A jeweller has bars of 18 carat gold and 12 carat gold. How much of each must be melted together to obtain a bar of 16 carat gold, weighing 120g (Given pure gold is of 24 carat.)?

Answer

To solve the problem of how much of each type of gold the jeweller needs to melt together to obtain a bar of 16 carat gold weighing 120 grams, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( x \) = grams of 18 carat gold
- \( y \) = grams of 12 carat gold

### Step 2: Set Up the Equations
We know that the total weight of the gold must equal 120 grams. Therefore, we can write our first equation as:
\[ x + y = 120 \]

Next, we need to consider the gold content in terms of carats. The carat value indicates the purity of the gold:
- 18 carat gold contains \( \frac{18}{24} \) pure gold.
- 12 carat gold contains \( \frac{12}{24} \) pure gold.
- 16 carat gold contains \( \frac{16}{24} \) pure gold.

The total amount of pure gold from both types of gold must equal the amount of pure gold in the 120 grams of 16 carat gold:
\[ \frac{18}{24}x + \frac{12}{24}y = \frac{16}{24} \times 120 \]

### Step 3: Simplify the Second Equation
We can simplify the second equation:
\[ \frac{18}{24}x + \frac{12}{24}y = \frac{16 \times 120}{24} \]
This simplifies to:
\[ \frac{3}{4}x + \frac{1}{2}y = 80 \]

To eliminate the fractions, we can multiply the entire equation by 4:
\[ 3x + 2y = 320 \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x + y = 120 \)
2. \( 3x + 2y = 320 \)

We can solve these equations simultaneously. From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 120 - x \]

Substituting \( y \) into the second equation:
\[ 3x + 2(120 - x) = 320 \]
Expanding this gives:
\[ 3x + 240 - 2x = 320 \]
Combining like terms:
\[ x + 240 = 320 \]
Subtracting 240 from both sides:
\[ x = 80 \]

### Step 5: Find the Value of \( y \)
Now, substitute \( x = 80 \) back into the equation for \( y \):
\[ y = 120 - 80 = 40 \]

### Conclusion
Thus, the jeweller needs:
- 80 grams of 18 carat gold
- 40 grams of 12 carat gold