Final answer:
The recurring decimal 0.5888... can be converted into a fraction by setting it equal to x, manipulating the equation to isolate the recurring part, and then simplifying. The fraction equivalent to 0.5888... is 583/990.
Step-by-step explanation:
To convert the recurring decimal 0.5888... to a fraction, you can use algebraic methods. Let's set x equal to the recurring decimal:
x = 0.5888...
Now, multiply x by 10000 to shift the decimal point four places to the right:
10000x = 5888.8888...
Also, multiply x by 100 to shift the decimal two places, which aligns the recurring digits:
100x = 58.8888...
Next, subtract the second equation from the first one to get rid of the recurring part:
9900x = 5830
Divide both sides of this equation by 9900 to solve for x:
x = 5830 / 9900
Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10:
x = 583 / 990
The simplified fraction of the recurring decimal 0.5888... is 583/990.