To determine a suitable combination of headbands and bracelets that Janelle and her friends could sell to achieve their goals, we can set up two equations based on the information provided:
Let:
- \( h \) = number of headbands sold
- \( b \) = number of bracelets sold
We have the following conditions:
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The total amount of money earned needs to be more than $200: \[ 4h + 2b > 200 \]
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The total number of items sold cannot exceed 100: \[ h + b \leq 100 \]
To find a suitable combination, we can manipulate both equations. First, let's simplify the first equation:
From \( 4h + 2b > 200 \), we can divide the entire equation by 2 to simplify it: \[ 2h + b > 100 \]
Now we can explore some possible combinations of \( h \) and \( b \) that satisfy both inequalities.
Example Combinations:
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Try \( h = 50 \) and \( b = 0 \):
- \( 2(50) + 0 > 100 \): \( 100 > 100 \) (not valid)
- Total items: \( 50 + 0 = 50 \) (valid)
- Total sales: \( 4(50) + 2(0) = 200 \) (not more than 200).
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Try \( h = 51 \) and \( b = 0 \):
- Total items: \( 51 + 0 = 51 \) (valid)
- Total sales: \( 4(51) + 2(0) = 204 \) (valid).
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Try \( h = 48 \) and \( b = 4 \):
- Total items: \( 48 + 4 = 52 \) (valid)
- Total sales: \( 4(48) + 2(4) = 192 + 8 = 200 \) (not more than 200).
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Try \( h = 47 \) and \( b = 8 \):
- Total items: \( 47 + 8 = 55 \) (valid)
- Total sales: \( 4(47) + 2(8) = 188 + 16 = 204 \) (valid).
Based on these findings, the combination of \( h = 51 \) and \( b = 0 \), or \( h = 47 \) and \( b = 8 \) would be suitable.
Thus, a valid combination Janelle can sell is:
- Headbands = 51, Bracelets = 0, or
- Headbands = 47, Bracelets = 8.
Both scenarios maintain the conditions of selling more than $200 while selling no more than 100 items total.
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