The value of `1.999`….. In the form of `p/q`, where p and q are integers and `q ne 0`, is

Options

A. `(19)/(10)`
B. `(1999)/(1000)`
C. `2`
D. `1/9`

Answer

To express the repeating decimal \(1.999...\) in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\), we can follow these steps:

### Step-by-Step Solution:

1. **Let \( x = 1.999...\)**
We start by defining the repeating decimal as a variable \(x\).

**Hint:** Define the repeating decimal as a variable to simplify the calculations.

2. **Multiply both sides by 10:**
Multiply \(x\) by 10 to shift the decimal point one place to the right.
\[ 10x = 19.999... \]

**Hint:** Multiplying by 10 helps to align the repeating decimals for subtraction.

3. **Rewrite the equation:**
Notice that \(19.999...\) can be rewritten as \(18 + 1.999...\). Thus, we have:
\[ 10x = 18 + x \]

**Hint:** Break down the repeating decimal to isolate \(x\).

4. **Rearrange the equation:**
Subtract \(x\) from both sides to isolate terms involving \(x\):
\[ 10x - x = 18 \]
This simplifies to:
\[ 9x = 18 \]

**Hint:** Isolate \(x\) by combining like terms.

5. **Solve for \(x\):**
Divide both sides by 9 to solve for \(x\):
\[ x = \frac{18}{9} \]
Simplifying gives:
\[ x = 2 \]

**Hint:** Simplifying fractions is key to finding the final value.

6. **Express in the form of \(\frac{p}{q}\):**
Since \(x = 2\), we can express this as:
\[ x = \frac{2}{1} \]
Here, \(p = 2\) and \(q = 1\).

**Hint:** Ensure your final answer is in the required form of \(\frac{p}{q}\).

### Final Answer:
Thus, the value of \(1.999...\) in the form of \(\frac{p}{q}\) is \(\frac{2}{1}\).