Multiplying and Dividing Fractions with Properties

Multiplying fractions. The product $\frac{2}{3} \times \frac{4}{5}$ means two-thirds of four-fifths. Split $\frac{4}{5}$ into $3$ equal parts: each part is $\frac{4}{15}$. Take $2$ of those parts and you have $\frac{8}{15}$. That is why you multiply the numerators ($2 \times 4 = 8$) and multiply the denominators ($3 \times 5 = 15$).

In symbols, $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Here $a$, $b$, $c$ and $d$ are integers, and the denominators $b$ and $d$ are not $0$.

Negative signs do not change this size work. When both fractions are nonzero, make two moves:

1. Sign. Count the negative factors. An odd count gives a negative product; an even count gives a positive product.
2. Size. Multiply the absolute values, numerator by numerator and denominator by denominator.

The absolute value of a number is its distance from $0$, so the absolute value of $-\frac{3}{4}$ is $\frac{3}{4}$. If either factor is $0$, the product is simply $0$.

Dividing fractions. In $6 \div 2 = 3$, the dividend is $6$, the divisor is $2$ and the quotient is $3$. A division asks a multiplication question: what number times the divisor gives the dividend? Here the answer is $3$, because $3 \times 2 = 6$.

To answer that question for fractions, you need the reciprocal. The reciprocal of a nonzero number is the number you multiply it by to get $1$. For a fraction $\frac{c}{d}$ with $c \neq 0$ and $d \neq 0$, the reciprocal is $\frac{d}{c}$, because $\frac{c}{d} \times \frac{d}{c} = \frac{c \times d}{d \times c} = 1$.

A reciprocal keeps the sign of its number. The reciprocal of $-\frac{2}{3}$ is $-\frac{3}{2}$, since $-\frac{2}{3} \times \left(-\frac{3}{2}\right) = \frac{6}{6} = 1$. Zero has no reciprocal, because $0$ times any number is $0$, never $1$.

Now the division rule makes sense: dividing by a nonzero number gives the same result as multiplying by its reciprocal.

Here is why, using $\frac{a}{b} \div \frac{c}{d}$ with $b$, $c$ and $d$ all nonzero. Try $\frac{a}{b} \times \frac{d}{c}$ as the quotient, and multiply it by the divisor $\frac{c}{d}$. The reciprocal pair $\frac{d}{c} \times \frac{c}{d}$ makes $1$, so you get back $\frac{a}{b} \times 1 = \frac{a}{b}$, the dividend. That is exactly what the division question asks for.

The divisor can be any nonzero rational number: a fraction, a mixed number or a decimal, positive or negative. It can never be $0$, because zero has no reciprocal to multiply by, so division by $0$ is undefined. A dividend of $0$ is fine: $0 \div \left(-\frac{2}{3}\right) = 0 \times \left(-\frac{3}{2}\right) = 0$.

Properties of operations are rules, such as $a \times b = b \times a$, that stay true whichever rational numbers you put in: positive, negative or zero. The table below lists the ones you need. In it, $a$, $b$ and $c$ stand for any rational numbers. One property, the inverse property, needs a nonzero number.

Two properties do most of the work. The commutative property lets you change the order of factors. The associative property lets you change the grouping of factors. Together they let you pair up factors that are easy to multiply, such as a number and its reciprocal.

Both belong to multiplication, not to division. Order matters in division: $8 \div 2 = 4$, but $2 \div 8 = \frac{1}{4}$. Grouping matters too: $(8 \div 4) \div 2 = 1$, but $8 \div (4 \div 2) = 4$. So rewrite each division by a nonzero number as multiplication by its reciprocal first. Then every number is a factor, and you may reorder and regroup.

The distributive property connects multiplication with addition, and it helps when a factor is a mixed number. A mixed number is a sum: $6\frac{3}{4} = 6 + \frac{3}{4}$. To find $-\frac{2}{3} \times 6\frac{3}{4}$, multiply each part by $-\frac{2}{3}$ and add the two results:

$-\frac{2}{3} \times \left(6 + \frac{3}{4}\right) = -\frac{2}{3} \times 6 + \left(-\frac{2}{3}\right) \times \frac{3}{4} = -4 + \left(-\frac{1}{2}\right) = -4\frac{1}{2}$

You get the same product by writing $6\frac{3}{4}$ as $\frac{27}{4}$ first: $-\frac{2}{3} \times \frac{27}{4} = -\frac{54}{12} = -4\frac{1}{2}$.

Takeaway. Multiplying and dividing fractions is sign work plus size work. First rewrite each division by a nonzero number as multiplication by its reciprocal. Then, when every factor is nonzero, count the negative factors to find the sign, and multiply the absolute values to find the size. If any factor is $0$, the product is $0$, and you never divide by $0$. Because everything is now multiplication, the commutative and associative properties let you put the factors in the friendliest order.

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