Using Properties to Add and Subtract Rational Numbers | Grade 7 Math | LevelKite

You can already add and subtract positive and negative decimals and fractions. In this lesson you will learn to choose a smarter order for that work.

Look at $-3.7 + 5.2 + 3.7$ . Working from left to right takes two careful decimal steps. But $-3.7$ and $3.7$ are opposites: two numbers the same distance from $0$ , on opposite sides of it. (Zero is its own opposite.) Opposites add to $0$ , so the work would be much shorter if you could add those two first.

You can, and a property of operations is the reason. A property of operations is a fact about an operation, such as addition, that stays true whichever rational numbers you use. A rational number is a number you can write as a fraction of two integers with a nonzero denominator, such as $-6$ , $2.5$ or $-\frac{3}{4}$ . (The integers are the whole numbers and their opposites.)

First, recall what a sum means. For rational numbers $p$ and $q$ , the sum $p + q$ is the number you reach when you start at $p$ and move a distance of $|q|$ in the direction of $q$ : the positive direction (toward greater numbers) when $q$ is positive, the negative direction (toward lesser numbers) when $q$ is negative, and no move at all when $q$ is $0$ . Here $|q|$ is the absolute value of $q$ , which is its distance from $0$ .

Now swap the two numbers in a sum and compare:

$5 + (-3)$ : start at $5$ and move $3$ in the negative direction. You reach $2$ .
$(-3) + 5$ : start at $-3$ and move $5$ in the positive direction. You reach $2$ .

The order changed, but the sum did not. That is not luck. It happens for every pair of rational numbers, and it is the first of four properties of addition in the table below. In the table, $a$ , $b$ and $c$ stand for any rational numbers, and $-a$ means the opposite of $a$ .

Four properties of addition, true for all rational numbers

a

,

b

and

c

Property	What it says	In symbols	Example
Commutative property of addition	Changing the order of the numbers you add does not change the sum.	$a + b = b + a$	$-2.5 + 4 = 1.5$ and $4 + (-2.5) = 1.5$
Associative property of addition	Changing the grouping (which numbers you add first) does not change the sum.	$(a + b) + c = a + (b + c)$	$(1.5 + (-0.5)) + (-3) = 1 + (-3) = -2$ and $1.5 + ((-0.5) + (-3)) = 1.5 + (-3.5) = -2$
Additive inverse property	A number and its opposite add to $0$ . The opposite is also called the additive inverse.	$a + (-a) = 0$	$\frac{3}{4} + \left(-\frac{3}{4}\right) = 0$
Additive identity property	Adding $0$ to a number leaves the number unchanged.	$a + 0 = a$	$-6.2 + 0 = -6.2$

Worked example

Bring opposites together

Find $-3.7 + 5.2 + 3.7$ .

Look before you compute. Every operation here is addition, so the properties of addition apply. The numbers $-3.7$ and $3.7$ are opposites, so the goal is to bring them together.
$-3.7 + 5.2 + 3.7$
Swap the first two numbers. The commutative property of addition says that changing the order does not change the sum.
$5.2 + (-3.7) + 3.7$
Group the last two numbers with brackets so that you add them first. The associative property of addition says that changing the grouping does not change the sum.
$5.2 + \left[(-3.7) + 3.7\right]$
Add inside the brackets. Opposites sum to $0$ . This is the additive inverse property.
$5.2 + 0$
Add $0$ . The additive identity property says that adding $0$ leaves $5.2$ unchanged.
$5.2 + 0 = 5.2$
Check by working the original problem from left to right, the standard order for a string of additions and subtractions. From $-3.7$ , move $5.2$ in the positive direction to reach $1.5$ . Then $1.5 + 3.7 = 5.2$ . Both orders give the same value.
$(-3.7 + 5.2) + 3.7 = 1.5 + 3.7 = 5.2$

Answer

The sum $-3.7 + 5.2 + 3.7$ is $5.2$ .

Be careful: the commutative and associative properties are about addition. Subtraction does not have them. Swap the numbers in $5 - 8$ and the value changes: $5 - 8 = -3$ , but $8 - 5 = 3$ .

So what can you do with an expression (numbers joined by operations) that has subtraction in it? Change each subtraction into addition. Subtracting a number gives the same result as adding its additive inverse, and this is true for all rational numbers. For example, $5 - 8 = 5 + (-8)$ . To find $5 + (-8)$ , start at $5$ and move $8$ in the negative direction. You reach $-3$ , the same value as $5 - 8$ .

After the rewrite, the negative sign belongs to the $8$ and travels with it. Now the swap is safe:

5 - 8 = 5 + (-8) = (-8) + 5 = -3

Key concept

Make it all addition, then reorder and regroup

Subtracting a number is the same as adding its additive inverse (its opposite). This is true for all rational numbers $a$ and $b$ . Here $-b$ means the opposite of $b$ , so when $b$ is negative, $-b$ is positive: $4 - (-1.5) = 4 + 1.5 = 5.5$ .

Once you rewrite every subtraction as addition, the expression is a sum and each number carries its own sign. The commutative and associative properties of addition then let you add the numbers in any order and any grouping without changing the value. Choose what makes the arithmetic easy: bring together opposites, decimals that combine into integers, and fractions with the same denominator.

a - b = a + (-b)

Worked example

Rewrite subtraction, then find partners

Find $6.4 - 9.5 - 2.4 + 1.5$ .

Rewrite each subtraction as adding the additive inverse, because the properties you want are properties of addition. Subtracting $9.5$ becomes adding $-9.5$ . Subtracting $2.4$ becomes adding $-2.4$ .
$6.4 + (-9.5) + (-2.4) + 1.5$
Look for partners. The numbers $6.4$ and $-2.4$ have opposite signs and the same digit after the decimal point, so their sum is an integer. The same is true of $-9.5$ and $1.5$ . Every operation is now addition, so the commutative and associative properties of addition let you reorder the sum. Swap the two middle numbers to put partners side by side.
$6.4 + (-2.4) + (-9.5) + 1.5$
Group each pair of partners with brackets. The associative property of addition says that changing the grouping does not change the sum.
$\left[6.4 + (-2.4)\right] + \left[(-9.5) + 1.5\right]$
Add inside each pair of brackets. From $6.4$ , move $2.4$ in the negative direction to reach $4$ . From $-9.5$ , move $1.5$ in the positive direction to reach $-8$ .
$4 + (-8)$
Add the two results. From $4$ , move $8$ in the negative direction. The first $4$ of that move reaches $0$ , and the other $4$ reaches $-4$ .
$4 + (-8) = -4$

Answer

The value of $6.4 - 9.5 - 2.4 + 1.5$ is $-4$ .

A common mistake: regrouping across a minus sign. In $8 - 2.5 + 2.5$ , it is tempting to add $2.5 + 2.5$ first, because that pair looks easy, and write $8 - (2.5 + 2.5) = 8 - 5 = 3$ . But the correct value is $8$ . In the original expression you subtract only the first $2.5$ , and you add the second one. The new parentheses subtract both. The associative property is a property of addition, so it does not allow regrouping across a subtraction.

Rewrite the subtraction first, and each number keeps its own sign: $8 + (-2.5) + 2.5 = 8 + \left[(-2.5) + 2.5\right] = 8 + 0 = 8$ . Working from left to right agrees: $8 - 2.5 = 5.5$ , and $5.5 + 2.5 = 8$ .

Worked example

Group fractions that share a denominator

Find $\frac{2}{3} - \frac{5}{4} - \left(-\frac{1}{3}\right) - \frac{3}{4}$ .

Rewrite each subtraction as adding the additive inverse. The additive inverse of $\frac{5}{4}$ is $-\frac{5}{4}$ , the additive inverse of $-\frac{1}{3}$ is $\frac{1}{3}$ , and the additive inverse of $\frac{3}{4}$ is $-\frac{3}{4}$ .
$\frac{2}{3} + \left(-\frac{5}{4}\right) + \frac{1}{3} + \left(-\frac{3}{4}\right)$
Every operation is now addition, so the commutative and associative properties of addition let you reorder the sum. Swap the two middle numbers to put thirds next to thirds and fourths next to fourths.
$\frac{2}{3} + \frac{1}{3} + \left(-\frac{5}{4}\right) + \left(-\frac{3}{4}\right)$
Group the fractions that share a denominator. The associative property of addition keeps the value the same.
$\left[\frac{2}{3} + \frac{1}{3}\right] + \left[\left(-\frac{5}{4}\right) + \left(-\frac{3}{4}\right)\right]$
Add inside each pair of brackets. The thirds give $\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$ . For the fourths, start at $-\frac{5}{4}$ and move $\frac{3}{4}$ in the negative direction to reach $-\frac{8}{4} = -2$ .
$1 + (-2)$
Add the two results. From $1$ , move $2$ in the negative direction to reach $-1$ .
$1 + (-2) = -1$

Answer

The value of $\frac{2}{3} - \frac{5}{4} - \left(-\frac{1}{3}\right) - \frac{3}{4}$ is $-1$ .

Here is the idea to keep. In a sum of rational numbers, the properties of addition let you choose the order and the grouping, and the value stays the same. Subtraction does not give you that freedom, so first rewrite each subtraction as adding the additive inverse. Then every number carries its own sign, and you can bring together opposites, decimals that combine into integers, and fractions that share a denominator.

Next, you will use these skills to solve word problems with rational numbers and to find and fix errors in worked solutions.