Question

A can do a work in 15 days. Both A and B together can do the same work in 6 days. Then B alone can do the same work in

• A. 14 days
• B. 12 days
• C. 8 days
• D. 10 days

Hint:

Shortcut formula for individual time when total joint time is given: \[ T_B = \frac{T_A \times T_{A+B}}{T_A - T_{A+B}} = \frac{15 \times 6}{15 - 6} = \frac{90}{9} = 10 days

Answer 1

The Correct Option is D

Concept: Work and time problems are efficiently solved using the concept of individual daily work efficiency rates. If a person completes an entire task in \(D\) days, their uniform work rate per single day is represented as \(\frac{1}{D}\) of the total work.

Step 1: Define individual work rates using the unit work approach.
Let the total work to be completed be denoted as 1 unit.
• Time taken by worker A alone = 15 days.
• Therefore, A's single-day work efficiency rate is: \[ W_A = \frac{1}{15} \]
• Time taken by both A and B working together = 6 days.
• Therefore, their combined single-day work efficiency rate is: \[ W_{A+B} = \frac{1}{6} \]

Step 2: Isolate the individual daily work rate of B.
The combined daily work is simply the sum of their individual daily work rates: \[ W_A + W_B = W_{A+B} \] Substitute our known values into this equation: \[ \frac{1}{15} + W_B = \frac{1}{6} \] Isolate \(W_B\) by subtracting \(\frac{1}{15}\) from both sides: \[ W_B = \frac{1}{6} - \frac{1}{15} \]

Step 3: Solve the fractional subtraction using a common denominator.
Find the Least Common Multiple (LCM) of the denominators 6 and 15:
• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 15: 15, 30, ...
• \(\text{LCM}(6, 15) = 30\) Convert both fractions to have the common denominator of 30: \[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \] \[ \frac{1}{15} = \frac{1 \times 2}{15 \times 2} = \frac{2}{30} \] Now subtract the numerators: \[ W_B = \frac{5}{30} - \frac{2}{30} = \frac{5 - 2}{30} = \frac{3}{30} \] Simplify the fraction to its lowest terms: \[ W_B = \frac{1}{10} \] This means worker B completes \(\frac{1}{10}\) of the total work profile in one day.

Step 4: Determine the total number of days required by B alone.
The number of days required is the reciprocal of the daily work efficiency rate: \[ \text{Days taken by B} = \frac{1}{W_B} = \frac{1}{\left(\frac{1}{10}\right)} = 10 \text{ days} \] Thus, worker B working entirely alone requires 10 days to finish the task. This matches Option (D).