Expert: Josh - 2/23/2009

Question
QUESTION: Hello:

I want to try to determine the amounts in the following calculation by using any one or more of the fractions 1/3, 2/3, 1/6, 5/6, 1/25 or 24/25. 1/3 is from $10,000/$30,000; 2/3 is from $20,000/$30,000; 1/6 is from $200/$1,200; 5/6 is from $1,000/$1,200, and 1/25 is from $1,200/$30,000, and 24/25 is from 28,800/$30,000.

Here is the calculation:  (2% X ___) + (5% X ___) = $1,200. The amounts that are missing must total $30,000.
For example, 2% X $10,000 + 5% X $20,000 = $1,200.
$10,000 + $20,000 = $30,000.

How can some of these fractions be determined without first knowing the answers of $10,000 and $20,000? Is it possible?

Try to think of a simple solution, one that does not require algebra. I already have an algebra solution.

I thank you for any helpful reply that you may send to me.



ANSWER: Kenneth,

As far as I know, there are no established methods for doing this other than using algebra. If you insist on avoiding algebra, it just makes things more difficult, not any easier. You will be looking at some sort of iterative process, like trial-and-error, perhaps assisted by intuition.

This is generally not advisable since a close form solution exists for this type of problem. I think it actually takes less time to solve 2x+5y=1200 and x+y=30000 using the substitution method, if you want an exact answer.

Actually, I guess you can consider the graphical method as an alternative. This involves plotting two straight lines 2x+5y=1200 and x+y=30000 on a piece of graph paper, and finding the point (X,Y) where they intersect. That gives you an approximate solution.

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply.

How is this equation 2x+5y=1200 and x+y=30000 solved? It has been a long time since I have had algebra.

I thank you for your follow-up reply.

Answer
Hi Kenneth,

For the graphical method, we rewrite both equations in gradient-intercept form:
Equation 1: 0.02x+0.05y=1200 becomes y=-0.4x+24000
Equation 2: x+y=30000  becomes y=-x+30000

First line has a slope of -0.4, it passes through (0,24000) and heads in the downward direction from left to right. Second line is similar. It has a steeper slope of -1, and it passes through (0,30000).

You have to draw a graph to see where the two lines intersect. Suppose they intersect at (X,Y), then x=X and y=Y are the solutions. Generally, you cannot know the answer exactly due to limited graphical precision.

For the algebra method, one way to proceed is to first eliminate the x variable. By this, I mean rewriting equation 2 to make x as the subject. Then, substitute x=30000-y in equation 1. (i.e., wherever you see x, you replace it with 30000-y). This gives
0.02(30000-y)+0.05y=1200
600+0.03y=1200
y=600/0.03=20000
Put this y value back into x=30000-y.
x=10000.