If `(1)/(1(1)/(4)) + ( 1)/( 6(2)/(3)) - ( 1)/( x) + ( 1)/( 10) = (11)/( 12)` , then find the value of x

Options

A. `(15)/( 3)`
B. `(20)/( 15)`
C. `(15)/(2)`
D. `(12)/(13)`

Answer

To solve the equation

\[ \frac{1}{\frac{1}{\frac{1}{4}}} + \frac{1}{6 \frac{2}{3}} - \frac{1}{x} + \frac{1}{10} = \frac{11}{12} \]

we will break it down step by step.

### Step 1: Simplify the fractions

First, we simplify the fractions on the left side of the equation.

1. **Simplifying \(\frac{1}{\frac{1}{4}}\)**:
\[ \frac{1}{\frac{1}{4}} = 4 \]

2. **Simplifying \(\frac{1}{6 \frac{2}{3}}\)**:
- First, calculate \(6 \frac{2}{3}\):
\[ 6 \frac{2}{3} = 6 + \frac{2}{3} = \frac{18}{3} + \frac{2}{3} = \frac{20}{3} \]
- Now, find \(\frac{1}{\frac{20}{3}}\):
\[ \frac{1}{\frac{20}{3}} = \frac{3}{20} \]

So, we can rewrite the equation as:
\[ 4 + \frac{3}{20} - \frac{1}{x} + \frac{1}{10} = \frac{11}{12} \]

### Step 2: Combine the constants

Next, we need to combine the constants \(4\), \(\frac{3}{20}\), and \(\frac{1}{10}\).

- Convert \(4\) to a fraction with a common denominator of \(20\):
\[ 4 = \frac{80}{20} \]

- Convert \(\frac{1}{10}\) to a fraction with a common denominator of \(20\):
\[ \frac{1}{10} = \frac{2}{20} \]

Now, combine:
\[ \frac{80}{20} + \frac{3}{20} + \frac{2}{20} = \frac{80 + 3 + 2}{20} = \frac{85}{20} \]

### Step 3: Substitute back into the equation

Now substitute back into the equation:
\[ \frac{85}{20} - \frac{1}{x} = \frac{11}{12} \]

### Step 4: Isolate \(\frac{1}{x}\)

Rearranging gives:
\[ -\frac{1}{x} = \frac{11}{12} - \frac{85}{20} \]

### Step 5: Find a common denominator

The common denominator for \(12\) and \(20\) is \(60\):
- Convert \(\frac{11}{12}\):
\[ \frac{11}{12} = \frac{55}{60} \]
- Convert \(\frac{85}{20}\):
\[ \frac{85}{20} = \frac{255}{60} \]

Now we can subtract:
\[ -\frac{1}{x} = \frac{55}{60} - \frac{255}{60} = \frac{55 - 255}{60} = \frac{-200}{60} = \frac{-10}{3} \]

### Step 6: Solve for \(x\)

Now, we can solve for \(x\):
\[ \frac{1}{x} = \frac{10}{3} \]

Taking the reciprocal gives:
\[ x = \frac{3}{10} \]

### Final Answer
The value of \(x\) is:
\[ \boxed{\frac{3}{10}} \]