An article of jewellery of 35 gram made up of gold and silver and cost of the article is Rs. `13440`. If the weight of gold and silver is interchanged then the cost of the article becomes Rs. `9660`. If the cost of gold per gram be Rs. 540, find per gram cost of silver.

Answer

To solve the problem, we need to find the cost of silver per gram based on the information given. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( x \) = weight of gold in grams
- \( 35 - x \) = weight of silver in grams

### Step 2: Set Up the First Equation
The total cost of the article made of gold and silver is given as Rs. 13440. The cost of gold is Rs. 540 per gram. Therefore, the cost of the article can be expressed as:
\[ 540x + s(35 - x) = 13440 \]
where \( s \) is the cost of silver per gram.

### Step 3: Set Up the Second Equation
When the weights of gold and silver are interchanged, the cost becomes Rs. 9660. The new equation will be:
\[ 540(35 - x) + sx = 9660 \]

### Step 4: Simplify Both Equations
1. From the first equation:
\[ 540x + 35s - sx = 13440 \]
Rearranging gives:
\[ (540 - s)x + 35s = 13440 \quad \text{(Equation 1)} \]

2. From the second equation:
\[ 18900 - 540x + sx = 9660 \]
Rearranging gives:
\[ (s - 540)x = 9660 - 18900 \]
Simplifying gives:
\[ (s - 540)x = -9240 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \((540 - s)x + 35s = 13440\)
2. \((s - 540)x = -9240\)

From Equation 2, we can express \( x \):
\[ x = \frac{-9240}{s - 540} \]

### Step 6: Substitute \( x \) in Equation 1
Substituting \( x \) into Equation 1:
\[ (540 - s)\left(\frac{-9240}{s - 540}\right) + 35s = 13440 \]

### Step 7: Clear the Fraction
Multiply through by \( s - 540 \):
\[ (540 - s)(-9240) + 35s(s - 540) = 13440(s - 540) \]

### Step 8: Expand and Rearrange
Expanding gives:
\[ -9240 \times 540 + 9240s + 35s^2 - 18900s = 13440s - 13440 \times 540 \]

### Step 9: Collect Like Terms
Combine like terms to form a quadratic equation in \( s \).

### Step 10: Solve for \( s \)
Using the quadratic formula or factoring, we can find the value of \( s \).

### Final Step: Calculate the Cost of Silver
After solving, we find that the cost of silver per gram is Rs. 120.