How to compare and order fractions, decimals, percents and negatives
TEAS 7 outline: Compare and order rational numbers
Also covers HESI A2 Math (Decimals)
- Put every number in one form, usually decimals, before you compare.
- Pad decimals with zeros to the same length: 0.6 = 0.60.
- On a number line, numbers grow from left to right.
- Among negatives, the one farther from zero is smaller: −2.5 < −2.
- "At most" means ≤ and "at least" means ≥.
Swipe the picture sideways to see all of it.
Four overnight lows: negative 2.5, negative 2 and three-fourths, one-fourth, and negative 0.4 degrees Celsius. To order mixed forms, first rewrite every number as a decimal.
Key facts
- On a number line, numbers increase from left to right; a < b means a is to the left of b.
- Easiest method for mixed forms: convert fractions and percents to decimals, then compare the decimals.
- Adding zeros to the end of a decimal does not change its value (0.6 = 0.60 = 0.600), so pad decimals to the same length before comparing.
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- Fractions with the same denominator compare by their numerators; fractions with different denominators can be rewritten with a common denominator first.
- Any positive number is greater than any negative number, and among negatives the one farther from zero is smaller (−2.5 < −2).
- Symbols: < less than, > greater than, ≤ less than or equal to ("no more than," "at most"), ≥ greater than or equal to ("at least").
Quick check1 question · 2 more at the end
Rain gauges at four weather stations recorded 0.09 inch, 0.1 inch, 0.089 inch and 0.099 inch of rain on the same day. Which amount is the greatest?
Correct answer: A.
Give every decimal the same number of places by adding zeros at the end: 0.090, 0.100, 0.089 and 0.099. Now compare them like whole numbers: 100 is greater than 99, 90 and 89. So 0.1 inch is the greatest, even though it has the fewest digits.
A Correct: written as 0.100, it is greater than 0.099, 0.090 and 0.089.
B 0.089 is the smallest amount; it may look big because it has three digits after the point.
C 0.099 has more digits and several 9s, but as 0.099 vs 0.100 it is smaller than 0.1.
D 0.09 is the same as 0.090, which is less than 0.100 and 0.099.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: To compare or order rational numbers, rewrite them all in one form, usually decimals. Pad the decimals with zeros so they have the same number of places, then compare digit by digit from the left. On the number line, numbers grow to the right, so a negative farther from zero is smaller.
1 What do the comparison symbols mean?Picture a number line, where every number to the right is bigger. 1 min
Picture a number line, where every number to the right is bigger. The symbols just say which side a number sits on.
| Symbol | Means | Everyday words |
|---|---|---|
| a < b | a is less than b (a is left of b) | "fewer than," "under" |
| a > b | a is greater than b (a is right of b) | "more than," "over" |
| a ≤ b | a is less than or equal to b | "at most," "no more than" |
| a ≥ b | a is greater than or equal to b | "at least," "no less than" |
Memory help: the open mouth of < or > always faces the bigger number.
2 How do I compare decimals?Give every decimal the same number of places, then compare them like whole numbers. 1 min
Give every decimal the same number of places, then compare them like whole numbers. Adding zeros at the end never changes a decimal's value, so 0.6 = 0.600.
Example 1. Four lab samples weigh 0.6 g, 0.58 g, 0.605 g and 0.065 g. Order them from least to greatest.
- Pad to three places: 0.600, 0.580, 0.605, 0.065.
- Compare as whole numbers: 65 < 580 < 600 < 605.
- Answer: 0.065, 0.58, 0.6, 0.605.
Classic wrong turn: thinking 0.58 is bigger than 0.6 because 58 is bigger than 6. Pad first, and you are comparing 580 with 600.
3 How do I compare fractions?When denominators match, the bigger numerator wins, and otherwise you change the fractions first. 1 min
When denominators match, the bigger numerator wins, and otherwise you change the fractions first. The denominator tells how many pieces the whole is cut into, so eighths are smaller pieces than fifths.
Example 2. One recipe uses 5/8 cup of sugar and another uses 3/5 cup. Which uses more?
- Way A, decimals: 5 ÷ 8 = 0.625 and 3 ÷ 5 = 0.6 (= 0.600).
- Way B, common denominator 40: 5/8 = 25/40 and 3/5 = 24/40.
- Both ways agree: 5/8 cup is more, by 1/40 cup.
Classic wrong turn: deciding by the denominators alone ("8 is bigger, so 5/8 is bigger" or "eighths are small, so 5/8 is smaller"). Only a full comparison tells you.
4 How do negatives change the order?Negatives run backward, so the one farther from zero is the smaller number. 1 min
Negatives run backward, so the one farther from zero is the smaller number. Any positive number beats any negative number.
Example 3. Four overnight lows were −2.5°C, −2 3/4°C, 1/4°C and −0.4°C. Order them from coldest to warmest.
- Convert: −2 3/4 = −2.75 and 1/4 = 0.25.
- Place on a number line: −2.75 is farthest left, then −2.5, then −0.4, then 0.25.
- Answer: −2 3/4, −2.5, −0.4, 1/4.
Classic wrong turn: ordering by the digits alone and putting −2.5 before −2 3/4 because 2.5 < 2.75. With negatives, the larger size is the colder, smaller number.
5 What if the list mixes fractions, decimals and percents?Turn everything into decimals first, because decimals are the easiest form to line up. 1 min
Turn everything into decimals first, because decimals are the easiest form to line up. Divide fractions top by bottom, and move a percent's decimal point two places left (62% = 0.62).
Use the calculator for awkward fractions like 7/12; benchmark fractions such as 1/2, 1/4 and 3/4 are faster from memory. Keep three decimal places so close values do not look equal.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: More digits does not mean bigger: 0.6 is greater than 0.58, because 0.60 > 0.58.
Watch out: Among negatives, a bigger-looking number is smaller: −0.9 is less than −0.2, not greater.
Watch out: A bigger denominator means smaller pieces, not a bigger fraction: 1/8 is less than 1/5.
Watch out: Convert before you compare, and do not compare a percent's digits with a decimal's digits (40% is 0.4, which is less than 0.45).
How to remember it
The one rule to remember is to make every number look alike before you compare. Same form, same number of decimal places, then read left to right.
Self-check:
- Order 0.3, 0.33, 0.303 and 1/3 from least to greatest.
- Explain why −1/2 is greater than −3/4.
- Write "the suitcase can weigh at most 23 kg" with a symbol, using w for the weight.
Show answers
- 0.3, 0.303, 0.33, 1/3.
- Change 1/3 to a decimal: 1 ÷ 3 = 0.333...
- Pad to three places: 0.300, 0.330, 0.303, 0.333.
- Compare as whole numbers: 300 < 303 < 330 < 333.
- −1/2 is closer to zero.
- As decimals, −1/2 = −0.5 and −3/4 = −0.75.
- −0.75 is farther from zero, so it sits farther left on the number line.
- Farther left means smaller, so −1/2 > −3/4.
- w ≤ 23.
- "At most" allows exactly 23 kg, so use ≤.