Final answer:
The repeating decimal 0.36363636... can be expressed as the ratio of integers 4/11 by setting up an equation to represent the decimal, then multiplying and subtracting to isolate the repeating part and simplify the resulting fraction.
Step-by-step explanation:
Expressing the number 0.36363636... as a ratio of integers involves identifying the repeating decimal pattern and converting it to a fraction. First, let's set up a variable to represent the repeating decimal:
Let x = 0.36363636...
Multiply x by 100 since the repeating pattern is two digits long:
100x = 36.363636...
Now, subtract the original equation (x = 0.363636...) from this new equation:
100x - x = 36.363636... - 0.363636...
99x = 36
Divide both sides by 99 to solve for x:
x = 36 / 99
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 9:
x = 4 / 11
So, the repeating decimal 0.36363636... can be expressed as the ratio of integers 4/11.