`5(1)/(4)+6(2)/(3)+7(1)/(6)=?`

Options

A. 19.5
B. `19(11)/(12)`
C. `19(1)/(12)`
D. 19

Explanation

To solve the problem \( 5\frac{1}{4} + 6\frac{2}{3} + 7\frac{1}{6} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Convert \( 5\frac{1}{4} \)**:
\[ 5\frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4} \]

2. **Convert \( 6\frac{2}{3} \)**:
\[ 6\frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3} \]

3. **Convert \( 7\frac{1}{6} \)**:
\[ 7\frac{1}{6} = \frac{7 \times 6 + 1}{6} = \frac{42 + 1}{6} = \frac{43}{6} \]

### Step 2: Find a Common Denominator
The denominators are 4, 3, and 6. The least common multiple (LCM) of these numbers is 12.

### Step 3: Convert Each Fraction to Have the Common Denominator
1. **Convert \( \frac{21}{4} \)**:
\[ \frac{21}{4} = \frac{21 \times 3}{4 \times 3} = \frac{63}{12} \]

2. **Convert \( \frac{20}{3} \)**:
\[ \frac{20}{3} = \frac{20 \times 4}{3 \times 4} = \frac{80}{12} \]

3. **Convert \( \frac{43}{6} \)**:
\[ \frac{43}{6} = \frac{43 \times 2}{6 \times 2} = \frac{86}{12} \]

### Step 4: Add the Improper Fractions
Now we can add the fractions:
\[ \frac{63}{12} + \frac{80}{12} + \frac{86}{12} = \frac{63 + 80 + 86}{12} = \frac{229}{12} \]

### Step 5: Convert the Improper Fraction to a Mixed Number
To convert \( \frac{229}{12} \) to a mixed number:
1. Divide 229 by 12:
- \( 12 \times 19 = 228 \) (the largest multiple of 12 less than 229)
- Remainder: \( 229 - 228 = 1 \)

Thus, we can express it as:
\[ \frac{229}{12} = 19 \frac{1}{12} \]

### Final Answer
The final answer is:
\[ 19 \frac{1}{12} \]

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