Should I use common denominators or convert to decimals?

Either works, so pick based on the numbers. Use a common denominator when denominators are small or related, such as 3/4 and 5/6, or when a fraction repeats like 2/3 and rounding could mislead you. Convert to decimals when the list already mixes fractions with decimals or percents, since one common language is faster. Both methods must give the same answer, so using the second one as a check is smart when two numbers are very close.

Why is -0.9 less than -0.4 when 9 is bigger than 4?

Because the digits tell you distance from zero, not value. Both numbers sit to the left of zero, and -0.9 is 0.9 units away while -0.4 is only 0.4 units away. Farther left always means less, so -0.9 0.4), then flip the sign of the inequality when both are negative.

How do I compare a mixed number to a decimal quickly?

Compare the whole-number parts first. Since 31/8 has a whole part of 3 and 2.95 has a whole part of 2, the mixed number is greater with no further work. Only if the whole parts match do you need to compare the fractional parts, which you do by turning the fraction into a decimal: 31/8=3.125 versus 3.09 gives 3.125>3.09. For negatives, remember the larger whole part means farther left, so -31/8<-2.95.

What if two numbers turn out to be equal?

Then you write = instead of , and on a number line they land on the exact same point. This happens often with equivalent forms: 3/4, 0.75 and 75\% are all the same number, and -6/8=-0.75. When ordering a list, equal values can be listed in either order or joined with an equals sign, but say clearly in your answer that they are equivalent so your reasoning is complete.

M7MATH-1.3

Comparing & Ordering Rational Numbers

Q: What if two numbers turn out to be equal?

Then you write = instead of , and on a number line they land on the exact same point. This happens often with equivalent forms: 3/4, 0.75 and 75\% are all the same number, and -6/8=-0.75. When ordering a list, equal values can be listed in either order or joined with an equals sign, but say clearly in your answer that they are equivalent so your reasoning is complete.

Compare rational numbers with common denominators or decimals, read the result on a number line, and order fractions, decimals and negatives from least to greatest.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Comparing & Ordering Rational Numbers, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Comparing & Ordering Rational Numbers free

What this lesson covers

You already know how to plot rational numbers on a number line and how to switch between fractions, decimals and percents. Now you put those skills to work answering the question that shows up constantly in real problems: which number is bigger?

With positive whole numbers this is easy. But once fractions, decimals and negatives are mixed together in the same list, your eyes can lie to you. Is

-\frac{5}{8}

greater or less than

-0.6

? Is

\frac{7}{9}

bigger than

0.8

? This lesson gives you two dependable methods — rewriting with a common denominator, or converting everything to decimals — plus the number-line picture that explains why the answer is what it is. By the end you will be able to take a scrambled set like

-1\frac{1}{4}

0.3

-\frac{2}{5}

\frac{1}{8}

and line it up from least to greatest without guessing.

Two Reliable Ways to Compare Any Two Rational Numbers

A rational number is any number you can write as a fraction of two integers, so fractions, terminating and repeating decimals, mixed numbers, percents and integers are all in the club. To compare two of them, you must put them in the same form. There are exactly two standard moves.

Common denominator. Rewrite both fractions with the same bottom number, then compare the numerators. To compare

\frac{5}{8}

and

\frac{7}{12}

, use the least common denominator

24

\frac{5}{8}=\frac{15}{24}

and

\frac{7}{12}=\frac{14}{24}

. Since

15>14

\frac{5}{8}>\frac{7}{12}

.

Decimal conversion. Divide numerator by denominator and compare place value by place value.

\frac{5}{8}=0.625

and

\frac{7}{12}=0.58\overline{3}

. Comparing tenths,

6>5

, so

\frac{5}{8}

wins again — the same answer, as it must be.

Situation	Better method	Why
Denominators are small or related (4 and 8)	Common denominator	Exact, quick mental work
List mixes fractions and decimals	Decimal conversion	Puts everything in one language
Fraction repeats, like $\frac{2}{3}$	Common denominator	Avoids rounding mistakes
Mixed numbers with different wholes	Compare wholes first	$2\frac{1}{9}>1\frac{7}{8}$ instantly

Where students go wrong: comparing numerators while ignoring denominators.

\frac{3}{4}

is not less than

\frac{5}{9}

just because

3<5

. Also watch decimal length —

0.7

is greater than

0.68

, even though

0.68

has more digits. Line up the decimal points and add zeros (

0.70

versus

0.68

) so place values match.

Negatives Flip Your Intuition

On a number line, numbers increase as you move right. That single fact settles every comparison: the number farther right is greater, the number farther left is less. For positives, farther right means "bigger looking." For negatives, it is the reverse — the number that looks bigger is actually less.

Compare

-3

and

-8

. Eight is more than three, but

-8

sits eight units left of zero while

-3

sits only three units left. So

-8<-3

. The same logic works for fractions:

-\frac{7}{8}

is farther from zero than

-\frac{1}{2}

, so

-\frac{7}{8}<-\frac{1}{2}

.

A clean rule: for two negative numbers, the one with the greater absolute value is the lesser number. Absolute value measures distance from zero, and going farther left means going lower in value.

A useful shortcut is to compare the two numbers as if they were positive, then reverse the inequality sign. Since

\frac{5}{8}>0.6

, it follows that

-\frac{5}{8}<-0.6

. Check it on the line:

-0.625

is left of

-0.6

.

Two more facts settle most mixed comparisons instantly. Every negative number is less than zero, and zero is less than every positive number, so any negative is less than any positive — no computation needed.

-100.5<0.001

.

The most common error in this lesson is writing

-0.4<-0.9

because "4 is smaller than 9." Whenever you compare negatives, sketch a quick number line with zero marked. Seeing

-0.9

sitting to the left of

-0.4

makes the correct statement

-0.9<-0.4

obvious.

Ordering a Mixed Set from Least to Greatest

When a problem hands you five or six numbers in different forms, use a repeatable routine instead of comparing them in random pairs.

First, split the list into negatives and non-negatives. Every negative goes to the left of every zero-or-positive, so you have already done half the ordering.

Second, convert each number to a decimal, rounding to enough places to break ties (three places is almost always plenty). Write the decimal next to the original number so you do not lose track of which is which.

Third, order the negatives from most negative to least negative — that means largest decimal distance from zero comes first. Then order the positives normally.

Fourth, and this is the step students skip, rewrite the final answer using the original forms given in the problem. If the question listed

\frac{3}{5}

, your answer should say

\frac{3}{5}

, not

0.6

, unless the directions say otherwise.

Example: order

\frac{1}{4}

-0.5

-\frac{3}{8}

0.2

-1

Number	Decimal	Group
$\frac{1}{4}$	$0.25$	positive
$-0.5$	$-0.5$	negative
$-\frac{3}{8}$	$-0.375$	negative
$0.2$	$0.2$	positive
$-1$	$-1.0$	negative

Negatives ordered:

-1

-0.5

-\frac{3}{8}

. Positives ordered:

0.2

\frac{1}{4}

. Final answer:

-1

-0.5

-\frac{3}{8}

0.2

\frac{1}{4}

.

If the directions say greatest to least, order it the easy way first and then reverse the whole list — that is safer than trying to think backwards while you compute.

Benchmarks, Number Lines and Sanity Checks

You do not always need exact arithmetic. Comparing each number to the benchmarks

0

\frac{1}{2}

and

1

often decides things in one step. A fraction is greater than

\frac{1}{2}

when the numerator is more than half the denominator:

\frac{7}{12}>\frac{1}{2}

because

7>6

. A fraction is less than

1

when the numerator is smaller than the denominator. So

\frac{7}{12}

and

\frac{4}{9}

need no common denominator at all — one is above one half, the other below, so

\frac{7}{12}>\frac{4}{9}

.

The number line is also your interpretation tool. Once you know

-\frac{3}{4}<\frac{1}{5}

, you can say more: the two numbers are

0.95

units apart, and

-\frac{3}{4}

is closer to

-1

than to

0

. Questions often ask you to place numbers between tick marks, so decide first which two whole numbers a value falls between.

-2\frac{3}{10}

lies between

-3

and

-2

, closer to

-2

.

Every inequality can be written two ways.

-5<-2

and

-2>-5

say exactly the same thing. Read the symbol out loud as "is less than" or "is greater than" and confirm the open end faces the greater number.

Finally, sanity-check with context. If a temperature drops from

-2.5

degrees to

-4.1

degrees, it got colder, which agrees with

-4.1<-2.5

. If one submarine is at

-30.5

meters and another at

-30\frac{1}{4}

meters, the first is deeper, because

-30.5<-30.25

. Real situations make the ordering feel obvious and catch sign slips fast.

Key terms

Rational number.: Any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ ; includes integers, terminating decimals and repeating decimals.
Common denominator.: A shared bottom number for two or more fractions, created by multiplying each fraction by a form of $1$ , so the numerators can be compared directly.
Least common denominator (LCD).: The smallest common multiple of the denominators, which keeps the numbers you compare as small as possible.
Absolute value.: The distance of a number from zero on the number line, always zero or positive; for two negatives, the greater absolute value belongs to the lesser number.
Inequality symbol.: The signs $<$ and $>$ , which point toward the lesser number; $-4<-1$ and $-1>-4$ state the same relationship.
Benchmark number.: A familiar reference value such as $0$ , $\frac{1}{2}$ or $1$ used to compare fractions quickly without exact computation.
Ascending order.: An arrangement from least to greatest, matching left-to-right order on a number line.
Terminating decimal.: A decimal that ends, such as $0.625$ ; a fraction terminates when its denominator in lowest terms has only factors of $2$ and $5$ .

Worked example

Order these numbers from least to greatest, then state which two of them are closest together:

-\frac{3}{4}

0.6

-1\frac{1}{5}

\frac{5}{8}

-0.7

Step 1: Sort into groups. The negatives are

-\frac{3}{4}

-1\frac{1}{5}

and

-0.7

. The positives are

0.6

and

\frac{5}{8}

. Every negative is less than every positive, so the three negatives fill the first three spots.

Step 2: Convert to decimals.

-\frac{3}{4} = -0.75

because

3 \div 4 = 0.75

-1\frac{1}{5} = -1.2

because

\frac{1}{5}=0.2

-0.7

is already a decimal.

\frac{5}{8}=0.625

because

5 \div 8 = 0.625

.

Step 3: Order the negatives. Their distances from zero are

0.75

1.2

and

0.7

. The largest distance is

1.2

, so

-1.2

is farthest left and therefore least. Next comes

-0.75

, then

-0.7

. Negatives in order:

-1\frac{1}{5}

-\frac{3}{4}

-0.7

.

Step 4: Order the positives. Compare

0.6

and

0.625

by place value: tenths are both

6

, but hundredths are

0

and

2

, so

0.6 < 0.625

. Positives in order:

0.6

\frac{5}{8}

.

Step 5: Write the full list in the original forms.

-1\tfrac{1}{5},\ -\tfrac{3}{4},\ -0.7,\ 0.6,\ \tfrac{5}{8}

Step 6: Find the closest pair. Gaps between neighbors are

0.45

0.05

1.3

and

0.025

. The smallest gap is

0.025

, so

0.6

and

\frac{5}{8}

are closest together.

Practice questions

Which list is arranged from least to greatest?

$-\frac{2}{3},\ -0.6,\ -\frac{1}{2},\ 0.55$
$-0.6,\ -\frac{2}{3},\ -\frac{1}{2},\ 0.55$
$-\frac{1}{2},\ -0.6,\ -\frac{2}{3},\ 0.55$
$0.55,\ -\frac{1}{2},\ -0.6,\ -\frac{2}{3}$

Answer: $-\frac{2}{3},\ -0.6,\ -\frac{1}{2},\ 0.55$

Convert:

-\frac{2}{3} \approx -0.667

-0.6

-\frac{1}{2}=-0.5

, and

0.55

. On a number line,

-0.667

is farthest left, then

-0.6

, then

-0.5

, then the only positive,

0.55

. The second list reverses the first two because

0.6

looks smaller than

\frac{2}{3}

, but the greater distance from zero belongs to

-\frac{2}{3}

, making it the lesser number. The third list orders the negatives as if they were positives, and the fourth is greatest to least.

Use a common denominator to decide which is greater,

-\frac{5}{8}

-0.6

. Show your reasoning and describe the positions of the two numbers on a number line.

Answer: $-0.6$ is greater, since $-\frac{5}{8}=-\frac{25}{40}$ and $-0.6=-\frac{24}{40}$ , so $-\frac{5}{8}<-0.6$ .

First write

-0.6

as a fraction:

-0.6=-\frac{6}{10}=-\frac{3}{5}

. The denominators

8

and

5

have least common denominator

40

. Rewrite each:

-\frac{5}{8}=-\frac{25}{40}

and

-\frac{3}{5}=-\frac{24}{40}

. Because

-25

is less than

-24

-\frac{5}{8}

is the lesser number. On a number line both sit between

-1

and

0

, with

-\frac{5}{8}=-0.625

just

0.025

units to the left of

-0.6

. The decimal check agrees, which is a good habit whenever the numbers are this close.

A diver is at

-12\frac{1}{2}

meters, a second diver is at

-12.35

meters, and a third is at

-12\frac{3}{5}

meters. Which diver is deepest, and write an inequality comparing the shallowest and deepest depths.

Answer: The third diver at $-12\frac{3}{5}$ meters is deepest; $-12\frac{3}{5} < -12.35$ .

Convert each depth to a decimal:

-12\frac{1}{2}=-12.5

-12.35

stays as is, and

-12\frac{3}{5}=-12.6

. Deeper means farther below zero, so the number farthest left on the number line is deepest:

-12.6

. The shallowest is

-12.35

, since it is closest to zero. Ordering all three from least to greatest gives

-12.6

-12.5

-12.35

. The inequality comparing deepest to shallowest is

-12\frac{3}{5} < -12.35

, which you can also write as

-12.35 > -12\frac{3}{5}

FAQ

Should I use common denominators or convert to decimals?: Either works, so pick based on the numbers. Use a common denominator when denominators are small or related, such as $\frac{3}{4}$ and $\frac{5}{6}$ , or when a fraction repeats like $\frac{2}{3}$ and rounding could mislead you. Convert to decimals when the list already mixes fractions with decimals or percents, since one common language is faster. Both methods must give the same answer, so using the second one as a check is smart when two numbers are very close.
Why is $-0.9$ less than $-0.4$ when 9 is bigger than 4?: Because the digits tell you distance from zero, not value. Both numbers sit to the left of zero, and $-0.9$ is $0.9$ units away while $-0.4$ is only $0.4$ units away. Farther left always means less, so $-0.9<-0.4$ . A shortcut: compare them as positives ( $0.9>0.4$ ), then flip the sign of the inequality when both are negative.
How do I compare a mixed number to a decimal quickly?: Compare the whole-number parts first. Since $3\frac{1}{8}$ has a whole part of $3$ and $2.95$ has a whole part of $2$ , the mixed number is greater with no further work. Only if the whole parts match do you need to compare the fractional parts, which you do by turning the fraction into a decimal: $3\frac{1}{8}=3.125$ versus $3.09$ gives $3.125>3.09$ . For negatives, remember the larger whole part means farther left, so $-3\frac{1}{8}<-2.95$ .
What if two numbers turn out to be equal?: Then you write $=$ instead of $<$ or $>$ , and on a number line they land on the exact same point. This happens often with equivalent forms: $\frac{3}{4}$ , $0.75$ and $75\%$ are all the same number, and $-\frac{6}{8}=-0.75$ . When ordering a list, equal values can be listed in either order or joined with an equals sign, but say clearly in your answer that they are equivalent so your reasoning is complete.

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The Crimsora tutor teaches Comparing & Ordering Rational Numbers live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

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