SOLUTION: Grapefruit costs 50 cents per 12 ounce can, and pineapple costs 85 cents per 12 ounce can. How many of each should she mix to obtain 84 ounces of a mixture that will cost 70 cents

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Question 1198083: Grapefruit costs 50 cents per 12 ounce can, and pineapple costs 85 cents per 12 ounce can. How many of each should she mix to obtain 84 ounces of a mixture that will cost 70 cents for every 12 ounces?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
https://www.algebra.com/my/mixture-price-two-part-both-parts-unknown.lesson?content_action=show_dev
https://www.algebra.com/my/Mixtures%3A-All-in-Symbols.lesson?content_action=show_dev

Grapefruit 50 cents per can
Pineapple 85 cents per can
MIXTURE is for 7 "cans" at 70 cents per can.

Account for cost.
w cans of the pineapple
7-w cans of the grapefuit
85w%2B50%287-w%29=70%2A7
Solve for w, the number of cans of the pineapple.

%2885-50%29w=70%2A7-50%2A7
highlight%28w=7%28%2870-50%29%2F%2885-50%29%29%29

w=7%2820%2F35%29=7%284%2F7%29=4
highlight%28w=4%29-----------four cans of the pineapple.

This means 8 cans of the grapefuit.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The response from the other tutor shows a standard algebraic solution method.

Here is a much faster and easier way to solve any 2-part mixture problem like this -- if you understand it, and if a formal algebraic solution is not required.

Look at the three costs per 12 ounces -- 50, 70, and 85 -- (on a number line, if it helps) and observe/calculate that 70 is 20/35 = 4/7 of the way from 50 to 85.

That means 4/7 of the mixture must be the higher priced ingredient. So...

ANSWERS:
pineapple: (4/7) of 84 ounces = 4*12 = 48 ounces
grapefruit: (3/7) of 84 ounces = 3*12 = 36 ounces

CHECK:
48(85)+36(50) = 4080+1800=5880
84(70) = 5880