The following table gives the calories, fat, and carbohydrates per ounce for three brands of cereal. How many ounces of each brand should be combined to get 659 calories, 5.9 g of fat and 164.8 g of carbohydrates?

CerealCaloriesFat (g)Carbohydrates (g)

All Fiber

 50

0   

22.0

Frosted Puffs

108

0.1

25.7

Natural Mixed Grain

127

5.5

18.0




All Fiber    

oz

Frosted Puffs    

oz

Natural Mixed Grain    

Let's assign variables to represent the number of ounces of each type of cereal to combine:

- Let \( x_1 \) be the number of ounces of **All Fiber** cereal.
- Let \( x_2 \) be the number of ounces of **Frosted Puffs** cereal.
- Let \( x_3 \) be the number of ounces of **Natural Mixed Grain** cereal.

We are given the following information for each cereal:

| Cereal              | Calories per ounce | Fat per ounce (g) | Carbohydrates per ounce (g) |
|---------------------|--------------------|-------------------|-----------------------------|
| **All Fiber**        | 50                 | 0.0               | 22.0                        |
| **Frosted Puffs**    | 108                | 0.1               | 25.7                        |
| **Natural Mixed Grain** | 127             | 5.5               | 18.0                        |

We need to combine these cereals so that the mixture has:
- **659 calories**, 
- **5.9 grams of fat**, and 
- **164.8 grams of carbohydrates**.

This gives us the following system of equations based on the total calories, fat, and carbohydrates:

1. **Calories equation**:
   \[
   50x_1 + 108x_2 + 127x_3 = 659
   \]

2. **Fat equation**:
   \[
   0.0x_1 + 0.1x_2 + 5.5x_3 = 5.9
   \]

3. **Carbohydrates equation**:
   \[
   22.0x_1 + 25.7x_2 + 18.0x_3 = 164.8
   \]

Now we can solve this system of equations to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).

Let me calculate that for you.The solution to the system of equations is:

- **2 ounces** of **All Fiber** cereal,
- **4 ounces** of **Frosted Puffs** cereal, and
- **1 ounce** of **Natural Mixed Grain** cereal.

This combination will give you exactly 659 calories, 5.9 grams of fat, and 164.8 grams of carbohydrates.

Would you like more details on the solution process or any further clarification?

Here are some related questions to explore:
1. How would the mixture change if you wanted fewer carbohydrates?
2. What if the goal was to lower the fat content while keeping the calorie count the same?
3. Can you adjust the mixture to include only two of the cereals while still meeting the nutrient goals?
4. How does each cereal contribute to the total calories, fat, and carbohydrates?
5. What is the ratio of the cereals in the mixture?

**Tip:** When working with systems of equations, it's helpful to check your answers by plugging them back into the original equations to ensure they satisfy all conditions.

The following table gives the calories, fat, and carbohydrates per ounce for three brands of cereal. How many ounces of each brand should be combined to get 659 calories, 5.9 g of fat and 164.8 g of carbohydrates?

| Cereal              | Calories per ounce | Fat per ounce (g) | Carbohydrates per ounce (g) |
|---------------------|--------------------|-------------------|-----------------------------|
| **All Fiber**        | 50                 | 0.0               | 22.0                        |
| **Frosted Puffs**    | 108                | 0.1               | 25.7                        |
| **Natural Mixed Grain** | 127             | 5.5               | 18.0                        |

How many ounces of each brand should be used?

Learn how to solve a system of linear equations to determine the exact amounts of All Fiber, Frosted Puffs, and Natural Mixed Grain cereals needed to achieve 659 calories, 5.9 grams of fat, and 164.8 grams of carbohydrates. This math problem covers essential algebraic concepts and is ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation