Express the following numbers in the form `(p)/(q)` where p and q are integers and `q != 0`. (i) 0.0875 (ii) 2.123456789 (iii) 0.181818….. (iv) 0.437 (v) .651

Answer

Let's express the given numbers in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

### (i) 0.0875
1. **Convert to Fraction**: Multiply by 10,000 to eliminate the decimal.
\[ 0.0875 = \frac{0.0875 \times 10000}{10000} = \frac{875}{10000} \]
2. **Simplify the Fraction**: Divide both the numerator and denominator by their greatest common divisor (GCD), which is 125.
\[ \frac{875 \div 125}{10000 \div 125} = \frac{7}{80} \]

### (ii) 2.123456789
1. **Convert to Fraction**: Multiply by \( 10^9 \) (since there are 9 digits after the decimal).
\[ 2.123456789 = \frac{2.123456789 \times 10^9}{10^9} = \frac{2123456789}{1000000000} \]
2. **Simplify the Fraction**: The fraction \( \frac{2123456789}{1000000000} \) cannot be simplified further as 2123456789 is a prime number.

### (iii) 0.181818…
1. **Let x = 0.181818…**: This is a repeating decimal.
2. **Multiply by 100**: To shift the decimal point.
\[ 100x = 18.181818… \]
3. **Set up the equation**: Subtract the original x from this equation.
\[ 100x - x = 18.181818… - 0.181818… \implies 99x = 18 \]
4. **Solve for x**:
\[ x = \frac{18}{99} \]
5. **Simplify the Fraction**: Divide both by 9.
\[ x = \frac{2}{11} \]

### (iv) 0.437
1. **Convert to Fraction**: Multiply by 1000 to eliminate the decimal.
\[ 0.437 = \frac{0.437 \times 1000}{1000} = \frac{437}{1000} \]
2. **Simplify the Fraction**: The fraction \( \frac{437}{1000} \) cannot be simplified further as 437 is a prime number.

### (v) 0.651
1. **Convert to Fraction**: Multiply by 1000 to eliminate the decimal.
\[ 0.651 = \frac{0.651 \times 1000}{1000} = \frac{651}{1000} \]
2. **Simplify the Fraction**: The fraction \( \frac{651}{1000} \) cannot be simplified further as 651 is not divisible by 2 or 5.

### Summary of Answers:
(i) \( \frac{7}{80} \)
(ii) \( \frac{2123456789}{1000000000} \)
(iii) \( \frac{2}{11} \)
(iv) \( \frac{437}{1000} \)
(v) \( \frac{651}{1000} \)