Do I have to find the least common denominator, or will any common denominator work?

Any common denominator works and will give a correct answer. Multiplying the two denominators together always produces one. Using the least common denominator just keeps the numbers smaller and often means less simplifying at the end. For example, 1/6 + (-1/4) can use 24 and give -2/24, which reduces to -1/12, the same answer you get directly with denominator 12.

How do I know whether my answer should be positive or negative?

Check the signs first. If both numbers have the same sign, the answer keeps that sign. If the signs differ, the answer takes the sign of the number with the larger absolute value — the one farther from zero. And if the two numbers are opposites, the answer is exactly zero, with no sign at all.

Can I turn every fraction into a decimal instead of finding common denominators?

Sometimes, but be careful. It works cleanly for fractions like 3/4 = 0.75 or 2/5 = 0.4 that terminate. Fractions like 1/3 or 2/7 repeat forever, so a rounded decimal gives only an approximate answer. When repeating decimals appear, work in fraction form to stay exact.

What does it mean that a number plus its opposite is zero, and why does it matter later?

It means the two numbers cancel each other out completely, like a 40 dollar debt paid off with a 40 dollar deposit. It matters because it lets you simplify long sums by spotting zero pairs, and because it is the foundation of subtraction: subtracting a number gives the same result as adding its opposite, which is the very next topic in this unit.

M7MATH-2.1

Adding Rational Numbers

Learn to add rational numbers — integers, unlike-denominator fractions, and decimals — using same-sign and different-sign rules, plus why a number and its opposite sum to zero.

What you'll do in this lesson

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

Start Adding Rational Numbers free

What this lesson covers

Adding rational numbers is the skill that ties together everything you already know about integers, fractions, and decimals. A rational number is any number you can write as a fraction of two integers, so

-7

\frac{3}{4}

-2.5

, and

0

all count. The good news is that you do not need a separate rule for each kind of number. One pair of rules — one for when the signs match and one for when they do not — handles every addition problem in this unit.

In this lesson you will learn to picture addition as movement on a number line, apply the same-sign and different-sign rules, rewrite fractions with a common denominator before adding, line up decimal points correctly, and use the fact that a number plus its opposite always equals zero. That last idea, called the additive inverse property, becomes the key to subtraction in the very next lesson.

What Rational Numbers Are and How Addition Moves on the Number Line

A rational number is any number that can be written as

\frac{a}{b}

where

a

and

b

are integers and

b \neq 0

. That definition includes integers like

-9

(since

-9 = \frac{-9}{1}

), fractions like

-\frac{2}{3}

, mixed numbers like

4\frac{1}{2}

, and terminating or repeating decimals like

0.75

and

0.\overline{3}

.

Adding on a number line means starting at the first number and moving. Adding a positive number moves you to the right; adding a negative number moves you to the left. So

-5 + 3

starts at

-5

and moves 3 units right, landing on

-2

. And

-5 + (-3)

starts at

-5

and moves 3 units left, landing on

-8

.

The absolute value of a number is its distance from zero, always positive or zero:

|-8| = 8

and

|8| = 8

. Absolute value is the tool that tells you how far to move, while the sign tells you which direction.

A common early mistake is reading

-5 + 3

as "negative eight" because both digits get lumped together. Slow down and ask two questions every time: Are the signs the same or different? Which number is farther from zero? Those two questions, not memorized answers, are what make the rules in the next section work on any problem.

Another useful check: adding a negative always makes a result smaller, and adding a positive always makes it larger. If your answer moved the wrong direction, you know something went wrong before you even check the arithmetic.

The Same-Sign and Different-Sign Rules

Two rules cover every addition of rational numbers.

Same signs: add the absolute values and keep the shared sign. Think of it as gaining twice or losing twice. So

-4.6 + (-2.1)

becomes

4.6 + 2.1 = 6.7

, and since both were negative, the answer is

-6.7

.

Different signs: subtract the smaller absolute value from the larger, then take the sign of the number with the larger absolute value. So

-11 + 4

becomes

11 - 4 = 7

, and since

|-11| > |4|

, the answer is negative:

-7

Problem	Signs	What you do to the absolute values	Sign of answer	Result
$-6 + (-9)$	same	$6 + 9 = 15$	negative	$-15$
$-6 + 9$	different	$9 - 6 = 3$	positive	$3$
$6 + (-9)$	different	$9 - 6 = 3$	negative	$-3$
$\frac{1}{2} + \frac{1}{4}$	same	$\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$	positive	$\frac{3}{4}$

Notice rows two and three: the same absolute values give answers that differ only in sign, because the sign is inherited from whichever number is farther from zero.

Where students go wrong most often is with different signs — they subtract correctly but then attach the wrong sign. Build the habit of circling the number with the bigger absolute value before you compute; that circled number donates its sign to the answer.

Addition is also commutative, so

-8 + 3

and

3 + (-8)

both equal

-5

. Rearranging terms to group same-sign numbers together often makes long sums easier.

Fractions with Unlike Denominators and Decimals

Fractions can only be added when the pieces are the same size, so unlike denominators must be rewritten using a common denominator — usually the least common multiple of the two denominators.

To compute

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

, the least common denominator of 4 and 6 is 12. Rewrite:

-\frac{3}{4} = -\frac{9}{12}

and

\frac{1}{6} = \frac{2}{12}

. Now the signs differ, so subtract absolute values:

\frac{9}{12} - \frac{2}{12} = \frac{7}{12}

. The larger absolute value belonged to

-\frac{9}{12}

, so the answer is

-\frac{7}{12}

.

A critical detail with negative fractions:

-\frac{3}{4}

means the whole fraction is negative. When you scale it up, the numerator becomes

-9

, not

9

. Writing

\frac{-3}{4} = \frac{-9}{12}

keeps the sign visible and prevents it from getting lost.

For mixed numbers, either convert to improper fractions or handle whole parts and fractional parts separately — but converting to improper fractions is safer with negatives, because

-2\frac{1}{3}

means

-\frac{7}{3}

, not

-2 + \frac{1}{3}

.

Decimals need place values lined up. To add

-8.4 + 3.75

, note the signs differ, so subtract absolute values: line up

8.40 - 3.75 = 4.65

. Since

|-8.4|

is larger, the answer is

-4.65

. Adding a placeholder zero to

8.4

makes the columns match and stops the common error of computing

8.4 - 3.75

5.35

.

You may also convert between forms when it helps:

-\frac{1}{2} + 0.3

can be handled as

-0.5 + 0.3 = -0.2

Opposites, Zero Pairs, and the Additive Inverse

Two numbers are opposites (also called additive inverses) if they are the same distance from zero on opposite sides. The additive inverse property says that for any rational number

a

a + (-a) = 0

-13 + 13 = 0

\frac{5}{8} + \left(-\frac{5}{8}\right) = 0

, and

-4.29 + 4.29 = 0

. Each of these pairs is called a zero pair, and spotting zero pairs inside a long sum is one of the fastest shortcuts in this unit.

For example, in

-7 + 12 + 7 + (-3)

, notice that

-7

and

7

form a zero pair. Since addition is commutative and associative, you can regroup as

(-7 + 7) + (12 + (-3)) = 0 + 9 = 9

. No number-line trudging required.

This property also explains real situations. If a bank account is overdrawn by 45 dollars and you deposit exactly 45 dollars, the balance is zero. If a hiker descends 120 feet and then climbs 120 feet, the net elevation change is zero.

One misconception to clear up: the opposite of a number is not always negative. The opposite of

-6

6

, because

-(-6) = 6

. The little dash means "opposite of," not "make it negative." And zero is its own opposite, since

0 + 0 = 0

.

This property is exactly why subtraction becomes addition in the next lesson: subtracting a number gives the same result as adding its opposite.

Modeling Real Situations with Sums

Word problems are where these rules earn their keep, so translate carefully before computing. Words like loss, debt, withdrawal, below, decrease, and descent signal negative values. Words like gain, deposit, above, increase, and rise signal positive values.

Suppose the temperature at dawn is

-3.5

degrees Celsius, rises

9.2

degrees by afternoon, then falls

4.8

degrees by evening. The evening temperature is

-3.5 + 9.2 + (-4.8)

. Work left to right:

-3.5 + 9.2 = 5.7

(different signs,

9.2 - 3.5 = 5.7

, positive because

9.2

is farther from zero). Then

5.7 + (-4.8) = 0.9

degrees Celsius.

A frequent stumble is treating the word "falls" as a subtraction sign and writing a negative number, which double-counts the direction. Choose one: either

5.7 - 4.8

5.7 + (-4.8)

. Both give

0.9

; writing

5.7 - (-4.8)

does not.

Another stumble is answering with a sign that makes no sense in context. Before you write down a final answer, ask whether it should be above or below zero. If a diver starts 30 feet below the surface and rises 12 feet, the answer must still be below the surface, so

-30 + 12 = -18

feet is reasonable, while

18

feet is not.

Finally, keep units and labels attached. "

-18

feet" or "18 feet below the surface" communicates the answer completely; a bare

-18

leaves your reader guessing.

Key terms

Rational number.: Any number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ ; includes integers, fractions, mixed numbers, and terminating or repeating decimals.
Absolute value.: The distance of a number from zero on the number line, written $|x|$ and never negative. For example, $|-4.5| = 4.5$ .
Opposite (additive inverse).: The number the same distance from zero but on the other side. The opposite of $-\frac{2}{5}$ is $\frac{2}{5}$ , and their sum is zero.
Additive inverse property.: The rule that $a + (-a) = 0$ for every rational number $a$ .
Zero pair.: A pair of opposite numbers inside a sum whose total is zero, such as $-11$ and $11$ , which can be removed to simplify the sum.
Least common denominator.: The smallest common multiple of two or more denominators, used to rewrite fractions so they can be added.
Commutative property of addition.: The rule that order does not change a sum: $a + b = b + a$ , which lets you regroup terms to find zero pairs.
Sum.: The result of an addition; its sign comes from the shared sign when signs match, or from the number with the larger absolute value when signs differ.

Worked example

Evaluate

-2\frac{1}{3} + \frac{3}{4} + 2\frac{1}{3} + (-1.5)

Step 1: Scan for a zero pair. The terms

-2\frac{1}{3}

and

2\frac{1}{3}

are opposites, so by the additive inverse property their sum is

0

. Using the commutative and associative properties, regroup as

\left(-2\frac{1}{3} + 2\frac{1}{3}\right) + \left(\frac{3}{4} + (-1.5)\right)

, which simplifies to

0 + \left(\frac{3}{4} + (-1.5)\right)

.

Step 2: Make the two remaining terms match in form. Since

\frac{3}{4} = 0.75

, the problem is now

0.75 + (-1.5)

. (Converting to fractions works too:

-1.5 = -\frac{3}{2} = -\frac{6}{4}

.)

Step 3: Check the signs. They are different, so subtract absolute values:

|-1.5| = 1.5

and

|0.75| = 0.75

, and

1.5 - 0.75 = 0.75

.

Step 4: Choose the sign. The number farther from zero is

-1.5

, which is negative, so the answer is negative.

Step 5: State the answer:

-0.75

, or equivalently

-\frac{3}{4}

.

Quick reasonableness check: the two large opposites cancel, and we added a bigger negative to a smaller positive, so the result should be a small negative number. It is.

Practice questions

What is

-\frac{5}{6} + \frac{1}{4}

$-\frac{7}{12}$
$-\frac{13}{12}$
$\frac{7}{12}$
$-\frac{4}{10}$

Answer: $-\frac{7}{12}$

The least common denominator of 6 and 4 is 12, so rewrite the fractions as

-\frac{10}{12}

and

\frac{3}{12}

. The signs are different, so subtract the absolute values:

\frac{10}{12} - \frac{3}{12} = \frac{7}{12}

. Because

\frac{10}{12}

is farther from zero and it was negative, the answer is

-\frac{7}{12}

. Choosing

-\frac{13}{12}

comes from adding the absolute values as if the signs matched, and

-\frac{4}{10}

comes from incorrectly adding numerators and denominators.

A submarine sits at

-85.5

meters relative to sea level. It rises

32.75

meters, then descends

14.25

meters. Write a single addition expression for its final depth and evaluate it.

Answer: $-85.5 + 32.75 + (-14.25) = -67$ meters

Rising is positive and descending is negative, so the expression is

-85.5 + 32.75 + (-14.25)

. Work left to right. For

-85.5 + 32.75

, the signs differ, so subtract absolute values:

85.50 - 32.75 = 52.75

, and since

-85.5

is farther from zero the result is

-52.75

. Next,

-52.75 + (-14.25)

has matching signs, so add absolute values,

52.75 + 14.25 = 67

, and keep the negative sign, giving

-67

. The final depth is 67 meters below sea level, which is reasonable because the submarine descended more than it rose overall.

Marcus says that

-9 + 9 = -18

because "you add the 9s and keep the negative." Explain the error and give the correct answer.

Answer: The signs are different, not the same, so the same-sign rule does not apply. Since $-9$ and $9$ are opposites, $-9 + 9 = 0$ .

The same-sign rule (add absolute values, keep the sign) only applies when both numbers have the same sign. Here one number is negative and one is positive, so the different-sign rule applies: subtract absolute values,

9 - 9 = 0

. Neither number is farther from zero, and zero needs no sign. This is a direct example of the additive inverse property,

a + (-a) = 0

: on a number line you start at

-9

, move 9 units right, and land exactly on zero.

FAQ

Do I have to find the least common denominator, or will any common denominator work?: Any common denominator works and will give a correct answer. Multiplying the two denominators together always produces one. Using the least common denominator just keeps the numbers smaller and often means less simplifying at the end. For example, $\frac{1}{6} + \left(-\frac{1}{4}\right)$ can use 24 and give $-\frac{2}{24}$ , which reduces to $-\frac{1}{12}$ , the same answer you get directly with denominator 12.
How do I know whether my answer should be positive or negative?: Check the signs first. If both numbers have the same sign, the answer keeps that sign. If the signs differ, the answer takes the sign of the number with the larger absolute value — the one farther from zero. And if the two numbers are opposites, the answer is exactly zero, with no sign at all.
Can I turn every fraction into a decimal instead of finding common denominators?: Sometimes, but be careful. It works cleanly for fractions like $\frac{3}{4} = 0.75$ or $\frac{2}{5} = 0.4$ that terminate. Fractions like $\frac{1}{3}$ or $\frac{2}{7}$ repeat forever, so a rounded decimal gives only an approximate answer. When repeating decimals appear, work in fraction form to stay exact.
What does it mean that a number plus its opposite is zero, and why does it matter later?: It means the two numbers cancel each other out completely, like a 40 dollar debt paid off with a 40 dollar deposit. It matters because it lets you simplify long sums by spotting zero pairs, and because it is the foundation of subtraction: subtracting a number gives the same result as adding its opposite, which is the very next topic in this unit.

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