Rational numbers are numbers that can be written as a fraction a/b, where a and b are integers and b is not 0. Fractions, decimals, and percents often describe the same value in different forms, so comparing them requires careful conversion. Ordering rational numbers helps you make sense of measurements, money, data, grades, and probabilities.
A number line is one of the clearest tools because every rational number has a position from least to greatest.
To compare rational numbers, change them into a common form such as decimals, fractions with common denominators, or percents. Once the values are in the same form, you can compare place values, numerators, or percent amounts directly. Negative rational numbers need extra attention because numbers farther left on the number line are smaller.
The main strategy is to convert, compare, then place each value in its correct position.
Key Facts
- A rational number can be written as a/b, where a and b are integers and b ≠ 0.
- To convert a fraction to a decimal, divide the numerator by the denominator: a/b = a ÷ b.
- To convert a decimal to a percent, multiply by 100: decimal × 100 = percent.
- To convert a percent to a decimal, divide by 100: percent ÷ 100 = decimal.
- To compare fractions with the same denominator, compare numerators: if a > c, then a/b > c/b.
- On a number line, values increase from left to right, so -0.8 < -0.3 < 0.2 < 1.
Vocabulary
- Rational number
- A number that can be written as a fraction of two integers with a nonzero denominator.
- Number line
- A straight line that shows numbers in order from least to greatest.
- Equivalent forms
- Different ways to write the same value, such as 1/2, 0.5, and 50%.
- Common denominator
- A shared denominator used to compare or combine fractions.
- Percent
- A way to describe a number as parts out of 100.
Common Mistakes to Avoid
- Comparing fractions by looking only at the denominator is wrong because a larger denominator can mean smaller pieces. For example, 1/8 is less than 1/4 even though 8 is greater than 4.
- Forgetting to convert percents before comparing is wrong because 75% is not the same as 75. Convert 75% to 0.75 or 75/100 before placing it on a number line.
- Ordering negative decimals as if they were positive is wrong because negative numbers get smaller as their absolute value increases. For example, -0.9 is less than -0.4 because -0.9 is farther left.
- Rounding too early is wrong because it can change the order of close values. Compare exact forms or use enough decimal places before deciding which number is greater.
Practice Questions
- 1 Order these numbers from least to greatest: 3/5, 0.72, 65%, 2/3.
- 2 Which is greater, -3/4 or -0.7? Show your comparison by converting one number to the other form.
- 3 Explain why placing 0.4, 4%, and 4/10 at the same point on a number line would be incorrect.