Question

Working together, two secretaries can stuff the envelopes for a political fund-raising letter in 4 hours. Working alone, it takes the slower worker 6 hours longer to do the job than the faster worker. How long does it take each to do the job alone

Answer 1

Answer: The faster one needs 6 hours, the slower one needs 12 hours.

Explanation:

Let's define Sa and Sb as the times that each worker needs to stuff the envelopes for a political fundraising letter.

Sa is the faster one

Sb is the slower one.

Let's define 1 as a complete task.

Then:

when they both work together, they need 4 hours:

(1/Sa + 1/Sb)*4h = 1.

The slower one needs 6 more hours than the faster one:

Sb = (Sa + 6h).

We can replace this in the first equation and get:

(1/Sa + 1/(Sa + 6h))*4h = 1.

let's solve this for Sa.

1/Sa + 1/(Sa + 6h) = 1/4h.

(Sa + 6h) + Sa = Sa*(Sa + 6h)/4h.

2*Sa + 6h = Sa^2/4h + Sa*(6/4)

Then we have a quadratic equation:

(1/4h)*Sa^2 - (2/4)*Sa - 6h = 0h

(0.25*1/h)*Sa^2 - 0.5*Sa - 6h = 0h

The solutions come from the Bhaskara equation:

`Sa = (0.5 +- √((0.5)^2 - 4*0.25h*(-6)) )/(2*0.25* 1/h) = (0.5 +- 2.5)/(0.5) h`

Then we have two solutions:

Sa = ((0.5 + 2.5)/0.5 )h = 6h.

Sb = ( (0.5 - 2.5)/0.5) = -4h

The one that makes sense is the positive option (the negative one has no physical meaning in this situation)

Then the faster worker needs 6 hours to stuff all the envelopes.

And the slower one needs 6h + 6h = 12hours to stuff all the envelopes.

So when they work together, the combined rate is:

(1/6h + 1/12h) = (2/12h + 1/12h) = (3/12h) = (1/4h)

So working together they need 4 hours to stuff all the envelopes.