ratio and proportion

Question

Apple and grape juice are combined in a ratio of 4 to 5 to make fruit punch. After 5 cups of apple juice are added to a pitcher of fruit punch, the amounts of apple and grape juice in the mixture are equal.

How many cups of fruit punch were in the pitcher to begin with?

Answer 1

I'm trying to find the right words here, but I'm having difficulty, so try to follow my line of reasoning as best as you can.

there are 4 parts of apple to 5 parts of grape. This means that there's 1 more part of grape amount to the apple amount

(5 - 4 = 1).

 

Since we're adding 5 parts of apple to the juice, and that end up causing the juice to have equal parts apple and grape, there had to be 5 more parts of grape to begin with.

 

The fruit juice is 4 parts apple and 5 parts grape, so the total number of fruit parts is 9

(4 + 5 = 9).

 

Now to have 5 more cups of grape and keep our ratio, we multiply the 9 by 5:

9 * 5 = 45

 

Lets check by looking at the total amount of apples and grapes in our fruit solution:

(4 * 5) + (5 * 5) = 45

20 + 25 = 45

 

And would you look at that! If we add 5 more cups to the apple, that would indeed equal the amount of grapes!

(20 + 5 = 25)

 

So there was 45 cups of fruit punch to begin with.