Question

A can do a piece of work in 5 days, \(B\) can do it in 10 days. With the help of \(C\), they finish the work in 2 days. In how many days \(C\) alone can do the whole work? (a) 3 (b) 4 (c) 5 (d) 6

Short answer

Answer: C can complete the work alone in 5 days.

Step-by-step solution

Determine the work rates of A and B

A can do the work in 5 days, so A's work rate is 1/5 of the work per day. B can do the work in 10 days, so B's work rate is 1/10 of the work per day.

Calculate the combined work rate of A, B, and C

Together, A, B, and C can finish the work in 2 days, which means their combined work rate is 1/2 of the work per day.

Find C's work rate

Since we know the combined work rate of A, B, and C, we can find the work rate of C by subtracting the work rates of A and B from the combined work rate:$$\text{Work rate of C} = \frac{1}{2} - \frac{1}{5} - \frac{1}{10}$$

Simplify the expression

To simplify the expression, find a common denominator for the fractions (in this case, 10). This gives:$$\text{Work rate of C} = \frac{5}{10} - \frac{2}{10} - \frac{1}{10}$$ Now, subtract the numerators:$$\text{Work rate of C} = \frac{2}{10}$$

Find how long it takes C to complete the work alone

C's work rate is 2/10 of the work per day, so to find how long it takes for C to complete the whole work, we need to take the reciprocal of this value:$$\text{Days for C to complete the work alone} = \frac{1}{\frac{2}{10}} = \frac{10}{2}$$

Calculate the result

Now, we can calculate the number of days it takes for C to complete the work alone: $$\text{Days for C to complete the work alone} = \frac{10}{2} = 5$$ C can complete the work alone in 5 days, which is option (c).

Key concepts

Work Rates

Understanding work rates is key to solving work and time problems. A work rate is simply the amount of work that can be completed in a given time. For instance, if someone can finish a task in 5 days, their work rate is \(\frac{1}{5}\). This means they complete \(\frac{1}{5}\) of the work each day.

Knowing individual work rates helps in understanding how much work each person contributes. For example:

• If A can complete a task in 5 days, their rate is \(\frac{1}{5}\).
• Similarly, if B can do it in 10 days, their rate is \(\frac{1}{10}\).

Recognizing these rates allows you to predict how quickly they can work together.

Combined Work Rates

When people work together, their efforts combine, impacting the total time to complete a task. To find the combined work rate, simply add individual rates.

The formula for combined work rate in this scenario would be:

• A's rate: \(\frac{1}{5}\)
• B's rate: \(\frac{1}{10}\)

If they finish the task in 2 days with help from C, their total combined work rate is \(\frac{1}{2}\).

This tells you how the additional help speeds up the work process. By knowing each rate and how they sum, you can calculate how well people work together to complete tasks faster.

Fraction Operations

Fraction operations are crucial in calculating work rates because individual contributions are often fractional. To handle these, you often need to add or subtract fractions.

In the task, three fractions needed simplification:

• Combined rate of A, B, and C: \(\frac{1}{2}\)
• A's rate: \(\frac{1}{5}\)
• B's rate: \(\frac{1}{10}\)

To subtract these, find a common denominator, which is 10 here, leading to:
\[ \frac{5}{10} - \frac{2}{10} - \frac{1}{10} \]
This step is essential to solve or simplify work and time problems accurately, especially when combining or separating work tasks.

Reciprocal Calculation

Understanding reciprocal calculations is essential to find out how long it takes one person to do an entire task alone, given their work rate. If you know someone completes \(\frac{2}{10}\) of a task per day, their rate is essentially a fraction of the total work.

To find the total time needed, you find the reciprocal of the work rate:

• C’s work rate: \(\frac{2}{10}\)

Thus, the number of days C would take alone is:
\[ \frac{1}{\frac{2}{10}} = \frac{10}{2} \]
This process helps determine the time required when individuals work independently, based on their contribution when working with others.