Question

You have a 10% sugar solution and a 30% sugar solution, but you need a 25% sugar solution to fill your

hummingbird feeders. You want to know how many cups of each solution you should mix together to make 8
cups of a 25% solution.
Let x represent the number of cups of the 10% solution and y represent the number of cups of the 30%
solution.

What equation represents the total volume?

X+y=____

What equation represents the total volume of sugar?

__x+__y=__

Solve the system formed by the two equations.
How many cups of each mixture should be combined?

__cups of the 10% sugar solution and __ cups of the 30% sugar solution.

Answer 1

Answers:

`x+y = \boxed{ \ 8 \ }`

`\boxed{ \ 0.10 \ }x+\boxed{ \ 0.30 \ }y = \boxed{ \ 2\ }`

`\boxed{ \ 2 }` cups of the 10% sugar solution

`\boxed{ \ 6 }` cups of the 30% sugar solution

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Work Shown:

x = number of cups of the 10% solution

y = number of cups of the 30% solution

The two mixes add together to get 8 cups since this is the total amount we want. Therefore, the first equation is simply x+y = 8

Solve for y to get y = 8-x. We'll use this equation later.

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The second equation is a bit more tricky.

0.10x = amount of pure sugar from just the 10% batch

0.30y = amount of pure sugar from just the 30% batch

0.10x+0.30y = total amount of pure sugar from both batches

25% of 8 = 0.25*8 = 2 = total amount of pure sugar required

0.10x+0.30y = 2 is the second equation

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Apply substitution to solve for x and y

0.10x+0.30y = 2 ..... start with the second equation

0.10x+0.30(8-x) = 2 ....... plug in y = 8-x

0.10x+2.4-0.30x = 2 ..... distribute

-0.20x + 2.4 = 2

-0.20x = 2-2.4 ....... subtract 2.4 from both sides

-0.20x = -0.4

x = -0.4/(-0.20) ..... divide both sides by -0.20

x = 2

Now that we know the value of x, we can find y.

y = 8-x

y = 8-2 ... plug in x = 2

y = 6

So we need 2 cups of the 10% solution and 6 cups of the 30% solution.

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As a check,

x+y = 2+6 = 8 works out

0.10*x+0.30*y = 0.10*2+0.30*6 = 0.2+1.8 = 2 works out as well