Question

a goldsmith combined an alloy that costs $4.20 per ounce with an alloy that costs $1.80 per ounce. how many ounces of each were used to make a mixture of 200 oz costing $2.76 per ounce? $4.20 alloy oz $1.80 alloy oz

Explanation:

Step1: Set up equations

Let $x$ be the amount (in ounces) of the $\$4.20$ - per - ounce alloy and $y$ be the amount (in ounces) of the $\$1.80$ - per - ounce alloy. We know two equations:

1. The total amount of the mixture: $x + y=200$, so $y = 200 - x$.
2. The cost - based equation: $4.2x+1.8y=2.76\times200$.

Step2: Substitute $y$ into the cost - based equation

Substitute $y = 200 - x$ into $4.2x + 1.8y=2.76\times200$.
$4.2x+1.8(200 - x)=2.76\times200$.
Expand the left - hand side: $4.2x+360-1.8x = 552$.
Combine like terms: $(4.2x-1.8x)+360=552$, $2.4x+360 = 552$.

Step3: Solve for $x$

Subtract 360 from both sides: $2.4x=552 - 360$, $2.4x=192$.
Divide both sides by 2.4: $x=\frac{192}{2.4}=80$.

Step4: Solve for $y$

Since $y = 200 - x$, substitute $x = 80$ into this equation, $y=200 - 80 = 120$.

Answer:

$4.20$ alloy: 80 oz
$1.80$ alloy: 120 oz