Three wheels can complete 40, 24 and 16 revolutions per minutes respectively. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again ?

A. $${\text{7}}\frac{1}{2}$$ sec.

B. 18 sec.

C. $$7\frac{1}{2}$$ min.

D. 18 min.

Answer: Option A

Solution(By Examveda Team)

For one complete revolution, the first, second and third wheels take
$$\eqalign{ & \frac{{60}}{{40}},\frac{{60}}{{24}},\frac{{60}}{{16}}\sec . \cr & {\text{i}}{\text{.e}}{\text{.,}}\frac{3}{2}{\text{,}}\frac{5}{2}{\text{,}}\frac{{15}}{4}{\text{ sec}}{\text{.}} \cr & {\text{respectively}} \cr} $$
Time taken for all red spots to touch the ground again simultaneously.
$$\eqalign{ & {\text{ = }}\left( {{\text{LCM of }}\frac{3}{2}{\text{,}}\frac{5}{2}{\text{,}}\frac{{15}}{4}{\text{ }}} \right)\sec . \cr & = \left( {\frac{{{\text{LCM of 3,5,15}}}}{{{\text{HCF of 2,2,4}}}}} \right)\sec . \cr & = \frac{{15}}{2}\sec . \cr & = 7\frac{1}{2}\sec . \cr} $$

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