`(2)1/2`+`(3)1/4`+`x`+`(2)1/2`=` (13)5/12`find the value of `x`

Options

A. A)`(5) 1/6`
B. B)`(3)1/5`
C. C)`(2)1/5`
D. D)`(5)1/7`

Answer

To solve the equation \( (2)1/2 + (3)1/4 + x + (2)1/2 = (13)5/12 \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert all mixed numbers into improper fractions.

1. \( 2 \frac{1}{2} = 2 \times 2 + 1 = \frac{5}{2} \)
2. \( 3 \frac{1}{4} = 3 \times 4 + 1 = \frac{13}{4} \)
3. \( 2 \frac{1}{2} = \frac{5}{2} \)
4. \( 13 \frac{5}{12} = 13 \times 12 + 5 = \frac{161}{12} \)

### Step 2: Substitute Improper Fractions into the Equation
Now we substitute these values back into the equation:

\[ \frac{5}{2} + \frac{13}{4} + x + \frac{5}{2} = \frac{161}{12} \]

### Step 3: Combine Like Terms
Combine the fractions on the left side:

\[ 2 \times \frac{5}{2} = \frac{10}{2} = 5 \]

So the equation becomes:

\[ 5 + \frac{13}{4} + x = \frac{161}{12} \]

### Step 4: Convert 5 to a Fraction
Next, we convert 5 into a fraction with a common denominator of 12:

\[ 5 = \frac{60}{12} \]

Now, substituting this back into the equation gives:

\[ \frac{60}{12} + \frac{13}{4} + x = \frac{161}{12} \]

### Step 5: Convert \(\frac{13}{4}\) to a Fraction with Denominator 12
To combine the fractions, we need to convert \(\frac{13}{4}\):

\[ \frac{13}{4} = \frac{39}{12} \]

### Step 6: Substitute Back into the Equation
Now the equation looks like this:

\[ \frac{60}{12} + \frac{39}{12} + x = \frac{161}{12} \]

### Step 7: Combine the Left Side
Combine the fractions on the left side:

\[ \frac{60 + 39}{12} + x = \frac{161}{12} \]
\[ \frac{99}{12} + x = \frac{161}{12} \]

### Step 8: Isolate x
Now, isolate \(x\):

\[ x = \frac{161}{12} - \frac{99}{12} \]
\[ x = \frac{161 - 99}{12} = \frac{62}{12} \]

### Step 9: Simplify the Fraction
Now simplify \(\frac{62}{12}\):

\[ x = \frac{31}{6} \]

### Step 10: Convert to Mixed Number
Convert \(\frac{31}{6}\) to a mixed number:

\[ 31 \div 6 = 5 \quad \text{(remainder 1)} \]
So, \(x = 5 \frac{1}{6}\).

### Final Answer:
Thus, the value of \(x\) is \(5 \frac{1}{6}\).

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