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Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job 12 days earlier. If the job needs to be completed 32 days earlier, how many additional workers should be hired?

Aug 10, 2024

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Let's denote the number of workers initially hired as $$n = 5$$, and the total work required to complete the job as $$W$$.

Let $$r$$ be the work rate of one worker (i.e., the amount of work one worker can do in one day). Then the work rate of $$n$$ workers is $$nr$$, and the time it takes $$n$$ workers to complete the job is:

$\text{Time} = \frac{W}{nr}$

### Step 1: Set up the equation for one additional worker

If one additional worker is hired, the total number of workers becomes $$n+1$$, and they can complete the job 12 days earlier. Therefore, the time it would take $$n+1$$ workers to complete the job is:

$\frac{W}{(n+1)r} = \frac{W}{nr} - 12$

We now substitute $$\frac{W}{nr}$$ with the initial time $$T$$:

$\frac{W}{(n+1)r} = T - 12$

Equating the two expressions for time:

$\frac{W}{(n+1)r} = \frac{W}{nr} - 12$

### Step 2: Solve for $$T$$

Substitute $$T = \frac{W}{nr}$$:

$\frac{W}{(n+1)r} = \frac{W}{nr} - 12$

Multiply both sides by $$(n+1)r$$ to clear the fractions:

$W = \frac{W(n+1)}{n} - 12(n+1)r$

Simplify and solve for $$T$$:

$W = \frac{Wn + W}{n} - 12(n+1)r$

Simplify the equation:

$W = W + 12nr - 12r = 12nr \quad \Rightarrow \quad T = \frac{W}{nr} = 12$

Now, let's calculate the number of additional workers needed to complete the job 32 days earlier.

### Step 3: Set up the equation for $$k$$ additional workers

If $$k$$ additional workers are hired, the total number of workers becomes $$n + k$$, and they can complete the job 32 days earlier. The equation for time is now:

$\frac{W}{(n+k)r} = T - 32$

Using the expression $$T = \frac{W}{nr}$$:

$\frac{W}{(n+k)r} = \frac{W}{nr} - 32$

Multiply both sides by $$(n+k)r$$:

$W = \frac{W(n+k)}{n} - 32(n+k)r$

Simplify the equation:

$W = W + 32(nr) - 32r = 32nr$

Now solve for $$k$$:

$\frac{W}{nr} - \frac{W}{(n+k)r} = 32 \quad \Rightarrow \quad k = 8$

$\boxed{8}$