Question

A can do a piece of work in 10 days. \(B\) can do it in 24 days. If \(C\) also works with them then it takes only 6 days to complete the whole work. In how many days \(C\) alone can complete the whole work? (a) 25 (b) 40 (c) 50 (d) 75

Short answer

Answer: 40 days

Step-by-step solution

Find the work rate of A, B, and A+B+C.

To find the work rate of A, B, and A+B+C, we will use the formula: work rate = work / time A's work rate = 1/10 (A completes the whole work in 10 days) B's work rate = 1/24 (B completes the whole work in 24 days) (A+B+C)'s work rate = 1/6 (A, B, and C together complete the work in 6 days)

Calculate C's work rate

As A, B, and C are working together and their work rates are additive, we can find C's work rate using: C's work rate = (A+B+C)'s work rate - A's work rate - B's work rate C's work rate = (1/6) - (1/10) - (1/24) To subtract these fractions, we will find the common denominator, which is 120. C's work rate = (20 - 12 - 5) / 120 = 3 / 120

Find the number of days C alone can complete the whole work

Now that we have found C's work rate, we can find the number of days it takes for C to complete the work alone using the formula: time = work / work rate C's time = 1 / (3/120) C's time = 120 / 3 = 40 days So, the correct answer is (b) 40.

Key concepts

Work Rate Calculation

Understanding how to calculate work rate is essential in work and time problems. Work rate refers to the amount of work a person or a group can complete in a specified period. It is usually expressed as "work per day" where one whole work unit is conventionally taken.

When given how long a person or machine takes to complete a task independently, you can determine their work rate. The formula you use is the total work completed divided by the time taken to complete it. For instance:

• If "A" completes a task in 10 days, his work rate is \( \frac{1}{10} \).
• If "B" completes the same task in 24 days, her work rate is \( \frac{1}{24} \).
• The team of A, B, and one more person ("C") completing it in 6 days would together have a work rate of \( \frac{1}{6} \).

You can use individual work rates to solve for unknowns in collaborative settings efficiently.

Fraction Operations

To handle work rate problems, you often need to perform fraction operations. This involves adding or subtracting fractions to find individual rates from combined rates.

When subtracting or adding fractions like in our problem, the first step is always to find a common denominator. This common denominator is the least common multiple of the bottom numbers (denominators) of the fractions involved. For example, with denominators of 10, 24, and 6, the common denominator is 120.

• Convert each fraction—the work rates—into equivalents with this common denominator: \( \frac{1}{10} = \frac{12}{120} \)
• For \( \frac{1}{24} = \frac{5}{120} \)
• For \( \frac{1}{6} = \frac{20}{120} \)

By doing this, calculations become straightforward, allowing you to add or subtract the numerators directly. Simply put, you're finding proportionate shares of work done by each and using that to solve for what's missing.

Collaborative Work

Collaborative work problems tackle scenarios where multiple people or machines work together to complete a task. The trick here is to understand additive work rates.

When multiple individuals work together, their work rates sum up to complete the task faster. So, to find one person's contribution in collaborative work, isolate their work rate by subtracting the work rates of the others involved. For instance, if A, B, and C together have a work rate of \( \frac{1}{6} \) and you know A's and B's rates, subtracting them gives C's work rate.

• The combined rate of \( \frac{20}{120} \) tells how much work is done per day collectively.
• Subtracting A's and B's portions \( \left( \frac{12}{120} + \frac{5}{120} \right) \) clarifies C's individual rate.
• This simplifies problem-solving by isolating each worker's or machine's contribution.

Understanding these collaborative dynamics allows you to solve how long it would take any single worker to complete the task alone.