Here,

We know Work = No. of individual * Rate * Time

so 60 = 12 * Rate * 9

or Rate =\(\frac{60}{(12 * 9)}\) = \(\frac{5}{9}\)

Now for 180 boxes ;

Work = No. of individual * Rate * Time

\(180 = \frac{5}{9} * 27 * Time\)

or Time = \(\frac{(180 * 9)}{(5 * 27)}\) = 12
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Explanation

To solve a Rates & Work problem with multiple workers, modify the standard formula Work = Rate × Time to this:

Work = Individual Rate × Number of Workers × Time

Use the first sentence to solve for an individual worker’s rate. Plug in the fact that 12 workers pack boxes at a constant rate of 60 boxes in 9 minutes:
Work = Individual Rate × Number of Workers × Time

60 boxes = (R)(12)(9 minutes)

\(R =\frac{5}{9}\) boxes per minute.


In other words, each worker can pack \(\frac{5}{9}\) of a box per minute. Plug that rate back into the formula, but use the details from the second sentence in the problem:

Work = Individual Rate × Number of Workers × Time

\(180 = \frac{5}{9} \times 27 \times T\)

\(180=15T\)

\(T=12\).
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Use the following equation:

\(\frac{workers * time}{output} = \frac{workers*time}{output}\)

In the equation above:
Workers and time are INVERSELY PROPORTIONAL.
As the number of workers increases, the amount of time required to produce the same output decreases.
Workers and output are DIRECTLY PROPORTIONAL.
As the number of workers increases, the amount of output also increases.
Time and output are also DIRECTLY PROPORTIONAL.
As the amount of time increases, the amount of output also increases.

In the problem above:
Since 12 workers take 9 minutes to pack 60 boxes, and we want to know the time required for 27 workers to pack 180 boxes, we get:
\(\frac{12*9}{60} = \frac{27t}{180}\)

\(\frac{9}{5} = \frac{3t}{20}\)

\(15t = 180\)

\(t = 12\)


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hard type question
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