Word Problem

RSK wrote:Hi guys,

I need help with this question:

If Bob produces 36 or fewer items in a week, he is paid x dollars per item. If Bob produces more than 36 items in a week, he is paid x dollars per item for the first 36 items and 1.5 times that amount for each additional item. How many items did Bob produce last week?

1) Last week Bob was paid a total of @480 for the items that he produced that week.

2) This week Bob produced 2 items more than last week and was paid a total of 510 for the items that he produced this week.

The stimulus tells us that as a function of the number u of units produced, the total amount Bob is compensated, T, is going to be:

T = ux when u =< 36
T = 36x + 1.5(u - 36)x when u > 36

Statement 1 is not sufficient, because not only can we not figure out which equation to use, but also we would still have u and x as unknowns with only one equation.

Statement 2 is not sufficient, for the same reason. Statement 2 would also put a second equation into the picture AND an extra T1 unknowns. We'd have too many unknowns and not enough equations.

But with statements 1 and 2 together, then we have two possible systems of equations, each with two unknowns:

480 = ux when u =< 36
510 = (u + 2)x

OR

480 = 36x + 1.5(u - 36)x when u > 36
510 = 36x + 1.5(u - 34)x

Let's solve the first pair. We get that x = 480/u and x = 510/(u + 2). So:

480/u = 510/(u + 2)
480(u[/i] + 2) = 510u
480u + 960 = 510u
960 = 30u
96 = 3u
32 = u

Okay. Let's see whether the second pair gives us anything different. Transforming our two equations, we'd get:

480 = x(36 + 1.5(u - 36))
510 = x(36 + 1.5(u - 34))

and then:

480 = x(1.5u - 18)
510 = x(1.5u - 15)

So we can transform these into x = 480/(1.5u - 18) and x = 510/(1.5u - 15). Then:

480/(1.5u - 18) = 510/(1.5u - 15)
480(1.5u - 15) = 510(1.5u - 18)
720u - 7200 = 765u - 7650
450 = 45u
10 = u

Logically this makes no sense, since this formula only applies to a situation in which u would be GREATER than 36. Thus it is irrelevant and impossible.

The answer is 32, so both statements together are sufficient.