Solving Work Efficiency Problem for A, B, and C

1. Calculate Individual Work Rates

Let the total amount of work be represented by units. We find the least common multiple (LCM) of the days given for combined work to establish a common unit base.

LCM of 54 and 56 is 1512 units. This is the total work.

• Rate of A, B, and C working together:

  $ \text{Rate}_{(A+B+C)} = \frac{1512 \text{ units}}{54 \text{ days}} = 28 \text{ units/day} $

• Rate of B and C working together:

  $ \text{Rate}_{(B+C)} = \frac{1512 \text{ units}}{56 \text{ days}} = 27 \text{ units/day} $

• Calculate Rate of A:

  $ \text{Rate}_{(A)} = \text{Rate}_{(A+B+C)} - \text{Rate}_{(B+C)} = 28 - 27 = 1 \text{ unit/day} $

2. Determine B's Work Rate using Efficiency

B is 75% more efficient than A. This means B's rate is A's rate plus 75% of A's rate.

• Efficiency calculation:

  $ \text{Rate}_{(B)} = \text{Rate}_{(A)} \times (1 + 0.75) = \text{Rate}_{(A)} \times 1.75 = \text{Rate}_{(A)} \times \frac{7}{4} $

• Substitute Rate of A:

  $ \text{Rate}_{(B)} = 1 \text{ unit/day} \times \frac{7}{4} = \frac{7}{4} \text{ units/day} $

3. Calculate Work Done and Remaining Work

A, B, and C work together for 52 days. Calculate the work completed during this period.

• Work done in 52 days:

  $ \text{Work done} = \text{Rate}_{(A+B+C)} \times \text{Days} = 28 \text{ units/day} \times 52 \text{ days} = 1456 \text{ units} $

• Calculate remaining work:

  $ \text{Remaining Work} = \text{Total Work} - \text{Work done} = 1512 \text{ units} - 1456 \text{ units} = 56 \text{ units} $

4. Calculate Time for B to Complete Remaining Work

Find how many days B will take to complete the remaining 56 units of work, using B's calculated work rate.

• Time calculation:

  $ \text{Days for B} = \frac{\text{Remaining Work}}{\text{Rate}_{(B)}} = \frac{56 \text{ units}}{\frac{7}{4} \text{ units/day}} $

  $ \text{Days for B} = 56 \times \frac{4}{7} = 8 \times 4 = 32 \text{ days} $

B alone will complete the remaining work in 32 days.