SOLUTION: Grapes cost $1.00 per pound and Oranges cost $.80 per pound. If Brandon buys 14 total pounds of fruit and it costs him $14.00, how many pounds of each did he buy?

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Question 201351: Grapes cost $1.00 per pound and Oranges cost $.80 per pound. If Brandon buys 14 total pounds of fruit and it costs him $14.00, how many pounds of each did he buy?
Found 2 solutions by Alan3354, jsmallt9:
Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
14 pounds of grapes.

Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Your problem is an example of what I would characterize as a "classic" word problem to be solved using a system of linear equations. I call them "Count and Value" problems because we get one equation from adding up the counts and one from adding up the "values". Counts are counts but the "values" can be any number associated one of the objects involved. "Values" are often amounts of money (like in your problem) or other things like weights.

So in your problem we will get one equation from adding the number of pounds of grapes and oranges and another from adding the values of these pounds of grapes and oranges.

Let x = the number of pounds of grapes bought by Brandon
Let y = the number of pounds of oranges bought by Brandon

The count equation then would be:
x + y = 14

Since x is the number of pounds of grapes and since $1 is the cost of one pound, then (1 * x) would be the number of dollars it would cost to buy "x" pounds of grapes. And since y is the number of pounds of oranges and since oranges cost $0.80 per pound, then (0.80 * y) would be what it cost to buy y"y" pounds of oranges.

(If you have trouble figuring out expressions like this, pretend you know how many pounds you are buying and ask yourself "What would I do to figure out the cost?". Let's pretend that we are buying 5.5 pounds of oranges. What would it cost to buy 5.5 pounds if it costs $0.80 per pound? Hopefully it is clear that the answer to this is 0.80 * 5.5 (multiply the cost per pound by the number of pounds). Once you figure out the correct logic then use that logic with the variable instead of the pretend number.)

So your "values" equation is:
1 * x + 0.80 * y = 14
or. more simply:
x + 0.80y = 14

Your system is now:
x + y = 14
x + 0.80y = 14

There are many methods to solve such a system:

Substitution
Elimination (aka Linear Combination)
Cramer's Rule (with determinants)
Various matrix methods
Perhaps the Elimination method is easiest on this system. Just subtract the second equation from the first. The x's will cancel out leaving:
0.20y = 0
Dividing by 0.20 we get
y = 0

Substituting 0 for y in either of the original equations gives x = 14. So Brandon bought 14 pounds of grapes and no (zero pounds) of oranges.