How do I know whether a fraction becomes a terminating or a repeating decimal without dividing?

Reduce the fraction to lowest terms first, then factor the denominator. If the only prime factors are 2 and 5, the decimal terminates, because 2 and 5 are the primes that build powers of ten. If any other prime appears, such as 3, 7, or 11, the decimal repeats. For instance 6/15 reduces to 2/5, which terminates as 0.4, while 5/14 has the factor 7 and repeats.

Is -21/3 between -2 and -1 or between -3 and -2?

Between -3 and -2. The mixed number means 21/3 units away from zero, and negatives move left, so you go past -2 by one-third of a unit toward -3. As a decimal it is -2.3, which confirms it is less than -2.

Can two different-looking numbers really share one dot on the number line?

Yes, and that is the whole point of equivalence. 1/2, 4/8, 50/100, and 0.5 all describe the same location using different-size pieces. A quick test: subtract the two forms. If the difference is 0, they are the same point.

Do I write the repeating bar over every digit after the decimal point?

No, only over the digits that actually cycle. In 5/12 = 0.416 the 4 and 1 appear once and only the 6 repeats. Putting the bar over all three digits would claim the pattern 416416, which is a different number entirely.

M7MATH-1.2

Rational Numbers on the Number Line

Learn to plot positive and negative fractions and decimals on a number line, see why equivalent forms share one point, and divide to get terminating or repeating decimals.

What you'll do in this lesson

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

Start Rational Numbers on the Number Line free

What this lesson covers

A number line is more than a ruler with tick marks — it is a picture of every rational number at once. Each rational number gets exactly one address on that line, and that address does not change just because you rewrite the number. The point one and a half units to the right of zero is the home of

1.5

\frac{3}{2}

\frac{6}{4}

, and

1\frac{1}{2}

all at the same time.

In this lesson you will practice three connected skills: partitioning the space between whole numbers so a fraction lands in exactly the right spot, plotting negatives by measuring the same distance to the left of zero, and using long division to turn any fraction into a decimal that either stops or repeats forever. By the end you should be able to look at

-\frac{11}{4}

and

-2.75

and know instantly that your pencil goes in the same place twice.

What a Rational Number Is, and Why Every One Has a Spot

A rational number is any number that can be written as a ratio

\frac{a}{b}

where

a

and

b

are integers and

b \neq 0

. That definition quietly includes a lot: every integer (

7 = \frac{7}{1}

), every fraction, every mixed number, every terminating decimal (

0.4 = \frac{4}{10}

), and every repeating decimal.

On the number line, the sign tells you which direction to travel from zero and the size tells you how far. Positive numbers sit to the right of

0

, negative numbers to the left. Zero itself is rational and is neither positive nor negative.

A key idea from your work with integers carries over: a number and its opposite sit the same distance from zero in opposite directions. So

\frac{5}{8}

and

-\frac{5}{8}

are mirror images across

0

.

Where students go wrong: treating a negative fraction's minus sign as if it were attached to only the numerator or only the denominator. In fact

-\frac{3}{4}

\frac{-3}{4}

, and

\frac{3}{-4}

are three ways of writing the same point, three-fourths of a unit to the left of zero. Another frequent slip is plotting

-2\frac{1}{3}

between

-2

and

-1

by counting one-third to the right of

-2

. Negative mixed numbers keep moving away from zero:

-2\frac{1}{3}

is one-third of a unit past

-2

heading toward

-3

, so it lands between

-3

and

-2

, closer to

-2

Partitioning: Getting the Tick Marks Right

To plot a fraction, first decide which two integers it falls between, then cut that unit interval into as many equal pieces as the denominator says.

For

\frac{7}{5}

: since

\frac{5}{5} = 1

and

\frac{10}{5} = 2

, the number is between

1

and

2

. Split the space from

1

2

into

5

equal parts and count

2

of them to the right of

1

, because

\frac{7}{5} = 1\frac{2}{5}

.

For

-\frac{7}{5} = -1\frac{2}{5}

: split the space from

-1

-2

into

5

equal parts and count

2

of them to the left of

-1

.

When a problem asks you to plot several numbers with different denominators, pick one partition that works for all of them. To graph

\frac{1}{2}

\frac{3}{4}

, and

\frac{5}{8}

on the same line, use eighths: they become

\frac{4}{8}

\frac{6}{8}

, and

\frac{5}{8}

, so the order is obvious.

Number	Between which integers?	How to partition	Count
$\frac{9}{4}$	$2$ and $3$	fourths	$1$ tick right of $2$
$-\frac{9}{4}$	$-3$ and $-2$	fourths	$1$ tick left of $-2$
$1.7$	$1$ and $2$	tenths	$7$ ticks right of $1$
$-0.35$	$-1$ and $0$	hundredths	$35$ ticks left of $0$

The most common partitioning error is counting tick marks instead of spaces. If you draw four marks between

0

and

1

you have created five equal spaces, so those marks are fifths, not fourths. Always count the gaps.

Equivalent Forms Name the Same Point

Two expressions are equivalent when they name the identical point on the number line. This is the reason a single dot can have many labels.

\frac{3}{4} = \frac{6}{8} = \frac{75}{100} = 0.75

Multiplying numerator and denominator by the same nonzero number does not move a point — it just describes the same location using smaller pieces. Six eighths and three fourths are the same distance from zero because eighths are half the size of fourths and you take twice as many.

This works for negatives too:

-\frac{6}{8} = -\frac{3}{4} = -0.75

, all one point three-quarters of a unit left of zero.

A useful classroom check: if two numbers are truly equivalent, their difference is

0

. If your two labels land on different dots, one of your conversions is wrong.

Students often assume that a longer decimal must be a bigger number, so they place

0.4

to the left of

0.375

. Rewrite with the same number of decimal places —

0.400

versus

0.375

— and it is clear that

0.4

is farther right. With negatives the trap flips:

-0.4

is farther left than

-0.375

, since a bigger distance from zero on the negative side means a smaller number.

One more pitfall:

0.5

and

\frac{1}{5}

are not equivalent, even though both use a

5

. Reading a fraction bar as "divide" prevents this:

\frac{1}{5} = 1 \div 5 = 0.2

Dividing to Convert: Terminating or Repeating

Every fraction bar means division, so

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

. Carry out that long division and exactly one of two things happens.

The division terminates when the remainder eventually becomes

0

. Example:

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

, so the point sits between

0.6

and

0.7

.

The division repeats when a remainder you have already seen comes back, which forces the same digits to cycle forever. Example:

\frac{2}{3} = 0.666\ldots = 0.\overline{6}

, and

\frac{5}{12} = 0.41666\ldots = 0.41\overline{6}

. The bar goes only over the digits that actually repeat.

There is a shortcut worth knowing. Reduce the fraction, then look at the prime factors of the denominator. If the only primes are

2

and

5

, the decimal terminates, because those are the factors of

10

. Any other prime factor forces repetition.

Fraction	Denominator factors	Decimal	Type
$\frac{7}{20}$	$2 \cdot 2 \cdot 5$	$0.35$	terminating
$\frac{9}{40}$	$2^3 \cdot 5$	$0.225$	terminating
$\frac{4}{9}$	$3 \cdot 3$	$0.\overline{4}$	repeating
$\frac{7}{6}$	$2 \cdot 3$	$1.1\overline{6}$	repeating

When the fraction is negative, divide the absolute values and then attach the minus sign:

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

.

Where students go wrong: stopping a repeating division after two digits and writing

\frac{1}{3} = 0.33

. That is only an approximation, and

0.33

is a genuinely different point on the line than

\frac{1}{3}

. Also watch the direction of division —

\frac{3}{8}

3 \div 8

, not

8 \div 3

Reading and Labeling an Unmarked Number Line

Many problems hand you a number line with a few labels and a mystery point. Work in this order.

First, find the scale. Look at two labeled points and count the spaces between them. If

0

and

1

have

8

spaces between them, each space is

\frac{1}{8}

. If

-2

and

0

have

10

spaces, each space is

0.2

, since

2 \div 10 = 0.2

.

Second, count spaces from the nearest labeled point, moving in the correct direction. Right means adding, left means subtracting.

Third, write the value in the form the problem asks for and sanity-check the sign. A point left of zero must come out negative.

For example, suppose the line is labeled

-1

and

0

with six equal spaces between them, and a point sits

4

spaces right of

-1

. Each space is

\frac{1}{6}

, so the point is

-1 + \frac{4}{6} = -\frac{2}{6} = -\frac{1}{3}

, which checks out as negative and close to zero.

The most common error here is assuming every number line is scaled in tenths or halves just because those are familiar. A second error is counting from the wrong end when the labeled numbers are negative. If you find a positive answer for a point drawn to the left of

0

, stop and recount.

This skill matters beyond this lesson: reading scales is exactly what you do with thermometers, elevation maps, bank balances, and later with coordinate graphs and inequalities.

Key terms

Rational number.: Any number that can be written as $\frac{a}{b}$ with $a$ and $b$ integers and $b \neq 0$ ; includes integers, fractions, terminating decimals, and repeating decimals.
Opposite.: Two numbers the same distance from $0$ on the number line but in opposite directions, such as $\frac{3}{5}$ and $-\frac{3}{5}$ .
Equivalent forms.: Different expressions that name the same point on the number line, such as $\frac{3}{4}$ , $\frac{6}{8}$ , and $0.75$ .
Partition.: To divide a unit interval on the number line into equal-size pieces; the denominator of the fraction tells you how many pieces.
Terminating decimal.: A decimal whose digits stop because the long division reaches a remainder of $0$ , such as $\frac{5}{8} = 0.625$ .
Repeating decimal.: A decimal in which one or more digits cycle forever, written with a bar over the repeating part, such as $\frac{5}{12} = 0.41\overline{6}$ .
Scale.: The value of one space between consecutive tick marks on a number line, found by dividing the distance between two labeled points by the number of spaces.
Mixed number.: A number written as an integer plus a fraction, such as $2\frac{3}{4}$ ; for negatives, $-2\frac{3}{4}$ means $2\frac{3}{4}$ units to the left of zero.

Worked example

Convert

-\frac{11}{6}

to a decimal, then plot both

-\frac{11}{6}

and

-1.25

on a number line from

-3

0

. Which point is farther left?

Step 1 — Divide, ignoring the sign for now.

11 \div 6

: six goes into eleven once with remainder

5

, so we have

1

and bring down a zero.

50 \div 6 = 8

remainder

2

, giving

1.8

. Then

20 \div 6 = 3

remainder

2

. The remainder

2

has repeated, so the digit

3

will repeat forever.

Step 2 — Write the result and reattach the sign.

\frac{11}{6} = 1.8\overline{3}

, so

-\frac{11}{6} = -1.8\overline{3}

. Note the bar covers only the

3

, not the

8

.

Step 3 — Choose a partition that fits both numbers. Writing

-\frac{11}{6}

-1\frac{5}{6}

suggests sixths, and

-1.25 = -1\frac{1}{4}

suggests fourths. Twelfths handle both:

-1\frac{5}{6} = -1\frac{10}{12}

and

-1\frac{1}{4} = -1\frac{3}{12}

.

Step 4 — Plot. Both points are between

-2

and

-1

. Cut that interval into

12

equal spaces. Counting left from

-1

, place

-1\frac{3}{12}

at the third space and

-1\frac{10}{12}

at the tenth space.

Step 5 — Compare. Ten twelfths past

-1

is farther from zero than three twelfths past

-1

, so

-\frac{11}{6}

is farther left. Decimal check:

-1.8\overline{3}

is less than

-1.25

, which matches. Answer:

-\frac{11}{6}

is farther left.

Practice questions

Which point on the number line is the same as

-\frac{9}{4}

$2.25$ units to the right of $0$
$2.25$ units to the left of $0$
$1.75$ units to the left of $0$
$9.4$ units to the left of $0$

Answer: $2.25$ units to the left of $0$

Divide to convert:

9 \div 4 = 2.25

, so

\frac{9}{4} = 2.25

and

-\frac{9}{4} = -2.25

. The negative sign means travel left from zero, and the size

2.25

tells how far. The choice with

1.75

comes from mistakenly counting one fourth to the right of

-2

instead of continuing away from zero, and

9.4

comes from reading the fraction bar as a decimal point.

A number line shows only the labels

-1

and

0

, with

8

equal spaces between them. Point

P

sits

5

spaces to the right of

-1

. Write the value of

P

as a fraction and as a decimal, and state whether the decimal terminates or repeats.

Answer: $P = -\frac{3}{8} = -0.375$ , a terminating decimal.

Each space is

\frac{1}{8}

of a unit because

8

equal spaces fill the distance from

-1

0

. Moving

5

spaces right of

-1

gives

-1 + \frac{5}{8} = -\frac{3}{8}

. Since the point lies between

-1

and

0

, the answer must be negative, which it is. Dividing,

3 \div 8 = 0.375

, so

P = -0.375

. The denominator

8 = 2^3

has only the prime factor

2

, so the decimal terminates.

Miguel says that

\frac{7}{12}

and

0.58

are the same point on the number line. Is he correct? Explain using division.

Answer: No. $\frac{7}{12} = 0.58\overline{3}$ , which is slightly to the right of $0.58$ .

Divide

7

12

70 \div 12 = 5

remainder

10

, then

100 \div 12 = 8

remainder

4

, then

40 \div 12 = 3

remainder

4

. The remainder

4

repeats, so the digit

3

repeats forever and

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

. Because

0.5833\ldots > 0.5800

, the fraction sits a tiny bit farther right. Miguel rounded rather than converted, and rounding produces a nearby point, not the same point.

FAQ

How do I know whether a fraction becomes a terminating or a repeating decimal without dividing?: Reduce the fraction to lowest terms first, then factor the denominator. If the only prime factors are $2$ and $5$ , the decimal terminates, because $2$ and $5$ are the primes that build powers of ten. If any other prime appears, such as $3$ , $7$ , or $11$ , the decimal repeats. For instance $\frac{6}{15}$ reduces to $\frac{2}{5}$ , which terminates as $0.4$ , while $\frac{5}{14}$ has the factor $7$ and repeats.
Is $-2\frac{1}{3}$ between $-2$ and $-1$ or between $-3$ and $-2$ ?: Between $-3$ and $-2$ . The mixed number means $2\frac{1}{3}$ units away from zero, and negatives move left, so you go past $-2$ by one-third of a unit toward $-3$ . As a decimal it is $-2.\overline{3}$ , which confirms it is less than $-2$ .
Can two different-looking numbers really share one dot on the number line?: Yes, and that is the whole point of equivalence. $\frac{1}{2}$ , $\frac{4}{8}$ , $\frac{50}{100}$ , and $0.5$ all describe the same location using different-size pieces. A quick test: subtract the two forms. If the difference is $0$ , they are the same point.
Do I write the repeating bar over every digit after the decimal point?: No, only over the digits that actually cycle. In $\frac{5}{12} = 0.41\overline{6}$ the $4$ and $1$ appear once and only the $6$ repeats. Putting the bar over all three digits would claim the pattern $416416\ldots$ , which is a different number entirely.

Learn this with a teacher, not a page

The Crimsora tutor teaches Rational Numbers on the Number Line live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free