Work and Time Problem Solution

This problem involves calculating the time taken by three individuals, A, B, and C, to complete a piece of work together, given the time taken by them in pairs.

Understanding Work Rate

The key concept here is the work rate. If a person or a group can complete a work in 'd' days, their work rate (the amount of work done in one day) is \( \frac{1}{d} \).

Calculating Individual and Combined Work Rates

Given the information:

• A and B together can do the work in 4 days.
• B and C together can do the work in 6 days.
• A and C together can do the work in 8 days.

Based on the work rate concept:

• Work done by (A + B) in 1 day = \( \frac{1}{4} \)
• Work done by (B + C) in 1 day = \( \frac{1}{6} \)
• Work done by (A + C) in 1 day = \( \frac{1}{8} \)

Now, if we add the work done by all pairs in one day, we get:

\[ (\text{Work by A in 1 day} + \text{Work by B in 1 day}) + (\text{Work by B in 1 day} + \text{Work by C in 1 day}) + (\text{Work by A in 1 day} + \text{Work by C in 1 day}) \] \[ = \frac{1}{4} + \frac{1}{6} + \frac{1}{8} \]

This sum is equal to twice the work done by A, B, and C together in one day, because each person's work rate is included twice (A is in A+B and A+C; B is in A+B and B+C; C is in B+C and A+C).

So, \( 2 \times (\text{Work by A + B + C in 1 day}) = \frac{1}{4} + \frac{1}{6} + \frac{1}{8} \)

Adding the Fractions

To add the fractions \( \frac{1}{4}, \frac{1}{6}, \) and \( \frac{1}{8} \), we find the least common multiple (LCM) of the denominators 4, 6, and 8. The LCM of 4, 6, and 8 is 24.

\[ \frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24} \] \[ \frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24} \] \[ \frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24} \]

Now, adding the fractions:

\[ \frac{6}{24} + \frac{4}{24} + \frac{3}{24} = \frac{6 + 4 + 3}{24} = \frac{13}{24} \]

So, \( 2 \times (\text{Work by A + B + C in 1 day}) = \frac{13}{24} \)

Calculating Work Rate of A, B, and C Together

To find the work done by A, B, and C together in 1 day, we divide the sum by 2:

\[ \text{Work by A + B + C in 1 day} = \frac{1}{2} \times \frac{13}{24} = \frac{13}{48} \]

Calculating Time Taken by A, B, and C Together

The total time taken by A, B, and C together to complete the work is the reciprocal of their combined work rate per day.

Time taken by A, B, and C together = \( \frac{1}{\text{Work by A + B + C in 1 day}} = \frac{1}{\frac{13}{48}} = \frac{48}{13} \) days.

Therefore, A, B, and C together can do the same work in \( \frac{48}{13} \) days.