If (3x - y) : (x + 5y) = 5 : 7, then the value of (x + y) : (x − y) is :

Q: If (3x - y) : (x + 5y) = 5 : 7, then the value of (x + y) : (x − y) is :

To solve the problem, we need to find the value of $(x + y) : (x - y)$ given that $(3x - y) : (x + 5y) = 5 : 7$. ### Step-by-step Explanation: 1. **Given Ratio:** \[ \frac{3x - y}{x + 5y} = \frac{5}{7} \] 2. **Cross-Multiplication:** \[ 7(3x - y) = 5(x + 5y) \] 3. **Expand Both Sides:** \[ 21x - 7y = 5x + 25y \] 4. **Rearrange Terms:** \[ 21x - 5x = 25y + 7y \] \[ 16x = 32y \] 5. **Simplify the Equation:** \[ x = 2y \] 6. **Substitute $x = 2y$ into $(x + y) : (x - y)$:** \[ (x + y) = (2y + y) = 3y \] \[ (x - y) = (2y - y) = y \] 7. **Calculate the Ratio:** \[ \frac{x + y}{x - y} = \frac{3y}{y} = 3 : 1 \] ### Conclusion: - The value of $(x + y) : (x - y)$ is indeed $3 : 1$, which corresponds to Option 3. ### Explanation of Incorrect Options: 1. **Option 1 (1 : 2):** - If $(x + y) : (x - y)$ were $1 : 2$, it would imply $\frac{3y}{y} = \frac{1}{2}$, which is incorrect as $\frac{3y}{y} = 3$. 2. **Option 2 (2 : 1):** - If $(x + y) : (x - y)$ were $2 : 1$, it would imply $\frac{3y}{y} = 2$, which is incorrect as $\frac{3y}{y} = 3$. 3. **Option 4 (1 : 3):** - If $(x + y) : (x - y)$ were $1 : 3$, it would imply $\frac{3y}{y} = \frac{1}{3}$, which is incorrect as $\frac{3y}{y} = 3$. Thus, Option 3 is the only correct answer, as it accurately reflects the calculated ratio of $(x + y) : (x - y)$.

Question

Accepted Answer

To solve the problem, we need to find the value of $(x + y) : (x - y)$ given that $(3x - y) : (x + 5y) = 5 : 7$.

### Step-by-step Explanation:

1. **Given Ratio:**
   $$
   \frac{3x - y}{x + 5y} = \frac{5}{7}
   $$

2. **Cross-Multiplication:**
   $$
   7(3x - y) = 5(x + 5y)
   $$

3. **Expand Both Sides:**
   $$
   21x - 7y = 5x + 25y
   $$

4. **Rearrange Terms:**
   $$
   21x - 5x = 25y + 7y
   $$
   $$
   16x = 32y
   $$

5. **Simplify the Equation:**
   $$
   x = 2y
   $$

6. **Substitute $x = 2y$ into $(x + y) : (x - y)$:**
   $$
   (x + y) = (2y + y) = 3y
   $$
   $$
   (x - y) = (2y - y) = y
   $$

7. **Calculate the Ratio:**
   $$
   \frac{x + y}{x - y} = \frac{3y}{y} = 3 : 1
   $$

### Conclusion:

- The value of $(x + y) : (x - y)$ is indeed $3 : 1$, which corresponds to Option 3.

### Explanation of Incorrect Options:

1. **Option 1 (1 : 2):**
   - If $(x + y) : (x - y)$ were $1 : 2$, it would imply $\frac{3y}{y} = \frac{1}{2}$, which is incorrect as $\frac{3y}{y} = 3$.

2. **Option 2 (2 : 1):**
   - If $(x + y) : (x - y)$ were $2 : 1$, it would imply $\frac{3y}{y} = 2$, which is incorrect as $\frac{3y}{y} = 3$.

3. **Option 4 (1 : 3):**
   - If $(x + y) : (x - y)$ were $1 : 3$, it would imply $\frac{3y}{y} = \frac{1}{3}$, which is incorrect as $\frac{3y}{y} = 3$.

Thus, Option 3 is the only correct answer, as it accurately reflects the calculated ratio of $(x + y) : (x - y)$.