A jeweler had a fixed amount of gold to make bracelets and necklaces. The amount of gold in each bracelet is 5 grams and the amount of gold in each necklace is 24 grams. The jeweler made a total of 12 bracelets and necklaces using 155 grams of gold. Determine the number of bracelets made and the number of necklaces made.
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To solve this problem, let's introduce variables: let \( x \) be the number of bracelets and \( y \) be the number of necklaces. We have two equations based on the information provided: 1. \( x + y = 12 \) (the total number of bracelets and necklaces) 2. \( 5x + 24y = 155 \) (the total amount of gold used in grams) Now, we'll solve these equations step by step. From the first equation, we can express \( y \) in terms of \( x \): \[ y = 12 - x \] Now, substitute this expression for \( y \) into the second equation: \[ 5x + 24(12 - x) = 155 \] Distributing the 24: \[ 5x + 288 - 24x = 155 \] Combine like terms: \[ -19x + 288 = 155 \] Subtract 288 from both sides: \[ -19x = 155 - 288 \] \[ -19x = -133 \] Now, divide by -19: \[ x = \frac{133}{19} = 7 \] Now that we have \( x \), we can find \( y \): \[ y = 12 - 7 = 5 \] Therefore, the jeweler made **7 bracelets** and **5 necklaces**.
