Dividing Integers and Equal Quotients | Grade 7 Math | LevelKite

You already know how to multiply integers. Dividing by a number that is not $0$ undoes multiplying by that number, so that knowledge is the main tool you need here.

In a division such as $-12 \div 3$ , the number being divided, $-12$ , is the dividend. The number you divide by, $3$ , is the divisor. The result is the quotient.

Whenever the divisor is not $0$ , you can read a division as a missing-factor question. (A factor is one of the numbers being multiplied.) So $-12 \div 3$ asks: what number times $3$ equals $-12$ ? You know that $-4 \times 3 = -12$ , so $-12 \div 3 = -4$ .

The table asks the same kind of question for all four sign combinations of $12$ and $3$ .

Four divisions, each answered by a multiplication fact

Division	Question it asks	Quotient	Check by multiplying
$12 \div 3$	What times $3$ equals $12$ ?	$4$	$4 \times 3 = 12$
$-12 \div 3$	What times $3$ equals $-12$ ?	$-4$	$-4 \times 3 = -12$
$12 \div (-3)$	What times $-3$ equals $12$ ?	$-4$	$-4 \times (-3) = 12$
$-12 \div (-3)$	What times $-3$ equals $-12$ ?	$4$	$4 \times (-3) = -12$

Key concept

A quotient is a missing factor

When the divisor $b$ is not $0$ , the quotient $a \div b$ is the one number $c$ that you multiply by $b$ to get the dividend $a$ . That check forces the sign. For example, $12 \div (-3)$ must be negative, because only a negative number times $-3$ gives a positive result.

So for two nonzero integers (neither one is $0$ ):

Size (how far the quotient is from $0$ ): divide the absolute values. The absolute value of a number is its distance from $0$ , so $|-12| = 12$ .
Sign: count the negative numbers among the dividend and the divisor. The quotient is negative exactly when that count is odd, which here means exactly one. With no negatives or two negatives, the quotient is positive.

a \div b = c \quad \text{means} \quad c \times b = a \qquad (b \neq 0)

Worked example

One negative number

Leo pays the same amount for a music app each month. After $8$ months, the payments have changed his account balance by $-56$ dollars in total. A negative change means the balance went down. What change does one payment make to the balance?

Write the division. The total change is split into 8 equal parts, one for each payment. The divisor, 8, is not 0, so the quotient exists.
$-56 \div 8$
Find the size. Divide the absolute values, because the signs of the two numbers change only the sign of the quotient, not its size.
$|-56| \div |8| = 56 \div 8 = 7$
Find the sign. Both numbers are nonzero, and exactly one of them, the dividend, is negative. One is an odd count, so the quotient is negative.
$-56 \div 8 = -7$
Check with multiplication. The quotient times the divisor must equal the dividend, and it does.
$-7 \times 8 = -56$

Answer

One payment changes Leo's balance by $-7$ dollars, so the balance goes down by $7$ dollars each month.

Worked example

Two negative numbers

What is $-96 \div (-12)$ ?

Check the divisor, then find the size. The divisor is not 0, so you can divide. Divide the absolute values: 12 times 8 is 96, so 96 divided by 12 is 8.
$|-96| \div |-12| = 96 \div 12 = 8$
Find the sign. Both numbers are nonzero and both are negative. Two is an even count, so the quotient is positive.
$-96 \div (-12) = 8$
Check with multiplication. A positive quotient times the negative divisor gives back the negative dividend, so the positive sign is correct.
$8 \times (-12) = -96$

Answer

The quotient of $-96$ and $-12$ is $8$ .

When $q \neq 0$ , a fraction bar is another way to write division: $\frac{p}{q}$ means $p \div q$ , with the dividend on top and the divisor on the bottom. A number that can be written this way with integers $p$ and $q$ , where $q \neq 0$ , is called a rational number.

So every quotient of two integers with a nonzero divisor is a rational number, even when it is not an integer. For example, $-3 \div 4 = \frac{-3}{4}$ .

In fraction form, a negative sign can sit in three places:

on the dividend, as in $\frac{-12}{4}$ ;
on the divisor, as in $\frac{12}{-4}$ ;
in front of the whole fraction, as in $-\frac{12}{4}$ . This means the opposite of $\frac{12}{4}$ . Two numbers are opposites when their sum is $0$ , like $3$ and $-3$ .

These quotients look different. Do they name the same number? The next table works out each one. For comparison, it also shows the quotient with no negative sign and two quotients with two negative signs.

Quotients built from 12 and 4

Quotient	Negative signs	Value	Why
$\frac{12}{4}$	0 (even)	$3$	$3 \times 4 = 12$
$\frac{-12}{4}$	1 (odd)	$-3$	$-3 \times 4 = -12$
$\frac{12}{-4}$	1 (odd)	$-3$	$-3 \times (-4) = 12$
$-\frac{12}{4}$	1 (odd)	$-3$	The opposite of $\frac{12}{4}$ , which is $3$
$\frac{-12}{-4}$	2 (even)	$3$	$3 \times (-4) = -12$
$-\frac{-12}{4}$	2 (even)	$3$	The opposite of $\frac{-12}{4}$ , which is $-3$

In the table, each negative sign switches the quotient to its opposite and leaves the size alone. A sign on the dividend or on the divisor changes the count of negative numbers by one, and a sign in front takes the opposite directly. So one negative sign gives $-3$ in all three places, and a second negative sign switches the value back to $3$ .

The same thing happens with any integers $p$ and $q$ , as long as $q \neq 0$ . Two negative signs cancel, so $\frac{-p}{-q} = \frac{p}{q}$ . One negative sign gives the same number whether it sits in front, on the dividend, or on the divisor:

-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q} \qquad (q \neq 0)

Worked example

Which quotients are equal?

Which of these quotients are equal to $-\frac{5}{8}$ ? (A) $\frac{-5}{8}$ (B) $\frac{5}{-8}$ (C) $\frac{-5}{-8}$ (D) $-\frac{-5}{-8}$

Check the divisors and compare the sizes. Each divisor is 8 or negative 8, so none is 0. Every quotient is built from 5 and 8, so they all have the same size. Only their signs can make them different.
$5 \div 8 = \frac{5}{8}$
Read the target, the quotient you are matching. It is the opposite of a positive quotient, so it is negative.
$-\frac{5}{8} = -(5 \div 8)$
Test A and B. Each has exactly one negative sign: on the dividend in A and on the divisor in B. When the divisor is not 0, one negative sign gives the same number in front, on the dividend, or on the divisor. So A and B both equal the target.
$\frac{-5}{8} = \frac{5}{-8} = -\frac{5}{8}$
Test C. It has two negative signs, one on the dividend and one on the divisor, and they cancel. C is positive, so it is the opposite of the target, not equal to it.
$\frac{-5}{-8} = \frac{5}{8}$
Test D. Its fraction part is C, which is positive. The negative sign in front takes the opposite, so D is negative and equals the target.
$-\frac{-5}{-8} = -\frac{5}{8}$

Answer

Quotients A, B and D are equal to $-\frac{5}{8}$ . Quotient C is not equal to it, because $\frac{-5}{-8} = \frac{5}{8}$ .

Here is the idea to keep. When the divisor is not $0$ , a quotient is the number that you multiply by the divisor to get the dividend. For two nonzero integers, this makes the quotient negative exactly when just one of the two is negative. It also explains why one negative sign gives the same quotient in front, on the dividend, or on the divisor: $-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$ for integers $p$ and $q$ with $q \neq 0$ .

Next, you will multiply and divide integers using properties of operations, and you will evaluate numerical expressions that involve integers.