Final answer:
To create a 6-karat gold alloy, an equation based on the karat parts of gold is used. After simplifying the equation, it is concluded that 8 ounces of 4-karat gold should be added to 8 ounces of 8-karat gold.
Step-by-step explanation:
To determine how many ounces of 4-karat gold should be added to 8 ounces of 8-karat gold to make an alloy that is 6-karat gold, we can set up an equation based on the karat values and total amounts of gold. The karat value of gold represents the gold content in 24 parts. Thus, 4-karat gold has 4/24 parts of gold, 8-karat gold has 8/24 parts of gold, and 6-karat gold has 6/24 parts of gold. The equation representing the mixture is: \(\frac{4}{24} \times x + \frac{8}{24} \times 8 = \frac{6}{24} \times (x + 8)\). Where x is the unknown amount of 4-karat gold to be added. Simplifying the equation: \(\frac{1}{6} \times x + \frac{1}{3} \times 8 = \frac{1}{4} \times (x + 8)\). Multiplying through by 12 (the least common multiple of 6, 4, and 3) to clear the fractions: \(2x + 32 = 3(x + 8)\), \(2x + 32 = 3x + 24\), \(x = 32 - 24 = 8\). Therefore, 8 ounces of 4-karat gold should be added to 8 ounces of 8-karat gold to make 6-karat gold.