Not Disvi yi ble To Higher Order Thinking Ines needs 237 wooden plaques for a craft project. The wooden plaques are sold in packs of 3 or 9 . What is the least number of packs she needs to buy? Explain.

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To determine the least number of packs Ines needs to buy to obtain exactly **237 wooden plaques**, where the plaques are sold in packs of **3** or **9**, we can use a methodical approach. Here's a step-by-step explanation:

### 1. **Understand the Problem**

- **Total Plaques Needed:** 237
- **Pack Sizes Available:** 3 plaques and 9 plaques

### 2. **Formulate the Equation**

Let:
- $$ x $$ = number of **3-packs**
- $$ y $$ = number of **9-packs**

The total number of plaques can be represented by the equation:
$$ 3x + 9y = 237 $$

### 3. **Simplify the Equation**

Notice that both coefficients and the total are divisible by **3**. Divide the entire equation by **3** to simplify:
$$ x + 3y = 79 $$

### 4. **Express One Variable in Terms of the Other**

Solve for $$ x $$:
$$ x = 79 - 3y $$

### 5. **Determine the Objective**

We aim to **minimize the total number of packs**, which is $$ x + y $$.

Substitute $$ x $$ from the previous step:
$$ x + y = (79 - 3y) + y = 79 - 2y $$

To **minimize $$ x + y $$**, we need to **maximize $$ y $$** because $$ x + y $$ decreases as $$ y $$ increases.

### 6. **Find the Maximum Possible $$ y $$**

Since both $$ x $$ and $$ y $$ must be non-negative integers:
$$ 79 - 3y \geq 0 $$
$$ 3y \leq 79 $$
$$ y \leq \frac{79}{3} $$
$$ y \leq 26.333 $$

Since $$ y $$ must be an integer, the maximum possible value for $$ y $$ is **26**.

### 7. **Calculate $$ x $$ Corresponding to $$ y = 26 $$**

$$ x = 79 - 3(26) $$
$$ x = 79 - 78 $$
$$ x = 1 $$

### 8. **Verify the Solution**

Ensure that the total number of plaques adds up to **237**:
$$ 3x + 9y = 3(1) + 9(26) = 3 + 234 = 237 $$

### 9. **Conclusion**

- **Number of 3-packs ($$ x $$)**: **1**
- **Number of 9-packs ($$ y $$)**: **26**
- **Total Number of Packs**: $$ 1 + 26 = 27 $$

Therefore, **Ines needs to buy a minimum of 27 packs**—1 pack of 3 plaques and 26 packs of 9 plaques—to obtain exactly 237 wooden plaques.
Ines needs to buy **27 packs** in total: **1 pack of 3 plaques** and **26 packs of 9 plaques** to get exactly 237 wooden plaques.

Not Disvi yi ble To Higher Order Thinking Ines needs 237 wooden plaques for a craft project. The wooden plaques are sold in packs of 3 or 9 . What is the least number of packs she needs to buy? Explain.

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