Multiplying and Dividing Rational Numbers | Grade 8 Math | LevelKite

You can already compare and order rational numbers. Now you will multiply and divide them, and you will see why a negative times a negative is positive.

A rational number is a number you can write as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers (whole numbers and their opposites) and $q \neq 0$ . Its decimal form either stops (terminates), like $-0.75$ , or repeats a block of digits forever, like $\frac{1}{3} = 0.\overline{3}$ . The bar marks the repeating digit. Rounding $0.\overline{3}$ gives only an approximation, so it takes the symbol $\approx$ , as in $\frac{1}{3} \approx 0.33$ . A number whose decimal form never terminates and never repeats is irrational, and this lesson does not use any.

Every nonzero number (every number except zero) has two parts. Its sign tells whether it is greater than zero (positive) or less than zero (negative). Its absolute value is its distance from zero on a number line, which you can think of as its size. To multiply or divide, you will find the sign and the size of the answer separately.

Start with what multiplication means. The numbers you multiply are factors, and the answer is their product. The product $3 \times (-2)$ means three groups of $-2$ :

$3 \times (-2) = (-2) + (-2) + (-2) = -6$

Now remove one group, so two groups are left. Removing a group of $-2$ means subtracting $-2$ . Subtracting a number is the same as adding its additive inverse: its opposite, the number that adds with it to make zero. The additive inverse of $-2$ is $2$ , so

$2 \times (-2) = -6 - (-2) = -6 + 2 = -4$

Each time the first factor goes down by 1, one group of $-2$ is removed and the product goes up by 2. This is the distributive property, which lets you multiply each part of a sum or difference separately. It holds for every rational number $n$ as the first factor, including zero and negative numbers:

$(n - 1) \times (-2) = n \times (-2) - 1 \times (-2) = n \times (-2) + 2$

The table follows this pattern down past zero.

Products with negative two as the first factor goes down by 1

First factor	Product	Change from the row above
$3$	$3 \times (-2) = -6$	starting row
$2$	$2 \times (-2) = -4$	up 2
$1$	$1 \times (-2) = -2$	up 2
$0$	$0 \times (-2) = 0$	up 2
$-1$	$(-1) \times (-2) = 2$	up 2
$-2$	$(-2) \times (-2) = 4$	up 2
$-3$	$(-3) \times (-2) = 6$	up 2

Read three facts from the table.

A positive factor times a negative factor gives a negative product: $2 \times (-2) = -4$ . The order of the two factors does not matter.
Zero times any number is zero, because zero groups of any number make zero: $0 \times (-2) = 0$ .
A negative factor times a negative factor gives a positive product: $(-3) \times (-2) = 6$ .

Fractions and decimals follow the same properties of operations as integers, so these facts hold for all rational numbers. Each pair of negative factors makes a positive product, which leads to one rule for any number of factors.

When every factor is nonzero, an odd number of negative factors gives a negative product, and an even number (including none) gives a positive product. If any factor is zero, the product is zero.

For example, $(-1) \times (-2) \times (-3) = 2 \times (-3) = -6$ . There are three negative factors, an odd number, so the product is negative.

The sign rule says nothing about size, so find the size separately: multiply the absolute values. With decimals, multiply as you do with positive decimals. With fractions, multiply the top numbers (numerators) and multiply the bottom numbers (denominators).

Here is why that works. $\frac{2}{3}$ is 2 pieces of size $\frac{1}{3}$ , and $\frac{4}{5}$ is 4 pieces of size $\frac{1}{5}$ . Since $\frac{1}{3}$ of $\frac{1}{5}$ is $\frac{1}{15}$ , the product $\frac{2}{3} \times \frac{4}{5}$ is $2 \times 4$ pieces of size $\frac{1}{15}$ , which is $\frac{8}{15}$ . In general, for integers $a$ , $b$ , $c$ and $d$ with $b \neq 0$ and $d \neq 0$ :

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Worked example

Multiply fractions with different signs

Find $-\frac{3}{4} \times \frac{2}{5}$ .

Find the sign first. Both factors are nonzero, and exactly one of them is negative. One is an odd number, so the product is negative.
Find the size by multiplying the absolute values, three-fourths and two-fifths. Multiply the numerators and multiply the denominators. Neither denominator is zero, so the rule for multiplying fractions applies.
$\frac{3}{4}\times\frac{2}{5}=\frac{3\times 2}{4\times 5}=\frac{6}{20}$
Simplify, which means write an equal fraction with smaller numbers. Both 6 and 20 divide evenly by 2. Dividing the numerator and the denominator by the same nonzero number does not change the value of the fraction.
$\frac{6}{20}=\frac{6\div 2}{20\div 2}=\frac{3}{10}$
Put the sign and the size together.
$-\frac{3}{4}\times\frac{2}{5}=-\frac{3}{10}$

Answer

The product of $-\frac{3}{4}$ and $\frac{2}{5}$ is $-\frac{3}{10}$ .

Do not borrow the sign rule from addition. Adding two negative numbers gives a negative sum, such as $-3 + (-4) = -7$ . So it is tempting to write $(-3) \times (-4) = -12$ .

Here is why that fails. You know $3 \times (-4) = -12$ , three groups of $-4$ . If $(-3) \times (-4)$ were also $-12$ , the two products would add to $-24$ . But the distributive property says they add to $(-3 + 3) \times (-4) = 0 \times (-4) = 0$ . Only $12$ adds with $-12$ to make $0$ , so $(-3) \times (-4) = 12$ .

When you multiply nonzero numbers, count the negative factors. Here there are two, an even number, so the product is positive.

Division undoes multiplication. In $-6 \div (-2) = 3$ , the number being divided, $-6$ , is the dividend. The number you divide by, $-2$ , is the divisor. The answer, $3$ , is the quotient. The quotient is the missing factor: it is the number that, multiplied by the divisor, gives the dividend. Here $3 \times (-2) = -6$ .

This meaning settles every question about zero.

Zero divided by a nonzero number is zero. For example, $0 \div (-2) = 0$ because $0 \times (-2) = 0$ . The same reasoning works for every nonzero divisor.
Division by zero is undefined, which means it has no answer. $5 \div 0$ would need a number that, multiplied by $0$ , gives $5$ , but every number times $0$ is $0$ . And $0 \div 0$ has no single answer, because every number times $0$ gives $0$ .

So every divisor must be nonzero.

To divide by a nonzero number, multiply by its reciprocal: the number that multiplies with it to make 1. For a fraction $\frac{c}{d}$ with integers $c \neq 0$ and $d \neq 0$ , the reciprocal is $\frac{d}{c}$ , because $\frac{c}{d} \times \frac{d}{c} = \frac{c \times d}{d \times c} = 1$ . Zero has no reciprocal, because zero times any number is zero, never 1.

Here is why the reciprocal finds the missing factor. $\frac{3}{4} \div \frac{2}{5}$ asks for the number that, multiplied by $\frac{2}{5}$ , gives $\frac{3}{4}$ . The product $\frac{3}{4} \times \frac{5}{2}$ does exactly that, because you can regroup factors without changing a product:

$\left(\frac{3}{4} \times \frac{5}{2}\right) \times \frac{2}{5} = \frac{3}{4} \times \left(\frac{5}{2} \times \frac{2}{5}\right) = \frac{3}{4} \times 1 = \frac{3}{4}$

A number and its reciprocal have the same sign, because their product, 1, is positive. So after you rewrite a division as a multiplication, the sign rule for products applies.

Key concept

Sign first, then size

Multiplying or dividing rational numbers takes two separate decisions.

Sign. Rewrite each division by a nonzero number as multiplication by its reciprocal. When every factor is nonzero, an odd number of negative factors gives a negative product, and an even number gives a positive product.

Size. Multiply the absolute values.

Zero. Zero times any number is zero. Zero divided by a nonzero number is zero. Division by zero is undefined.

In the formula, $a$ , $b$ , $c$ and $d$ are integers.

\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{a\times d}{b\times c}\qquad (b\neq 0,\ c\neq 0,\ d\neq 0)

Worked example

Divide a negative fraction by a negative fraction

Find $-\frac{5}{6} \div \left(-\frac{2}{3}\right)$ .

Check the divisor. Negative two-thirds is not zero, so the quotient exists and the divisor has a reciprocal. To find it, swap the numerator and denominator and keep the negative sign. The product of the divisor and this number is 1, which confirms it is the reciprocal.
$-\frac{2}{3}\times\left(-\frac{3}{2}\right)=\frac{6}{6}=1$
Rewrite the division as multiplication by that reciprocal. Dividing by a nonzero number and multiplying by its reciprocal give the same result.
$-\frac{5}{6}\div\left(-\frac{2}{3}\right)=-\frac{5}{6}\times\left(-\frac{3}{2}\right)$
Find the sign. Both factors are nonzero and both are negative. Two is an even number, so the product is positive.
Find the size. Multiply the numerators and multiply the denominators of the absolute values. Then simplify: 15 and 12 both divide evenly by 3, a nonzero number, so divide both by 3.
$\frac{5}{6}\times\frac{3}{2}=\frac{15}{12}=\frac{15\div 3}{12\div 3}=\frac{5}{4}$
Check with the meaning of division: the quotient times the nonzero divisor must give the dividend. Both factors are nonzero and exactly one is negative, so this product is negative. It matches the dividend.
$\frac{5}{4}\times\left(-\frac{2}{3}\right)=-\frac{10}{12}=-\frac{5}{6}$

Answer

The quotient $-\frac{5}{6} \div \left(-\frac{2}{3}\right)$ is $\frac{5}{4}$ .

Worked example

Multiply and divide three negative decimals

Find $(-1.5) \times (-0.4) \div (-0.08)$ .

Check the divisor: negative 0.08 is not zero, so the expression is defined. Dividing by it is multiplying by its reciprocal, which is also negative. That makes three nonzero factors, all negative. Three is an odd number, so the answer is negative.
Find the size, working from left to right. Multiply the absolute values 1.5 and 0.4. Since 15 times 4 is 60, and tenths times tenths are hundredths, the product is 0.60.
$1.5\times 0.4=0.60=0.6$
Divide by the absolute value of the divisor. Multiply the dividend and the divisor by 100, a nonzero number, so the divisor becomes a whole number. Multiplying both by the same nonzero number does not change the quotient.
$0.6\div 0.08=60\div 8=7.5$
Attach the negative sign from the first step.
$(-1.5)\times(-0.4)\div(-0.08)=-7.5$
Check: the quotient times the nonzero divisor must give the dividend, which is the product of the first two numbers, 0.6. Both factors are nonzero and both are negative, so the product is positive. It matches.
$-7.5\times(-0.08)=0.6$

Answer

The value of $(-1.5) \times (-0.4) \div (-0.08)$ is $-7.5$ .

Takeaway. You can multiply or divide rational numbers by deciding the sign and the size separately. Rewrite each division by a nonzero number as multiplication by its reciprocal. Then, when every factor is nonzero, an odd number of negative factors gives a negative product and an even number gives a positive product. Multiply the absolute values to get the size. Zero times any number is zero, zero divided by a nonzero number is zero, and division by zero is undefined.

Next, you will convert between fractions and decimals, so you can write answers like these in either form.