If $\frac{x}{y}=\frac{5}{3}$, then $\frac{x+y}{x-y}$ is equal to

Q: If $\frac{x}{y}=\frac{5}{3}$, then $\frac{x+y}{x-y}$ is equal to

To solve the problem, we need to determine the value of $\frac{x+y}{x-y}$ given that $\frac{x}{y} = \frac{5}{3}$. ### Step-by-step Explanation: 1. **Express $x$ in terms of $y$:** Given $\frac{x}{y} = \frac{5}{3}$, we can express $x$ as: \[ x = \frac{5}{3}y \] 2. **Substitute $x$ in the expression $\frac{x+y}{x-y}$:** We need to find $\frac{x+y}{x-y}$. Substitute $x = \frac{5}{3}y$ into the expression: \[ \frac{x+y}{x-y} = \frac{\frac{5}{3}y + y}{\frac{5}{3}y - y} \] 3. **Simplify the expression:** - The numerator becomes: \[ \frac{5}{3}y + y = \frac{5}{3}y + \frac{3}{3}y = \frac{8}{3}y \] - The denominator becomes: \[ \frac{5}{3}y - y = \frac{5}{3}y - \frac{3}{3}y = \frac{2}{3}y \] - Thus, the expression simplifies to: \[ \frac{\frac{8}{3}y}{\frac{2}{3}y} = \frac{8}{3} \times \frac{3}{2} = \frac{8}{2} = 4 \] 4. **Conclusion:** Therefore, $\frac{x+y}{x-y} = 4$, which corresponds to Option 1. ### Explanation of Incorrect Options: - **Option 2 (2):** This option is incorrect because the simplification of the expression $\frac{x+y}{x-y}$ results in 4, not 2. The calculations clearly show that the ratio simplifies to 4. - **Option 3 (-2):** This option is incorrect because the expression does not yield a negative value. The calculations demonstrate that the result is a positive 4. - **Option 4 (-4):** This option is incorrect for similar reasons as Option 3. The expression simplifies to a positive value, specifically 4, not a negative value. By following the logical steps and calculations, we confirm that Option 1 is indeed the correct answer.

Question

Accepted Answer

To solve the problem, we need to determine the value of $\frac{x+y}{x-y}$ given that $\frac{x}{y} = \frac{5}{3}$.

### Step-by-step Explanation:

1. **Express $x$ in terms of $y$:**

Given $\frac{x}{y} = \frac{5}{3}$, we can express $x$ as:
   $$
   x = \frac{5}{3}y
   $$

2. **Substitute $x$ in the expression $\frac{x+y}{x-y}$:**

We need to find $\frac{x+y}{x-y}$. Substitute $x = \frac{5}{3}y$ into the expression:
   $$
   \frac{x+y}{x-y} = \frac{\frac{5}{3}y + y}{\frac{5}{3}y - y}
   $$

3. **Simplify the expression:**

- The numerator becomes:
     $$
     \frac{5}{3}y + y = \frac{5}{3}y + \frac{3}{3}y = \frac{8}{3}y
     $$

- The denominator becomes:
     $$
     \frac{5}{3}y - y = \frac{5}{3}y - \frac{3}{3}y = \frac{2}{3}y
     $$

- Thus, the expression simplifies to:
     $$
     \frac{\frac{8}{3}y}{\frac{2}{3}y} = \frac{8}{3} \times \frac{3}{2} = \frac{8}{2} = 4
     $$

4. **Conclusion:**

Therefore, $\frac{x+y}{x-y} = 4$, which corresponds to Option 1.

### Explanation of Incorrect Options:

- **Option 2 (2):** This option is incorrect because the simplification of the expression $\frac{x+y}{x-y}$ results in 4, not 2. The calculations clearly show that the ratio simplifies to 4.

- **Option 3 (-2):** This option is incorrect because the expression does not yield a negative value. The calculations demonstrate that the result is a positive 4.

- **Option 4 (-4):** This option is incorrect for similar reasons as Option 3. The expression simplifies to a positive value, specifically 4, not a negative value.

By following the logical steps and calculations, we confirm that Option 1 is indeed the correct answer.