How Can You Compare and Order Rational Numbers?

Understanding how to compare and order rational numbers is a fundamental math skill. This guide provides a step-by-step approach, using clear examples and explanations, to help you master this concept. We’ll cover converting between fractions, decimals, percentages, and even estimating square roots to accurately arrange rational numbers.

Converting Rational Numbers for Comparison

The easiest way to compare rational numbers is to convert them into a common format: decimals. Let’s break down the conversion process for different types of rational numbers:

Percentages to Decimals

Converting a percentage to a decimal is straightforward:

1. Replace the percent sign (%) with a decimal point (.).
2. Move the decimal point two places to the left.

Example:

• 13% becomes 0.13
• 213% becomes 2.13

Fractions to Decimals

To convert a fraction to a decimal, remember that the fraction bar represents division:

1. Divide the numerator (top number) by the denominator (bottom number).
2. Use long division if necessary.

Example:

To convert ¾ to a decimal, divide 3 by 4:

   0.75
4 | 3.00
   2.8
   ----
     20
     20
   ----
      0

Therefore, ¾ = 0.75.

Note: Some fractions result in repeating decimals (e.g., ⅓ = 0.333…). In these cases, round to three or four decimal places for easier comparison.

Estimating Square Roots

Comparing square roots without a calculator can be done through estimation:

1. Identify the two perfect squares that the number falls between.
2. Determine the square roots of these perfect squares. The square root of the original number lies between these two values.

Example:

To estimate √41:

• The perfect squares surrounding 41 are 36 and 49.
• √36 = 6 and √49 = 7.
• Therefore, 6 < √41 < 7. This tells us √41 is between 6 and 7.

Ordering Rational Numbers: A Practical Example

Let’s put it all together and order the following rational numbers from least to greatest:

87%, ⅖, √78, 45/4, 6.743

1. Convert to Decimals:

   • 87% = 0.87
   • ⅖ = 0.4 (2 divided by 5)
   • √78 ≈ 8.83 (estimated between 8 and 9)
   • 45/4 = 11.25
   • 6.743 (already a decimal)

2. Order the Decimals:

0.4, 0.87, 6.743, 8.83, 11.25

3. Replace with Original Values:

⅖, 87%, 6.743, √78, 45/4

Therefore, the numbers ordered from least to greatest are ⅖, 87%, 6.743, √78, and 45/4.

Conclusion

By converting rational numbers to a common format (decimals), you can easily compare and order them. Remember the specific rules for converting percentages, fractions, and estimating square roots. With practice, you’ll be able to confidently tackle any rational number comparison problem.