Arithmetic

Time & Work

Medium

Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take thirty days to finish the job, whereas five humans and fifteen robots working together take sixty days to finish it. How many days will fifteen humans working together (without any robot) take to finish it?

$45$

$36$

$40$

$32$

✅ Correct Option: 4

We have two different work scenarios and need to find how long 15 humans alone would take.

In work problems, we think in terms of "rate of work" - how much of the job gets completed per day.

Let's define:

H = work rate of one human (fraction of job completed per day)

R = work rate of one robot (fraction of job completed per day)

When multiple workers work together, their rates add up. So 15 humans working together complete 15H of the job per day.

Scenario 1: 15 humans + 5 robots finish the job in 30 days

Combined rate per day = 15H + 5R

Since they complete the entire job in 30 days, their daily rate = $\frac{1}{30}$ of the job

Equation 1: $15H + 5R = \frac{1}{30}$

Scenario 2: 5 humans + 15 robots finish the job in 60 days

Combined rate per day = 5H + 15R

Since they complete the entire job in 60 days, their daily rate = $\frac{1}{60}$ of the job

Equation 2: $5H + 15R = \frac{1}{60}$

We have:

$15H + 5R = \frac{1}{30}$ ... (1)

$5H + 15R = \frac{1}{60}$ ... (2)

Multiply equation (1) by 3:

$3(15H + 5R) = 3(\frac{1}{30})$

$45H + 15R = \frac{1}{10}$ ... (3)

Subtract equation (2) from equation (3):

$(45H + 15R) - (5H + 15R) = \frac{1}{10} - \frac{1}{60}$

$45H + 15R - 5H - 15R = 40H$

$\frac{1}{10} - \frac{1}{60} = \frac{6}{60} - \frac{1}{60} = \frac{5}{60} = \frac{1}{12}$

$40H = \frac{1}{12}$

$H = \frac{1}{12} ÷ 40 = \frac{1}{480}$

Now we know one human completes $\frac{1}{480}$ of the job per day.

For 15 humans working together:

Combined rate = $15 × \frac{1}{480} = \frac{15}{480} = \frac{1}{32}$ of the job per day

Time to complete the entire job:

If they complete $\frac{1}{32}$ of the job per day, they need 32 days to complete the whole job

15 humans working together will take 32 days to finish the job.

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