https://file.aiquickdraw.com/mathbot/file/17289695751258781ue75.jpg

The problem you're dealing with is a classic mixture problem where two different concentrations of fruit juice need to be combined to produce a mixture with a desired concentration.

Let's define:
- \( x \) = the number of pints of the 45% pure fruit juice.
- \( y \) = the number of pints of the 95% pure fruit juice.

We are given:
- The total amount of the mixture should be 140 pints.
- The concentration of the final mixture should be 90% pure fruit juice.

From this, we have two equations:
1. The total number of pints:  
   \[ x + y = 140 \]
2. The total amount of pure fruit juice in the mixture:  
   \[ 0.45x + 0.95y = 0.90 \times 140 \]
   \[ 0.45x + 0.95y = 126 \]

### Solving the system of equations:

1. From the first equation:
   \[ y = 140 - x \]

2. Substitute \( y = 140 - x \) into the second equation:
   \[ 0.45x + 0.95(140 - x) = 126 \]
   Expanding this equation:
   \[ 0.45x + 0.95 \times 140 - 0.95x = 126 \]
   \[ 0.45x + 133 - 0.95x = 126 \]
   Combine like terms:
   \[ -0.50x + 133 = 126 \]
   \[ -0.50x = 126 - 133 \]
   \[ -0.50x = -7 \]
   \[ x = \frac{-7}{-0.50} = 14 \]

So, \( x = 14 \) pints of the 45% pure fruit juice.

3. Now, substitute \( x = 14 \) back into the first equation to find \( y \):
   \[ y = 140 - 14 = 126 \]

Thus, you need:
- 14 pints of the 45% pure fruit juice.
- 126 pints of the 95% pure fruit juice.

Would you like further clarification or additional details?

Here are 5 related questions to explore:

1. How do we check the correctness of the solution for mixture problems?
2. How can we generalize this method for other concentration mixture problems?
3. What are other real-life applications of mixture problems?
4. Can we use graphical methods to solve systems of equations like this one?
5. How do the proportions change if we alter the desired concentration?

**Tip:** Always ensure the total volume and the desired concentration match when combining mixtures.

The Royal Fruit Company produces two types of fruit drinks. The first type is 45% 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 140 pints of a mixture that is 90% pure fruit juice?

Learn how to solve a classic mixture problem where two drinks, one with 45% and the other with 95% pure fruit juice, are combined to create 140 pints of a 90% fruit juice mixture. This detailed solution walks through the algebraic steps and system of equations necessary for solving the problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation