$2\frac{1}{2} + 3\frac{1}{4} + x + 2\frac{1}{3} = 13\frac{5}{12}$, find the value of 'x':

Q: $2\frac{1}{2} + 3\frac{1}{4} + x + 2\frac{1}{3} = 13\frac{5}{12}$, find the value of 'x':

To find the value of $ x $ in the equation $ 2\frac{1}{2} + 3\frac{1}{4} + x + 2\frac{1}{3} = 13\frac{5}{12} $, we need to first convert all the mixed numbers into improper fractions and then solve for $ x $. 1. Convert the mixed numbers to improper fractions: - $ 2\frac{1}{2} = \frac{5}{2} $ - $ 3\frac{1}{4} = \frac{13}{4} $ - $ 2\frac{1}{3} = \frac{7}{3} $ - $ 13\frac{5}{12} = \frac{161}{12} $ 2. Add the fractions on the left side of the equation: - First, find a common denominator for $\frac{5}{2}$, $\frac{13}{4}$, and $\frac{7}{3}$. The least common denominator is 12. - Convert each fraction: - $\frac{5}{2} = \frac{30}{12}$ - $\frac{13}{4} = \frac{39}{12}$ - $\frac{7}{3} = \frac{28}{12}$ 3. Add these fractions: \[ \frac{30}{12} + \frac{39}{12} + \frac{28}{12} = \frac{97}{12} \] 4. Substitute back into the equation: \[ \frac{97}{12} + x = \frac{161}{12} \] 5. Solve for $ x $: \[ x = \frac{161}{12} - \frac{97}{12} = \frac{64}{12} = \frac{16}{3} = 5\frac{1}{3} \] Thus, the value of $ x $ is $ 5\frac{1}{3} $, which corresponds to Option 1. **Explanation for why other options are incorrect:** - **Option 2: $ 3\frac{1}{5} $**: Converting $ 3\frac{1}{5} $ to an improper fraction gives $\frac{16}{5}$. If we substitute $\frac{16}{5}$ for $ x $, the equation does not balance to $\frac{161}{12}$. - **Option 3: $ 2\frac{1}{5} $**: Converting $ 2\frac{1}{5} $ to an improper fraction gives $\frac{11}{5}$. Substituting $\frac{11}{5}$ for $ x $ also fails to satisfy the equation. - **Option 4: $ 5\frac{1}{7} $**: Converting $ 5\frac{1}{7} $ to an improper fraction gives $\frac{36}{7}$. Substituting $\frac{36}{7}$ for $ x $ does not result in the correct sum of $\frac{161}{12}$. Therefore, Option 1 is the only correct answer, as it satisfies the equation perfectly.

Question

Accepted Answer

To find the value of $ x $ in the equation $ 2\frac{1}{2} + 3\frac{1}{4} + x + 2\frac{1}{3} = 13\frac{5}{12} $, we need to first convert all the mixed numbers into improper fractions and then solve for $ x $.

1. Convert the mixed numbers to improper fractions:
   - $ 2\frac{1}{2} = \frac{5}{2} $
   - $ 3\frac{1}{4} = \frac{13}{4} $
   - $ 2\frac{1}{3} = \frac{7}{3} $
   - $ 13\frac{5}{12} = \frac{161}{12} $

2. Add the fractions on the left side of the equation:
   - First, find a common denominator for $\frac{5}{2}$, $\frac{13}{4}$, and $\frac{7}{3}$. The least common denominator is 12.
   - Convert each fraction:
     - $\frac{5}{2} = \frac{30}{12}$
     - $\frac{13}{4} = \frac{39}{12}$
     - $\frac{7}{3} = \frac{28}{12}$

3. Add these fractions:
   $$
   \frac{30}{12} + \frac{39}{12} + \frac{28}{12} = \frac{97}{12}
   $$

4. Substitute back into the equation:
   $$
   \frac{97}{12} + x = \frac{161}{12}
   $$

5. Solve for $ x $:
   $$
   x = \frac{161}{12} - \frac{97}{12} = \frac{64}{12} = \frac{16}{3} = 5\frac{1}{3}
   $$

Thus, the value of $ x $ is $ 5\frac{1}{3} $, which corresponds to Option 1.

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

- **Option 2: $ 3\frac{1}{5} $**: Converting $ 3\frac{1}{5} $ to an improper fraction gives $\frac{16}{5}$. If we substitute $\frac{16}{5}$ for $ x $, the equation does not balance to $\frac{161}{12}$.

- **Option 3: $ 2\frac{1}{5} $**: Converting $ 2\frac{1}{5} $ to an improper fraction gives $\frac{11}{5}$. Substituting $\frac{11}{5}$ for $ x $ also fails to satisfy the equation.

- **Option 4: $ 5\frac{1}{7} $**: Converting $ 5\frac{1}{7} $ to an improper fraction gives $\frac{36}{7}$. Substituting $\frac{36}{7}$ for $ x $ does not result in the correct sum of $\frac{161}{12}$.

Therefore, Option 1 is the only correct answer, as it satisfies the equation perfectly.