SOLUTION: You have 5 gallons of apple cider made from different varieties of apples, 55% of which were McIntosh apples. The cider is too sweet. You have cider made from 100% McIntosh apple

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: You have 5 gallons of apple cider made from different varieties of apples, 55% of which were McIntosh apples. The cider is too sweet. You have cider made from 100% McIntosh apple      Log On


   

Question 1103342: You have 5 gallons of apple cider made from different varieties of apples, 55% of which were McIntosh apples. The cider is too sweet. You have cider made from 100% McIntosh apples, but this is too tart. How much of the pure McIntosh cider must be added to the 55% mixed cider to get a cider that is 70% McIntosh apples?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39621) About Me  (Show Source):
Answer by greenestamps(13203) About Me  (Show Source):
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5 gallons of 55% McIntosh apples, plus x gallons of 100% McIntosh apples, equals (5+x) gallons of 70% McIntosh apples:

.55%285%29%2B1.00%28x%29+=+.70%285%2Bx%29
2.75%2Bx+=+3.5%2B.7x
.3x+=+.75
x+=+2.5

You need to add 2.5 gallons of the pure McIntosh cider to get a mixture that is 70% McIntosh apples.

That is the traditional algebraic approach to mixture problems.

Here is a much faster way to solve problems like this....

(1) Think of the percentages on a number line -- 55, 70, and 100; you are mixing 55% with 100% to get 70%.
(2) The 70 is one-third of the way from 55 to 100. (70-55=15; 100-55=45; 15/45 = 1/3.)
(3) The 70% begin 1/3 of the way from 55% to 100% means 1/3 of the mixture must be the 100% McIntosh apples, which means 2/3 must be the original cider.
(4) 1/3 is half of 2/3; that means the amount of 100% McIntosh cider to be added is half the amount of the original cider. Since you started with 5 gallons of cider, the amount of 100% McIntosh cider you need to add is 2.5 gallons.

With all the words of explanation, that looks like a lot of work. But here are all the calculations needed, without the words:

70-55=15; 100-55=45; 15/45 = 1/3

The mixture needs to be 1/3 the 100% McIntosh cider and 2/3 the original cider.

Here is a diagram that can be used to solve the problem by a very similar method:

matrix%283%2C3%2C55%2C%22%22%2C30%2C%22%22%2C70%2C%22%22%2C100%2C%22%22%2C15%29

This diagram shows the percentages of the two ingredients in the first column and the percentage of the mixture in the second column. The third column shows the differences, computed diagonally, between the percentage of the mixture and the percentages of each ingredient (70-55=15; 100-70=30).

When the calculations are done this way, the numbers in the third column tell the ratio in which the two ingredients must be mixed. The ratio is 30:15, or 2:1, telling us that the mixture must be 2 parts the 55% McIntosh cider to 1 part 100% McIntosh cider.