A circular field has a circumference of 360km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?
Given: A circular field has a circumference of 360km. Three cyclists start together and can cycle 48, 60 and 72 km a day.
To find: Here we have to find at what time they will meet again.
Solution:
Time taken by 1st cyclist to complete one round $=\ \frac{360}{48}\ =\ 7.5 days\ =\ 180 hours$
Time taken by 2nd cyclist to complete one round $=\ \frac{360}{60}\ =\ 6 days\ =\ 144 hours$
Time taken by 3rd cyclist to complete one round $=\ \frac{360}{72}\ =\ 5 days\ =\ 120 hours$
Now, we just have to find the LCM of 180, 144 and 120.
Writing the numbers as a product of their prime factors:
Prime factorisation of 180:
- $2\ \times\ 2\ \times\ 3\ \times\ 3\ \times\ 5\ =\ 2^2\ \times\ 3^2\ \times\ 5^1$
Prime factorisation of 144:
- $2\ \times\ 2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ 3\ =\ 2^4\ \times\ 3^2$
Prime factorisation of 120:
- $2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ 5\ =\ 2^3\ \times\ 3^1\ \times\ 5^1$
Multiplying the highest power of each prime number:
- $2^4\ \times\ 3^2\ \times\ 5^1\ =\ 720$
LCM(180, 144, 120) $=$ 720
This means that all three cyclists will meet again after 720 hours.
720 hours $=$ 30 days
So, all three cyclists will meet again after 30 days.
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