Math problem is very hard

Question

The Royal Fruit Company produces two types of fruit drinks. The first type is 60% pure fruit juice, and the second type is 85% pure fruit juice. The company is attempting to produce a fruit drink that contains 70% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 50% pints of a mixture that is pure fruit juice?

Answer 1

Well, first, let's write down what we know and work with that.

We want to make 50 pints of something.

We make those 50 pints using pints of two different drinks.

Let's call those drinks x and y (x for the 60% fruit juice and y for the 85% fruit juice).

So,

x+y=50 (the total pints from both drinks is 50)

Now, let's nail down the mixture.

We get 0.6 (60% in decimal form) of a pint of pure fruit juice for each pint of 60% fruit juice and 0.85 (85% in decimal form) of a pint of pure fruit juice for each pint of 85% fruit juice.

We want 50 pints of a drink that is 70% fruit juice, so 0.7 (70% in decimal form) of a pint of pure fruit juice for each pint of 70% fruit juice.

Notice that (pure juice per pint) x (number of pints) = pure juice. So,

0.6x + 0.85y = 0.7(50) = 35 (pure juice in the mixture).

Since we need both equations to hold, we get the resulting system of equations:

x+y=50

0.6x+0.85y=35

Solving for y in the first equation, we have

y=50-x

Substituting this y into the second equation, we have

0.6x+0.85(50-x)=35

-0.25x+42.5=35 (simplified the x coefficient)

-0.25x=-7.5 (subtracted 42.5 from both sides)

x=30 (divided both sides by -0.25)

Now that we have x=30, we can go back to our first equation:

x+y=50

30+y=50 (substituting x=30)

y=20 (subtracted 30 from both sides)

Therefore, 30 pints of the 60% drink and 20 pints of the 85% drink.

To check, we use the second equation:

0.6x+0.85y=0.6(30)+0.85(20)=35=35 check.

Note that this process didn't really depend on the numbers. It works for general mixtures (such as mixing up cleaner for around the house).