# CAT Practice : Averages, Ratios, Mixtures

When we add one of the two components of an alloy to another alloy, how does the mixture change.

## Mixtures - Alloys

Q.16: 100 kgs of an alloy of tin and lead in the ratio 1:3 is mixed with x kgs of an alloy of tin and lead in the ratio 3:2. If the overall alloy should contain between 40% and 50% tin, what is the range of values x can take?
1. 100 kgs ≤ x ≤ 200 kgs
2. 80 kgs ≤ x ≤ 240 kgs
3. 110 kgs ≤ x ≤ 220 kgs
4. 75 kgs ≤ x ≤ 250 kgs

Choice D. 75 kgs ≤ x ≤ 250 kgs

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## Detailed Solution

Alloy 1 Tin 25 Lead 75 Total 100 kg
Alloy 2 Tin
3x / 5
2x / 5
x kg

The table shows the composition of the two alloys. The total percentage of tin is the total weight of the tin as a percentage of the total weight of the two alloys put together.
Percentage of Tin overall = $\left( {{\rm{25 + }}{{{\rm{3x}}} \over {\rm{5}}}} \right){\rm{ x }}{{{\rm{100}}} \over {{\rm{100 + x}}}}$

This number should be between 40% and 50%
40 ≤ $\left( {{\rm{25 + }}{{{\rm{3x}}} \over {\rm{5}}}} \right){\rm{ x }}{{{\rm{100}}} \over {{\rm{100 + x}}}}$ ≤ 50

Solving the inequality, x has to lie between 75 and 250 kg.

Alternatively, we can find the range by thinking of it as two mixtures problems for the two boundary conditions.
(i) x kgs of 25% Tin mixed with 100 kgs of 60% Tin to give 40% Tin → This gives x = 75 kgs.
(ii) x kgs of 25% Tin mixed with 100 kgs of 60% Tin to give 50% Tin → This gives x = 250 kgs.

Correct Answer: 75 kgs ≤ x ≤ 250 kgs

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