An alloy contains 40% of silver, 30% of copper, and 30% of nickel. How much silver (in kg) should be added to 25 kg of the alloy so that the new alloy contains 50% of silver?
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Understanding the Alloy and Silver Percentage
The problem asks us to determine how much pure silver needs to be added to an existing alloy to increase the percentage of silver in the final mixture. We start with a known quantity and composition of the alloy.
Initial Alloy Composition
We are given an initial alloy weighing 25 kg. The composition is specified as:
- Silver: 40%
- Copper: 30%
- Nickel: 30%
Calculating Initial Weights
First, let's calculate the weight of each metal in the initial 25 kg alloy:
- Weight of Silver = $$ 40\% \times 25 \text{ kg} $$ = $$ 0.40 \times 25 \text{ kg} $$ = 10 kg
- Weight of Copper = $$ 30\% \times 25 \text{ kg} $$ = $$ 0.30 \times 25 \text{ kg} $$ = 7.5 kg
- Weight of Nickel = $$ 30\% \times 25 \text{ kg} $$ = $$ 0.30 \times 25 \text{ kg} $$ = 7.5 kg
The total weight is $$ 10 \text{ kg} + 7.5 \text{ kg} + 7.5 \text{ kg} = 25 \text{ kg} $$, which matches the given total weight.
| Metal | Percentage | Weight (kg) |
|---|---|---|
| Silver | 40% | 10 |
| Copper | 30% | 7.5 |
| Nickel | 30% | 7.5 |
| Total | 100% | 25 |
Calculating Added Silver for Target Percentage
We want the new alloy to contain 50% silver. Let 'x' represent the weight of pure silver (in kg) that needs to be added.
Setting up the Final Composition Equation
When 'x' kg of silver is added:
- The new weight of silver will be the initial weight plus the added amount: $$ (10 + x) \text{ kg} $$
- The new total weight of the alloy will be the initial weight plus the added silver: $$ (25 + x) \text{ kg} $$
The problem requires the new alloy to be 50% silver. This means the ratio of the new silver weight to the new total alloy weight must be equal to 0.50 (which is 50%):
$$ \frac{\text{New Silver Weight}}{\text{New Total Alloy Weight}} = 0.50 $$Substituting the expressions:
$$ \frac{10 + x}{25 + x} = 0.5 $$Solving the Equation for 'x'
We can solve this algebraic equation for 'x':
- Multiply both sides by $$ (25 + x) $$ to eliminate the denominator: $$ 10 + x = 0.5 \times (25 + x) $$
- Distribute the 0.5 on the right side: $$ 10 + x = 12.5 + 0.5x $$
- To isolate 'x', first subtract $$ 0.5x $$ from both sides: $$ 10 + x - 0.5x = 12.5 $$ $$ 10 + 0.5x = 12.5 $$
- Next, subtract 10 from both sides to isolate the term with 'x': $$ 0.5x = 12.5 - 10 $$ $$ 0.5x = 2.5 $$
- Finally, divide by 0.5 (or multiply by 2) to find 'x': $$ x = \frac{2.5}{0.5} $$ $$ x = 5 $$
Verifying the Result
Let's check if adding 5 kg of silver satisfies the condition:
- New Silver Weight = $$ 10 \text{ kg} + 5 \text{ kg} = 15 \text{ kg} $$
- New Total Alloy Weight = $$ 25 \text{ kg} + 5 \text{ kg} = 30 \text{ kg} $$
- Percentage of Silver = $$ \frac{15 \text{ kg}}{30 \text{ kg}} \times 100\% = 50\% $$
The result matches the required 50% silver content.
Conclusion
To achieve a new alloy containing 50% silver, 5 kg of silver must be added.
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