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Ordering Fractions and Decimals on the Number Line

Compare and order fractions, decimals and negative numbers, and read or place them on a number line.

Practice this skill 22 min Order fractions and decimals, including on a number line Leer en español
Key idea

What you'll learn

In this lesson you will learn to compare and order fractions, decimals and mixed numbers — including negative ones — and to read and place them on a number line.

Why it matters on the GED:

  • Many questions ask you to pick the greatest, least, or the correct order of a list that mixes forms, like \tfrac{5}{8}, 0.6 and \tfrac23.
  • Number lines appear in drag-and-drop and click-on-the-graph (hot spot) questions, and later in inequalities and graphs.
  • Ordering is hidden inside word problems: "Which board is shortest?", "Who ran fastest?", "Which store has the lowest price per ounce?"

In real life you use this when you compare prices ($2.09 vs $2.10), read a tape measure (\tfrac{5}{16} inch vs \tfrac38 inch), or check a bank balance that has gone below zero.

The good news: there is one reliable method that always works — put every number in the same form, then compare. The rest of this lesson shows you how, plus shortcuts that save time on the no-calculator part.

Words to know

  • Fraction — a part of a whole written \tfrac{a}{b} (\tfrac34 means 3 of 4 equal parts).
  • Numerator — the top number of a fraction (the 3 in \tfrac34): how many parts you have.
  • Denominator — the bottom number (the 4 in \tfrac34): how many equal parts make one whole.
  • Decimal — a number written with a decimal point, using place value (0.75 = 75 hundredths).
  • Place value — the value of a digit's position: tenths (0.1), hundredths (0.01), thousandths (0.001).
  • Equivalent fractions — fractions with the same value (\tfrac12 = \tfrac24 = \tfrac{50}{100}).
  • Common denominator — a shared bottom number used to compare or add fractions (\tfrac23 = \tfrac{8}{12}, \tfrac34 = \tfrac{9}{12}).
  • Mixed number — a whole number plus a fraction (1\tfrac14).
  • Improper fraction — a fraction whose numerator is at least its denominator (\tfrac54 = 1\tfrac14).
  • Number line — a line where every point is a number; values increase to the right.
  • Tick mark — a short line marking an equal step on a number line.
  • Benchmark — an easy reference number such as 0, \tfrac12 or 1 used to estimate.
  • Inequality symbols — < "is less than", > "is greater than" (0.3 < 0.5). The small, pointed end faces the smaller number.
  • Negative number — a number less than zero, to the left of 0 (-2.5).

All Math words in the glossary

Comparing decimals: line up, pad, compare left to right

To compare decimals, compare place by place from left to right — just like alphabetizing words.

Method

  1. Line up the decimal points.
  2. Add zeros on the right so all numbers have the same number of decimal places (this does not change their values: 0.6 = 0.60 = 0.600).
  3. Compare the whole-number parts. If they are equal, compare tenths, then hundredths, then thousandths. The first place where the digits differ decides.

Example: Compare 0.7 and 0.65. Pad: 0.70 vs 0.65. Tenths: 7 > 6, so 0.7 > 0.65.

Example: Compare 3.09 and 3.1. Pad: 3.09 vs 3.10. Wholes equal; tenths: 0 < 1, so 3.09 < 3.1.

Example: Order 0.305, 0.35, 0.3. Pad: 0.305, 0.350, 0.300. Order: 0.3 < 0.305 < 0.35.

Zoomed-in number line from 0.29 to 0.36 with a tick every 0.005 and labels every 0.01. Three points are marked: 0.3 (blue), 0.305 (red) and 0.35 (green). Reading left to right gives 0.3 < 0.305 < 0.35. 0.3 < 0.305 < 0.35 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.3 0.305 0.35
NumberTenthsHundredthsThousandths
0.300300
0.305305
0.350350

Watch out: a longer decimal is not automatically bigger. 0.125 has three digits but is less than 0.5.

Negative decimals. On the left side of zero, the order flips. The number farther from zero is the smaller one. Since 0.8 > 0.3, we get -0.8 < -0.3. Think of temperature: -0.8 °F is colder than -0.3 °F.

Number line from -1 to 0 marked in tenths. Points at -0.8 (red) and -0.3 (blue). Brackets show that -0.8 is 0.8 from zero and -0.3 is 0.3 from zero. Because -0.8 is farther left, -0.8 < -0.3. −1.0 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 −0.8 −0.3 0.3 from 0 0.8 from 0 −0.8 is farther left (farther from 0), so −0.8 < −0.3

Comparing fractions: four tools

Tool 1 — Same denominator: compare numerators. \tfrac{5}{9} > \tfrac{4}{9} (5 ninths is more than 4 ninths).

Tool 2 — Same numerator: the smaller denominator is the bigger fraction, because the whole is cut into fewer, larger pieces. \tfrac{1}{4} > \tfrac{1}{8} and \tfrac{3}{5} > \tfrac{3}{7}.

Tool 3 — Common denominator: rewrite both fractions over a shared denominator (a common multiple of the bottoms). Example: \tfrac23 vs \tfrac34. Use 12: \tfrac{8}{12} vs \tfrac{9}{12}. So \tfrac23 < \tfrac34.

Two equal bars, each cut into 12 equal parts. The top bar is marked in thirds and 8 of 12 parts are shaded: 2/3 = 8/12. The bottom bar is marked in fourths and 9 of 12 parts are shaded: 3/4 = 9/12. Since 8/12 < 9/12, 2/3 < 3/4. 2 3 = 8 12 3 4 = 9 12 8 twelfths < 9 twelfths, so 2/3 < 3/4

Tool 4 — Convert to decimals: divide numerator by denominator. \tfrac{3}{8} = 3 \div 8 = 0.375; \tfrac{2}{5} = 0.4. So \tfrac38 < \tfrac25. This is the best tool when the list mixes fractions and decimals.

Shortcut — cross-multiply (positive fractions): to compare \tfrac{a}{b} and \tfrac{c}{d}, compare a \times d with c \times b. Example: \tfrac{5}{6} vs \tfrac{4}{5}: 5 \times 5 = 25 and 4 \times 6 = 24. Since 25 > 24, \tfrac56 > \tfrac45.

Benchmarks let you estimate fast. Is the fraction more or less than \tfrac12? (Compare the numerator to half the denominator.) \tfrac{3}{7}: half of 7 is 3.5, and 3 < 3.5, so \tfrac37 < \tfrac12. \tfrac{5}{9}: 5 > 4.5, so \tfrac59 > \tfrac12. Done — no division needed.

Mixed numbers: compare the whole parts first. 2\tfrac{7}{8} < 3\tfrac{1}{10} because 2 < 3. Only if the wholes match do you compare the fractions.

Fraction–decimal pairs worth memorizing:

Fraction\tfrac18\tfrac15\tfrac14\tfrac13\tfrac38\tfrac12\tfrac58\tfrac23\tfrac34\tfrac78
Decimal0.1250.20.250.333…0.3750.50.6250.666…0.750.875

Reading and placing numbers on a number line

On a number line, numbers increase to the right and decrease to the left. A number to the left is always smaller: -3 < -1 < 0 < 0.5 < 2.

Step 1 — Find the value of one space. Pick two labeled numbers, then

\text{one space} = \frac{\text{right label} - \text{left label}}{\text{number of spaces between them}}

Count spaces (gaps), not tick marks. A line from 0 to 1 with 4 gaps has 5 tick marks, but each gap is \tfrac14.

Example: between 2 and 3 there are 5 spaces → each space is \tfrac{1}{5} = 0.2. The 3rd tick after 2 is 2 + 3(0.2) = 2.6.

Number line from 2 to 3 split into 5 equal spaces, so each space is 0.2. Ticks are labeled 2, 2.2, 2.4, 2.6, 2.8 and 3. Three hops of +0.2 go from 2 to the third tick, which is marked in red at 2.6. One space = (3 − 2) ÷ 5 = 0.2 2 2.2 2.4 2.8 3 +0.2 +0.2 +0.2 2.6 3rd tick

Step 2 — Place a number: change it to a form that matches the spaces. To place 1\tfrac38 on a line marked in eighths, start at 1 and count 3 spaces right. To place \tfrac{7}{4} on a line marked in fourths, rewrite it as 1\tfrac34.

Step 3 — Read a point: count spaces from the nearest labeled number and multiply by the space value.

Negative side: -\tfrac34 is three quarter-spaces left of 0. -1.25 is one quarter-space left of -1.

The figure below shows a line from -1 to 1 in quarters. Notice that -\tfrac34 is left of -0.2, and \tfrac12 is just left of \tfrac58.

Number line from -1 to 1 with tick marks every 0.25. Four points are marked: -3/4 at -0.75, -0.2, 1/2 at 0.5, and 5/8 at 0.625. −1 0 1 −3/4 −0.2 1/2 5/8

Number line from -1 to 1 with tick marks every 0.25. Points: -\tfrac34, -0.2, \tfrac12, \tfrac58. Reading left to right gives the order from least to greatest: -\tfrac34 < -0.2 < \tfrac12 < \tfrac58.

Ordering a mixed list (the GED way)

When a list mixes fractions, decimals, mixed numbers and negatives, use this routine:

  1. Sort by sign. Negatives go first (left), then zero, then positives.
  2. Convert every number to a decimal with the same number of places (3 is usually enough; use 4 if numbers are very close).
  3. Order the positives by comparing place by place.
  4. Order the negatives by size, then reverse them (biggest size = smallest number).
  5. Rewrite the list using the original forms — the answer choices use them.

Example: Order \tfrac{1}{3},\ -0.5,\ 0.3,\ -\tfrac{3}{4} from least to greatest.

  • Negatives: -0.5 and -0.75. Sizes: 0.75 > 0.5, so -0.75 < -0.5.
  • Positives: 0.300 and 0.333. So 0.3 < \tfrac13.
  • Order: -\tfrac34,\ -0.5,\ 0.3,\ \tfrac13.
Number line from -1 to 1 with ticks every 0.25. Points: -3/4 at -0.75, -0.5, 0.3 and 1/3 (just right of 0.3). Read left to right: -3/4 < -0.5 < 0.3 < 1/3. −1 −0.5 0 0.5 1 − 3 4 −0.5 0.3 1 3 Least to greatest: −3/4, −0.5, 0.3, 1/3

"Between" questions: "Which number is between \tfrac12 and \tfrac34?" means 0.5 < x < 0.75. The endpoints themselves are not between.

Greatest-to-least questions: read the question twice — many wrong answer choices are simply the right order written backwards.

Formula

Rules to memorize (not on the formula sheet)

None of these are on the GED formula sheet — you must know them.

Value of one space on a number line:

\text{space} = \frac{b - a}{n}

where a is the left label, b is the right label, and n is the number of equal spaces between them.

Fraction to decimal:

\frac{a}{b} = a \div b

Cross-multiplication test for positive fractions:

\frac{a}{b} < \frac{c}{d} \quad\text{exactly when}\quad a \times d < c \times b

Order of negatives: if 0 < x < y, then -y < -x.

Worked example

Example 1 (easy): order decimals

Order 0.4, 0.38, 0.405 from least to greatest.

  1. Pad to three places: 0.400, 0.380, 0.405.
  2. Tenths: 4, 3, 4. The smallest is 3, so 0.380 comes first.
  3. Compare 0.400 and 0.405 at the thousandths: 0 < 5.
  4. Order: 0.380 < 0.400 < 0.405.
Zoomed-in number line from 0.37 to 0.41 with ticks every 0.005. Points at 0.38 (blue), 0.4 (red) and 0.405 (green). Order from least to greatest: 0.38, 0.4, 0.405. 0.38 < 0.4 < 0.405 0.37 0.38 0.39 0.40 0.41 0.38 0.4 0.405

Answer: 0.38,\ 0.4,\ 0.405.

Check: 0.405 - 0.4 = 0.005 > 0 and 0.4 - 0.38 = 0.02 > 0. ✓

Worked example

Example 2: which fraction is larger?

Which is larger, \tfrac{5}{6} or \tfrac{7}{9}?

  1. A common denominator of 6 and 9 is 18.
  2. \tfrac56 = \tfrac{5 \times 3}{6 \times 3} = \tfrac{15}{18} and \tfrac79 = \tfrac{7 \times 2}{9 \times 2} = \tfrac{14}{18}.
  3. 15 > 14, so \tfrac{15}{18} > \tfrac{14}{18}.
Two equal bars, each cut into 18 equal parts. The top bar is marked in sixths with 15 of 18 parts shaded: 5/6 = 15/18. The bottom bar is marked in ninths with 14 of 18 parts shaded: 7/9 = 14/18. The top bar has one more part shaded, so 5/6 is larger. 5 6 = 15 18 7 9 = 14 18 15/18 > 14/18, so 5/6 is larger

Answer: \tfrac56 is larger.

Check (cross-multiply): 5 \times 9 = 45 and 7 \times 6 = 42; 45 > 42. ✓

Worked example

Example 3 (word problem): drill bits

A carpenter has drill bits of \tfrac38 in, \tfrac{5}{16} in, \tfrac14 in and \tfrac{11}{32} in. Put them in order from smallest to largest.

  1. The largest denominator is 32, and 8, 16 and 4 all divide into it. Convert:
    • \tfrac38 = \tfrac{12}{32}
    • \tfrac{5}{16} = \tfrac{10}{32}
    • \tfrac14 = \tfrac{8}{32}
    • \tfrac{11}{32} stays the same
  2. Compare numerators: 8 < 10 < 11 < 12.
  3. Translate back.
Ruler from 0 to 1/2 inch marked in 32nds of an inch. The four drill bits are marked: 1/4 = 8/32, 5/16 = 10/32, 11/32, and 3/8 = 12/32. From smallest to largest: 1/4, 5/16, 11/32, 3/8 inch. 0 1 8 1 4 3 8 1 2 in 1 4 8 32 5 16 10 32 11 32 11 32 3 8 12 32 in 32nds: 8/32 < 10/32 < 11/32 < 12/32

Answer: \tfrac14,\ \tfrac{5}{16},\ \tfrac{11}{32},\ \tfrac38 inch.

Check: as decimals, 0.25 < 0.3125 < 0.34375 < 0.375. ✓

Worked example

Example 4 (work backwards): reading and placing on a number line

A number line goes from 1 to 2 and is split into 8 equal spaces. (a) What number is at the 5th tick mark after 1? (b) Where should 1.75 go?

  1. Value of one space: \dfrac{2 - 1}{8} = \dfrac18 = 0.125.
  2. (a) Five spaces past 1: 1 + 5 \times \tfrac18 = 1\tfrac58 = 1.625.
  3. (b) Work backwards: 1.75 - 1 = 0.75, and 0.75 \div 0.125 = 6 spaces.
Number line from 1 to 2 split into 8 equal spaces of 1/8 = 0.125. Ticks are labeled 1, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, 2. The 5th tick after 1 is marked in red at 1 5/8 = 1.625; the 6th tick is marked in blue at 1.75. One space = (2 − 1) ÷ 8 = 1/8 = 0.125 1 1.125 1.25 1.375 1.5 1.625 1.75 1.875 2 (a) 5th tick: 1.625 (b) 6th tick: 1.75

Answers: (a) 1\tfrac58 = 1.625; (b) the 6th tick mark after 1.

Check: 1 + \tfrac68 = 1\tfrac34 = 1.75. ✓

Worked example

Example 5 (GED-hard): stock price changes

The change in a stock's price (in dollars) on four days was: Monday -0.45, Tuesday -\tfrac38, Wednesday \tfrac14, Thursday -\tfrac12. Order the days from the lowest (most negative) change to the highest, and name the day with the smallest loss.

  1. Convert to decimals: Mon -0.450, Tue -0.375, Wed +0.250, Thu -0.500.
  2. The only positive is Wednesday — it goes last.
  3. Negatives by size: 0.500 > 0.450 > 0.375. Reverse for negatives: -0.500 < -0.450 < -0.375.
  4. Order: Thursday, Monday, Tuesday, Wednesday.
  5. Losses are the negative days. The smallest loss is the negative number closest to zero: Tuesday (-\tfrac38, a loss of $0.375).
Number line from -0.6 to 0.3 with labels every 0.1. Stock changes are marked: Thursday -1/2 at -0.5, Monday -0.45, Tuesday -3/8 at -0.375 (all red, losses) and Wednesday 1/4 at 0.25 (green, a gain). Left to right: Thursday, Monday, Tuesday, Wednesday. Tuesday is the loss closest to zero. −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 Thu − 1 2 Tue − 3 8 Wed 1 4 Mon −0.45 Least to greatest: Thu, Mon, Tue, Wed — smallest loss: Tue

Answer: Thu, Mon, Tue, Wed; smallest loss on Tuesday.

Check: on a number line, -0.5 is farthest left and -0.375 is the closest negative to 0. ✓

Calculator tip

TI-30XS MultiView tips

  • Fraction to decimal: type the fraction with the n/d key (for example 5 n/d 8), press enter, then press toggle to switch between \tfrac58 and 0.625.
  • Or just divide: 5 ÷ 8 enter gives 0.625.
  • Negative numbers: use the (−) key, not the subtraction key: (−) 3 n/d 4.
  • Write each decimal down on your erasable note board before comparing — you will compare more accurately on paper than in your head.

When not to use it: simple comparisons like 0.6 vs 0.59, or \tfrac14 vs \tfrac18, are often in the calculator-free part of the test. Practice padding with zeros, the benchmark \tfrac12, and the memorized fraction table so you do not need the calculator.

Common mistake

Common mistakes

  • "Longer decimal = bigger." Wrong: 0.125 > 0.5. Right: pad to 0.125 vs 0.500; 1 < 5 in the tenths, so 0.125 < 0.5.
  • "Bigger denominator = bigger fraction." Wrong: \tfrac18 > \tfrac14. Right: eighths are smaller pieces, so \tfrac18 < \tfrac14.
  • Comparing only numerators when denominators differ. Wrong: \tfrac{3}{8} > \tfrac{2}{3} because 3 > 2. Right: 0.375 < 0.667.
  • Forgetting that negatives flip. Wrong: -0.8 > -0.3 because 8 > 3. Right: -0.8 is farther left, so -0.8 < -0.3.
  • Counting tick marks instead of spaces. Wrong: "5 tick marks from 0 to 1, so each is \tfrac15." Right: 5 tick marks make 4 spaces, so each is \tfrac14.
  • Writing fraction digits as a decimal. Wrong: \tfrac58 = 0.58. Right: \tfrac58 = 5 \div 8 = 0.625.
Test strategy

GED test strategy

  • Read the question twice. Underline "least", "greatest", "least to greatest" or "greatest to least". A reversed list is almost always one of the wrong choices.
  • Convert once, then compare. Write every number as a decimal with the same number of places in a neat column on your erasable note board.
  • Eliminate by sign. If the question asks for the least number and there is a negative, all positive choices are out.
  • Use benchmarks to estimate. Is it more or less than 0? Than \tfrac12? Than 1? This often eliminates two choices instantly.
  • Number lines: find the space value before anything else, and count spaces from the nearest label to avoid long counts.
  • Check units in word problems (inches vs feet, seconds) — all values must be in the same unit before comparing.
  • Time: these should be 1-minute questions. If you are stuck and the calculator is allowed, convert to decimals with it and move on.

Check yourself

Order fractions and decimals, including on a number lineNo calculator

Which number is the greatest?

Answer choices
Hint

Line up the decimal points and give every number the same number of decimal places by adding zeros on the right.

Check yourself

Order fractions and decimals, including on a number lineNo calculatorHot spot

Click the point on the number line that represents \dfrac{5}{4}.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Number line from 0 to 2. Only 0, 1 and 2 are labeled. There are tick marks every one-fourth (0.25) between the labels, so each whole is split into 4 equal spaces. 0 1 2
Hint

First figure out how much each small space is worth: how many spaces are between 0 and 1?

Summary

  • To compare decimals, pad with zeros and compare place by place from left to right; more digits does not mean bigger.
  • To compare fractions, use a common denominator, convert to decimals, or cross-multiply; benchmarks like \tfrac12 give quick estimates.
  • On a number line, one space = \dfrac{\text{difference of labels}}{\text{number of spaces}}; count spaces, not tick marks.
  • Numbers increase to the right. For negatives, the one farther from zero is smaller.
  • For mixed lists: sort by sign, convert to decimals, order, then rewrite in the original forms.
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