Anil alone can do a job in $20$ days while Sunil alone can do it in $40$ days. Anil starts the job, and after $3$ days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done $10\%$ of the job, then in how many days was the job done?

$15$

$12$

$13$

$14$

✅ Correct Option: 3

This is a classic work rate problem where different people join at different times. We need to track who worked for how long and use the fact that all work portions must add up to 100%.

When someone can complete a job in a certain number of days, their work rate is the fraction of job they complete per day.

Anil's rate: Can do job in 20 days → Does $\frac{1}{20}$ of job per day

Sunil's rate: Can do job in 40 days → Does $\frac{1}{40}$ of job per day

We say the total job takes $x$ days to complete.

Who worked for how long?

Anil: Started from day 1, worked all $x$ days

Sunil: Joined after 3 days, worked $(x-3)$ days

Bimal: We don't know when he joined, but he did 10% of the job

Since Bimal did 10% of the job, this means Anil and Sunil together did 90% of the job.

Work done = Work rate × Time worked

Anil's work: $\frac{1}{20} \times x = \frac{x}{20}$ of the job

Sunil's work: $\frac{1}{40} \times (x-3) = \frac{x-3}{40}$ of the job

Since Anil and Sunil together completed 90% of the job:

$\frac{x}{20} + \frac{x-3}{40} = \frac{9}{10}$

We multiply everything by 40 to clear the fractions:

$40 \times \frac{x}{20} + 40 \times \frac{x-3}{40} = 40 \times \frac{9}{10}$

$2x + (x-3) = 36$

$2x + x - 3 = 36$

$3x - 3 = 36$

$3x = 39$

$x = 13$

Therefore, the job was completed in 13 days.

In work rate problems, always remember that work done = rate × time, and all individual contributions must sum to 100% of the job.

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