$\frac{1}{24}$
This problem requires calculating the final proportion of copper in a gold alloy formed by mixing equal weights of 22 carat gold and 24 carat gold.
Analyzing Gold Carat Compositions
- 22 Carat Gold: This alloy consists of 22 parts gold and 2 parts copper by weight, making a total of 24 parts. The proportion of copper in 22 carat gold is $\frac{2}{24}$.
- 24 Carat Gold: This is pure gold, meaning it contains 0 parts copper. The proportion of copper is 0.
Calculating Copper in the Mixture
Assume we mix equal weights of both alloys, say $W$ kg of 22 carat gold and $W$ kg of 24 carat gold.
- The total weight of the final alloy is $W + W = 2W$ kg.
- The weight of copper contributed by the 22 carat gold is $W \times \frac{2}{24} = \frac{2W}{24}$ kg.
- The weight of copper contributed by the 24 carat gold is $W \times 0 = 0$ kg.
- The total weight of copper in the final alloy is $\frac{2W}{24} + 0 = \frac{2W}{24}$ kg.
Determining Final Copper Proportion
The weight proportion of copper in the final alloy is the total weight of copper divided by the total weight of the alloy:
Proportion of Copper = $\frac{\text{Total Copper Weight}}{\text{Total Alloy Weight}}$
Proportion of Copper = $\frac{\frac{2W}{24}}{2W}$
Simplify the expression:
Proportion of Copper = $\frac{2W}{24 \times 2W} = \frac{1}{24}$
Therefore, the weight proportion of copper in the final alloy is $\frac{1}{24}$.
Important Questions from Mixture Problems (Notes)
- A has a container containing 60 litres of pure milk. He takes out 4 litres of milk and replaces it with the same quantity of water. He sells this mixture to B. B sells 30 litres of the mixture and added 5 litres of water in the remaining mixture. The ratio of milk to water in the remaining mixture is:
- In 80 litres mixture of milk and water, the ratio of amount of milk to that of amount of water is 7 : 3. In order to make this ratio 2 : 1, how many litres of water should be added ?
- A $595$ litre of mixture contains milk and water in the ratio $17:18$. How much milk must be added to the mixture so that it contains milk and water in the proportion of $3:2$?
- How many liters of water should be added to a 30-liter mixture containing milk and water in the ratio 7:3 such that the resultant mixture has 40% water in it?
Two solutions $X$ and $Y$ containing ingredients $A, B$ and $C$, in proportions $a:b:c$ and $c:b:a$, respectively, are mixed. For the resultant mixture to have $A, B$ and $C$ in $1:1:1$ proportion, it is necessary that $a:b:c$ is
