Arithmetic

Mixture & Alligation

Medium

If a certain weight of an alloy of silver and copper is mixed with $3$ kg of pure silver, the resulting alloy will have $90\%$ silver by weight. If the same weight of the initial alloy is mixed with $2$ kg of another alloy which has $90\%$ silver by weight, the resulting alloy will have $84\%$ silver by weight. Then, the weight of the initial alloy, in kg, is

$3$

$2.5$

$3.5$

$4$

✅ Correct Option: 1

We have an initial alloy of silver and copper with unknown weight = x kg and unknown percentage of silver = p%.

This alloy is mixed in two different scenarios, and we need to find its weight.

When mixing alloys, the total amount of silver before mixing equals the total amount of silver after mixing.

Amount of silver = (Percentage of silver) × (Weight of alloy)

x kg of initial alloy + 3 kg of pure silver = (x + 3) kg of 90% silver alloy

Silver from initial alloy = p% of x = $\frac{px}{100}$

Silver from pure silver = 100% of 3 = 3 kg

Silver in final mixture = 90% of (x + 3) = $\frac{90(x + 3)}{100}$

Since silver is conserved:

$\frac{px}{100} + 3 = \frac{90(x + 3)}{100}$

$px + 300 = 90(x + 3)$

$px + 300 = 90x + 270$

$90x - px = 300 - 270 = 30$ ... (Equation 1)

x kg of initial alloy + 2 kg of 90% silver alloy = (x + 2) kg of 84% silver alloy

Silver from initial alloy = $\frac{px}{100}$

Silver from 90% alloy = 90% of 2 = 1.8 kg

Silver in final mixture = 84% of (x + 2) = $\frac{84(x + 2)}{100}$

Since silver is conserved:

$\frac{px}{100} + 1.8 = \frac{84(x + 2)}{100}$

$px + 180 = 84(x + 2)$

$px + 180 = 84x + 168$

$84x - px = 180 - 168 = 12$ ... (Equation 2)

From our equations:

Equation 1: $90x - px = 30$

Equation 2: $84x - px = 12$

Both equations have the term "px", so subtracting eliminates this unknown.

$(90x - px) - (84x - px) = 30 - 12$

$90x - px - 84x + px = 18$

$6x = 18$

$x = 3$

The weight of the initial alloy is 3 kg.

In mixture problems, always remember that the total amount of each component remains constant before and after mixing. This principle helps you set up the conservation equations correctly.

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