How much pure gold should be added to a 4-ounce alloy that is presently 20% gold to make it 55% gold? Round your answer to two decimal places.
A. 3.36-ounce
B. 3.11-ounce
C. 2.20-ounce
D. 1.65-ounce
E. 1.40-ounce
Answer (1)
Let x be the amount of pure gold to add.
Set up the equation: 4 + x 0.8 + x = 0.55 .
Solve for x : 0.8 + x = 2.2 + 0.55 x ⇒ 0.45 x = 1.4 ⇒ x = 0.45 1.4 .
Calculate x and round to two decimal places: x ≈ 3.11 ounces. The final answer is 3.11
Explanation
Problem Analysis Let's analyze the problem. We have a 4-ounce alloy that is 20% gold, and we want to add pure gold to increase the gold percentage to 55%. We need to find the amount of pure gold to add.
Setting up the equation Let x be the amount of pure gold (100% gold) to be added, in ounces. The initial amount of gold in the alloy is 4 × 0.20 = 0.8 ounces. The total weight of the new alloy will be 4 + x ounces. The total amount of gold in the new alloy will be 0.8 + x ounces. The percentage of gold in the new alloy is 4 + x 0.8 + x , which should be equal to 0.55.
The equation Now, we set up the equation: 4 + x 0.8 + x = 0.55
Solving for x Next, we solve the equation for x : 0.8 + x = 0.55 ( 4 + x ) 0.8 + x = 2.2 + 0.55 x x − 0.55 x = 2.2 − 0.8 0.45 x = 1.4 x = 0.45 1.4
Calculating x Now, we calculate the value of x : x = 0.45 1.4 ≈ 3.1111 Rounding to two decimal places, we get x ≈ 3.11 ounces.
Final Answer Therefore, we need to add approximately 3.11 ounces of pure gold to the alloy to make it 55% gold.
Examples
Imagine you are a jeweler creating a custom piece. You have some gold alloy, but it's not the right purity for your design. This problem helps you calculate exactly how much pure gold you need to add to achieve the desired gold content, ensuring your final piece meets the required standards. This kind of calculation is crucial in jewelry making, electronics manufacturing (where gold is used in connectors), and even in the creation of specialized alloys for aerospace applications.
