The Four Operations on Rational Numbers

How to add, subtract, multiply and divide negative fractions, mixed numbers and decimals, how to work out a calculation with several operations, and how to use them in word problems.

Before you start

A rational number is a number that can be written as $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ . Integers, fractions, mixed numbers, and decimals that terminate or recur are all rational.

The negative sign of a fraction can be written in three places: $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$ . The negative sign of a mixed number belongs to the whole of it: $-2\frac{3}{5}$ means $-\left(2 + \frac{3}{5}\right) = -\frac{13}{5}$ .

To add or subtract fractions, use a common denominator: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ . To multiply, multiply the numerators and multiply the denominators: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ . To divide by a fraction, exchange its numerator and its denominator and multiply: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ .

To add two numbers with the same sign, add their numerical values (the numbers without their signs) and keep the sign: $-6 + (-5) = -11$ . To add two numbers with different signs, subtract the smaller numerical value from the larger, and take the sign of the number with the larger numerical value: $-8 + 5 = -3$ .

Subtracting a number is the same as adding its opposite. So subtracting a negative number is the same as adding the positive number: $3 - (-5) = 3 + 5 = 8$ .

In multiplication and division, two numbers with the same sign give a positive answer, and two numbers with different signs give a negative answer: $-6 \times (-7) = 42$ and $20 \div (-4) = -5$ .

A negative number that follows an operation sign ( $+$ , $-$ , $\times$ or $\div$ ) is written in brackets, as in $3 - (-5)$ . At the start of an expression the brackets may be left out: $-8 + 5$ means the same as $(-8) + 5$ .

Adding and Subtracting Rational Numbers

The rules for adding and subtracting integers hold for all rational numbers. The working with fractions is the same as in primary school. What is new is the signs.

Method

Change each mixed number to an improper fraction, keeping its sign.
Subtracting a negative number becomes adding the positive number. Adding a negative number becomes subtracting the positive number.
Write the fractions with a common denominator, the LCM of the denominators.
Add or subtract the numerators as integers. Keep the denominator, and simplify.

Example 1: Adding and subtracting negative fractions

Evaluate each of the following.
(a) $-\frac{3}{4} + \frac{1}{6}$
(b) $1\frac{3}{4} - \left(-\frac{5}{6}\right) + \left(-\frac{15}{8}\right)$

Solution

(a) The LCM of 4 and 6 is 12: $-\frac{3}{4} + \frac{1}{6} = -\frac{9}{12} + \frac{2}{12} = \frac{-9 + 2}{12}$

As with integers, $-9 + 2 = -7$ , so the answer is $\frac{-7}{12} = -\frac{7}{12}$ .

(b) $1\frac{3}{4} = \frac{7}{4}$ . Subtracting $-\frac{5}{6}$ is adding $\frac{5}{6}$ , and adding $-\frac{15}{8}$ is subtracting $\frac{15}{8}$ . This gives $\frac{7}{4} + \frac{5}{6} - \frac{15}{8}$ .

The LCM of 4, 6 and 8 is 24: $\frac{42}{24} + \frac{20}{24} - \frac{45}{24} = \frac{42 + 20 - 45}{24} = \frac{17}{24}$

The values are (a) $-\frac{7}{12}$ and (b) $\frac{17}{24}$ .

Example 2: Negative mixed numbers

Evaluate each of the following.
(a) $-2\frac{1}{3} - 1\frac{1}{2}$
(b) $1\frac{1}{4} - 3\frac{2}{3}$

Solution

(a) $-2\frac{1}{3} = -\frac{7}{3}$ and $1\frac{1}{2} = \frac{3}{2}$ .

The LCM of 3 and 2 is 6: $-\frac{7}{3} - \frac{3}{2} = -\frac{14}{6} - \frac{9}{6} = \frac{-14 - 9}{6}$

As with integers, $-14 - 9 = -23$ , so the answer is $\frac{-23}{6} = -\frac{23}{6} = -3\frac{5}{6}$ .

Check on the number line: start at $-2\frac{1}{3}$ and move $1\frac{1}{2}$ units to the left. You stop at $-3\frac{5}{6}$ , which is further from 0 than where you started.

(b) $1\frac{1}{4} = \frac{5}{4}$ and $3\frac{2}{3} = \frac{11}{3}$ . The LCM of 4 and 3 is 12: $\frac{5}{4} - \frac{11}{3} = \frac{15}{12} - \frac{44}{12} = \frac{15 - 44}{12}$

$15 - 44 = 15 + (-44) = -29$ , so the answer is $\frac{-29}{12} = -\frac{29}{12} = -2\frac{5}{12}$ . It is negative because the number subtracted is the greater of the two.

The values are (a) $-3\frac{5}{6}$ and (b) $-2\frac{5}{12}$ .

Multiplying and Dividing Rational Numbers

The rules of signs for multiplying and dividing integers also hold for all rational numbers.

Reciprocals

Two numbers whose product is 1 are reciprocals of each other. $\frac{3}{20} \times \frac{20}{3} = 1$ , so the reciprocal of $\frac{3}{20}$ is $\frac{20}{3}$ .

To find the reciprocal of a fraction, exchange its numerator and its denominator and keep its sign. The reciprocal of $-\frac{3}{20}$ is $-\frac{20}{3}$ , because $-\frac{3}{20} \times \left(-\frac{20}{3}\right) = 1$ . A number and its reciprocal have the same sign.

Write an integer over 1, and a mixed number as an improper fraction, first. The reciprocal of $8 = \frac{8}{1}$ is $\frac{1}{8}$ , and the reciprocal of $1\frac{1}{2} = \frac{3}{2}$ is $\frac{2}{3}$ .

Method

Change each mixed number to an improper fraction.
Find the sign of the answer. For two numbers, the same signs give a positive answer and different signs give a negative answer. For three or more factors, none of them 0, the product is positive when the number of negative factors is even and negative when it is odd. Then work with the fractions without their signs.
To multiply, cancel any common factors, then multiply the numerators and multiply the denominators.
To divide, multiply by the reciprocal of the divisor.
Write the result with the sign that you found.

0 has no reciprocal, and division by 0 is not defined.

Example 3: Multiplying two rational numbers

Evaluate each of the following.
(a) $-\frac{5}{12} \times \frac{8}{15}$
(b) $-1\frac{1}{2} \times \left(-2\frac{2}{9}\right)$

Solution

(a) Sign: negative $\times$ positive is negative.

Without the signs, the product is $\frac{5}{12} \times \frac{8}{15}$ . Divide 5 and 15 by 5, and divide 8 and 12 by 4: $\frac{1}{3} \times \frac{2}{3} = \frac{2}{9}$ . The sign is negative, so the answer is $-\frac{2}{9}$ .

(b) $1\frac{1}{2} = \frac{3}{2}$ and $2\frac{2}{9} = \frac{20}{9}$ . Sign: negative $\times$ negative is positive.

Without the signs, the product is $\frac{3}{2} \times \frac{20}{9}$ . Divide 3 and 9 by 3, and divide 20 and 2 by 2: $\frac{1}{1} \times \frac{10}{3} = \frac{10}{3} = 3\frac{1}{3}$ . The sign is positive, so the answer is $3\frac{1}{3}$ .

The values are (a) $-\frac{2}{9}$ and (b) $3\frac{1}{3}$ .

Example 4: A product of three factors

Evaluate $4\frac{1}{5} \times \left(-\frac{7}{3}\right) \times \left(-\frac{1}{7}\right)$ .

Solution

$4\frac{1}{5} = \frac{21}{5}$ .

Sign: there are two negative factors, an even number, so the answer is positive. Taking the factors two at a time gives the same sign: positive $\times$ negative is negative, and then negative $\times$ negative is positive.

Without the signs, the product is $\frac{21}{5} \times \frac{7}{3} \times \frac{1}{7}$ . Cancel the 7 in a numerator with the 7 in a denominator, and divide 21 and 3 by 3: $\frac{7}{5} \times \frac{1}{1} \times \frac{1}{1} = \frac{7}{5} = 1\frac{2}{5}$

$4\frac{1}{5} \times \left(-\frac{7}{3}\right) \times \left(-\frac{1}{7}\right) = 1\frac{2}{5}$

Example 5: Dividing

Evaluate each of the following.
(a) $7\frac{1}{2} \div \left(-\frac{3}{20}\right)$
(b) $-2\frac{2}{5} \div 8$

Solution

(a) $7\frac{1}{2} = \frac{15}{2}$ , and the reciprocal of $-\frac{3}{20}$ is $-\frac{20}{3}$ . So $\frac{15}{2} \div \left(-\frac{3}{20}\right) = \frac{15}{2} \times \left(-\frac{20}{3}\right)$ .

Sign: positive $\times$ negative is negative.

Without the signs, the product is $\frac{15}{2} \times \frac{20}{3}$ . Divide 15 and 3 by 3, and divide 20 and 2 by 2: $\frac{5}{1} \times \frac{10}{1} = 50$ . The sign is negative, so the answer is $-50$ .

Check: $-50 \times \left(-\frac{3}{20}\right) = \frac{150}{20} = 7\frac{1}{2}$

(b) $-2\frac{2}{5} = -\frac{12}{5}$ , and the reciprocal of 8 is $\frac{1}{8}$ . So $-\frac{12}{5} \div 8 = -\frac{12}{5} \times \frac{1}{8}$ .

Sign: negative $\times$ positive is negative. Without the signs, divide 12 and 8 by 4: $\frac{12}{5} \times \frac{1}{8} = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$ . The answer is $-\frac{3}{10}$ .

The values are (a) $-50$ and (b) $-\frac{3}{10}$ .

The Four Operations on Negative Decimals

Negative decimals follow the same rules of signs. Use the rules to find the sign, and do the decimal working without the signs. To add or subtract, line up the decimal points: write $2.4$ as $2.40$ when the other number has two decimal places.

Example 6: Adding and subtracting negative decimals

Evaluate each of the following.
(a) $3.5 - 7.2$
(b) $-2.4 - 1.35$
(c) $1.2 - (-0.75)$

Solution

(a) $3.5 - 7.2 = 3.5 + (-7.2)$ . The signs are different, so subtract the numerical values: $7.2 - 3.5 = 3.7$ . The answer takes the sign of $-7.2$ , which has the larger numerical value: $-3.7$ .

(b) $-2.4 - 1.35 = -2.4 + (-1.35)$ . The signs are the same, so add the numerical values: $2.40 + 1.35 = 3.75$ . Both numbers are negative, so the answer is negative: $-3.75$ .

The values are (a) $-3.7$ , (b) $-3.75$ and (c) $1.95$ .

Example 7: Multiplying and dividing negative decimals

Evaluate each of the following.
(a) $-1.5 \times 2.3$
(b) $-1.17 \div (-0.9)$

Solution

(a) Sign: negative $\times$ positive is negative. Without the signs, $15 \times 23 = 345$ . Together $1.5$ and $2.3$ have $1 + 1 = 2$ decimal places, so the product has 2 decimal places: $1.5 \times 2.3 = 3.45$ . The sign is negative, so the answer is $-3.45$ .

(b) Sign: negative $\div$ negative is positive. Without the signs, multiply both numbers by 10 so that the divisor is a whole number. This does not change the quotient: $1.17 \div 0.9 = 11.7 \div 9 = 1.3$ .

The values are (a) $-3.45$ and (b) $1.3$ .

Several Operations in One Calculation

Rational numbers follow the same order of operations as integers.

Order of operations

Work out what is inside brackets, starting with the innermost brackets.
Work out powers and roots.
Multiply and divide, from left to right.
Add and subtract, from left to right.

Brackets round a single negative number, as in $\left(-\frac{2}{3}\right)$ , only hold its sign. There is nothing to work out inside them.

A power of a fraction or of a decimal is a product of equal factors, as it is for an integer: $\left(-\frac{2}{3}\right)^{2} = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) = \frac{4}{9}$ and $(-0.2)^{3} = (-0.2) \times (-0.2) \times (-0.2) = -0.008$ . The brackets show that all of $-\frac{2}{3}$ , its sign included, is squared, and all of $-0.2$ is cubed. The square root sign still means the positive square root: $\sqrt{\frac{4}{9}} = \frac{2}{3}$ , because $\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$ .

Example 8: Brackets, powers and roots

Evaluate each of the following.
(a) $-0.25 - \frac{3}{4} \div \left(\frac{1}{2} - 2\right)$
(b) $\left(-\frac{1}{2}\right)^{3} + \sqrt{\frac{9}{16}} \times \frac{2}{3}$

Solution

(a) Write the decimal as a fraction: $-0.25 = -\frac{1}{4}$ .

Brackets first: $\frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}$

Division before subtraction: $\frac{3}{4} \div \left(-\frac{3}{2}\right) = \frac{3}{4} \times \left(-\frac{2}{3}\right) = -\frac{1}{2}$

Subtraction last: $-\frac{1}{4} - \left(-\frac{1}{2}\right) = -\frac{1}{4} + \frac{2}{4} = \frac{1}{4}$

(b) Powers and roots first. $\left(-\frac{1}{2}\right)^{3} = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right)$ . There are three negative factors, an odd number, so the power is negative. Without the signs, $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$ , so the power is $-\frac{1}{8}$ .

$\sqrt{\frac{9}{16}} = \frac{3}{4}$ , because $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$ .

Multiplication before addition. Cancel the 3 in a numerator with the 3 in a denominator, and divide 2 and 4 by 2: $\frac{3}{4} \times \frac{2}{3} = \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$

Addition last: $-\frac{1}{8} + \frac{1}{2} = -\frac{1}{8} + \frac{4}{8} = \frac{3}{8}$

The values are (a) $\frac{1}{4}$ and (b) $\frac{3}{8}$ .

When a calculation has both fractions and decimals, first write them all as fractions or all as decimals. Decimals are quick when every fraction gives a terminating decimal. When a fraction gives a recurring decimal, as $\frac{1}{3} = 0.333\ldots$ does, change the decimals to fractions instead. The digits of a recurring decimal never end, so it cannot be written out in full for a calculation, and a rounded value may not give the exact answer.

Example 9: Choosing fractions or decimals

Evaluate each of the following.
(a) $1.5 \times \left(-\frac{2}{5}\right) - (-0.75) \div \frac{1}{4}$
(b) $\left(-\frac{1}{3} + 0.5\right) \times (-1.2)$

Solution

(a) Both fractions give terminating decimals, so use decimals: $-\frac{2}{5} = -0.4$ and $\frac{1}{4} = 0.25$ . The calculation is $1.5 \times (-0.4) - (-0.75) \div 0.25$ .

Multiply and divide first: $1.5 \times (-0.4) = -0.6$ and $-0.75 \div 0.25 = -3$ .

Then subtract: $-0.6 - (-3) = -0.6 + 3 = 2.4$

Check with fractions: $\frac{3}{2} \times \left(-\frac{2}{5}\right) = -\frac{3}{5}$ and $-\frac{3}{4} \div \frac{1}{4} = -\frac{3}{4} \times \frac{4}{1} = -3$ . Then $-\frac{3}{5} - (-3) = -\frac{3}{5} + 3 = 2\frac{2}{5}$ , which is $2.4$ .

(b) $\frac{1}{3}$ gives a recurring decimal, so use fractions: $0.5 = \frac{1}{2}$ and $-1.2 = -\frac{12}{10} = -\frac{6}{5}$ .

Brackets first: $-\frac{1}{3} + \frac{1}{2} = -\frac{2}{6} + \frac{3}{6} = \frac{1}{6}$

Then multiply. Positive $\times$ negative is negative. Without the signs, cancel the 6 in a numerator with the 6 in a denominator: $\frac{1}{6} \times \frac{6}{5} = \frac{1}{5}$ . The sign is negative, so the answer is $-\frac{1}{5}$ .

The values are (a) $2.4$ and (b) $-\frac{1}{5}$ .

Word Problems

A fall, a loss or a position below a starting level is written as a negative number. The same change made several times is a multiplication. Write the calculation with its signs, work it out, then say what the answer means.

Example 10: A falling water level

In a dry month the water level in a pond falls by $\frac{3}{4}$ cm every day, a change of $-\frac{3}{4}$ cm each day.
(a) Find the change in the water level after 10 days.
(b) How many days does it take for the water level to change by $-13\frac{1}{2}$ cm?

Solution

(a) The change is 10 lots of $-\frac{3}{4}$ : $10 \times \left(-\frac{3}{4}\right) = -\frac{30}{4} = -\frac{15}{2} = -7\frac{1}{2}$

(b) Divide the total change by the change in one day: $-13\frac{1}{2} \div \left(-\frac{3}{4}\right) = -\frac{27}{2} \times \left(-\frac{4}{3}\right)$

Negative $\times$ negative is positive. Divide 27 and 3 by 3, and divide 4 and 2 by 2: $\frac{27}{2} \times \frac{4}{3} = \frac{9}{1} \times \frac{2}{1} = 18$

(a) The water level changes by $-7\frac{1}{2}$ cm: it falls by $7\frac{1}{2}$ cm. (b) It takes 18 days.

Example 11: Profit and loss

Mr Ong runs a drinks stall. In one week it rains on 3 days, and on each of those days he makes a loss of $38.50, which is a profit of −$38.50. On each of the other 4 days he makes a profit of $55.30.
(a) Find his total profit for the week.
(b) Find his average profit per day.

Solution

(a) Total profit in dollars: $3 \times (-38.50) + 4 \times 55.30$

Multiply first: $3 \times (-38.50) = -115.50$ and $4 \times 55.30 = 221.20$ .

Then add: $-115.50 + 221.20$ . The signs are different, so subtract the numerical values: $221.20 - 115.50 = 105.70$ . The answer takes the sign of $221.20$ , which has the larger numerical value, so it is positive.

(b) Divide the total profit by the 7 days: $105.70 \div 7 = 15.10$

(a) His total profit for the week is $105.70. (b) His average profit is $15.10 per day.

Common Mistakes

Using "two negatives make a positive" when adding or subtracting. That rule is for multiplying or dividing two negative numbers. $-\frac{1}{2} - \frac{1}{4} = -\frac{3}{4}$ , not $\frac{3}{4}$ .
Reading $-2\frac{1}{3}$ as $-2 + \frac{1}{3}$ . The sign belongs to the whole mixed number: $-2\frac{1}{3} = -\frac{7}{3}$ , not $-\frac{5}{3}$ .
Changing the sign when taking a reciprocal. The reciprocal of $-\frac{3}{20}$ is $-\frac{20}{3}$ , not $\frac{20}{3}$ . Dividing by a negative number is multiplying by a negative number.
Cancelling across an addition or a subtraction. Common factors are cancelled only in a multiplication. In $\frac{3}{4} \times \frac{4}{9}$ the 4s cancel. In $\frac{3}{4} + \frac{4}{9}$ they do not.
Rounding a recurring decimal in the middle of a calculation. $\frac{1}{3} \times 0.6$ is $\frac{1}{3} \times \frac{3}{5} = \frac{1}{5} = 0.2$ exactly. Using $0.33$ for $\frac{1}{3}$ gives $0.198$ .

Practice

Work each question on paper before you open its solution.

Question 1

Evaluate each of the following.
(a) $-\frac{2}{3} + \frac{1}{4}$
(b) $-\frac{5}{6} - \frac{3}{8}$
(c) $\frac{7}{10} - \left(-\frac{4}{15}\right)$

ShowHide solution

(a) The LCM of 3 and 4 is 12: $-\frac{8}{12} + \frac{3}{12} = \frac{-8 + 3}{12} = -\frac{5}{12}$

(b) The LCM of 6 and 8 is 24: $-\frac{20}{24} - \frac{9}{24} = \frac{-20 - 9}{24} = -\frac{29}{24} = -1\frac{5}{24}$

(c) Subtracting $-\frac{4}{15}$ is adding $\frac{4}{15}$ . The LCM of 10 and 15 is 30: $\frac{7}{10} + \frac{4}{15} = \frac{21}{30} + \frac{8}{30} = \frac{29}{30}$

The values are (a) $-\frac{5}{12}$ , (b) $-1\frac{5}{24}$ and (c) $\frac{29}{30}$ .

Question 2

Evaluate each of the following.
(a) $-1\frac{3}{4} - \left(-2\frac{1}{6}\right)$
(b) $2\frac{1}{2} - 4\frac{1}{3}$
(c) $-1\frac{2}{5} + \left(-3\frac{1}{2}\right) - \left(-\frac{9}{10}\right)$

ShowHide solution

(a) Subtracting $-2\frac{1}{6}$ is adding $2\frac{1}{6}$ : $-1\frac{3}{4} + 2\frac{1}{6} = -\frac{7}{4} + \frac{13}{6}$ . The LCM of 4 and 6 is 12: $-\frac{21}{12} + \frac{26}{12} = \frac{-21 + 26}{12} = \frac{5}{12}$

(b) $2\frac{1}{2} = \frac{5}{2}$ and $4\frac{1}{3} = \frac{13}{3}$ . The LCM of 2 and 3 is 6: $\frac{15}{6} - \frac{26}{6} = \frac{15 - 26}{6} = -\frac{11}{6} = -1\frac{5}{6}$

(c) Adding $-3\frac{1}{2}$ is subtracting $3\frac{1}{2}$ , and subtracting $-\frac{9}{10}$ is adding $\frac{9}{10}$ . This gives $-\frac{7}{5} - \frac{7}{2} + \frac{9}{10}$ . The LCM of 5, 2 and 10 is 10: $-\frac{14}{10} - \frac{35}{10} + \frac{9}{10} = \frac{-14 - 35 + 9}{10} = -\frac{40}{10} = -4$

The values are (a) $\frac{5}{12}$ , (b) $-1\frac{5}{6}$ and (c) $-4$ .

Question 3

Evaluate each of the following.
(a) $-\frac{3}{8} \times \frac{4}{9}$
(b) $-\frac{2}{3} \times 1\frac{1}{8} \times \left(-\frac{4}{5}\right)$
(c) $-1\frac{1}{3} \times \left(-2\frac{1}{4}\right) \times \left(-\frac{5}{6}\right)$

ShowHide solution

(a) Negative $\times$ positive is negative. Divide 3 and 9 by 3, and divide 4 and 8 by 4: $\frac{3}{8} \times \frac{4}{9} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ . The answer is $-\frac{1}{6}$ .

(b) $1\frac{1}{8} = \frac{9}{8}$ . There are two negative factors, an even number, so the answer is positive. Without the signs, the product is $\frac{2}{3} \times \frac{9}{8} \times \frac{4}{5}$ . Divide 9 and 3 by 3, and divide 4 and 8 by 4: $\frac{2}{1} \times \frac{3}{2} \times \frac{1}{5}$ . Then cancel the 2 in a numerator with the 2 in a denominator: $\frac{1}{1} \times \frac{3}{1} \times \frac{1}{5} = \frac{3}{5}$

(c) $1\frac{1}{3} = \frac{4}{3}$ and $2\frac{1}{4} = \frac{9}{4}$ . There are three negative factors, an odd number, so the answer is negative. Without the signs, the product is $\frac{4}{3} \times \frac{9}{4} \times \frac{5}{6}$ . Cancel the 4 in a numerator with the 4 in a denominator, which leaves $\frac{9}{3} \times \frac{5}{6} = 3 \times \frac{5}{6}$ . Divide 3 and 6 by 3: $\frac{5}{2} = 2\frac{1}{2}$ . The answer is $-2\frac{1}{2}$ .

The values are (a) $-\frac{1}{6}$ , (b) $\frac{3}{5}$ and (c) $-2\frac{1}{2}$ .

Question 4

(a) Write down the reciprocal of each number: $-\frac{7}{9}$ , $-4$ and $2\frac{1}{3}$ .
(b) Evaluate $-2\frac{1}{4} \div \left(-1\frac{1}{2}\right)$ .
(c) Evaluate $-\frac{5}{12} \div 10$ .

ShowHide solution

(a) Exchange the numerator and the denominator and keep the sign. $-4 = -\frac{4}{1}$ and $2\frac{1}{3} = \frac{7}{3}$ .

(b) $-2\frac{1}{4} = -\frac{9}{4}$ and $-1\frac{1}{2} = -\frac{3}{2}$ . Negative $\div$ negative is positive, and the reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$ : $\frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3}$ . Divide 9 and 3 by 3, and divide 2 and 4 by 2: $\frac{3}{2} \times \frac{1}{1} = \frac{3}{2} = 1\frac{1}{2}$

(c) The reciprocal of 10 is $\frac{1}{10}$ , and negative $\times$ positive is negative. Without the signs, divide 5 and 10 by 5: $\frac{5}{12} \times \frac{1}{10} = \frac{1}{12} \times \frac{1}{2} = \frac{1}{24}$ . The answer is $-\frac{1}{24}$ .

(a) $-\frac{9}{7}$ , $-\frac{1}{4}$ and $\frac{3}{7}$ (b) $1\frac{1}{2}$ (c) $-\frac{1}{24}$

Question 5

Evaluate each of the following.
(a) $-4.6 + 1.85$
(b) $-0.8 - (-2.35)$
(c) $-0.6 \times (-0.25)$
(d) $1.815 \div (-0.15)$

ShowHide solution

(a) The signs are different: $4.60 - 1.85 = 2.75$ , and $-4.6$ has the larger numerical value, so the answer is negative: $-2.75$ .

(b) $-0.8 - (-2.35) = -0.8 + 2.35$ . The signs are different: $2.35 - 0.80 = 1.55$ , and $2.35$ has the larger numerical value, so the answer is positive: $1.55$ .

(c) Negative $\times$ negative is positive. $6 \times 25 = 150$ , and there are $1 + 2 = 3$ decimal places: $0.150 = 0.15$ .

(d) Positive $\div$ negative is negative. Multiply both numbers by 100: $1.815 \div 0.15 = 181.5 \div 15 = 12.1$ . The answer is $-12.1$ .

The values are (a) $-2.75$ , (b) $1.55$ , (c) $0.15$ and (d) $-12.1$ .

Question 6

Evaluate each of the following.
(a) $-\frac{7}{11} \div \left(\frac{2}{11} - \frac{5}{3}\right)$
(b) $(-1.2 + 0.45) \times (-0.4) - 0.5$
(c) $\left(-1\frac{1}{2}\right)^{2} - \sqrt{\frac{25}{36}} \div \frac{1}{3}$
(d) $(-0.3)^{3} \div 0.9 + 0.1$

ShowHide solution

(a) Brackets first, with the common denominator 33: $\frac{2}{11} - \frac{5}{3} = \frac{6}{33} - \frac{55}{33} = -\frac{49}{33}$

Then divide. Negative $\div$ negative is positive, so work out $\frac{7}{11} \div \frac{49}{33} = \frac{7}{11} \times \frac{33}{49}$ . Divide 7 and 49 by 7, and divide 33 and 11 by 11: $\frac{1}{1} \times \frac{3}{7} = \frac{3}{7}$

(b) Brackets first: $-1.2 + 0.45 = -0.75$ . Then multiply: $-0.75 \times (-0.4) = 0.3$ . Then subtract: $0.3 - 0.5 = -0.2$ .

(c) Power and root first. $-1\frac{1}{2} = -\frac{3}{2}$ , and $\left(-\frac{3}{2}\right)^{2} = \left(-\frac{3}{2}\right) \times \left(-\frac{3}{2}\right) = \frac{9}{4}$ . $\sqrt{\frac{25}{36}} = \frac{5}{6}$ , because $\frac{5}{6} \times \frac{5}{6} = \frac{25}{36}$ .

Then divide: $\frac{5}{6} \div \frac{1}{3} = \frac{5}{6} \times \frac{3}{1}$ . Divide 3 and 6 by 3: $\frac{5}{2} \times \frac{1}{1} = \frac{5}{2}$

Then subtract: $\frac{9}{4} - \frac{5}{2} = \frac{9}{4} - \frac{10}{4} = -\frac{1}{4}$

(d) Power first: $(-0.3)^{3} = (-0.3) \times (-0.3) \times (-0.3)$ . Three negative factors give a negative product, $3 \times 3 \times 3 = 27$ , and there are $1 + 1 + 1 = 3$ decimal places, so $(-0.3)^{3} = -0.027$ .

Then divide. Negative $\div$ positive is negative, and $0.027 \div 0.9 = 0.27 \div 9 = 0.03$ , so the quotient is $-0.03$ . Then add: $-0.03 + 0.1 = 0.07$

The values are (a) $\frac{3}{7}$ , (b) $-0.2$ , (c) $-\frac{1}{4}$ and (d) $0.07$ .

Question 7

Evaluate each of the following.
(a) $0.5 - 1\frac{1}{4} \times (-0.6)$
(b) $\left(\frac{2}{3} - 1.5\right) \div (-0.25)$

ShowHide solution

(a) $1\frac{1}{4} = 1.25$ , a terminating decimal, so use decimals. Multiply first: positive $\times$ negative is negative, $125 \times 6 = 750$ , and there are $2 + 1 = 3$ decimal places, so $1.25 \times (-0.6) = -0.750 = -0.75$ .

Then subtract: $0.5 - (-0.75) = 0.50 + 0.75 = 1.25$

(b) $\frac{2}{3}$ gives a recurring decimal, so use fractions: $1.5 = \frac{3}{2}$ and $-0.25 = -\frac{1}{4}$ .

Brackets first: $\frac{2}{3} - \frac{3}{2} = \frac{4}{6} - \frac{9}{6} = -\frac{5}{6}$

Then divide. Negative $\div$ negative is positive, and the reciprocal of $\frac{1}{4}$ is 4: $\frac{5}{6} \times \frac{4}{1}$ . Divide 4 and 6 by 2: $\frac{5}{3} \times \frac{2}{1} = \frac{10}{3} = 3\frac{1}{3}$

The values are (a) $1.25$ , which is $1\frac{1}{4}$ , and (b) $3\frac{1}{3}$ .

Question 8

Jun Hao works out $-3\frac{1}{2} + 1\frac{1}{4}$ like this: $-3 + \frac{1}{2} + 1\frac{1}{4} = -1\frac{1}{4}$ . Explain his mistake and find the correct value.

ShowHide solution

He has read $-3\frac{1}{2}$ as $-3 + \frac{1}{2}$ . The negative sign belongs to the whole mixed number: $-3\frac{1}{2} = -\left(3 + \frac{1}{2}\right) = -\frac{7}{2}$ .

$1\frac{1}{4} = \frac{5}{4}$ , and the LCM of 2 and 4 is 4: $-\frac{7}{2} + \frac{5}{4} = -\frac{14}{4} + \frac{5}{4} = \frac{-14 + 5}{4} = -\frac{9}{4} = -2\frac{1}{4}$

The correct value is $-2\frac{1}{4}$ . Jun Hao made the $\frac{1}{2}$ positive, but it is part of the negative number $-3\frac{1}{2}$ .

Question 9

The lowest temperatures in Sapporo, in Japan, on four days in January were $-6.5$ °C, $-2.8$ °C, $1.4$ °C and $-4.1$ °C.
(a) Find the average of the four temperatures.
(b) Find the difference between the highest and the lowest of the four temperatures.

ShowHide solution

(a) Sum: $-6.5 + (-2.8) + 1.4 + (-4.1)$ . The negative numbers add to $-(6.5 + 2.8 + 4.1) = -13.4$ , and $-13.4 + 1.4 = -12$ .

Average: $-12 \div 4 = -3$

(b) The highest is $1.4$ and the lowest is $-6.5$ . Difference: $1.4 - (-6.5) = 1.4 + 6.5 = 7.9$

(a) The average temperature is $-3$ °C. (b) The difference is 7.9 °C.

Question 10

A sea turtle is at $-12\frac{1}{2}$ m, which means $12\frac{1}{2}$ m below the surface of the sea. It swims up $4\frac{3}{4}$ m and then down $2\frac{2}{3}$ m. Find its new position.

ShowHide solution

Going up is added and going down is subtracted: $-12\frac{1}{2} + 4\frac{3}{4} - 2\frac{2}{3} = -\frac{25}{2} + \frac{19}{4} - \frac{8}{3}$

The LCM of 2, 4 and 3 is 12: $-\frac{150}{12} + \frac{57}{12} - \frac{32}{12} = \frac{-150 + 57 - 32}{12} = -\frac{125}{12} = -10\frac{5}{12}$

Its new position is $-10\frac{5}{12}$ m: the turtle is $10\frac{5}{12}$ m below the surface.

Question 11

A submarine is diving. Its position changes by $-2\frac{1}{2}$ m each minute.
(a) Find the change in its position after 6 minutes.
(b) How many minutes does it take for its position to change by $-22\frac{1}{2}$ m?
(c) The submarine starts at $-40$ m. Find its position after 6 minutes.

ShowHide solution

(a) $-2\frac{1}{2} = -\frac{5}{2}$ . The change is 6 lots of it: $6 \times \left(-\frac{5}{2}\right) = -\frac{30}{2} = -15$

(b) Divide the total change by the change in one minute: $-22\frac{1}{2} \div \left(-2\frac{1}{2}\right) = -\frac{45}{2} \times \left(-\frac{2}{5}\right)$ . Negative $\times$ negative is positive. Cancel the 2s, and divide 45 and 5 by 5: $\frac{45}{2} \times \frac{2}{5} = 9$

(a) Its position changes by $-15$ m: it goes 15 m deeper. (b) It takes 9 minutes. (c) Its position is $-55$ m, which is 55 m below the surface.

Question 12

Water is let out of a tank, and the water level changes by $-1.2$ cm each hour for 5 hours. Then rain makes the level rise by $\frac{3}{4}$ cm each hour for 4 hours. Find the change in the water level over the whole 9 hours.

ShowHide solution

Total change: $5 \times (-1.2) + 4 \times \frac{3}{4}$

Multiply first: $5 \times (-1.2) = -6$ and $4 \times \frac{3}{4} = 3$ .

Then add: $-6 + 3 = -3$

The water level changes by $-3$ cm: after the 9 hours it is 3 cm lower than at the start.

Key Points

The rules of signs for integers hold for fractions, mixed numbers and decimals.
Change a mixed number to an improper fraction, keeping its sign: $-2\frac{1}{3} = -\frac{7}{3}$ .
To add or subtract fractions, use a common denominator, then add or subtract the numerators as integers.
To multiply, find the sign, cancel common factors, then multiply the numerators and multiply the denominators. To divide, multiply by the reciprocal.
A number and its reciprocal have the same sign. 0 has no reciprocal, and division by 0 is not defined.
With negative decimals, find the sign by the rules and do the decimal working without the signs.
When a calculation has both fractions and decimals, write them all as fractions or all as decimals. Use fractions when a decimal would recur.
A power is a product of equal factors: $\left(-\frac{1}{2}\right)^{3} = -\frac{1}{8}$ . The square root sign means the positive square root: $\sqrt{\frac{9}{16}} = \frac{3}{4}$ .
Order of operations: brackets; then powers and roots; then multiplication and division from left to right; then addition and subtraction from left to right.