Simplify `1 (5)/(6) + 2 (3)/(5) + 4(2)/(3) = ?`

Options

A. `2 (1)/(10)`
B. `7 (1)/(10)`
C. `6 (1)/(10)`
D. `9 (1)/(10)`

Answer

To simplify the expression \(1 \frac{5}{6} + 2 \frac{3}{5} + 4 \frac{2}{3}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
We start by converting each mixed number into an improper fraction.

1. For \(1 \frac{5}{6}\):
\[ 1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6} \]

2. For \(2 \frac{3}{5}\):
\[ 2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5} \]

3. For \(4 \frac{2}{3}\):
\[ 4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3} \]

### Step 2: Add the Improper Fractions
Now we will add the three improper fractions:
\[ \frac{11}{6} + \frac{13}{5} + \frac{14}{3} \]

### Step 3: Find the Least Common Multiple (LCM)
To add these fractions, we need a common denominator. The denominators are 6, 5, and 3. The LCM of 6, 5, and 3 is 30.

### Step 4: Convert Each Fraction to Have the Common Denominator
Now we convert each fraction to have the denominator of 30:

1. For \(\frac{11}{6}\):
\[ \frac{11}{6} = \frac{11 \times 5}{6 \times 5} = \frac{55}{30} \]

2. For \(\frac{13}{5}\):
\[ \frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30} \]

3. For \(\frac{14}{3}\):
\[ \frac{14}{3} = \frac{14 \times 10}{3 \times 10} = \frac{140}{30} \]

### Step 5: Add the Converted Fractions
Now we can add the fractions:
\[ \frac{55}{30} + \frac{78}{30} + \frac{140}{30} = \frac{55 + 78 + 140}{30} = \frac{273}{30} \]

### Step 6: Simplify the Result
Now we simplify \(\frac{273}{30}\). We can divide both the numerator and the denominator by 3:
\[ \frac{273 \div 3}{30 \div 3} = \frac{91}{10} \]

### Step 7: Convert Back to Mixed Number
Now we convert \(\frac{91}{10}\) back to a mixed number:
\[ 91 \div 10 = 9 \quad \text{(remainder 1)} \]
Thus, we can write:
\[ \frac{91}{10} = 9 \frac{1}{10} \]

### Final Answer
The simplified result is:
\[ 9 \frac{1}{10} \]

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