Question

Javier is making a special fertilizer for his garden. The ingredients are mixed in the ratios shown below. • 3/8of the fertilizer is cottonseed. • 1/4of the fertilizer is phosphate. • 1/4of the fertilizer is wood ash. • 1/8of the fertilizer is limestone. Part A Javier makes 24 cups of fertilizer every week. How many cups of each ingredient does he use each week? Enter your answers and your work in the box provided. Assessment 1 12 Part B Javier experiments by mixing 24 cups of his special fertilizer with some kelp meal. The new mixture is 1/3kelp meal. Javier claims that the ratio of cottonseed, phosphate, wood ash, and limestone per cup are the same as in the original recipe. Is he correct? If not, what is the correct ratio of each ingredient per cup? Explain your reasoning. Enter your answers and your work or explanation in the box provided.

Answer 1

To solve this problem, we will work through both parts step-by-step.

### Part A: Ingredient Calculation

Javier makes a total of **24 cups** of fertilizer each week. We will calculate the amount of each ingredient based on the ratios provided.

1. **Cottonseed:**
- Fraction: \( \frac{3}{8} \)
- Calculation: \( 24 \times \frac{3}{8} = 24 \times 0.375 = 9 \) cups

2. **Phosphate:**
- Fraction: \( \frac{1}{4} \)
- Calculation: \( 24 \times \frac{1}{4} = 24 \times 0.25 = 6 \) cups

3. **Wood Ash:**
- Fraction: \( \frac{1}{4} \)
- Calculation: \( 24 \times \frac{1}{4} = 24 \times 0.25 = 6 \) cups

4. **Limestone:**
- Fraction: \( \frac{1}{8} \)
- Calculation: \( 24 \times \frac{1}{8} = 24 \times 0.125 = 3 \) cups

### Summary for Part A
- **Cottonseed:** 9 cups
- **Phosphate:** 6 cups
- **Wood Ash:** 6 cups
- **Limestone:** 3 cups

---

### Part B: New Mixture Ratio

Now, Javier mixes his special fertilizer with kelp meal, where the new mixture consists of **1/3 kelp meal**. This means that the fertilizer makes up **2/3 of the new mixture** (since 1 - 1/3 = 2/3).

To analyze the new ratios, we can calculate the total volume of fertilizer in the new mixture:

- If the total volume of the new mixture is made up of \( 24 \) cups of fertilizer, then the total mixture volume can be calculated by stating that \( \frac{2}{3} \) of the total mixture amount corresponds to \( 24 \) cups.

Let \( x \) be the total amount of the new mixture. We set up the equation:
\[
\frac{2}{3}x = 24
\]

Solving for \( x \):
\[
x = 24 \times \frac{3}{2} = 36 \text{ cups (total new mixture)}
\]

#### Ratios in the New Mixture

Now let's determine the proportions of each ingredient in the mixture:

The fertilizer still is comprised of:
- Cottonseed: 9 cups
- Phosphate: 6 cups
- Wood Ash: 6 cups
- Limestone: 3 cups

Now, to find the amount of each ingredient per cup in the total new mixture, we divide the amount of each ingredient by the total amount of the new mixture.

1. **Cottonseed per cup:**
\[
\frac{9}{36} = \frac{1}{4}
\]

2. **Phosphate per cup:**
\[
\frac{6}{36} = \frac{1}{6}
\]

3. **Wood Ash per cup:**
\[
\frac{6}{36} = \frac{1}{6}
\]

4. **Limestone per cup:**
\[
\frac{3}{36} = \frac{1}{12}
\]

### Conclusion for Part B

The new ratio of each ingredient per cup is:
- Cottonseed: \( \frac{1}{4} \)
- Phosphate: \( \frac{1}{6} \)
- Wood Ash: \( \frac{1}{6} \)
- Limestone: \( \frac{1}{12} \)

**Javier's claim is incorrect.** This new ratio shows that it changes compared to the original recipe.

### Final Ratio Representation
To ensure clarity:
- Original Ratio: \( \frac{3/8}, \frac{1/4}, \frac{1/4}, \frac{1/8} \)
- New Ratio per cup: \( \frac{1}{4}, \frac{1}{6}, \frac{1}{6}, \frac{1}{12} \)

So Javier's assertion that the ratios remain the same is not true based on the mixing of the kelp meal.