To find the correct order of the numbers [tex]\(5.\overline{2}, -5 \frac{1}{3}, 520\%,\)[/tex] and [tex]\(-5.7\)[/tex] from least to greatest, we need to convert each form into a comparable numerical value.
1. Repeating decimal: [tex]\(5.\overline{2}\)[/tex]
- This is a repeating decimal where the digit 2 repeats indefinitely. It is equivalent to:
[tex]\[ 5 + \frac{2}{9} = 5 + 0.2222\ldots \approx 5.2222\ldots \][/tex]
2. Mixed fraction: [tex]\(-5 \frac{1}{3}\)[/tex]
- This mixed fraction can be converted to an improper fraction or decimal:
[tex]\[ -5 - \frac{1}{3} = -5 - 0.3333\ldots \approx -5.3333\ldots \][/tex]
3. Percentage: [tex]\(520\%\)[/tex]
- This percentage can be converted to a decimal by dividing by 100:
[tex]\[ 520\% = \frac{520}{100} = 5.2 \][/tex]
4. Decimal: [tex]\(-5.7\)[/tex]
- This is already in decimal form.
Now we compare these numerical values:
- [tex]\(5.\overline{2} \approx 5.2222\ldots\)[/tex]
- [tex]\(-5 \frac{1}{3} \approx -5.3333\ldots\)[/tex]
- [tex]\(520\% = 5.2\)[/tex]
- [tex]\(-5.7\)[/tex]
From least to greatest, the numbers are:
1. [tex]\(-5.7\)[/tex]
2. [tex]\(-5 \frac{1}{3}\)[/tex]
3. [tex]\(5.2\)[/tex] (or [tex]\(520\%\)[/tex])
4. [tex]\(5.\overline{2}\)[/tex]
So, the order from least to greatest is:
[tex]\[ -5.7, -5 \frac{1}{3}, 520\%, 5.\overline{2} \][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{-5.7,-5 \frac{1}{3}, 520 \%, 5 . \overline{2}}
\][/tex]
