Math 7 · 7.NS.A.2 · 7.NS.A.2A

Multiplying and dividing rational numbers

You can multiply -3 by 4, and you can multiply \displaystyle \frac{2}{3} by \displaystyle \frac{3}{4}. What happens when a number is negative and a fraction at the same time, and do the sign rules for integers still hold?

You need

Operations with rational numbersMath 7 Multiplying and dividing integersMath 7 Dividing fractionsGrade 6

Where you are 10 in Rational numbers

Number9

Rational numbers10

▴ 5 above Operations with rational numbers Terminating and repeating decimals

Multiplying and dividing rational numbers

Rational numbers in real situations Turning a repeating decimal into a fraction

When you reach for it

You already know two separate things. From multiplying and dividing integers, (-3)(4) = -12 and (-12) \div (-4) = 3: the signs decide whether the answer is positive or negative. From multiplying fractions and dividing fractions, \displaystyle \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2} and \displaystyle \frac{1}{2} \div \frac{1}{4} = 2: the numerators and denominators decide the size. This lesson puts the two together, for products and quotients such as

\begin{gathered} \left(-\frac{3}{5}\right) \cdot \frac{10}{9} \\ (-1.35) \div (-0.9) \end{gathered}

in which a number is a rational number that is negative and not a whole number at once.

Reach for it whenever a quantity that can go down is scaled or shared out: a snow depth that shrinks by \displaystyle 2\frac{2}{5} centimeters in three quarters of an hour, a balance that drops by 12.75 dollars every week, a rate found by dividing a change by a time. You need it too whenever a formula is evaluated at a negative fraction, which is most of algebra from here on.

The whole skill rests on one fact: the sign and the size of a product or a quotient can be found separately. The size is ordinary fraction or decimal arithmetic. The sign follows the same rules as for integers. The procedure below shows why the integer rules carry over unchanged, which is not obvious: (-1)(-1) = 1 was a fact about integers, and nothing so far says that \displaystyle \left(-\frac{1}{2}\right)\left(-\frac{1}{2}\right) must be positive too.

Adding and subtracting rational numbers is a separate move with different rules. There the sizes help decide the sign, as in -3 + 5; for products and quotients they never do.

The procedure

The lesson is taught in two parts: multiplying first, then dividing, which is built on multiplying.

Part 1: multiplying

Multiplying by -1 gives the opposite. For every rational number a,

(-1) \cdot a = -a

This is not a new rule. It follows from properties that every rational number has. Add a and (-1) \cdot a:

\begin{aligned} a + (-1) \cdot a &= 1 \cdot a + (-1) \cdot a \\ &= \big(1 + (-1)\big) \cdot a \\ &= 0 \cdot a = 0 \end{aligned}

The first line uses the identity property, 1 \cdot a = a. The second uses the distributive property, read from right to left. The third uses 1 + (-1) = 0 and the fact that any number times 0 is 0. So (-1) \cdot a is the number that adds to a to make 0, and that is exactly what the opposite of a is. Taking a = -1 gives (-1)(-1) = -(-1) = 1, for the same reason.

The sign rule for two factors. Let p and q be positive rational numbers. A negative rational number is the opposite of a positive one, so -p = (-1) \cdot p. The commutative and associative properties let the factors of a product be reordered and regrouped, so

\begin{aligned} (-p) \cdot q &= (-1) \cdot (pq) = -(pq) \\ (-p) \cdot (-q) &= (-1)(-1) \cdot (pq) = pq \end{aligned}

So the product of two numbers with different signs is negative, the product of two numbers with the same sign is positive, and either way the size is the product of the sizes, pq. These are word for word the rules for integers. The argument used only properties that every rational number has, so the rules hold for every fraction and every decimal, not just for whole numbers and their opposites.

Many factors. Each negative factor brings one factor of -1. Two of them multiply to 1, so what matters is whether the count of negative factors is even or odd: an even count gives a positive product, an odd count a negative one. If any factor is 0, the product is 0, which is neither positive nor negative.

The size. With the sign settled, multiply the sizes the way you already know. For fractions, numerator times numerator over denominator times denominator:

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

For decimals, use the standard algorithm. A factor shared by any numerator and any denominator can be canceled before multiplying, because all the numerators end up in one product and all the denominators in another, and the commutative property lets you pair them however you like. Settling the sign first means the canceling only ever deals with positive numbers.

What a negative factor does on the number line. Multiplying by \displaystyle -\frac{2}{3} is multiplying by \displaystyle \frac{2}{3} and then by -1, since \displaystyle -\frac{2}{3} = (-1) \cdot \frac{2}{3}. The first step scales the distance from zero by \displaystyle \frac{2}{3}. The second reflects the point to the other side of zero.

Figure 1Multiplying \displaystyle \frac{3}{2} by \displaystyle -\frac{2}{3}. The size \displaystyle \frac{2}{3} shrinks it to 1, and the sign flips it across zero to -1.

Starting at \displaystyle \frac{3}{2}, the size takes it to \displaystyle \frac{3}{2} \cdot \frac{2}{3} = 1, and the sign flips it to -1. So \displaystyle \frac{3}{2} \cdot \left(-\frac{2}{3}\right) = -1.

Part 2: dividing

Division undoes multiplication. For b \ne 0, a \div b is the number q for which q \cdot b = a. As with fractions, dividing by b is multiplying by its reciprocal \displaystyle \frac{1}{b}, the number whose product with b is 1:

a \div b = a \cdot \frac{1}{b}

The reciprocal of a negative number is negative. The reciprocal of \displaystyle -\frac{2}{5} is \displaystyle -\frac{5}{2}, because

\left(-\frac{2}{5}\right)\left(-\frac{5}{2}\right) = \frac{10}{10} = 1

It has to be negative. A number times its reciprocal is 1, which is positive, and by the sign rule only two numbers with the same sign have a positive product. Turning a fraction upside down never changes its sign.

The sign rule for quotients is the sign rule for products. Since \displaystyle a \div b = a \cdot \frac{1}{b} and \displaystyle \frac{1}{b} has the same sign as b, the quotient has the sign that a \cdot b would have: positive when a and b have the same sign, negative when they differ. The size is the quotient of the sizes. Every quotient can be checked by multiplying back: if a \div b = q, then q \cdot b must give a, sign included.

Where the minus sign goes in a fraction. A fraction is a division, numerator by denominator. For integers p and q with q \ne 0, the quotients (-p) \div q and p \div (-q) each have exactly one negative part, so each is the opposite of p \div q:

\frac{-p}{q} = \frac{p}{-q} = -\frac{p}{q}

A minus sign on top, underneath or in front means the same number. Two of them cancel: \displaystyle \frac{-p}{-q} = \frac{p}{q}, a quotient of two numbers with the same sign. This is also why every quotient of integers with a non-zero divisor is a rational number, the negative ones included: \displaystyle \frac{15}{-6} is \displaystyle -\frac{5}{2}, one integer over another.

Zero. Zero divided by a non-zero number is 0, since 0 \cdot b = 0. Dividing by zero has no answer at all. If a \ne 0, no q gives q \cdot 0 = a, because q \cdot 0 is always 0. If a = 0, every q gives q \cdot 0 = 0, so there is no single answer. Put another way, 0 has no reciprocal.

What a quotient counts. For a positive divisor, a \div b still asks how many copies of b make a, and the sign of the answer says which way the copies point.

Figure 2\displaystyle -\frac{3}{4} is three steps of \displaystyle \frac{1}{4} to the left of zero, so \displaystyle -\frac{3}{4} \div \frac{1}{4} = -3.

\displaystyle -\frac{3}{4} is three quarter-steps to the left of zero, so \displaystyle -\frac{3}{4} \div \frac{1}{4} = -3, and multiplying back, \displaystyle (-3) \cdot \frac{1}{4} = -\frac{3}{4}. In a story the same reading turns a change divided by a time into a rate: a snow depth that changes by -6 centimeters in 2 hours changes by -6 \div 2 = -3 centimeters per hour, which is to say it shrinks by 3 centimeters each hour.

Worked examples

Examples 1 to 5 belong to the first part, multiplying. Examples 6 to 11 belong to the second, dividing.

Example 1

Signs first, then sizes

Find each product.

(a) \displaystyle -\frac{3}{5} \cdot \frac{10}{9} \qquad (b) \displaystyle \left(-\frac{7}{8}\right)\left(-\frac{4}{21}\right) \qquad (c) \displaystyle \left(-\frac{5}{6}\right) \cdot 0

Solution

(a) The factors have different signs, so the product is negative. For the size, cancel before multiplying: 3 and 9 share the factor 3, and 10 and 5 share the factor 5.

\frac{3}{5} \cdot \frac{10}{9} = \frac{1 \cdot 2}{1 \cdot 3} = \frac{2}{3}

So \displaystyle -\frac{3}{5} \cdot \frac{10}{9} = -\frac{2}{3}. Multiplying first gives \displaystyle \frac{30}{45}, which reduces to the same size.

(b) Both factors are negative, so the product is positive. 7 and 21 share 7, and 4 and 8 share 4:

\left(-\frac{7}{8}\right)\left(-\frac{4}{21}\right) = \frac{1 \cdot 1}{2 \cdot 3} = \frac{1}{6}

(c) One factor is 0, so the product is 0: \displaystyle \left(-\frac{5}{6}\right) \cdot 0 = 0. The sign rule has nothing to say here, since 0 is neither positive nor negative.

Example 2

Decimals, and choosing a form

Find each product.

(a) (-2.5)(1.4) \qquad (b) (-0.3)(-0.12) \qquad (c) \displaystyle -0.75 \cdot \frac{8}{9}

Solution

(a) Different signs, so the product is negative. For the size, 25 \times 14 = 350, and the two factors have two decimal places between them, so 2.5 \times 1.4 = 3.50. So (-2.5)(1.4) = -3.5.

(b) Same signs, so the product is positive. 3 \times 12 = 36, and the factors have 1 + 2 = 3 decimal places between them, so the size is 0.036 and (-0.3)(-0.12) = 0.036.

(c) As a decimal, \displaystyle \frac{8}{9} = 0.888\ldots never ends, so a product worked in decimals could only be approximate. Write 0.75 as a fraction instead, \displaystyle \frac{3}{4}. The signs differ, so the product is negative; the 3 cancels against the 9 and the 4 against the 8:

-\frac{3}{4} \cdot \frac{8}{9} = -\frac{1 \cdot 2}{1 \cdot 3} = -\frac{2}{3}

When fractions and decimals are mixed, the fraction form is the one that is always exact.

Example 3

Several factors and powers

(a) Find \displaystyle \left(-\frac{1}{2}\right) \cdot \frac{4}{5} \cdot \left(-\frac{5}{6}\right) \cdot (-3).

(b) Find \displaystyle \left(-\frac{2}{3}\right)^3 and \displaystyle \left(-\frac{2}{3}\right)^4.

Solution

(a) Three of the four factors are negative. Three is odd, so the product is negative. For the size, put every numerator on top and every denominator below, writing 3 as \displaystyle \frac{3}{1}:

\frac{1 \cdot 4 \cdot 5 \cdot 3}{2 \cdot 5 \cdot 6 \cdot 1} = \frac{60}{60} = 1

So the product is -1.

(b) A power is a repeated product. \displaystyle \left(-\frac{2}{3}\right)^3 has three negative factors, an odd count, so it is negative. \displaystyle \left(-\frac{2}{3}\right)^4 has four, an even count, so it is positive:

\begin{aligned} \left(-\frac{2}{3}\right)^3 &= -\frac{8}{27} \\ \left(-\frac{2}{3}\right)^4 &= \frac{16}{81} \end{aligned}

This is the rule for powers of a negative integer, unchanged: an odd power of a negative number is negative and an even power is positive.

(c) An exponent applies only to what it is attached to, the order of operations with negative numbers. In \displaystyle -\left(\frac{3}{4}\right)^2 the square is taken first and the opposite afterwards, so \displaystyle -\left(\frac{3}{4}\right)^2 = -\frac{9}{16}. In \displaystyle \left(-\frac{3}{4}\right)^2 the brackets make the negative number itself the base, so \displaystyle \left(-\frac{3}{4}\right)^2 = \left(-\frac{3}{4}\right)\left(-\frac{3}{4}\right) = \frac{9}{16}.

Example 4

Mixed numbers

Find (a) \displaystyle \left(-2\frac{1}{4}\right)\left(1\frac{1}{3}\right) \qquad (b) \displaystyle \left(-1\frac{1}{2}\right)^2

Solution

The minus sign in front of a mixed number belongs to the whole of it: \displaystyle -2\frac{1}{4} means \displaystyle -\left(2 + \frac{1}{4}\right), not \displaystyle -2 + \frac{1}{4}. So write each one as an improper fraction with the sign in front, \displaystyle -2\frac{1}{4} = -\frac{9}{4}, and multiply as fractions.

(a) Different signs, so the product is negative. Cancel the 4 against the 4, and 3 out of the 9 and the 3:

-\frac{9}{4} \cdot \frac{4}{3} = -\frac{3 \cdot 1}{1 \cdot 1} = -3

(b) \displaystyle -1\frac{1}{2} = -\frac{3}{2}, and two negative factors give a positive product:

\left(-\frac{3}{2}\right)\left(-\frac{3}{2}\right) = \frac{9}{4} = 2\frac{1}{4}

Example 5

Using the properties to save work

Find each value.

(a) \displaystyle \left(-\frac{3}{8}\right) \cdot 13 \cdot \left(-\frac{8}{3}\right) \qquad (b) \displaystyle -\frac{5}{6}\left(12 - \frac{3}{5}\right) \qquad (c) (-1.2)(4.5) + (-1.2)(5.5)

Solution

(a) The first and last factors are reciprocals. The commutative property moves them next to each other and the associative property groups them:

\left[\left(-\frac{3}{8}\right)\left(-\frac{8}{3}\right)\right] \cdot 13 = 1 \cdot 13 = 13

Two negative factors, so the sign is positive, as it must be.

(b) The distributive property multiplies each term in the brackets by \displaystyle -\frac{5}{6}. A difference is the sum of the first term and the opposite of the second, so the second product is \displaystyle \left(-\frac{5}{6}\right)\left(-\frac{3}{5}\right):

\begin{aligned} -\frac{5}{6}\left(12 - \frac{3}{5}\right) &= \left(-\frac{5}{6}\right)(12) \\ &\qquad + \left(-\frac{5}{6}\right)\left(-\frac{3}{5}\right) \\ &= -10 + \frac{1}{2} = -9\frac{1}{2} \end{aligned}

Working the brackets first gives the same: \displaystyle 12 - \frac{3}{5} = \frac{57}{5}, and \displaystyle -\frac{5}{6} \cdot \frac{57}{5} = -\frac{57}{6} = -\frac{19}{2}, which is \displaystyle -9\frac{1}{2}.

\begin{aligned} &(-1.2)(4.5) + (-1.2)(5.5) \\ &\qquad = (-1.2)(4.5 + 5.5) \\ &\qquad = (-1.2)(10) = -12 \end{aligned}

Two awkward products have become one easy one.

Example 6

Dividing signed fractions

Find each quotient.

(a) \displaystyle -\frac{5}{8} \div \frac{15}{16} \qquad (b) \displaystyle \left(-\frac{9}{10}\right) \div \left(-\frac{3}{4}\right) \qquad (c) \displaystyle \frac{7}{3} \div (-14)

Solution

(a) Different signs, so the quotient is negative. Multiply by the reciprocal of \displaystyle \frac{15}{16}, then cancel 5 out of 5 and 15, and 8 out of 8 and 16:

-\frac{5}{8} \cdot \frac{16}{15} = -\frac{1 \cdot 2}{1 \cdot 3} = -\frac{2}{3}

(b) Same signs, so the quotient is positive. The reciprocal of \displaystyle -\frac{3}{4} is \displaystyle -\frac{4}{3}, and

\begin{aligned} \left(-\frac{9}{10}\right) \div \left(-\frac{3}{4}\right) &= \frac{9}{10} \cdot \frac{4}{3} \\ &= \frac{36}{30} = \frac{6}{5} \end{aligned}

Check by multiplying back: \displaystyle \frac{6}{5} \cdot \left(-\frac{3}{4}\right) = -\frac{18}{20} = -\frac{9}{10}, the number we started with.

\begin{aligned} \frac{7}{3} \div (-14) &= \frac{7}{3} \cdot \left(-\frac{1}{14}\right) \\ &= -\frac{7}{42} = -\frac{1}{6} \end{aligned}

Example 7

Where the minus sign sits

(a) Simplify \displaystyle \frac{-12}{-18}. \qquad (b) Simplify \displaystyle \frac{15}{-6}.

(c) Which of \displaystyle \frac{-7}{2}, \displaystyle \frac{7}{-2}, \displaystyle \frac{-7}{-2} and \displaystyle -\frac{-7}{-2} are equal to -3.5?

Solution

(a) A negative divided by a negative is positive, so \displaystyle \frac{-12}{-18} = \frac{12}{18} = \frac{2}{3}.

(b) A positive divided by a negative is negative, so \displaystyle \frac{15}{-6} = -\frac{15}{6} = -\frac{5}{2}.

(c) Count the minus signs in each, since each one is a factor of -1. \displaystyle \frac{-7}{2} and \displaystyle \frac{7}{-2} each have one, so both are \displaystyle -\frac{7}{2} = -3.5. \displaystyle \frac{-7}{-2} has two, which cancel, so it is \displaystyle \frac{7}{2} = 3.5. \displaystyle -\frac{-7}{-2} has three, an odd count, so it is the opposite of 3.5, which is -3.5. Three of the four equal -3.5: all except \displaystyle \frac{-7}{-2}.

Example 8

Dividing decimals

Find (a) -4.2 \div 0.06 \qquad (b) (-1.35) \div (-0.9) \qquad (c) \displaystyle -\frac{2}{3} \div 0.4

Solution

(a) Different signs, so the quotient is negative. For the size, write the division as a fraction and multiply top and bottom by 100, which leaves a fraction’s value unchanged because it is multiplying by \displaystyle \frac{100}{100} = 1:

\frac{4.2}{0.06} = \frac{420}{6} = 70

So -4.2 \div 0.06 = -70. Check: (-70)(0.06) = -4.2.

(b) Same signs, so the quotient is positive. Multiply top and bottom by 10: \displaystyle \frac{1.35}{0.9} = \frac{13.5}{9} = 1.5. So (-1.35) \div (-0.9) = 1.5. Check: (1.5)(-0.9) = -1.35.

(c) \displaystyle \frac{2}{3} does not end as a decimal, so work in fractions: \displaystyle 0.4 = \frac{2}{5}, whose reciprocal is \displaystyle \frac{5}{2}. The signs differ, so

-\frac{2}{3} \div 0.4 = -\frac{2}{3} \cdot \frac{5}{2} = -\frac{5}{3}

Example 9

A fraction bar over a difference

Find \displaystyle \dfrac{\frac{1}{2} - \frac{5}{6}}{-\frac{2}{9}}.

Solution

A long fraction bar groups everything above it and everything below it, so the numerator is worked out first, as if it were in brackets. Over the common denominator 6,

\frac{1}{2} - \frac{5}{6} = \frac{3}{6} - \frac{5}{6} = -\frac{2}{6} = -\frac{1}{3}

Now divide. Both numbers are negative, so the quotient is positive:

\begin{aligned} -\frac{1}{3} \div \left(-\frac{2}{9}\right) &= \frac{1}{3} \cdot \frac{9}{2} \\ &= \frac{9}{6} = \frac{3}{2} \end{aligned}

Example 10

Evaluating expressions at negative fractions

For \displaystyle x = -\frac{1}{2} and \displaystyle y = \frac{1}{3}, find (a) \dfrac{xy}{x - y} \qquad (b) \dfrac{x}{y} - \dfrac{y}{x}

Solution

Put each value in brackets as it goes in, so its sign stays attached to it.

(a) The numerator is \displaystyle xy = \left(-\frac{1}{2}\right)\left(\frac{1}{3}\right) = -\frac{1}{6}. The denominator is \displaystyle x - y = -\frac{1}{2} - \frac{1}{3} = -\frac{3}{6} - \frac{2}{6} = -\frac{5}{6}. Both are negative, so the quotient is positive:

\begin{aligned} \frac{xy}{x - y} &= -\frac{1}{6} \div \left(-\frac{5}{6}\right) \\ &= \frac{1}{6} \cdot \frac{6}{5} = \frac{1}{5} \end{aligned}

(b) \displaystyle \frac{x}{y} = -\frac{1}{2} \div \frac{1}{3} = -\frac{1}{2} \cdot 3 = -\frac{3}{2}, and \displaystyle \frac{y}{x} = \frac{1}{3} \div \left(-\frac{1}{2}\right) = \frac{1}{3} \cdot (-2) = -\frac{2}{3}. The two quotients are reciprocals of each other, and so they have the same sign. Then

\begin{aligned} \frac{x}{y} - \frac{y}{x} &= -\frac{3}{2} - \left(-\frac{2}{3}\right) \\ &= -\frac{9}{6} + \frac{4}{6} = -\frac{5}{6} \end{aligned}

Example 11

Melting snow

Over three quarters of an hour, the depth of snow on a flat roof changes by \displaystyle -2\frac{2}{5} centimeters, shrinking at a steady rate.

(a) By how much does the depth change each hour?

(b) By how much does it change in 2.5 hours?

(d) Suppose it was already shrinking at the same rate two hours ago. How did the depth then compare with the depth now?

Solution

(a) A rate is a change divided by a time. Here the change is \displaystyle -2\frac{2}{5} = -\frac{12}{5} centimeters and the time is \displaystyle \frac{3}{4} of an hour. The signs differ, so the rate is negative:

\begin{aligned} -\frac{12}{5} \div \frac{3}{4} &= -\frac{12}{5} \cdot \frac{4}{3} \\ &= -\frac{16}{5} = -3.2 \end{aligned}

The depth changes by -3.2 centimeters each hour. The negative sign says it is shrinking.

(b) Change is rate times time: (-3.2)(2.5) = -8. The depth changes by -8 centimeters, so the snow is 8 centimeters shallower.

(c) Time is change divided by rate: (-20) \div (-3.2) = 6.25. Both are negative, so the quotient is positive, as a length of time should be. It takes 6.25 hours, which is 6 hours and 15 minutes.

(d) Measure time in hours from now, with negative times in the past. At the steady rate, the change from now to time t is -3.2t centimeters. Two hours ago is t = -2, and

(-3.2)(-2) = 6.4

The snow two hours ago was 6.4 centimeters deeper than it is now. A negative rate times a negative time is positive, and the story agrees: snow that is melting was deeper in the past.

Boundaries

The divisor must not be zero. a \div 0 has no value for any a, since 0 has no reciprocal. Zero may be the number divided: 0 \div b = 0 for every b \ne 0. In an expression with a variable in a denominator, such as \displaystyle \frac{xy}{x - y}, the values that make the denominator zero (here x = y) are not allowed.

Zero has no sign. The sign rule is about non-zero factors. A product with a zero factor is 0 whatever the other signs are, and 0 counts as neither positive nor negative.

Each minus sign is one factor of -1. It may sit on the numerator, on the denominator or in front of the fraction, and one of them means the same number in any of those places. Two of them cancel, so \displaystyle \frac{-p}{-q} = \frac{p}{q}.

The sign of a mixed number covers both parts. \displaystyle -2\frac{1}{4} = -\frac{9}{4}, never \displaystyle -2 + \frac{1}{4}.

Canceling works across a product, never across a sum or difference. In \displaystyle \frac{3}{5} \cdot \frac{10}{9} the 3 and the 9 may be divided by 3. In \displaystyle \dfrac{\frac{1}{2} - \frac{5}{6}}{-\frac{2}{9}} nothing cancels until the numerator has been worked out to a single number.

Division is neither commutative nor associative. a \div b and b \div a are different numbers in general. They are reciprocals of each other, so they share a sign, but only their sign. And \displaystyle \frac{3}{4} \div \frac{1}{2} \div \frac{1}{2} means \displaystyle \left(\frac{3}{4} \div \frac{1}{2}\right) \div \frac{1}{2} = 3, worked from left to right, not \displaystyle \frac{3}{4} \div \left(\frac{1}{2} \div \frac{1}{2}\right) = \frac{3}{4}.

A decimal that does not end gives only an approximate answer. For exact results, write such a number as a fraction first, as in Examples 2 and 8. The answer is still a rational number: products and quotients of rational numbers (with a non-zero divisor) are always rational, which is the closure of the rational numbers under these two operations.

Where this goes next

With all four operations on signed fractions and decimals in hand, the numbers in real situations can be worked with directly: temperatures that drop by fractions of a degree per hour, debts shared among several people, changes averaged over a week. Putting several operations in one expression is the order of operations with negative numbers, and turning a quotient of two integers into a decimal is long division, which also shows why some quotients end and others repeat.

The quotient of two signed changes is the idea behind slope: rise divided by run, where either may be negative, and a negative slope is a line going downhill exactly as the snow depth in Example 11 went down. Multiplying by a decimal such as 0.8 or 1.2 is how a percent change is applied in one step. The sign rules for powers, odd powers of negatives negative and even powers positive, carry straight into the laws of exponents.

Exercises

Part 1: products

\displaystyle -\frac{4}{9} \cdot \frac{3}{8}
\displaystyle \left(-\frac{10}{21}\right)\left(-\frac{14}{15}\right)
(-1.6)(0.25)
(-0.7)(-0.09)
\displaystyle \left(-1\frac{1}{2}\right)\left(2\frac{2}{3}\right)
\displaystyle \left(-\frac{1}{3}\right)\left(-\frac{3}{5}\right)\left(-\frac{5}{7}\right)
\displaystyle \left(-\frac{3}{2}\right)^3
Find -0.4^2 and (-0.4)^2.

Part 1: using the properties

Use the commutative, associative or distributive property to make each one easy.

\displaystyle \left(-\frac{5}{9}\right)(-17)\left(\frac{9}{5}\right)
\displaystyle -\frac{3}{4}\left(8 + \frac{2}{3}\right)
(-2.5)(3.7) + (-2.5)(6.3)

Part 2: quotients

\displaystyle -\frac{7}{12} \div \frac{14}{9}
\displaystyle \left(-\frac{4}{15}\right) \div \left(-\frac{2}{5}\right)
\displaystyle -6 \div \frac{3}{4}
\displaystyle \frac{5}{8} \div (-10)
-2.88 \div 0.12
(-0.45) \div (-1.5)
\displaystyle -3\frac{1}{3} \div 1\frac{1}{4}

Part 2: signs, fraction bars and letters

Find \displaystyle \dfrac{\frac{1}{4} - \frac{2}{3}}{-\frac{5}{6}}.
Which of \displaystyle \frac{-2}{5}, \displaystyle \frac{2}{-5}, \displaystyle \frac{-2}{-5} and -0.4 are equal to \displaystyle -\frac{2}{5}?
For \displaystyle x = \frac{2}{3} and \displaystyle y = -\frac{1}{2}, find \dfrac{x - y}{xy}.

Part 2: problems in words

On a steady descent, a hiker’s elevation changes by -450 meters in \displaystyle 2\frac{1}{2} hours. (a) What is the change in elevation per hour? (b) What is the change in the first \displaystyle \frac{3}{4} of an hour?
A savings balance changes by -12.75 dollars every week. (a) What is the total change over 8 weeks? (b) After how many weeks has the total change reached -191.25 dollars?

Reasoning

Use the distributive property to explain why \displaystyle (-1) \cdot \left(-\frac{2}{3}\right) = \frac{2}{3}.
Explain why the reciprocal of a negative number is negative, and why that makes the sign rule for quotients the same as the sign rule for products.
For integers a and b with b \ne 0, explain why \displaystyle \frac{-a}{b} and \displaystyle \frac{a}{-b} are both equal to \displaystyle -\frac{a}{b}.
Explain why \displaystyle \frac{0}{5} has a value and \displaystyle \frac{5}{0} does not.
The product of nine non-zero rational numbers is negative. What can you say about how many of the nine are negative? Explain.

Answers

\displaystyle -\frac{1}{6}. Different signs, so negative; cancel 4 with 8 and 3 with 9 to get \displaystyle \frac{1 \cdot 1}{3 \cdot 2}.
\displaystyle \frac{4}{9}. Same signs, so positive; \displaystyle \frac{10}{15} = \frac{2}{3} and \displaystyle \frac{14}{21} = \frac{2}{3}, and \displaystyle \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9}.
-0.4. Different signs; 1.6 \times 0.25 = 0.4, a quarter of 1.6.
0.063. Same signs; 7 \times 9 = 63, with 1 + 2 = 3 decimal places.
-4. \displaystyle -\frac{3}{2} \cdot \frac{8}{3} = -\frac{8}{2} = -4.
\displaystyle -\frac{1}{7}. Three negative factors, an odd count, so negative; the 3s and 5s cancel, leaving \displaystyle \frac{1}{7}.
\displaystyle -\frac{27}{8}. Three negative factors; \displaystyle \frac{3 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 2} = \frac{27}{8}.
-0.4^2 = -0.16 and (-0.4)^2 = 0.16. Without brackets the square acts on 0.4 only, and the opposite is taken afterwards.
17. Group the reciprocals: \displaystyle \left(-\frac{5}{9}\right)\left(\frac{9}{5}\right) = -1, and (-1)(-17) = 17.
\displaystyle -\frac{13}{2}, which is \displaystyle -6\frac{1}{2}. Distribute: \displaystyle -\frac{3}{4} \cdot 8 = -6 and \displaystyle -\frac{3}{4} \cdot \frac{2}{3} = -\frac{1}{2}, and \displaystyle -6 + \left(-\frac{1}{2}\right) = -6\frac{1}{2}.
-25. Pull out the common factor: (-2.5)(3.7 + 6.3) = (-2.5)(10) = -25.
\displaystyle -\frac{3}{8}. Different signs; \displaystyle \frac{7}{12} \cdot \frac{9}{14} = \frac{1 \cdot 3}{4 \cdot 2} after canceling 7 and 3.
\displaystyle \frac{2}{3}. Same signs; \displaystyle \frac{4}{15} \cdot \frac{5}{2} = \frac{20}{30}.
-8. \displaystyle -6 \cdot \frac{4}{3} = -\frac{24}{3} = -8. Check: \displaystyle (-8) \cdot \frac{3}{4} = -6.
\displaystyle -\frac{1}{16}. \displaystyle \frac{5}{8} \cdot \left(-\frac{1}{10}\right) = -\frac{5}{80}.
-24. Different signs; \displaystyle \frac{2.88}{0.12} = \frac{288}{12} = 24.
0.3. Same signs; \displaystyle \frac{0.45}{1.5} = \frac{4.5}{15} = 0.3.
\displaystyle -\frac{8}{3}, which is \displaystyle -2\frac{2}{3}. \displaystyle -\frac{10}{3} \div \frac{5}{4} = -\frac{10}{3} \cdot \frac{4}{5} = -\frac{40}{15}.
\displaystyle \frac{1}{2}. The numerator is \displaystyle \frac{3}{12} - \frac{8}{12} = -\frac{5}{12}, and \displaystyle -\frac{5}{12} \div \left(-\frac{5}{6}\right) = \frac{5}{12} \cdot \frac{6}{5} = \frac{1}{2}.
\displaystyle \frac{-2}{5}, \displaystyle \frac{2}{-5} and -0.4. Each has one minus sign; \displaystyle \frac{-2}{-5} has two, which cancel, so it equals \displaystyle \frac{2}{5}.
\displaystyle -\frac{7}{2}. \displaystyle x - y = \frac{2}{3} + \frac{1}{2} = \frac{7}{6} and \displaystyle xy = -\frac{1}{3}, so the quotient is \displaystyle \frac{7}{6} \cdot (-3) = -\frac{7}{2}.
(a) -180 meters per hour: \displaystyle -450 \div \frac{5}{2} = -450 \cdot \frac{2}{5} = -180. (b) -135 meters: \displaystyle (-180) \cdot \frac{3}{4} = -135.
(a) -102 dollars: (-12.75)(8) = -102. (b) 15 weeks: (-191.25) \div (-12.75) = 15, positive because both are negative.
Add \displaystyle -\frac{2}{3} and \displaystyle (-1)\left(-\frac{2}{3}\right). Writing the first as \displaystyle 1 \cdot \left(-\frac{2}{3}\right), the distributive property turns the sum into \displaystyle \big(1 + (-1)\big)\left(-\frac{2}{3}\right), which is \displaystyle 0 \cdot \left(-\frac{2}{3}\right) = 0. So \displaystyle (-1)\left(-\frac{2}{3}\right) is the number that adds to \displaystyle -\frac{2}{3} to give 0, which is the opposite of \displaystyle -\frac{2}{3}, namely \displaystyle \frac{2}{3}.
A number times its reciprocal is 1, which is positive, and a product of two numbers is positive only when they have the same sign. So a negative number’s reciprocal is negative. Then \displaystyle a \div b = a \cdot \frac{1}{b}, and \displaystyle \frac{1}{b} has the same sign as b, so the quotient has the same sign as the product a \cdot b.
\displaystyle \frac{-a}{b} is the number that gives -a when multiplied by b, and so is \displaystyle -\frac{a}{b}, since \displaystyle \left(-\frac{a}{b}\right) \cdot b = -\left(\frac{a}{b} \cdot b\right) = -a. Likewise \displaystyle \frac{a}{-b} is the number that gives a when multiplied by -b, and so is \displaystyle -\frac{a}{b}, since \displaystyle \left(-\frac{a}{b}\right)(-b) = \frac{a}{b} \cdot b = a. Division by a non-zero number has only one answer, so the numbers are equal.
\displaystyle \frac{0}{5} asks for the number that gives 0 when multiplied by 5, and 0 is that number and the only one. \displaystyle \frac{5}{0} asks for a number that gives 5 when multiplied by 0, but every number times 0 is 0, so there is none.
An odd number of them: 1, 3, 5, 7 or 9. Every pair of negative factors multiplies to a positive number, so a negative product needs one negative factor left over after pairing.