Comparing rational numbers involves determining which number is larger or smaller. This article provides a step-by-step guide on how to compare rational numbers, including fractions, decimals, percentages, and square roots.
Converting to a Common Format
The most effective way to compare rational numbers is to convert them into a common format, typically decimals. This allows for easy comparison using place value.
Converting Percentages to Decimals
To convert a percentage to a decimal, move the decimal point two places to the left. For example:
- 13% becomes 0.13
- 213% becomes 2.13
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. For example:
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To convert ¾ to a decimal, perform the long division 3 ÷ 4.
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The result is 0.75.
Some fractions result in repeating decimals. In such cases, round the decimal to three or four decimal places for easier comparison.
Comparing Square Roots
Comparing square roots can be challenging without a calculator. One method is to estimate the square root by identifying the two perfect squares it falls between. For example:
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To estimate √41, note that 41 lies between the perfect squares 36 (6²) and 49 (7²).
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Therefore, √41 falls between 6 and 7. This provides an approximate range for comparison purposes.
Example: Ordering Rational Numbers
Let’s order the following rational numbers from least to greatest:
87%, ⅖, √78, 45/4, 6.743
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Convert to Decimals:
- 87% = 0.87
- ⅖ = 0.4
- √78 ≈ 8.83 (since 78 is between 8²=64 and 9²=81)
- 45/4 = 11.25
- 6.743 remains as is.
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Order the Decimals:
0.4, 0.87, 6.743, 8.83, 11.25. -
Replace with Original Values:
⅖, 87%, 6.743, √78, 45/4.
Therefore, the numbers in ascending order are ⅖, 87%, 6.743, √78, and 45/4.
Conclusion
Comparing rational numbers requires a systematic approach involving conversion to a common format, typically decimals. By following these steps, you can confidently determine the order of rational numbers, regardless of their initial representation as fractions, decimals, percentages, or square roots. Remember to consider rounding for repeating decimals and estimating for square roots to simplify the comparison process.
