Arithmetic

Time & Work

Medium

When they work alone, $B$ needs $25 \%$ more time to finish a job than $A$ does. They two finish the job in $13$ days in the following manner: $A$ works alone till half the job is done, then $A$ and $B$ work together for four days, and finally $B$ works alone to complete the remaining $5\%$ of the job. In how many days can $B$ alone finish the entire job?

$16$

$20$

$18$

$22$

✅ Correct Option: 2

Let's break down what happens in this 13-day work sequence:

A works alone until half the job is done

A and B work together for 4 days

B works alone to complete the remaining 5% of the job

We need to find how long B takes to finish the entire job alone.

Let's say A can finish the entire job alone in a days.

Since B needs 25% more time than A:

B's time = A's time + 25% of A's time

B's time = a + 0.25a = 1.25a = $\dfrac{5a}{4}$ days

Work Rates (portion of job completed per day):

A's rate = $\dfrac{1}{a}$ jobs per day

B's rate = $\dfrac{1}{\dfrac{5a}{4}} = \dfrac{4}{5a}$ jobs per day

Here's the key insight: What portion of the job do A and B complete together?

A completes: 50% of the job alone

B completes: 5% of the job alone

A and B together complete: 100% - 50% - 5% = 45% of the job

So A and B working together for 4 days complete $\dfrac{45}{100} = \dfrac{9}{20}$ of the job.

When A and B work together for 4 days:

Combined work rate × Time = Work completed

$4 \times \left(\dfrac{1}{a} + \dfrac{4}{5a}\right) = \dfrac{9}{20}$

Let's simplify the left side:

$4 \times \left(\dfrac{1}{a} + \dfrac{4}{5a}\right) = 4 \times \left(\dfrac{5}{5a} + \dfrac{4}{5a}\right)$

$= 4 \times \dfrac{9}{5a} = \dfrac{36}{5a}$

So our equation becomes:

$\dfrac{36}{5a} = \dfrac{9}{20}$

Cross-multiplying:

$36 \times 20 = 9 \times 5a$

$720 = 45a$

$a = \dfrac{720}{45} = 16$

A takes 16 days to complete the job alone.

B takes $\dfrac{5a}{4} = \dfrac{5 \times 16}{4} = \dfrac{80}{4} = 20$ days to complete the job alone.

B alone can finish the entire job in 20 days.

In work and time problems, always identify what portion of work each person/combination completes. This helps you set up the correct equations without getting confused by the different working arrangements.

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