Expert: Josh - 8/4/2008

Question
Hi,

I was hoping that you could help me solve this problem?

Working alone, Bill can finish the job one hours faster than Adam. It would take Carl three times as long as Bill to do the job alone. Working together, it takes them 1 hour. How long would it take each of them to finish the job if they were working alone.

Thank you for your time,

Amanda

Answer
Hi Amanda,

What would you do to get started? Before we attempt to solve this problem, we should assign a variable to each of the unknown. We may use "a", "b" and "c" to represent the work rates of Adam, Bill and Carl.

To formulate the problem, we write formulas to capture the relationship between these variables. Finally, we use algebraic techniques to solve for the unknown qualities.

The basic principle is "work=time*rate". Suppose t is the time it takes Adam to finish the work. Translating the information we have been given, "Bill can finish the job one hours faster than Adam" is equivalent to b*(t-1)=a*t. This means that Bill takes one hour LESS to do the same amount of work as Adam.

Likewise, "Carl three times as long as Bill" is turned into the following equation: c*(3(t-1))=a*t. Recall that "t-1" is the time it takes Bill to finish the work; and it takes Carl 3 times that to do the same work that Adam and Bill had undertaken. [we can let W=a*t to represent the fixed amount of work to be done]

Working together, the guys take an hour to finish the job. The combined work rate is given by (a+b+c), while the combined time is 1. So, (a+b+c)*1=W.

The equations thus far:
a*t=W       ...[1]
b*(t-1)=W   ...[2]
3c*(t-1)=W  ...[3]
W=a+b+c     ...[4]

Observe that we have 5 unknowns, but only 4 equations. This point is important and we will revisit this issue later.

Q: What are the unknowns?
A: t := amount of time it takes Adam to finish the job on his own.
  Implicitly,  we also have
  a, b, c:= the work rates of Adam, Bill and Carl, respectively.
  W := treat this as a dummy variable; it's only symbolic, we don't really care what value it has as long as it remains constant.

Our task: To solve for "t". Knowing "t", we also know how long it takes for each person to finish the job alone.

Algebra:
There are various ways to go about this. The most complex part appears to involve equation [2] and [3]. To simplify things, equating [2] with [3], we obtain 3c=b after cancellation.

[Aside: It is helpful to pause for a moment to see if this actually makes sense. It's easy to get caught up in the algebra in more difficult problems without much feeling for what we are doing. I would like you to get in the habit of checking your answers. It's one thing to get the answers slightly off, but it is another if the answers (numbers, or equations) make no sense. Here, "3c=b" tells us that Bill works 3 times as fast as Carl. This agrees with the fact that Carl takes 3 times as long to complete the job.]

Now we are left with 3 unknowns,
a*t=W       ...[1]
3c=b        ...[23]
W=a+b+c     ...[4]

--------------------------------------------------------------------
The general idea is to continue eliminating the unknown variables, one by one, until we have the answer to one of the unknown. Then, we use back-substitution to determine the rest using the equations we have established.
--------------------------------------------------------------------

Equating [1] & [4], the system reduces to 2 equations
3c=b        ...[23]
t=(a+b+c)/a ...[14]

Substituting for b, the system reduces to 1 equation
t=(a+4c)/a ...[1234]

We don't get anywhere unless we come up with an additional equation (a fifth equation for our five unknowns -- see earlier observation). Returning to the four original equations:

a*t=W       ...[1]
b*(t-1)=W   ...[2]
3c*(t-1)=W  ...[3]
W=a+b+c     ...[4]

we observe that ([1]+[2]+[3])/3 = [4], viz.,
[a*t+b*(t-1)+3c*(t-1)]/3 = a+b+c ...we seek to separate t from a,b,c
(at+bt-b+3ct-3c) = 3(a+b+c)
t(a+b+3c)=3a+4b+6c
t=(3a+4b+6c)/(a+b+3c)

Just as things seem overly complicated, we exhaust all information when we make the substitution for b, using b=3c from [23],

At last, we find
t=(3a+18c)/(a+6c)
t=3 [hour]

Back substitution begins:
From [2], Bill takes (t-1)=(3-1)=2 [hour]
From [3], Carl takes 3(t-1)=6 [hour]

You should check that the solution makes sense, for instance, by checking the work rates.

a=W/3
b=W/2
c=W/6

Does b=3c ?
Does a+b+c=W ?

Smile :)