Find the sum
`2+(3)/(4)+1(5)/(8)+3(7)/(16)`

Answer

To find the sum of the expression \( 2 + \frac{3}{4} + 1 \cdot \frac{5}{8} + 3 \cdot \frac{7}{16} \), we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ 2 + \frac{3}{4} + \frac{5}{8} + \frac{21}{16} \]
(Note: \( 1 \cdot \frac{5}{8} = \frac{5}{8} \) and \( 3 \cdot \frac{7}{16} = \frac{21}{16} \))

### Step 2: Find the LCM of the denominators
The denominators we have are \( 1, 4, 8, \) and \( 16 \). The least common multiple (LCM) of these numbers is \( 16 \).

### Step 3: Convert each term to have the same denominator
Now we will convert each term to have a denominator of \( 16 \):
- \( 2 = \frac{2 \cdot 16}{16} = \frac{32}{16} \)
- \( \frac{3}{4} = \frac{3 \cdot 4}{4 \cdot 4} = \frac{12}{16} \)
- \( \frac{5}{8} = \frac{5 \cdot 2}{8 \cdot 2} = \frac{10}{16} \)
- \( \frac{21}{16} \) remains as \( \frac{21}{16} \)

### Step 4: Add the fractions
Now we can add all the fractions together:
\[ \frac{32}{16} + \frac{12}{16} + \frac{10}{16} + \frac{21}{16} = \frac{32 + 12 + 10 + 21}{16} = \frac{75}{16} \]

### Step 5: Simplify the result
The result \( \frac{75}{16} \) is already in its simplest form. We can also express it as a mixed number:
\[ \frac{75}{16} = 4 \frac{11}{16} \]

### Final Answer
Thus, the sum of the expression \( 2 + \frac{3}{4} + 1 \cdot \frac{5}{8} + 3 \cdot \frac{7}{16} \) is:
\[ \frac{75}{16} \text{ or } 4 \frac{11}{16} \]

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