Final answer:
To convert the repeating decimal 3.152... to an improper fraction, we express it as x = 3.1525252..., multiply by 100 to get 100x = 315.252525..., subtract the original from this, and simplify to get x = 3121/990.
Step-by-step explanation:
To convert a repeating decimal to an improper fraction, we use algebraic methods. Let's denote the repeating decimal 3.152... by 'x'. Since the digits 52 are repeating, we can express it as:
x = 3.1525252...
Now, to isolate the repeating part, we'll multiply 'x' by 100 since there are two digits in the repeating section:
100x = 315.252525...
Subtract the original 'x' from this equation to get rid of the repeating part:
100x - x = 315.252525... - 3.152525...
99x = 312.1
Now divide both sides by 99 to solve for 'x':
x = 312.1 / 99
Finally, simplify the fraction by dividing both numerator and denominator by their greatest common factor if necessary, but in this case, it's already in simplest form. Hence, the improper fraction for 3.152... is 3121/990.