Final answer:
The correct answer is A. 6.2 lbs of each metal. This is found by creating a system of equations based on the total weight and gold content required and solving for the quantities of the two differing gold content alloys.
Step-by-step explanation:
To create an alloy containing 50% gold weighing 12.4 lbs, the metallurgist must combine two alloys: one with 60% gold and one with 40% gold. Let's designate x as the weight in pounds of the 60% gold alloy and y as the weight in pounds of the 40% gold alloy. The sum of these weights must equal 12.4 lbs, which is the total weight of the desired alloy. Hence, the equation x + y = 12.4.
Next, considering the gold content in each type of alloy, we get another equation based on the percentage of gold in the final alloy. We multiply the weight of each alloy by its gold percentage and set their sum equal to the total amount of gold in the finished alloy, which is 50% of 12.4 lbs, or 6.2 lbs. Our second equation is then 0.60x + 0.40y = 6.2.
To solve this system of equations, we can use either substitution or elimination. Let's use elimination in this example:
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- Multiply the second equation by 10 to eliminate the decimals: 6x + 4y = 62
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- Multiply the first equation by 4 and subtract it from the modified second equation: 6x + 4y - (4x + 4y) = 62 - 49.6
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- Simplify to find x: 2x = 12.4, so x = 6.2 lbs
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- Substitute x into the first equation to find y: 6.2 + y = 12.4, thus y = 6.2 lbs
Therefore, the metallurgist needs 6.2 lbs of the 60% gold alloy and 6.2 lbs of the 40% gold alloy, which makes option A the correct answer.