Let's solve the problem step by step.

Problem Breakdown:

1. Types of drinks:

   • Drink 1: 45% pure fruit juice.
   • Drink 2: 70% pure fruit juice.

2. Goal: Mix the two drinks to create 70 pints of a mixture that is 60% pure fruit juice.

Variables:

• Let $x$ = the number of pints of the first drink (45% pure).
• Let $y$ = the number of pints of the second drink (70% pure).

The total volume of the mixture is 70 pints: $x + y = 70$

The percentage of pure fruit juice in the mixture should be 60%: $0.45x + 0.70y = 0.60 \cdot 70$

System of Equations:

1. $x + y = 70$
2. $0.45x + 0.70y = 42$ (since $0.60 \times 70 = 42$).

Solving the System:

From the first equation, express $y$ in terms of $x$: $y = 70 - x$

Substitute $y = 70 - x$ into the second equation: $0.45x + 0.70(70 - x) = 42$

Expand and simplify: $0.45x + 49 - 0.70x = 42$ $-0.25x + 49 = 42$ $-0.25x = -7$ $x = 28$

Substitute $x = 28$ back into $y = 70 - x$: $y = 70 - 28 = 42$

Solution:

• First drink (45% pure): $\mathbf{28}$ pints.
• Second drink (70% pure): $\mathbf{42}$ pints.

Would you like a detailed explanation of the calculations or further clarification?

Here are some related questions to practice:

1. What happens if the desired mixture is 50% pure fruit juice instead?
2. How do you verify that the solution satisfies both equations?
3. Can this method be applied to mixtures with more than two components?
4. How would the solution change if the total volume were 100 pints instead of 70?
5. What is the significance of the percentages in determining the proportions?

Tip: Always verify your solution by substituting it back into the original equations!