The art club designed and made banners of the school colors, blue and white, for their fund-raiser. Each banner required `1/4` yard of blue material and `3/8` yard of white material. The club originally planned to purchase exactly enough material to make 500 banners, but found the material to be cheaper if purchased i full bolts - the blue material in 10 yard bolts and the white material in 12-yard bolts. How many extra banners was the club able to make if they purchased enough full bolts to make at least 500 banners?

Options

A. 12
B. 13
C. 15
D. 16

Answer

To solve the problem step by step, we will follow the outlined procedure to determine how many extra banners the art club can make after purchasing full bolts of blue and white materials.

### Step 1: Calculate the material needed for 500 banners
Each banner requires:
- Blue material: \( \frac{1}{4} \) yard
- White material: \( \frac{3}{8} \) yard

For 500 banners:
- Blue material needed = \( 500 \times \frac{1}{4} = 125 \) yards
- White material needed = \( 500 \times \frac{3}{8} = 187.5 \) yards

### Step 2: Determine the number of bolts needed
The materials are purchased in full bolts:
- Blue material comes in 10-yard bolts.
- White material comes in 12-yard bolts.

**For Blue Material:**
- Total blue material needed = 125 yards
- Number of bolts required = \( \lceil \frac{125}{10} \rceil = 13 \) bolts (since \( 13 \times 10 = 130 \) yards)

**For White Material:**
- Total white material needed = 187.5 yards
- Number of bolts required = \( \lceil \frac{187.5}{12} \rceil = 16 \) bolts (since \( 16 \times 12 = 192 \) yards)

### Step 3: Calculate the extra material available
**Extra Blue Material:**
- Total blue material purchased = 130 yards
- Extra blue material = \( 130 - 125 = 5 \) yards

**Extra White Material:**
- Total white material purchased = 192 yards
- Extra white material = \( 192 - 187.5 = 4.5 \) yards

### Step 4: Calculate the number of extra banners that can be made
**Using Extra Blue Material:**
- Each banner requires \( \frac{1}{4} \) yard of blue material.
- Number of extra banners from blue material = \( \frac{5}{\frac{1}{4}} = 5 \times 4 = 20 \) banners

**Using Extra White Material:**
- Each banner requires \( \frac{3}{8} \) yard of white material.
- Number of extra banners from white material = \( \frac{4.5}{\frac{3}{8}} = 4.5 \times \frac{8}{3} = 12 \) banners

### Step 5: Determine the limiting factor
Since each banner requires both blue and white material, we take the minimum of the two calculated values:
- Extra banners from blue material = 20
- Extra banners from white material = 12

The limiting factor is the white material, so the maximum number of extra banners that can be made is **12**.

### Final Answer
The art club can make **12 extra banners**.

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