Question 750467
Change to a rational number the given number:
(1) n = 0.8222...
First we "fix" the decimal to be a whole number plus a repeating decimal. In this case
(2) n = 8.222.../10 or
(3) n = (8 + .222...)/10
We know that a single digit m divided by 9 is
(4) m/9 = 0.mmm... or .m repeated
In the given case in (3) we have
(5) 0.222.... which by applying (4) becomes
(6) 2/9 = 0.222...
Now place the identity of (6) into (3) and get
(7) n = (8 + 2/9)/10
Now simplify (7) to get
(8) n = ((72+2)/9)/10 or
(9) n = (74/9)/10 or
(10) n = 74/90 or
(11) n = 37/45
To ckeck the answer do the following
Is (0.8222.. = 37/45)?
Is (0.8222.. = 74/90)?
Is (90*0.8222...) = 74)?
Is (9*8.222... = 74)?
Is (72 + 9*0.222... = 74)?
Is (72 + 9*2/9 = 74)?
Is (72 + 2 = 74)?
Is (74 = 74)? Yes
Answer: 37/45 is the rational number equivalent of the decimal 0.8222...
Hint: If you have a two digit repeating number, such as
(12) 0.ababab.... it is the rational number
(13) ab/99 and if you have a three digit repeating number, such as
(14) 0.abcabcabc... it is the rational number
(15) abc/999 and four
(16) abcd/9999 etc.
PS You can do all of the above on a calculator.