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Problem 1229

# If A can do a job in 8 days and $$B$$ can do the same job in 12 days, how long would it take the two men working together?

Expert verified
It would take A and B $$\frac{24}{5}$$ days to complete the job working together.
See the step by step solution

## Step 1: Find the working rate of A

A can finish the job in 8 days, so his working rate is the fraction of the job he can complete in one day. This fraction can be calculated as follows:  A's working rate = $$\frac{1}{8}$$ 

## Step 2: Find the working rate of B

B can finish the job in 12 days, so his working rate is the fraction of the job he can complete in one day. This fraction can be calculated as follows:  B's working rate = $$\frac{1}{12}$$ 

## Step 3: Find the combined working rate of A and B

To find the combined working rate, we'll add the working rates of A and B.  A and B's combined working rate = A's working rate + B's working rate = $$\frac{1}{8} + \frac{1}{12}$$ 

## Step 4: Simplify the combined working rate

To add the fractions, we need a common denominator, which in this case is 24.  A and B's combined working rate = $$\frac{3}{24} + \frac{2}{24}$$ = $$\frac{3+2}{24}$$ = $$\frac{5}{24}$$ 

## Step 5: Find the time it takes for A and B to complete the job together

The combined working rate is the fraction of the job A and B can complete in one day. To find out how long it would take them to complete the entire job, we need to find the reciprocal of their combined working rate.  Time taken for A and B to complete the job together = $$\frac{1}{(A and B's combined working rate)}$$ = $$\frac{1}{\frac{5}{24}}$$ 

## Step 6: Simplify the expression

To simplify the expression, we can multiply the numerator and the denominator by 24.  Time taken for A and B to complete the job together = $$\frac{1 \times 24}{5}$$ = $$\frac{24}{5}$$  So, it would take A and B $$\frac{24}{5}$$ days to complete the job working together.

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