If $2 = x + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}$ then value of x is :

Q: If $2 = x + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}$ then value of x is :

To solve the given equation $2 = x + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}$, we need to simplify the expression on the right-hand side. 1. **Simplify the innermost fraction:** \[ 3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4} \] 2. **Simplify the next fraction:** \[ 1 + \frac{1}{\frac{13}{4}} = 1 + \frac{4}{13} = \frac{13}{13} + \frac{4}{13} = \frac{17}{13} \] 3. **Simplify the outermost fraction:** \[ \frac{1}{\frac{17}{13}} = \frac{13}{17} \] 4. **Substitute back into the equation:** \[ 2 = x + \frac{13}{17} \] 5. **Solve for $x$:** \[ x = 2 - \frac{13}{17} = \frac{34}{17} - \frac{13}{17} = \frac{21}{17} \] Thus, the value of $x$ is $\frac{21}{17}$, which corresponds to Option 2. **Explanation for why other options are incorrect:** - **Option 1: $ \frac{18}{17} $** - If $x$ were $\frac{18}{17}$, then substituting back into the equation would not satisfy the original equation $2 = x + \frac{13}{17}$. - **Option 3: $ \frac{16}{17} $** - Similarly, substituting $\frac{16}{17}$ for $x$ would not satisfy the equation, as it would result in a value less than 2. - **Option 4: $ \frac{20}{17} $** - Using $\frac{20}{17}$ for $x$ would also not satisfy the equation, as it would result in a value slightly less than 2. Therefore, the only value of $x$ that satisfies the equation is $\frac{21}{17}$, making Option 2 the correct answer.

Question

Accepted Answer

To solve the given equation $2 = x + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}$, we need to simplify the expression on the right-hand side.

1. **Simplify the innermost fraction:**
   $$
   3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4}
   $$

2. **Simplify the next fraction:**
   $$
   1 + \frac{1}{\frac{13}{4}} = 1 + \frac{4}{13} = \frac{13}{13} + \frac{4}{13} = \frac{17}{13}
   $$

3. **Simplify the outermost fraction:**
   $$
   \frac{1}{\frac{17}{13}} = \frac{13}{17}
   $$

4. **Substitute back into the equation:**
   $$
   2 = x + \frac{13}{17}
   $$

5. **Solve for $x$:**
   $$
   x = 2 - \frac{13}{17} = \frac{34}{17} - \frac{13}{17} = \frac{21}{17}
   $$

Thus, the value of $x$ is $\frac{21}{17}$, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

- **Option 1: $ \frac{18}{17} $**
  - If $x$ were $\frac{18}{17}$, then substituting back into the equation would not satisfy the original equation $2 = x + \frac{13}{17}$.

- **Option 3: $ \frac{16}{17} $**
  - Similarly, substituting $\frac{16}{17}$ for $x$ would not satisfy the equation, as it would result in a value less than 2.

- **Option 4: $ \frac{20}{17} $**
  - Using $\frac{20}{17}$ for $x$ would also not satisfy the equation, as it would result in a value slightly less than 2.

Therefore, the only value of $x$ that satisfies the equation is $\frac{21}{17}$, making Option 2 the correct answer.