A candy distributor needs to mix a 10% fat-content chocolate with a 30% fat-content chocolate to create 200 kilograms of a 17% fat-content chocolate. How many kilograms of each kind of chocolate must they use?

Answer: They must mix
 Kilograms of the 10% chocolate and
 Kilograms of the 30% chocolate.

We can solve this problem using a system of equations. Let:

- \( x \) be the kilograms of 10% fat-content chocolate.
- \( y \) be the kilograms of 30% fat-content chocolate.

We have two conditions:

1. The total weight of the mixture is 200 kilograms:
   \[
   x + y = 200
   \]

2. The total fat content of the mixture must be 17%, which gives us the equation for the fat content:
   \[
   0.10x + 0.30y = 0.17 \times 200
   \]
   Simplifying the right-hand side:
   \[
   0.10x + 0.30y = 34
   \]

Now, we solve the system of equations:

1. \( x + y = 200 \)
2. \( 0.10x + 0.30y = 34 \)

### Step 1: Solve for one variable
From the first equation, solve for \( y \) in terms of \( x \):
\[
y = 200 - x
\]

### Step 2: Substitute into the second equation
Substitute \( y = 200 - x \) into the second equation:
\[
0.10x + 0.30(200 - x) = 34
\]
Simplifying:
\[
0.10x + 60 - 0.30x = 34
\]
\[
-0.20x + 60 = 34
\]
\[
-0.20x = 34 - 60
\]
\[
-0.20x = -26
\]
\[
x = \frac{-26}{-0.20} = 130
\]

### Step 3: Find \( y \)
Now that we know \( x = 130 \), substitute into the equation \( y = 200 - x \):
\[
y = 200 - 130 = 70
\]

### Final Answer:
- \( x = 130 \) kilograms of 10% fat-content chocolate.
- \( y = 70 \) kilograms of 30% fat-content chocolate.

The distributor needs to mix **130 kilograms of the 10% chocolate** and **70 kilograms of the 30% chocolate**.

Would you like more details or have any questions? 

Here are some related questions:
1. How would the mixture change if the desired fat content was 20% instead of 17%?
2. What happens if only 150 kilograms of chocolate are required with the same fat percentages?
3. How much fat is there in total in the 200 kilograms of the final mixture?
4. If we switch the percentages, using 30% fat chocolate as the primary one, how does the result change?
5. Can you solve for a scenario where the 10% chocolate isn't available?

**Tip**: When working with mixtures, use systems of linear equations to keep track of both the total quantity and the desired concentration!

A candy distributor needs to mix a 10% fat-content chocolate with a 30% fat-content chocolate to create 200 kilograms of a 17% fat-content chocolate. How many kilograms of each kind of chocolate must they use?

Learn how to mix 10% and 30% fat-content chocolates to create a 200 kg mixture with a 17% fat content. This detailed guide includes step-by-step calculations using linear equations. Ideal for Grades 8-10 students studying mixture problems in algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation