Express 8.54545454545... as a rational number in the form p/q, where p and q are positive integers with no common factors.

To express the repeating decimal 8.54545454545... as a rational number in the form p/q, where p and q are positive integers with no common factors, we can follow these steps:

Step 1: Let's denote the repeating decimal by x: \[ x = 8.54545454545... \]

Step 2: Since the digits "54" are repeating, we can multiply x by 100 to shift the decimal two places to the right to line up with the same repeating pattern: \[ 100x = 854.54545454545... \]

Step 3: Now we'll subtract the original x from 100x to remove the repeating part: \[ 100x - x = 854.54545454545... - 8.54545454545... \]

Step 4: Perform the subtraction: \[ 99x = 846 \]

Step 5: Solve for x to get the value as a fraction: \[ x = \frac{846}{99} \]

Step 6: Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 846 and 99 is 3: \[ x = \frac{846 \div 3}{99 \div 3} = \frac{282}{33} \]

Step 7: Further simplifying, we divide both by 3 again: \[ x = \frac{282 \div 3}{33 \div 3} = \frac{94}{11} \]

So, the repeating decimal 8.54545454545... expressed as a rational number in simplest form is \(\frac{94}{11}\).

Extra: In mathematics, we often come across repeating decimals, which are decimals that have a pattern of one or more digits repeating infinitely. Any repeating decimal can be converted into a rational number because a rational number is simply a number that can be expressed as the division of two integers.

This conversion process involves algebraic manipulation where we represent the repeating decimal as a variable (like x), create an equation by shifting the decimal point, and then solve for x. The result is a value represented as a fraction where the numerator and denominator are integers with no common factors. Simplifying the fraction involves dividing both the numerator and denominator by their greatest common divisor so that we arrive at the simplest form of the fraction. This simplest form is the rational number equivalent to the original repeating decimal.