Question

Two cogged wheels of which one has 32 cogs and other 54 cogs, work into each other. If the latter turns 80 times in three quarters of a minute, how often does the other turn in 8 seconds?
$
  (a){\text{ 48}} \\
  (b){\text{ 135}} \\
  (c){\text{ 24}} \\
  (d){\text{ 36}} \\
 $

Answer 1

Hint: Three quarter of a minute means we are talking about $\dfrac{3}{4}$ of a minute and a minute has 60 seconds so the time in which the wheel with 32 cogs has a revolution of 80 turns can be calculated. Use a unitary method along these concepts to get the answer.

Complete step-by-step answer:
There are two cogged wheels one has 32 cogs and the other has 54 cogs.
Now it is given that the latter turns 80 times in three quarters of a minute.
As we know in a minute there are 60 seconds.
Therefore in three quarters of a minute = $\dfrac{3}{4} \times 60 = 45$ seconds.
Therefore the wheel which has 32 cogs revolves 80 times in 45 seconds.
Therefore the wheel which has 1 cog revolves $\dfrac{{80}}{{32}}$ times in 45 seconds.
Therefore the wheel which has 1 cog revolves $\dfrac{{80}}{{32 \times 45}}$ times in 1 seconds.
Therefore the wheel which has 54 cogs revolves $\dfrac{{80 \times 54}}{{32 \times 45}}$ times in 1 seconds.
Therefore the wheel which has 54 cogs revolves $\dfrac{{80 \times 54 \times 8}}{{32 \times 45}}$ times in 8 seconds.
So on simplifying we have,
$ \Rightarrow \dfrac{{80 \times 54 \times 8}}{{32 \times 45}} = \dfrac{{20 \times 54}}{{45}} = \dfrac{{20 \times 6}}{5} = 4 \times 6 = 24$
Therefore the wheel which has 54 cogs revolves 24 times in 8 seconds.
So this is the required answer.
Hence option (C) is correct.

Note: Whenever we face such types of problems the only thing which is the key is the unitary method, unitary method involves breaking down the entity to unity so that any other value can be evaluated. This helps to get on the right track to get the answer.