Final answer:
The repeating decimal 0.52525252... can be shown to be rational by expressing it as the ratio of two integers, specifically 52/99.
Step-by-step explanation:
To show that the number 0.52525252... is rational, we can write it as a ratio of two integers. Since the digits 52 are repeating, we can denote the original number as x and write down the repeating decimal:
x = 0.52525252...
Multiply both sides of the equation by 100 (since the repeating part is two digits long) to shift the decimal places to the right twice:
100x = 52.52525252...
Now subtract the original equation (x) from this new equation (100x) to eliminate the repeating part:
100x - x = 52.52525252... - 0.52525252...
99x = 52
Now you can solve for x by dividing both sides by 99:
x = 52 / 99
Therefore, the repeating decimal 0.52525252... can be written as the rational number 52/99.