4 kg of a metal contains `1/5` copper and rest is zinc. Another 5 kg of metal contains `1/6` copper and rest is zinc. The ratio of copper and zinc into the mixture of these two metals :

Options

A. 49 : 221
B. 39 : 231
C. 94 : 181
D. none of these

Answer

To solve the problem of finding the ratio of copper and zinc in the mixture of two metals, we will follow these steps:

### Step 1: Calculate the amount of copper in the first metal.
The first metal weighs 4 kg and contains \( \frac{1}{5} \) copper.

\[ \text{Amount of copper in the first metal} = 4 \times \frac{1}{5} = \frac{4}{5} \text{ kg} \]

### Step 2: Calculate the amount of zinc in the first metal.
The rest of the first metal is zinc. Therefore, the amount of zinc is:

\[ \text{Amount of zinc in the first metal} = 4 - \frac{4}{5} = \frac{20}{5} - \frac{4}{5} = \frac{16}{5} \text{ kg} \]

### Step 3: Calculate the amount of copper in the second metal.
The second metal weighs 5 kg and contains \( \frac{1}{6} \) copper.

\[ \text{Amount of copper in the second metal} = 5 \times \frac{1}{6} = \frac{5}{6} \text{ kg} \]

### Step 4: Calculate the amount of zinc in the second metal.
The rest of the second metal is zinc. Therefore, the amount of zinc is:

\[ \text{Amount of zinc in the second metal} = 5 - \frac{5}{6} = \frac{30}{6} - \frac{5}{6} = \frac{25}{6} \text{ kg} \]

### Step 5: Calculate the total amount of copper in the mixture.
Now, we combine the amounts of copper from both metals:

\[ \text{Total copper} = \frac{4}{5} + \frac{5}{6} \]

To add these fractions, we need a common denominator. The least common multiple of 5 and 6 is 30.

\[ \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \]
\[ \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \]

Now, adding them together:

\[ \text{Total copper} = \frac{24}{30} + \frac{25}{30} = \frac{49}{30} \text{ kg} \]

### Step 6: Calculate the total amount of zinc in the mixture.
Now, we combine the amounts of zinc from both metals:

\[ \text{Total zinc} = \frac{16}{5} + \frac{25}{6} \]

Again, we need a common denominator. The least common multiple of 5 and 6 is 30.

\[ \frac{16}{5} = \frac{16 \times 6}{5 \times 6} = \frac{96}{30} \]
\[ \frac{25}{6} = \frac{25 \times 5}{6 \times 5} = \frac{125}{30} \]

Now, adding them together:

\[ \text{Total zinc} = \frac{96}{30} + \frac{125}{30} = \frac{221}{30} \text{ kg} \]

### Step 7: Find the ratio of copper to zinc.
Now we can find the ratio of copper to zinc:

\[ \text{Ratio of copper to zinc} = \frac{\text{Total copper}}{\text{Total zinc}} = \frac{\frac{49}{30}}{\frac{221}{30}} = \frac{49}{221} \]

Thus, the final ratio of copper to zinc in the mixture is:

\[ \text{Ratio of copper to zinc} = 49 : 221 \]