Educational math worksheet on ratios, rates, and unit rates with visual examples.

Problem Analysis:

The task involves solving a problem related to rates, ratios, and unit rates. The specific problem is:

> Problem:
> A recipe calls for 3 cups of flour for every 2 cups of sugar. If you want to make a batch that uses 9 cups of flour, how many cups of sugar will you need?

Step-by-Step Solution:

1. Understand the Given Ratio:
- The recipe specifies a ratio of flour to sugar as \( 3 \text{ cups of flour} : 2 \text{ cups of sugar} \).
- This can be written as a fraction: \( \frac{\text{flour}}{\text{sugar}} = \frac{3}{2} \).

2. Set Up the Proportion:
- We are asked to find the amount of sugar needed when using 9 cups of flour.
- Let \( x \) represent the number of cups of sugar needed.
- According to the ratio, the proportion can be set up as:
\[
\frac{\text{flour}}{\text{sugar}} = \frac{3}{2} = \frac{9}{x}
\]

3. Solve the Proportion:
- Cross-multiply to solve for \( x \):
\[
3 \cdot x = 2 \cdot 9
\]
\[
3x = 18
\]
- Divide both sides by 3:
\[
x = \frac{18}{3} = 6
\]

4. Interpret the Result:
- If you use 9 cups of flour, you will need 6 cups of sugar to maintain the same ratio as the original recipe.

Final Answer:

\[
\boxed{6}
\]

Explanation:

The key to solving this problem is understanding and applying the concept of proportions. The given ratio \( \frac{3}{2} \) ensures that the relationship between flour and sugar remains consistent. By setting up the proportion \( \frac{3}{2} = \frac{9}{x} \) and solving for \( x \), we ensure that the quantities are scaled correctly while maintaining the original ratio. This method is fundamental in problems involving rates and ratios.