SOLUTION: Alpine Trail Mix is 40% nuts and Meadows Trail mix is 25% nuts. How much Alpine should be mixed with Meadows to form a 10 lb batch of trail mix that is 32% nuts?

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Question 1189801: Alpine Trail Mix is 40% nuts and Meadows Trail mix is 25% nuts. How much Alpine should be mixed with Meadows to form a 10 lb batch of trail mix that is 32% nuts?
Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52864) About Me  (Show Source):
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Alpine Trail Mix is 40% nuts and Meadows Trail mix is 25% nuts.
How much Alpine should be mixed with Meadows to form a 10 lb batch of trail mix that is 32% nuts?
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x pounds of the 40% nuts and

(10-x) pounds of the 25% nuts.


Now write the nuts balance equation for ingredients and for final mixture

    0.4x + 0.25*(10-x) = 0.32*10   pounds.


From the equation

    x = %280.32%2A10-0.25%2A10%29%2F%280.4-0.25%29 = %28%280.32-0.25%29%2A10%29%2F0.15 = %280.07%2A10%29%2F0.15 = 0.7%2F0.15 = 4 10%2F15 = 4 2%2F3 pounds.


ANSWER.  Take 4 2%2F3 pounds of the 40% nuts and the rest, 5 1%2F3 pounds of the 25% nuts.

Solved.



Answer by greenestamps(13206) About Me  (Show Source):
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(1) A standard formal algebraic method....

x = pounds of 40% nuts
10-x = pounds of 25% nuts

The nuts make up 40% of the x pounds and 25% of the (10-x) pounds; and they make up 32% of the 10 pounds:

.40%28x%29%2B.25%2810-x%29=.32%2810%29
.40x%2B2.5-.25x=3.2
.15x=0.7
x=0.7%2F.15=70%2F15=14%2F3

ANSWER: 14/3 pounds, or 4 2/3 pounds, of the 40% nut mixture, and the other 5 1/3 pounds of the 25% nut mixture.

CHECK:
.4(14/3)+.25(16/3)=(5.6+4)/3=9.6/3=3.2
.32(10)=3.2

(2) A quick informal method for solving this type of 2-part mixture problem, if formal algebra is not required....

On a number line, observe/calculate that 32% is 7/15 of the way from 25% to 40%.
That means 7/15 of the mixture is the mix with 40% nuts.
7/15 of 10 pounds is 70/15 = 14/3 pounds or 4 2/3 pounds

ANSWER (again, of course): 4 2/3 pounds of the 40% nut mix; 5 1/3 pounds of the 25% nut mix.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

A = amount of Alpine trail mix
M = amount of Meadows trail mix
The amounts are in pounds

We want a 10 pound batch after mixing the two bags, so A+M = 10
Solving for M gets us M = 10-A

If we use A pounds of alpine, then there are 0.40A pounds of nuts in the batch (since it's 40% nuts)
Using M pounds of Meadows involves 0.25M pounds of nuts for similar reasoning.

Overall we'll have 0.40A+0.25M pounds of nuts when combining the two batches.

Dividing that amount of nuts over A+M = 10 will get us the percentage of nuts we want to aim for (32%)

(amount of nuts)/(amount of trail mix) = 32%
(0.40A+0.25M)/(10) = 0.32
0.40A+0.25M = 0.32*10
0.40A+0.25M = 3.2
Effectively, we want the final batch to have 3.2 pounds of nuts.

Now let's plug in M = 10-A to solve for A
0.40A+0.25M = 3.2
0.40A+0.25(10-A) = 3.2
0.40A+2.5-0.25A = 3.2
0.15A+2.5 = 3.2
0.15A = 3.2-2.5
0.15A = 0.7
A = 0.7/0.15
A = 70/15
A = (14*5)/(3*5)
A = 14/3
A = (12+2)/3
A = (12/3)+(2/3)
A = 4+(2/3)
A = 4 & 2/3
You should use 4 & 2/3 pounds of Alpine trail mix
In decimal form, this is roughly equal to 4.667 pounds.

M = 10-A
M = 10-(14/3)
M = (30/3)-(14/3)
M = (30-14)/3
M = 16/3
M = (15+1)/3
M = (15/3)+(1/3)
M = 5+(1/3)
M = 5 & 1/3
And you should mix in 5 & 1/3 pounds of Meadows trail mix
In decimal form, this is roughly equal to 5.333 pounds.