Question

A manufacturing crew needs to assemble 1000 boxes per day, divided equally among the workers. One day, three workers call in sick, and the remaining members each need to assemble 75 more boxes than usual. How many workers are on the manufacturing crew?

Answer 1

I would start by writing as much as you can as equations.

Answer 2

How?

Answer 3

Well, the easiest one is \(\frac{1000}{W} = B\), where W = number of workers and B = boxes per worker.

Answer 4

You also know that \(\frac{1000}{W - 3} = B + 75\)

Answer 5

So, plug \(B\) from the first equation into the second, and solve.

Answer 6

Why did you close the question? Did you somehow figure it out?

Don't waste our time if you're not serious with your question.

Answer 7

Excuse me, I am serious about the question. I just didn't understand it.

Answer 8

Okay, so why close it?

Did you substitute the first formula into the second?

Answer 9

That is, put: \(B = \frac{1000}{W}\) into \(B + 75 = \frac{1000}{W - 3}\)

Answer 10

\[\frac{ 1000 }{ w-3 }=\frac{ 1000 }{ w }+75 ?\]

Answer 11

Yes, good!

Answer 12

Now you need to solve for \(w\).

Answer 13

Okay, this is ridiculous but that's my problem. How exactly do you solve for w?

Answer 14

Not ridiculous at all.

You need to isolate it on one side of the equation. You do that by doing the same thing to both sides of the equal sign.

Answer 15

So, you could multiply both sides by \(w - 3\) to "get rid of it" on the left side.

Answer 16

\[(w - 3)\frac{ 1000 }{ w-3 }=(w - 3) \frac{ 1000 }{ w }+75\]

\[1000=(w - 3)(\frac{1000}{ w }+75)\]

Answer 17

Now expand your right side using the FOIL method.

Answer 18

I am so sorry but I have no idea what I'm doing.

Answer 19

Okay, the FOIL method is like this:

If you have \((w - x)(y + z)\), you multiply each of the terms in the left brackets by the terms in the right brackets, as follows:

w times y
w times z
x times y
x times z

So, you would end up with:

\(wy + wz - xy - xz\)

Notice the plus and minus signs. Those are important.

So, a more common example would be something like this:

\((x + 5)(x - 2)\)

You would do the following:

x times x
x times -2
5 times x
5 times -2

Which would give you:

\(x^{2} - 2x + 5x -10\) which, simplified, is \(x^{2} + 3x - 10\).

Answer 20

We have to do the same thing, but with \((w - 3)(\frac{1000}{ w }+75)\).

So, we do:

w times 1000/w
w times 75
-3 times 1000/w
-3 times 75

And get a final formula of:

\(\frac{1000w}{w} + 75w + \frac{(-3)(1000)}{w} - 225\)

or:

\(1000 + 75w - \frac{3000}{w} - 225\)

which is:

\(75w - \frac{3000}{w} + 775\)

Remember, that all equals 1000. So, the equation we're working with is:

\(75w - \frac{3000}{w} + 775 = 1000\)

Let's get rid of that fraction. We can do that by multiplying both sides of the equation by \(w\). We now have:

\(75w^{2} - 3000 + 775w = 1000w\)

or

\(75w^{2} - 225w = 3000\)

Simplify by dividing everything by 75.

\(w^{2} - 3w = 40\)

\(w^{2} - 3w - 40 = 0\)

Factor that, and you get:

\((w - 8)(w + 5) = 0\)

Meaning your roots are 8 and -5. Since you can't have -5 workers, \(w\) must equal 8. Plug it into your original equation to confirm.

Answer 21

Sorry for being such a pain but I really appreciate you taking the time to do step-by-step for me.

Answer 22

Not a pain at all. No problem. :)