A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then what fraction of the work is left?
Understand the Problem
The question is asking for the fraction of work remaining after A and B work together for 4 days, given their individual work rates. We will calculate the amount of work done by A and B together in those 4 days and then determine how much of the total work is left.
Answer
The remaining work after 4 days of work by A and B is given by $$1 - 4 \left( \frac{1}{a} + \frac{1}{b} \right)$$.
Answer for screen readers
The remaining work after A and B work together for 4 days is given by the expression:
$$\text{Remaining work} = 1 - 4 \left( \frac{1}{a} + \frac{1}{b} \right)$$
Steps to Solve
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Identify the work rates
Let the total work be represented by 1 (or 100%).
Assume A can complete the work in $a$ days, thus A's work rate is $\frac{1}{a}$ work per day.
Assume B can complete the work in $b$ days, so B's work rate is $\frac{1}{b}$ work per day. -
Calculate combined work rate
To find the total work rate when A and B work together, we add their individual work rates:
$$\text{Combined work rate} = \frac{1}{a} + \frac{1}{b}$$ -
Find work done in 4 days
To find the work completed in 4 days, multiply the combined work rate by the time (4 days):
$$\text{Work done in 4 days} = 4 \left( \frac{1}{a} + \frac{1}{b} \right)$$ -
Calculate remaining work
Subtract the work done in 4 days from the total work:
$$\text{Remaining work} = 1 - \text{Work done in 4 days}$$ -
Express remaining work as a fraction
After calculating the remaining work, express it as a fraction of the total work.
The remaining work after A and B work together for 4 days is given by the expression:
$$\text{Remaining work} = 1 - 4 \left( \frac{1}{a} + \frac{1}{b} \right)$$
More Information
The answer represents the fraction of work that has not been completed after A and B have worked together for a certain period. Understanding how to calculate work rates is crucial in many real-world situations such as project management.
Tips
- A common mistake is forgetting to convert the days into the right rate when combining work rates. To avoid this, always double-check that the rates are calculated correctly before combining them.
- Another mistake is to not account for the total work being equal to 1. Ensure that the remaining work is calculated as the total work minus the completed work.
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