Express the repeating decimal as a quotient of two integers. 4.09 (9 is repeating)

1. Let \( x = 4.099999\ldots \).

2. Multiply by 10 to shift the decimal point for the non-repeating part: \( 10x = 40.99999\ldots \).

3. Multiply by 100 to shift the decimal point so that the repeating part aligns: \( 100x = 409.99999\ldots \).

4. Subtract the equation of step 2 from step 3:

100x - 10x = 409.9999\ldots - 40.9999\ldots

90x = 369

5. Solve for \( x \) to get the fraction:

x = \frac{369}{90}

6. Simplify the fraction:

Divide both numerator and denominator by 9:

x=\frac{41}{10}