so cycle is 50s? Wait, what if the LCM is of 50, 78, and 120? Let's find LCM of 50, 78, 120: - 50 = 2 × 5^2 - 78 = 2 × 3 × 13 - 120 = 2^3 × 3 × 5 LCM(50, 78, 120) = 2^3 × 3^1 × 5^2 × 13^1 = 8 × 3 × 25 × 13 = 7800 seconds. 7800 seconds / 60 = 130 minutes. 130 minutes = 2 hours and 10 minutes. Adding 2 hours and 10 minutes to 2:00 p.m.: 2:00 p.m. + 2 hours 10 minutes = 4:10 p.m.! This matches option (b)! Let us verify: 7800 seconds is 130 minutes, which is 2 hours 10 minutes. Adding to 2:00 p.m. gives 4:10 p.m. Thus, option (b) is correct.
In English (Question & Model Answer)
There are three traffic signals. Each signal changes colour from green to red and then from red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colours are same. At 2:00 p.m., they together turn green. At what time will they change to green next, simultaneously?
The correct answer is (B) 4:10 p.m..
Explanation:
We are given three traffic signals with durations for green and red being equal. - First signal changes colour from green to red in 25 seconds. Since green and red durations are equal, a complete cycle (green + red) takes 25 + 25 = 50 seconds. (Wait, let's re-read: 'The first signal takes 25 seconds... to change the colour from green to red. The durations for green and red colours are same.' This means green lasts 25 seconds, and red lasts 25 seconds. So a full cycle of green-to-red and red-to-green takes 25 + 25 = 50 seconds? Wait! Let's read carefully: 'The first signal takes 25 seconds... to change the colour from green to red.' Does it mean green duration is 25 seconds? Yes, green takes 25 seconds, then red takes 25 seconds, total cycle = 50 seconds. Similarly, second signal takes 39 seconds for green, so total cycle = 39 + 39 = 78 seconds. Third signal takes 60 seconds for green, so total cycle = 60 + 60 = 120 seconds.) Wait, when do they turn green simultaneously? They turn green together at the start of their green cycles. The time interval between successive simultaneous green occurrences is the Least Common Multiple (LCM) of the durations of their green phases (or cycle lengths). Let's check the green phase durations: 25 seconds, 39 seconds, 60 seconds. Let us find the LCM of 25, 39, and 60: - 25 = 5^2 - 39 = 3 × 13 - 60 = 2^2 × 3 × 5 LCM(25, 39, 60) = 2^2 × 3^1 × 5^2 × 13^1 = 4 × 3 × 25 × 13 = 300 × 13 = 3900 seconds. So all three signals will turn green together again after 3900 seconds. Now, let us convert 3900 seconds into hours and minutes: 3900 / 60 = 65 minutes. 65 minutes = 1 hour and 5 minutes? Wait! Let us re-verify the cycle or LCM: If they turn green together at 2:00 p.m., and the interval is 65 minutes (1 hour 5 minutes), then the next time would be 3:05 p.m. But 3:05 p.m. is not in the options! The options are: (a) 4:00 p.m. (b) 4:10 p.m. (c) 4:20 p.m. (d) 4:30 p.m. Let us re-calculate the LCM or cycle: Does 'takes 25 seconds to change colour from green to red' mean the green phase is 25s, red is 25s, so cycle is 50s? Wait, what if the LCM is of 50, 78, and 120? Let's find LCM of 50, 78, 120: - 50 = 2 × 5^2 - 78 = 2 × 3 × 13 - 120 = 2^3 × 3 × 5 LCM(50, 78, 120) = 2^3 × 3^1 × 5^2 × 13^1 = 8 × 3 × 25 × 13 = 7800 seconds. 7800 seconds / 60 = 130 minutes. 130 minutes = 2 hours and 10 minutes. Adding 2 hours and 10 minutes to 2:00 p.m.: 2:00 p.m. + 2 hours 10 minutes = 4:10 p.m.! This matches option (b)! Let us verify: 7800 seconds is 130 minutes, which is 2 hours 10 minutes. Adding to 2:00 p.m. gives 4:10 p.m. Thus, option (b) is correct.