The question says express each repeating decimal as a fraction:
5.23232323
This is my work but my answer doesn't seem right:
100x= 523.23232323
- 10x= 52.23232323
90x= 470.909090
/ 90x. \90x
x= 5.23232323
What am I doing wrong?
And how am I suppose to know when to multiply x by 10,100, or 1000?
Thank you!!
I'd read the decimal as a fraction.
5.23 = 5 2323/10000 = 5 23/100
just use your decimal
let x = .232323...
100x = 23.232323...
subtract them:
99x = 23
x = 23/99
so 5.2323.... = 5 23/99 or 518/99
check with your calculator to verify it to be correct.
To express a repeating decimal as a fraction, you need to set up an equation to eliminate the repeating part.
Let's assume the repeating part is represented by "0.23" (the digits after the decimal point).
To solve this, you'll want to shift the decimal point so that the repeating part aligns with the non-repeating part (i.e., "23.23").
So, let's call the given number "x":
100x = 523.232323...
Now, multiply the equation by a suitable power of 10 to eliminate the repeating part. Since there are two digits in the repeating part, multiply by 100:
10000x = 52323.232323...
Next, subtract the original equation from the multiplied equation to eliminate the repeating part:
10000x - 100x = 52323.232323... - 523.232323...
Simplifying:
9900x = 51700
Divide both sides by 9900 to solve for "x":
x = 51700 / 9900
Simplifying this fraction:
x = 517 / 99
So, the fraction equivalent of the repeating decimal 5.232323... is 517/99.
To determine which power of 10 to multiply by, count the number of digits in the repeating part. In this case, there are two digits (23). Therefore, we multiply by 100 to align the repeating part with the non-repeating part ("23.23"). The power of 10 will depend on the number of digits in the repeating part.
To express a repeating decimal as a fraction, you can use a method called Geometric Series. Let's start by representing the repeating decimal as follows:
5.23232323 = 5 + 0.23 + 0.0023 + ...
Next, let's multiply the repeating part by the appropriate power of 10 so that the decimal places align and we can subtract them:
100x = 523.23232323
- x = 5.23232323
Now, we can subtract the equation:
99x = 523.23232323 - 5.23232323
= 518
To isolate x, we divide both sides of the equation by 99:
x = 518 / 99
Now, we simplify the fraction:
x = (518 / 99)
To answer your question about when to multiply by 10, 100, or 1000: You multiply by a power of 10 that has the same number of repeating digits as the decimal. In this case, since there are two repeating digits (23), we multiply by 100. If there were three repeating digits, we would multiply by 1000, and so on.
To recap, the correct expression of 5.23232323 as a fraction is: 518/99