A bead shop sell wooden beads for `$0.20` each and crystal beads for `$0.50` each. If a jewelry artist buys 127 beads total and pays `$41` for them, how much more did she spend on crystal beads than wooden beads ?

Options

A. `$11`
B. `$15`
C. `$23`
D. `$26`

Explanation

To solve the problem step by step, we will set up a system of equations based on the information given.

### Step 1: Define Variables
Let:
- \( c \) = number of crystal beads
- \( w \) = number of wooden beads

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of beads is 127:
\[ c + w = 127 \quad \text{(Equation 1)} \]

2. The total cost of the beads is $41. The cost of wooden beads is $0.20 each and the cost of crystal beads is $0.50 each:
\[ 0.20w + 0.50c = 41 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
To eliminate the decimals, we can multiply Equation 2 by 10:
\[ 2w + 5c = 410 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have the following system of equations:
1. \( c + w = 127 \)
2. \( 2w + 5c = 410 \)

From Equation 1, we can express \( w \) in terms of \( c \):
\[ w = 127 - c \]

### Step 5: Substitute \( w \) in Equation 3
Now substitute \( w \) in Equation 3:
\[ 2(127 - c) + 5c = 410 \]
Expanding this gives:
\[ 254 - 2c + 5c = 410 \]
Combine like terms:
\[ 3c + 254 = 410 \]

### Step 6: Solve for \( c \)
Subtract 254 from both sides:
\[ 3c = 410 - 254 \]
\[ 3c = 156 \]
Divide by 3:
\[ c = 52 \]

### Step 7: Find \( w \)
Now substitute \( c \) back into the equation for \( w \):
\[ w = 127 - 52 = 75 \]

### Step 8: Calculate the Total Cost for Each Type of Bead
Now we can find out how much was spent on each type of bead:
- Cost of crystal beads:
\[ \text{Cost of crystal beads} = 0.50 \times c = 0.50 \times 52 = 26 \]
- Cost of wooden beads:
\[ \text{Cost of wooden beads} = 0.20 \times w = 0.20 \times 75 = 15 \]

### Step 9: Find the Difference in Spending
To find out how much more was spent on crystal beads than wooden beads:
\[ \text{Difference} = \text{Cost of crystal beads} - \text{Cost of wooden beads} = 26 - 15 = 11 \]

### Final Answer
The jewelry artist spent $11 more on crystal beads than on wooden beads.

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