In two alloys A and B, the ratio of copper and zinc is 5: 2 and 3:4 respectively. 7 kg of alloy A and 21 kg of the alloy B are mixed together to form a new alloy. What will be the ratio of copper and zinc in the new alloy?

Question

Accepted Answer

The correct option is 4: 1:1

To determine the ratio of copper and zinc in the new alloy, we must first calculate the total quantity of copper and the total quantity of zinc contributed by both alloys A and B. This problem involves weighted averages of ratios, a fundamental concept in mixtures and alligation.

**Detailed explanation of why it is correct:**

We are given two alloys, A and B, and their respective copper-to-zinc ratios.

**Alloy A:**  * Ratio of Copper to Zinc is $5:2$.
  * Total parts in the ratio are $5 + 2 = 7$.
  * The total mass of alloy A mixed is $7 \text{ kg}$.
  * The quantity of Copper in Alloy A is $\left(\frac{5}{7}\right) \times 7 \text{ kg} = 5 \text{ kg}$.
  * The quantity of Zinc in Alloy A is $\left(\frac{2}{7}\right) \times 7 \text{ kg} = 2 \text{ kg}$.**Alloy B:**  * Ratio of Copper to Zinc is $3:4$.
  * Total parts in the ratio are $3 + 4 = 7$.
  * The total mass of alloy B mixed is $21 \text{ kg}$.
  * The quantity of Copper in Alloy B is $\left(\frac{3}{7}\right) \times 21 \text{ kg} = 3 \times 3 \text{ kg} = 9 \text{ kg}$.
  * The quantity of Zinc in Alloy B is $\left(\frac{4}{7}\right) \times 21 \text{ kg} = 4 \times 3 \text{ kg} = 12 \text{ kg}$.**New Alloy (A + B):**

To find the new ratio, we sum the total quantities of copper and zinc from both alloys:  * Total Copper in new alloy = Copper in A + Copper in B = $5 \text{ kg} + 9 \text{ kg} = 14 \text{ kg}$.
  * Total Zinc in new alloy = Zinc in A + Zinc in B = $2 \text{ kg} + 12 \text{ kg} = 14 \text{ kg}$.The ratio of Copper to Zinc in the new alloy is:
$$\text{Ratio} = \frac{\text{Total Copper}}{\text{Total Zinc}} = \frac{14 \text{ kg}}{14 \text{ kg}} = \frac{1}{1}$$
The ratio is $1:1$. This means the new alloy contains equal amounts of copper and zinc.

**The incorrect option is 1: 1:2**

If the ratio were $1:2$, it would imply that the total amount of copper ($14 	ext{ kg}$) is half the total amount of zinc ($14 	ext{ kg}$), meaning the total zinc would be $28 	ext{ kg}$. However, as calculated, the total zinc is $14 	ext{ kg}$. A ratio of $1:2$ would only be correct if, after mixing, the total amount of zinc was double the total amount of copper.

**The incorrect option is 2: 2:1**

A ratio of $2:1$ would mean that the total amount of copper ($14 	ext{ kg}$) is twice the total amount of zinc ($14 	ext{ kg}$). For this to be true, the total zinc would have to be $7 	ext{ kg}$. Since the calculated total zinc is $14 	ext{ kg}$, this option is incorrect. This ratio represents the reciprocal of the incorrect ratio $1:2$ and would imply a majority of copper.

**The incorrect option is 3: 2:3**

If the ratio were $2:3$, the total copper to zinc would be in this proportion. Based on our calculated total copper of $14 	ext{ kg}$, the total zinc required for a $2:3$ ratio would be $\frac{3}{2} 	imes 14 	ext{ kg} = 21 	ext{ kg}$. Since the total zinc is only $14 	ext{ kg}$, this option is incorrect. This ratio is incorrect because the quantity of copper is not two-thirds of the quantity of zinc.