Answer:
3 8/33
Explanation:
You want the mixed-number equivalent of the repeating decimal 3.2424....
Repeating decimal
A repeating decimal fraction that starts at the decimal point can be converted to a ratio of integers by expressing the repeating digits over the same number of 9s.
Here, the two repeating digits "24" mean the fraction equivalent is ...
0.2424... = 24/99 = 8/33
The mixed number equivalent of 3.2424... is 3 8/33.
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Additional comment
If the repeating portion does not start at the decimal point, you can use the following technique to do the conversion.
- Multiply the original number by 10^n, where n is the number of repeating digits.
- Subtract the original number from this product. Repeating digits should cancel to zeros.
- Divide this difference by (10^n) -1 and simplify the resulting fraction.
Effectively, this multiplies and divides the number by ((10^n) -1)/((10^n) -1). In the case here, that would be multiplication by 1 in the form 99/99.
3.242424... × 99/99 = 321/99 = 3 8/33
The "321" will be manifested on a calculator as 320.99999.... If you multiply by (100 -1) with pencil and paper, you can see that you get ...
324.2424... - 3.2424... = 321 . . . . . . repeating digits cancel
If the repeat starts somewhere else, you can still use the "fraction with equal number of 9s" technique, but with a multiplier:
3.6242424... = 3.6 + 0.0242424 ... = 3.6 + (1/10)(24/99)
= 36/10 + 8/330 = 3 103/165
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