Twelve workers pack boxes at a constant rate of 60 boxes in 9 minutes.

Since 12 workers take 9 minutes to pack 60 boxes,
27 workers(which is \(\frac{9}{4}\) times the workers) take 9 minutes to pack \(\frac{9}{4}*60 = 135\) boxes.

Now, we know that 27 workers pack 135 boxes in 9 minutes.

In order to pack 180 boxes, they would need \(\frac{180}{135}*9 = \frac{4}{3}*9 = 12\) minutes.

Therefore, the workers(working at the same constant rate) take 12 minutes(Option A) to pack 180 boxes.

The rate of 12 workers is 60/9 = 20/3.

We can let n = the rate of 27 workers and create the proportion:

12/(20/3) = 27/n

36/20 = 27/n

9/5 = 27/n

9n = 5 x 27

n = 15

So it will take 27 workers 180/15 = 12 minutes to pack all the boxes.

Alternate Solution:

Since twelve workers pack boxes at a constant rate of 60 boxes in 9 minutes, it will take them 9 x 3 = 27 minutes to pack 180 boxes.

Since the number of minutes to pack 180 boxes will decrease as the number of workers increases, we can set up an inverse proportion to determine how much time it will take 27 workers to pack 180 boxes. Let t denote the time, in minutes, required for 27 workers to pack 180 boxes.

(12)(27) = (27)(t)

t = 12

Answer: A

Simple formula. An find answer for this problem

(Men1 *Time1)/work1 = (Men2 * Time2)/work2

As per question

12*9/60 = 27* time/180

Time=12

Give kudos if it helps

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Your approach is nice but I feel flipping the ratio on both sides will make it easy to understand the logic:-

work1/(Men1 *Time1) = work2/(Men2 * Time2)

i.e. rate of work is equal. Work done upon time taken in minutes (i.e. no. of boxes packed upon total time in minutes)

60 boxes packed/(12 men*9 minutes) = 180 boxes packed/(27 men* total time in minutes for work 2)

total time in minutes for work 2 = 12 minutes.

PS: For computing speed the formula is distance traveled upon time taken. We could compare reciprocals of constant speed to determine the time taken but why not compare the equate constant speed for earlier interpretation.

Let's Apply Work Rate Formula: \(W=RT\)

Current Rate\(= \frac{60}{12*9}\)

Now equating with the expecting: \(\frac{60}{12*9}=\frac{180}{27*T}\) [We are working to figure out the time for 180 boxes by 27 workers]

\(T=12\)

The answer is\( A.\)

We can solve this using proportion / variation as well. By solving using variation, we can solve without deeper thinking and cross multiplication.

The given quantities are: 12 Workers pack 60 Boxes in 9 Minutes.
We need to find Time taken by 27 Workers to pack 180 Boxes?

Think about each quantity one at a time vis-à-vis the value we need to find. Here we need to find the Time.

So the thinking should be:
Workers: Initially, there were 12 workers, and in the second situation, there are 27 workers. More workers means less time. So the fraction for Workers should be 12/27.
Boxes: Initially, there were 60 boxes, and in the second situation, there are 180 boxes. More boxes, means more time. So the fraction for the Boxes should be 180/60.
Time: Initially, the Time is 9 minutes, and for the second situation, we need to find the time. So for time, we should just use 9 minutes, as numerator.

Now multiply the above three fractions derived. Time = 9 * 12/27 * 180/60 = 12 Minutes (Option A).

This method should work in any such problems be it any quantity.

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