`(p)/(q)` form of 3.777 … is

Options

A. `(3)/(10)`
B. `(3)/(11)`
C. `(34)/(9)`
D. `(37)/(10)`

Answer

To convert the repeating decimal \(3.777...\) into a fraction in the form \(\frac{p}{q}\), we can follow these steps:

### Step 1: Let \(x\) equal the repeating decimal
Let \(x = 3.777...\)

### Step 2: Multiply by 10
To eliminate the decimal, multiply both sides of the equation by 10:
\[ 10x = 37.777... \]

### Step 3: Set up the equations
Now we have two equations:
1. \(x = 3.777...\) (Equation 1)
2. \(10x = 37.777...\) (Equation 2)

### Step 4: Subtract Equation 1 from Equation 2
Now, subtract Equation 1 from Equation 2:
\[ 10x - x = 37.777... - 3.777... \]
This simplifies to:
\[ 9x = 34 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 9 to solve for \(x\):
\[ x = \frac{34}{9} \]

### Conclusion
Thus, the repeating decimal \(3.777...\) can be expressed as the fraction:
\[ \frac{34}{9} \]