Question Find the amount of pure gold to add to a 4-ounce alloy that is $20\%$ gold to make it $55\%$ gold. Round to 2 decimal places.

Studdy Solution

STEP 1

1. The 4-ounce alloy contains $20\%$ gold, which means it contains $0.20 \times 4$ ounces of gold.
2. We are adding pure gold, which is $100\%$ gold.
3. The final mixture should be $55\%$ gold.
4. The weight of the gold in the final mixture is the sum of the gold in the original alloy and the pure gold added.
5. The total weight of the final mixture is the sum of the original alloy's weight and the weight of the pure gold added.

STEP 2

1. Determine the amount of gold in the original alloy.
2. Set up an equation based on the percentage of gold in the final mixture.
3. Solve the equation to find the amount of pure gold to be added.
4. Round the answer to two decimal places.

STEP 3

Calculate the amount of gold in the original 4-ounce alloy.
$\text{Amount of gold in original alloy} = 0.20 \times 4$

STEP 4

Perform the multiplication to find the amount of gold.
$\text{Amount of gold in original alloy} = 0.80 \text{ ounces}$

STEP 5

Let $x$ be the amount of pure gold to be added. Write an equation based on the percentage of gold in the final mixture.
$\frac{0.80 + x}{4 + x} = 0.55$

STEP 6

Multiply both sides of the equation by $(4 + x)$ to clear the denominator.
$0.80 + x = 0.55(4 + x)$

STEP 7

Distribute $0.55$ on the right side of the equation.
$0.80 + x = 0.55 \times 4 + 0.55x$

STEP 8

Perform the multiplication on the right side.
$0.80 + x = 2.20 + 0.55x$

STEP 9

Subtract $0.55x$ from both sides to get all terms involving $x$ on one side.
$0.80 + x - 0.55x = 2.20$

STEP 10

Combine like terms.
$0.80 + 0.45x = 2.20$

STEP 11

Subtract $0.80$ from both sides to isolate the term with $x$ .
$0.45x = 2.20 - 0.80$

STEP 12

Perform the subtraction on the right side.
$0.45x = 1.40$

STEP 13

Divide both sides by $0.45$ to solve for $x$ .
$x = \frac{1.40}{0.45}$

STEP 14

Perform the division to find the value of $x$ .
$x \approx 3.11$

STEP 15

Round the answer to two decimal places.
$x \approx 3.11 \text{ ounces}$
The amount of pure gold that should be added to the 4-ounce alloy to make it $55\%$ gold is approximately $3.11$ ounces.

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