Arithmetic

Time & Work

Medium

Gautam and Suhani, working together, can finish a job in $20$ days. If Gautam does only $60 \%$ of his usual work on a day, Suhani must do $150 \%$ of her usual work on that day to exactly make up for it. Then, the number of days required by the faster worker to complete the job working alone is

Entered answer:

✅ Correct Answer: 36

We have two workers, Gautam and Suhani, and we need to find how long the faster worker takes to complete the job alone.

Key Information:

Together: 20 days to complete the job

Special condition: If Gautam works at 60% efficiency, Suhani must work at 150% efficiency to maintain the same daily output

Work rate = Amount of work completed per day

If someone completes a job in 'd' days, their work rate = $\tfrac{1}{d}$ (fraction of job completed per day)

Let's define:

G = Gautam's normal work rate (fraction of job he completes per day)

S = Suhani's normal work rate (fraction of job she completes per day)

Since they complete the job together in 20 days:

Their combined daily work rate = $\tfrac{1}{20}$

Therefore: $G + S = \tfrac{1}{20}$ ... (1)

The special condition tells us something crucial:

Gautam at 60% efficiency = 0.6G

Suhani at 150% efficiency = 1.5S

This combination produces the same daily output as their normal combined work

Therefore: $0.6G + 1.5S = G + S$ ... (2)

This is true because the problem states this modified work pattern "exactly makes up for" the reduction in Gautam's work, meaning the total daily output remains the same.

From equation (2):

$0.6G + 1.5S = G + S$

$1.5S - S = G - 0.6G$

$0.5S = 0.4G$

$5S = 4G$

Therefore: $G : S = 5 : 4$

This means Gautam is more efficient than Suhani.

Since $G : S = 5 : 4$ , we can write:

$G = 5k$ (for some constant k)

$S = 4k$

From equation (1): $G + S = \tfrac{1}{20}$

$5k + 4k = \tfrac{1}{20}$

$9k = \tfrac{1}{20}$

$k = \tfrac{1}{180}$

Therefore:

$G = \tfrac{5}{180} = \tfrac{1}{36}$

$S = \tfrac{4}{180} = \tfrac{1}{45}$

Work rate = $\tfrac{1}{36}$ means Gautam takes 36 days alone

Work rate = $\tfrac{1}{45}$ means Suhani takes 45 days alone

Since 36 < 45, Gautam is the faster worker.

Answer: 36 days

In work rate problems, we remember:

Work rate = $\tfrac{1}{\text{time taken alone}}$
Combined work rate = sum of individual work rates
The person with higher work rate is faster (takes less time)

This problem cleverly uses a compensation scenario to help us find the efficiency ratio between the two workers.

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