Solving Work and Time Problems: A, B, and C Working Together

This question asks us to find out how many days it will take for three individuals, A, B, and C, to complete a specific work when they work together. We are given the time taken by A and B together, and the time taken by C alone.

Understanding Work Rate

In work and time problems, the key idea is to think about the 'rate' at which work is done. The rate is the amount of work completed per unit of time (in this case, per day). If someone completes a work in 'd' days, their rate is $\frac{1}{d}$ of the work per day.

Calculating Individual and Combined Rates

We are given the following information:

• A and B together can complete the work in 10 days.
• C alone can complete the same work in 15 days.

From this, we can find their rates:

• Combined rate of A and B per day = $\frac{1}{10}$ of the work.
• Rate of C per day = $\frac{1}{15}$ of the work.

Finding the Combined Rate of A, B, and C

When A, B, and C work together, their individual rates (or in this case, the combined rate of A+B and the rate of C) add up to give their total combined rate per day.

Combined rate of (A + B + C) per day = (Rate of A + B) + (Rate of C)

Combined rate of (A + B + C) per day = $\frac{1}{10} + \frac{1}{15}$

Calculating the Total Combined Rate

To add the fractions $\frac{1}{10}$ and $\frac{1}{15}$, we need to find a common denominator. The least common multiple (LCM) of 10 and 15 is 30.

$\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}$

$\frac{1}{15} = \frac{1 \times 2}{15 \times 2} = \frac{2}{30}$

Now, add the fractions:

Combined rate of (A + B + C) per day = $\frac{3}{30} + \frac{2}{30} = \frac{3 + 2}{30} = \frac{5}{30}$

Simplify the fraction:

Combined rate of (A + B + C) per day = $\frac{5}{30} = \frac{1}{6}$ of the work.

Calculating the Time Taken When Working Together

If the combined rate of A, B, and C is $\frac{1}{6}$ of the work per day, it means they complete $\frac{1}{6}$ of the work each day. To complete the whole work (which is 1 unit), the time taken is the reciprocal of their combined rate.

Time taken by (A + B + C) together = $\frac{1}{\text{Combined rate per day}}$

Time taken by (A + B + C) together = $\frac{1}{\frac{1}{6}} = 1 \times \frac{6}{1} = 6$ days.

Therefore, A, B, and C can complete the work together in 6 days.

Revision Table: Work and Time Concepts

Additional Information: Work and Time Problem Tips

Work and time problems often involve calculating rates and combining them. Here are some useful tips:

• Always convert the given time into a daily rate by taking the reciprocal.
• If people work together, add their individual rates to find their combined rate.
• If people work on different parts or have different efficiencies, adjust their rates accordingly (though not needed in this specific problem).
• The total work is usually considered as 1 unit.
• If units of time are different (e.g., hours and days), convert them to a common unit before calculation.