Question 991632: A fruit grower can use two types of fertilizer in an orange grove, brand A and brand B. Each bag of brand A contains 8 pounds of nitrogen and 4 pounds of phosphoric acid. Each bag of Brand B contains 7 pounds of nitrogen and 7 pounds of phosphoric acid. Test indicate that tHE grove needs 720 pounds of nitrogen and 500 pounds of phosphoric acid. How many bags of each brand should be used to provide the required amounts of nitrogen and phosporic acid
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the equation for nitrogen is:
8a + 7b = 720
a is the number of bags of brand A.
b is the number of bags of brand B.
the total number of bags for brand A and brand B combined must contain 720 pounds of nitrogen.
the equation for phosphorus is:
4a + 7b = 500
a is the number of bags of brand A.
b is the number of bags of brand B.
the total number of bags for brand A and brand B combined must contain 500 pounds of phosphorus.
since the same bag contains nitrogen and phosphorus, then you have two equations that need to be solved simultaneously so that the number of bags of brand A and brand B is the same in both equations.
the two equations are:
8a + 7b = 720
4a + 7b = 500
multiply the second equation by 2 to get:
8a + 7b = 720
8a + 14b = 1000
subtract the first equation from the second equation to get:
7b = 280
replace 7b in the first equation to get:
8a + 280 = 720
subtract 280 from both sides of that equation to get:
8a = 440.
you have:
7b = 280
8a = 440.
solve for a and b to get:
b = 40
a = 55
that's your solution.
you need 55 bags of brand A and 40 bags of brand B.
55 bags of brand A will get you 8 * 55 = 440 pounds of nitrogen and 4 * 55 = 220 bags of phosphorus.
40 bags of brand B will get you 7 * 40 = 280 pounds of nitrogen and 7 * 40 = 280 pounds of phosphorus,.
the nitrogen total is 440 + 280 = 720 pounds.
the phosphorus total is 220 + 280 = 500 pounds.
solution is confirmed.
it is useful to create a two dimensional table when you set up the problem.
the table would look something like this:
brand A brand B total
nitrogen 8 * a 7 * b 720
phosphorus 4 * a 7 * b 500
a represents the number of bags of brand A.
b represents the number of bags of brand B.
this table is where you get 8a + 7b = 720 and 4a + 7b = 500
since you now know that a = 55 and b = 40, you can replace a and b in this table to get:
brand A brand B total
nitrogen 8 * 55 7 * 40 720
phosphorus 4 * 55 7 * 40 500
this shows you the same result in tabular form.
as long as you know what you're doing, this step isn't really necessary.
i just did it to show you how it looks in the table that you set up to solve the problem.
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