What is the value of $$ \frac{11 / 2-1 \frac{2}{3}+1 \frac{1}{3}}{1 \frac{1}{3}+1 / 6 \div 1 / 2} $$

Question

UpStudy AI · Accepted Answer

To evaluate the expression:

$$
\frac{\frac{11}{2} - 1\,\frac{2}{3} + 1\,\frac{1}{3}}{1\,\frac{1}{3} + \frac{1}{6} \div \frac{1}{2}}
$$

Let's break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions

- $$1\,\frac{2}{3} = \frac{5}{3}$$
- $$1\,\frac{1}{3} = \frac{4}{3}$$
- $$1\,\frac{1}{3} = \frac{4}{3}$$ (in the denominator)

### Step 2: Simplify the Numerator

$$
\frac{11}{2} - \frac{5}{3} + \frac{4}{3}
$$

To combine these, find a common denominator, which is 6:

$$
\frac{11}{2} = \frac{33}{6}, \quad \frac{5}{3} = \frac{10}{6}, \quad \frac{4}{3} = \frac{8}{6}
$$

Now, perform the subtraction and addition:

$$
\frac{33}{6} - \frac{10}{6} + \frac{8}{6} = \frac{31}{6}
$$

### Step 3: Simplify the Denominator

$$
\frac{4}{3} + \left(\frac{1}{6} \div \frac{1}{2}\right)
$$

First, perform the division:

$$
\frac{1}{6} \div \frac{1}{2} = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3}
$$

Now, add the fractions:

$$
\frac{4}{3} + \frac{1}{3} = \frac{5}{3}
$$

### Step 4: Divide the Numerator by the Denominator

$$
\frac{\frac{31}{6}}{\frac{5}{3}} = \frac{31}{6} \times \frac{3}{5} = \frac{93}{30}
$$

Simplify the fraction by dividing numerator and denominator by 3:

$$
\frac{93}{30} = \frac{31}{10} = 3\,\frac{1}{10}
$$

### Final Answer

$$\boxed{\dfrac{31}{10}}$$
The value of the expression is $$ \frac{31}{10} $$ or $$ 3\,\frac{1}{10} $$.

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