Simplify : `8 3/4 - 3 7/8 - 4 1/16 - 3/4` .

Options

A. `9/16`
B. `3/4`
C. `1/16`
D. `3/8`

Answer

To simplify the expression \(8 \frac{3}{4} - 3 \frac{7}{8} - 4 \frac{1}{16} - \frac{3}{4}\), we will follow these steps:

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

1. For \(8 \frac{3}{4}\):
\[ 8 \frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \]

2. For \(3 \frac{7}{8}\):
\[ 3 \frac{7}{8} = \frac{3 \times 8 + 7}{8} = \frac{24 + 7}{8} = \frac{31}{8} \]

3. For \(4 \frac{1}{16}\):
\[ 4 \frac{1}{16} = \frac{4 \times 16 + 1}{16} = \frac{64 + 1}{16} = \frac{65}{16} \]

4. The fraction \(\frac{3}{4}\) remains as it is.

Now we rewrite the expression:
\[ \frac{35}{4} - \frac{31}{8} - \frac{65}{16} - \frac{3}{4} \]

### Step 2: Find a Common Denominator
The denominators are \(4\), \(8\), and \(16\). The least common multiple (LCM) of these denominators is \(16\).

### Step 3: Convert Each Fraction to Have the Common Denominator
1. Convert \(\frac{35}{4}\) to a denominator of \(16\):
\[ \frac{35}{4} = \frac{35 \times 4}{4 \times 4} = \frac{140}{16} \]

2. Convert \(\frac{31}{8}\) to a denominator of \(16\):
\[ \frac{31}{8} = \frac{31 \times 2}{8 \times 2} = \frac{62}{16} \]

3. \(\frac{65}{16}\) remains the same.

4. Convert \(\frac{3}{4}\) to a denominator of \(16\):
\[ \frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16} \]

Now the expression looks like this:
\[ \frac{140}{16} - \frac{62}{16} - \frac{65}{16} - \frac{12}{16} \]

### Step 4: Combine the Fractions
Now we can combine the fractions since they have the same denominator:
\[ \frac{140 - 62 - 65 - 12}{16} \]

### Step 5: Perform the Subtraction
1. First, calculate \(140 - 62\):
\[ 140 - 62 = 78 \]

2. Next, subtract \(65\):
\[ 78 - 65 = 13 \]

3. Finally, subtract \(12\):
\[ 13 - 12 = 1 \]

So, we have:
\[ \frac{1}{16} \]

### Final Answer
The simplified result of the expression \(8 \frac{3}{4} - 3 \frac{7}{8} - 4 \frac{1}{16} - \frac{3}{4}\) is:
\[ \frac{1}{16} \]