Adding and Subtracting Signed Rational Numbers | Grade 8 Math | LevelKite

You can already place rational numbers on a number line and put them in order. In this lesson you will combine them: you will add and subtract them, even when some are negative.

A quick reminder first. A rational number is a number you can write as $\frac{p}{q}$ , where $p$ and $q$ are integers ( $\ldots, -2, -1, 0, 1, 2, \ldots$ ) and $q$ is not $0$ , because division by $0$ is undefined. As a decimal, a rational number either ends (terminates) or repeats. So $-4.7$ , $\frac{3}{4}$ , $-2$ and $0.\overline{3}$ are all rational. The bar over the $3$ shows that this digit repeats forever.

Adding is moving along the number line. Let $a$ and $b$ be rational numbers. To find $a + b$ , start at $a$ . Then look at $b$ :

If $b$ is positive, move right.
If $b$ is negative, move left.
If $b$ is $0$ , stay where you are.

How far do you move? The distance is the absolute value of $b$ , written $\lvert b \rvert$ . A number's absolute value is its distance from $0$ , so it is never negative. For example, $\lvert -4 \rvert = 4$ and $\lvert 2.5 \rvert = 2.5$ .

Try $1.5 + (-4)$ . Start at $1.5$ and move $4$ units left. The first $1.5$ units bring you to $0$ . The other $2.5$ units carry you on to $-2.5$ . So $1.5 + (-4) = -2.5$ .

You do not have to trace the walk every time. Reaching $a$ is also a move from $0$ , so $a + b$ is really two moves in a row that start at $0$ .

Two moves in the same direction build on each other.
Two moves in opposite directions cancel, partly or completely.

The table turns these ideas into four cases. Together they cover every pair of rational numbers $a$ and $b$ .

Four cases for adding two rational numbers

The two numbers	What to do	Example
Same sign: both positive or both negative	Add the absolute values. The sum has the sign the two numbers share.	$-1.5 + (-2.25) = -3.75$
Different signs and different distances from $0$	Subtract the smaller absolute value from the larger. The sum has the sign of the number farther from $0$ .	$\frac{3}{8} + \left(-\frac{7}{8}\right) = -\frac{4}{8} = -\frac{1}{2}$
Opposites: different signs and the same distance from $0$	The two moves cancel completely. The sum is $0$ .	$2.6 + (-2.6) = 0$
At least one number is $0$	The sum equals the other number.	$0 + \left(-\frac{2}{3}\right) = -\frac{2}{3}$

Worked example

Adding decimals with different signs

Find $-4.7 + 1.9$ .

Check the signs first, because they tell you which case applies. The first number is negative and the second is positive, so the signs differ. On the number line you start at negative 4.7 and move 1.9 units right, toward 0.
Find each absolute value, which is the distance from 0. The move of 1.9 units is shorter than the 4.7 units between your start and 0, so you stop before you reach 0.
$\lvert -4.7 \rvert = 4.7, \qquad \lvert 1.9 \rvert = 1.9, \qquad 4.7 > 1.9$
Subtract the smaller absolute value from the larger. This gives the distance that is still left between you and 0.
$4.7 - 1.9 = 2.8$
Choose the sign. The negative number is farther from 0, so you finish on the negative side of 0 and the sum is negative.
$-4.7 + 1.9 = -2.8$

Answer

The sum $-4.7 + 1.9$ is $-2.8$ .

What does subtracting mean? For rational numbers $a$ and $b$ , the difference $a - b$ is the number you add to $b$ to get $a$ . You know that $7 - 2 = 5$ because $2 + 5 = 7$ . The same meaning works when numbers are negative.

Take $5 - (-2)$ . What do you add to $-2$ to get $5$ ? From $-2$ , move $2$ units right to reach $0$ , then $5$ more units right to reach $5$ . That is $7$ units right in all, so $5 - (-2) = 7$ . Notice that $5 + 2 = 7$ as well. Subtracting $-2$ gave the same result as adding $2$ .

Key concept

Subtracting adds the additive inverse

For all rational numbers $a$ and $b$ , subtracting $b$ gives the same result as adding $-b$ , the additive inverse of $b$ .

Why it works: $a - b$ is the number that adds to $b$ to give $a$ . The number $a + (-b)$ does exactly that. Because you can add in any order and grouping, $b + \left(a + (-b)\right) = a + \left(b + (-b)\right) = a + 0 = a$ .

How to use it: change two things together. The subtraction becomes addition, and the number after it becomes its opposite. The number before the subtraction sign stays the same.

a - b = a + (-b)

Worked example

Subtracting fractions with unlike denominators

Find $-\frac{5}{6} - \frac{3}{4}$ .

Rewrite the subtraction as adding the additive inverse. The number being subtracted is three fourths, so you add its opposite, negative three fourths. The first number, negative five sixths, does not change.
$-\frac{5}{6} - \frac{3}{4} = -\frac{5}{6} + \left(-\frac{3}{4}\right)$
Rename both fractions in twelfths, because you can only combine parts of the same size and both 6 and 4 divide evenly into 12. Multiplying a numerator and its denominator by the same nonzero number does not change the fraction's value.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}, \qquad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
Both numbers are negative: negative 10 twelfths and negative 9 twelfths. This is the same-sign case, so add the absolute values to find how far the sum is from 0.
$\frac{10}{12} + \frac{9}{12} = \frac{10 + 9}{12} = \frac{19}{12}$
Give the sum the sign the two numbers share. Both moves go left, so the sum is negative. The numbers 19 and 12 have no common factor greater than 1, so the fraction is in simplest form.
$-\frac{10}{12} + \left(-\frac{9}{12}\right) = -\frac{19}{12}$
Check with the meaning of subtraction: the answer added to three fourths must give negative five sixths. The signs differ, so subtract 9 twelfths from 19 twelfths and use the sign of the number farther from 0, which is negative.
$\frac{9}{12} + \left(-\frac{19}{12}\right) = -\frac{10}{12} = -\frac{5}{6}$

Answer

The difference $-\frac{5}{6} - \frac{3}{4}$ is $-\frac{19}{12}$ .

Worked example

Mixing a fraction with decimals

Find the exact value of $0.5 - \left(-\frac{2}{3}\right) - 1.75$ .

Rewrite each subtraction as adding the additive inverse. The opposite of negative two thirds is two thirds, and the opposite of 1.75 is negative 1.75. The first number, 0.5, does not change.
$0.5 - \left(-\frac{2}{3}\right) - 1.75 = 0.5 + \frac{2}{3} + (-1.75)$
Choose one form for all three numbers. Two thirds is a repeating decimal, so decimals would force you to round, while fractions stay exact. Write each decimal that ends as tenths or hundredths. Then simplify by dividing the numerator and denominator by the same nonzero number: 5 for the first fraction and 25 for the second.
$\frac{2}{3} = 0.\overline{6}, \qquad 0.5 = \frac{5}{10} = \frac{1}{2}, \qquad 1.75 = \frac{175}{100} = \frac{7}{4}$
Rename the fractions in twelfths, because 2, 3 and 4 each divide evenly into 12. Multiply each numerator and denominator by the same nonzero number: 6 for one half, 4 for two thirds and 3 for seven fourths.
$\frac{1}{2} = \frac{6}{12}, \qquad \frac{2}{3} = \frac{8}{12}, \qquad \frac{7}{4} = \frac{21}{12}$
Add the first two numbers. Both are positive, so add the absolute values and keep the positive sign. Leave the result in twelfths for the next step.
$\frac{6}{12} + \frac{8}{12} = \frac{6 + 8}{12} = \frac{14}{12}$
Add the last number. The signs differ, and negative 21 twelfths is farther from 0 than 14 twelfths. Subtract the smaller absolute value from the larger and make the sum negative. The numbers 7 and 12 share no factor greater than 1, so the result is in simplest form.
$\frac{14}{12} + \left(-\frac{21}{12}\right) = -\left(\frac{21}{12} - \frac{14}{12}\right) = -\frac{7}{12}$
Check the size with rounded decimals. Two thirds rounds to 0.67, so that step needs the approximately-equal sign. The estimate matches the exact answer rounded to hundredths, so the result is reasonable.
$0.5 + \frac{2}{3} - 1.75 \approx 0.5 + 0.67 - 1.75 = 1.17 - 1.75 = -0.58, \qquad -\frac{7}{12} = -0.58\overline{3} \approx -0.58$

Answer

The exact value of $0.5 - \left(-\frac{2}{3}\right) - 1.75$ is $-\frac{7}{12}$ .

Takeaway. Adding rational numbers is moving along the number line: right for a positive number, left for a negative one. When the signs match, add the absolute values and keep the shared sign. When the signs differ, subtract the smaller absolute value from the larger and use the sign of the number farther from $0$ ; if the numbers are opposites, the sum is $0$ . Subtracting is adding the additive inverse, so $a - b = a + (-b)$ for all rational numbers $a$ and $b$ .

Next, you will multiply and divide rational numbers.