Math Core

A stock that falls $1.50 a day for $2.5$ days changes by $2.5 \times (-1.50)$ dollars. Now that you know the sign rules for integers, you can multiply and divide any rational numbers: fractions, decimals and mixed numbers, positive or negative.

One idea, two steps

The sign rules don't care whether the numbers are whole. Every product or quotient of rational numbers is found the same way.

Multiplying and dividing rational numbers

Find the sign. Same signs give a positive answer. Different signs give a negative answer. (With more than two factors, count the negatives: even means positive, odd means negative.)
Work with the absolute values using the fraction and decimal skills you already have.

As a reminder of those skills:

To multiply fractions, multiply the numerators and multiply the denominators.
To divide by a fraction, multiply by its reciprocal (flip the second fraction).
Change mixed numbers to fractions before you multiply or divide.

Signs and reciprocals

The reciprocal of a negative number is also negative, because their product has to be positive $1$ . The reciprocal of $-\dfrac{2}{7}$ is $-\dfrac{7}{2}$ , since

\left(-\frac{2}{7}\right)\left(-\frac{7}{2}\right) = \frac{14}{14} = 1

The number $0$ has no reciprocal, which is another way of saying you can't divide by zero.

Worked example: Multiplying fractions

Find $-\dfrac{5}{6} \times \dfrac{9}{10}$ .

Different signs, so the answer is negative. Simplify before multiplying: $5$ and $10$ share a factor of $5$ , and $9$ and $6$ share a factor of $3$ .

-\frac{5}{6} \times \frac{9}{10} = -\frac{1}{2} \times \frac{3}{2} = -\frac{3}{4}

Worked example: Dividing with mixed numbers

Find $-2\tfrac{1}{4} \div \left(-1\tfrac{1}{2}\right)$ .

Same signs, so the answer is positive. Change to fractions: $2\tfrac{1}{4} = \dfrac{9}{4}$ and $1\tfrac{1}{2} = \dfrac{3}{2}$ .

\frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = \frac{3}{2} = 1\tfrac{1}{2}

Check: $1\tfrac{1}{2} \times \left(-1\tfrac{1}{2}\right) = \dfrac{3}{2} \times \left(-\dfrac{3}{2}\right) = -\dfrac{9}{4} = -2\tfrac{1}{4}$ . Correct.

Worked example: Decimals

$-1.5 \times 2.4$
$-7.2 \div (-0.9)$

Solutions.

Different signs, so negative. $1.5 \times 2.4 = 3.6$ , so the answer is $-3.6$ .
Same signs, so positive. Multiply both numbers by $10$ to get $72 \div 9 = 8$ . The answer is $8$ .

Order of operations and properties

The order of operations still applies: parentheses, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

The distributive property can save a lot of work: $a(b + c) = ab + ac$ . For example,

-8 \times 3\tfrac{1}{4} = -8 \times \left(3 + \tfrac{1}{4}\right) = -24 + (-2) = -26

Worked example: Putting it together

Evaluate $\dfrac{-3}{5} \times 10 + 12 \div \left(-\dfrac{4}{3}\right)$ .

Do both the multiplication and the division first:

\begin{aligned} \frac{-3}{5} \times 10 &= \frac{-30}{5} = -6 \\ 12 \div \left(-\frac{4}{3}\right) &= 12 \times \left(-\frac{3}{4}\right) = -9 \end{aligned}

Then add: $-6 + (-9) = -15$ .

Common mistake

When you divide by a negative fraction, flip the fraction but keep its negative sign. The reciprocal of $-\dfrac{4}{3}$ is $-\dfrac{3}{4}$ , not $\dfrac{3}{4}$ . Flipping a fraction never changes its sign.

Tip

A quick sign check: before calculating, write a small $+$ or $-$ next to the problem for the sign of the answer. At the end, compare.