Question

An air conditioner can cool the hall in $40$ minutes while another takes $45$ minutes to cool under similar conditions. If both air conditioners are switched on at the same time then how long will it take to cool the room approximately?
(A) $18$ minutes
(B) $19$ minutes
(C) $22$ minutes
(D) $24$ minutes

Answer 1

Hint: To solve the problem, air conditioners have capacity, which is the inverse of time taken by it to cool the hall, that means,
${\text{Capacity = }}\dfrac{1}{{{\text{Time taken to cool the hall}}}}$
So, when both the air conditioners are turned on at the same time, the combined capacity will be the addition of the capacity of both the air conditioners. So, the time will required will be the reciprocal of the combined capacity, that means,
${\text{Time required = }}\dfrac{1}{{{\text{Combined Capacity}}}}$

Complete answer: Let the first air conditioner be named A and the second air conditioner be named B.
Given, time taken by A to cool the hall, ${T_A} = 40$minutes
And, time taken by B to cool the hall ${T_B} = 45$minutes
Now, the capacity of A,${C_A} = \dfrac{1}{{{\text{Time taken by A to cool}}}} = \dfrac{1}{{{T_A}}}$
$ \Rightarrow {C_A} = \dfrac{1}{{40}}$
And, the capacity of$B$, ${C_B} = \dfrac{1}{{{\text{Time taken by B to cool}}}} = \dfrac{1}{{{T_B}}}$
$ \Rightarrow {C_B} = \dfrac{1}{{45}}$
Now, when the two air conditioners are switched on at the same instance, both $A$and $B$ together cool the room.
Therefore, the combined capacity of A and B, ${C_{A + B}} = {C_A} + {C_B} - - - (1)$
Now, substituting the values of ${C_A}$and ${C_B}$in equation $(1)$,
$ \Rightarrow {C_{A + B}} = \dfrac{1}{{40}} + \dfrac{1}{{45}}$
Taking LCM of the denominators, we get,
$ \Rightarrow {C_{A + B}} = \dfrac{{45 + 40}}{{40 \times 45}}$
Simplifying the fraction, we get,
$ \Rightarrow {C_{A + B}} = \dfrac{{85}}{{1800}}$
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow {C_{A + B}} = \dfrac{{17}}{{360}}$
Therefore, the time required to cool the room by both A and B, ${T_{A + B}} = \dfrac{1}{{{\text{Combined Capacity}}}} = \dfrac{1}{{{C_{A + B}}}}$
$ \Rightarrow {T_{A + B}} = \dfrac{1}{{\dfrac{{17}}{{360}}}}$
$ \Rightarrow {T_{A + B}} = \dfrac{{360}}{{17}}$
Simplifying the calculations, we get,
$ \Rightarrow {T_{A + B}} = 21.176$
$ \Rightarrow {T_{A + B}} \approx 22$ minutes (approximately)
Therefore, the combined time required by the two air conditioners is $22$ minutes.
Hence, the correct answer is option (C).

Note:
The other approach to this problem is by unitary method, in which we are to find how much each air conditioner cools in $1$ minute. Then we find how much the two air conditioners can cool in $1$ minute together. Then, finding the reciprocal of how much the two air conditioners cool together, we get the required time in which both the air conditioners cool the room. We must take care of calculations as the question involves tedious calculative steps.