Percent mixture problem using a system of linear equations.

Question

The Royal Fruit Company produces two types of fruit drinks. The first type is 70% pure fruit juice, and the second type is 95% pure fruit juice. The company is attempting to produce a fruit drink that contains 90% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 170 pints of a mixture that is 90% pure fruit juice?

Answer 1

Mixture problems like this can all be solved using the basic idea of having a system of linear equations.

First Equation:

(amount A)(value A) + (amount B)(value B) = (amount of mixture)(value of mixture)

Second Equation:

(amount A) + (amount B) = (amount of mixture)

For your problem we have the following:

1. amount of A = X (we don't know yet)
2. value of A = 70% = 0.70
3. amount of B = Y (we don't know yet)
4. value of B = 95% = 0.95
5. amount of mixture = 170 pints
6. value of mixture = 90% = 0.90

So the two equations become:

1. X(0.70) + Y(0.95) = (170)(0.90), and
2. X + Y = 170

From the second equation we cans that X = 170 - Y.

We can substitute this into the first equation to get:

1. (170 - Y)(0.70) +Y(0.95) = (170)(0.90), and we can distribute and eliminate the parenthesis to get:
2. 119 - 0.70Y + 0.95Y = 153, which can be simplified by subtracting 119 from both sides and combining like terms to get us:
3. 0.25Y = 34, now diving both sides by 0.25 will get us:
4. Y = 34 / 0.25 = 136
5. Since X = 170 - Y, we have that X = 170 - 136 = 34

And so, in order to get 170 pints of 90% juice drink you need to mix 34 pints of 70% juice with 136 pints of 95% juice.

I hope this helps you with similar problems you encounter in the future. Let me know if you found this helpful or if you need some clarification.

David

Answer 2

x = pints of 70% fruit juice

170-x = pints of 95% fruit juice

170 = pints of 90% fruit juice

x*0.70 + (170-x)*0.95 = 170(0.9)

0.7x + 161.95 - 0.95x =153

-0.25x = -8.95

x = 35.8 pints 70% fruit juice

170-x = 136.2 pints 95% fruit juice