Simplify `1(1)/(3) - 1(1)/(9) + 1(1)/(6) = ?`

Options

A. `1 (5)/(18)`
B. `1 (7)/(18)`
C. `1 (1)/(9)`
D. `1 (4)/(9)`

Answer

To simplify the expression \(1 \frac{1}{3} - 1 \frac{1}{9} + 1 \frac{1}{6}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert each mixed number into an improper fraction.

1. For \(1 \frac{1}{3}\):
\[ 1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3} \]

2. For \(1 \frac{1}{9}\):
\[ 1 \frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{10}{9} \]

3. For \(1 \frac{1}{6}\):
\[ 1 \frac{1}{6} = \frac{1 \times 6 + 1}{6} = \frac{7}{6} \]

So, we can rewrite the expression as:
\[ \frac{4}{3} - \frac{10}{9} + \frac{7}{6} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(3\), \(9\), and \(6\).

- The LCM of \(3\), \(9\), and \(6\) is \(18\).

### Step 3: Rewrite Each Fraction with the Common Denominator
Now, we will rewrite each fraction with the common denominator \(18\).

1. For \(\frac{4}{3}\):
\[ \frac{4}{3} = \frac{4 \times 6}{3 \times 6} = \frac{24}{18} \]

2. For \(\frac{10}{9}\):
\[ \frac{10}{9} = \frac{10 \times 2}{9 \times 2} = \frac{20}{18} \]

3. For \(\frac{7}{6}\):
\[ \frac{7}{6} = \frac{7 \times 3}{6 \times 3} = \frac{21}{18} \]

Now, we can rewrite the expression as:
\[ \frac{24}{18} - \frac{20}{18} + \frac{21}{18} \]

### Step 4: Combine the Fractions
Now, we can combine the fractions:
\[ \frac{24 - 20 + 21}{18} = \frac{25}{18} \]

### Step 5: Convert to Mixed Number
To convert \(\frac{25}{18}\) into a mixed number:
1. Divide \(25\) by \(18\):
- \(18\) goes into \(25\) once (1), with a remainder of \(7\).

So, we can express \(\frac{25}{18}\) as:
\[ 1 \frac{7}{18} \]

### Final Answer
Thus, the simplified form of the expression \(1 \frac{1}{3} - 1 \frac{1}{9} + 1 \frac{1}{6}\) is:
\[ 1 \frac{7}{18} \]

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