`(99+(1)/(7)+99(2)/(7)+99(3)/(7)+99(4)/(7)+99(5)/(7)+99(6)/(7))` is equal to

Options

A. 603
B. 600
C. 598
D. 597

Answer

To solve the expression \( 99 + \frac{1}{7} + 99 \cdot \frac{2}{7} + 99 \cdot \frac{3}{7} + 99 \cdot \frac{4}{7} + 99 \cdot \frac{5}{7} + 99 \cdot \frac{6}{7} \), we can follow these steps:

### Step 1: Group the terms
We can group the terms involving 99 together:
\[ 99 + 99 \cdot \frac{1}{7} + 99 \cdot \frac{2}{7} + 99 \cdot \frac{3}{7} + 99 \cdot \frac{4}{7} + 99 \cdot \frac{5}{7} + 99 \cdot \frac{6}{7} \]
This can be simplified as:
\[ 99 + 99 \left( \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} \right) \]

### Step 2: Factor out 99
Now, we can factor out 99 from the grouped terms:
\[ 99 \left( 1 + \left( \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} \right) \right) \]

### Step 3: Simplify the fraction
Next, we simplify the fraction inside the parentheses:
\[ \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} = \frac{1 + 2 + 3 + 4 + 5 + 6}{7} \]
Calculating the sum of the integers:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
Thus, we have:
\[ \frac{21}{7} = 3 \]

### Step 4: Substitute back into the equation
Now substitute back into the equation:
\[ 99 \left( 1 + 3 \right) = 99 \cdot 4 \]

### Step 5: Calculate the final result
Now multiply:
\[ 99 \cdot 4 = 396 \]

### Final Answer
The value of the expression is:
\[ \boxed{396} \]