Answer
The jeweler should need $\text{96 grams}$ of $18-\text{ karat}$ gold and $\text{204 grams}$ of $12-\text{ karat}$ gold.
Work Step by Step
Let $x$ represent the grams of $18-\text{karat}$ gold,
Let $y$ represent the grams of $12-\text{karat}$ gold,
The amount of pure gold in each solution is found by multiplying the amount of karat by the concentration rate. This information can be organized in a table.
There are two unknown quantities; therefore, a system of two independent equations relating $x$ and $y$ is set up.
Amount of 75 percent gold+Amount of 28 percent gold=Amount of 25 percent gold
And:
Amount of pure 75 percent gold+Amount of pure 28 percent gold=Amount of pure 25 percent gold
Consider the equation,
$\begin{align}
& x+y=300 \\
& x=300-y
\end{align}$ …… (1)
And
$0.75x+0.50y=174$ …… (2)
Substitute $300-y$for $x$ in equation $\left( 2 \right)$ to get,
$\begin{align}
& 0.75\left( 300-y \right)+0.50y=174 \\
& 225-0.75y+0.50y=174 \\
& -0.25y=-51
\end{align}$
Divide above equation by $-0.25$ to get,
$\begin{align}
& \frac{-0.25y}{-0.25}=\frac{-51}{-0.25} \\
& y=204
\end{align}$
Substitute $204$for $y$ in equation $\left( 1 \right)$ to get,
$\begin{align}
& x=300-204 \\
& x=96 \\
\end{align}$
Check: $\left( 96,204 \right)$
Put $x=96$and $y=204$ in the equation (1),
$\begin{align}
\left( 96 \right)+\left( 204 \right)\overset{?}{\mathop{=}}\,300 & \\
300=300 & \\
\end{align}$
And Put $x=96$and $y=204$ in the equation (2),
$\begin{align}
0.75\left( 96 \right)+0.50\left( 204 \right)\overset{?}{\mathop{=}}\,174 & \\
72+102\overset{?}{\mathop{=}}\,174 & \\
174=174 & \\
\end{align}$
The ordered pair $\left( 96,204 \right)$ satisfies both equations.
Hence, the jeweler should need $\text{96 grams}$ of $18-\text{ karat}$ gold and $\text{204 grams}$ of $12-\text{ karat}$ gold.
