 # 1.01 Calculations with whole numbers

Lesson

## Order of operations

An operation is a mathematical process, such as addition, subtraction, multiplication and division. Other operations include raising a number to a power and taking a root of a number. An operator is a symbol that indicates the type of operation. For addition, subtraction, multiplication and division, the corresponding operators are $+$+, $-$, $\times$× and $\div$÷​.

There are a number of conventions (or rules) which we follow in order to correctly interpret and work with expressions that include operations. In particular, there is a priority order for evaluating these operations, which is as follows:

Order of operations
1. Anything enclosed by grouping symbols is evaluated first. Grouping symbols include parentheses $\left(\ldots\right)$(), brackets $\left[\ldots\right]$[], and braces $\left\{\ldots\right\}${}.
2. Powers and roots, in order from left to right.
3. Multiplication and division, in order from left to right.
4. Addition and subtraction, in order from left to right.

#### Worked examples

##### Example 1

Evaluate the expression $5\times\left(6+6\right)$5×(6+6).

Think: According to the order of operations we should evaluate the sum $6+6$6+6 first, since it is inside grouping symbols. We can then multiply by $5$5 afterwards.

Do:

 $5\times\left(6+6\right)$5×(6+6) $=$= $5\times12$5×12 Evaluating the addition inside the parentheses first $=$= $60$60 Evaluating the multiplication next

##### example 2

Evaluate the expression $100-9\times6+18\div6$1009×6+18÷​6.

Think: This time there are no grouping symbols involved. Using the order of operations we should evaluate the multiplication and division first, from left to right. We can then evaluate the subtraction and the addition, again from left to right.

Do:

 $100-9\times6+18\div6$100−9×6+18÷​6 $=$= $100-54+18\div6$100−54+18÷​6 Evaluating the multiplication first, since it is further left $=$= $100-54+3$100−54+3 Evaluating the division next $=$= $46+3$46+3 Evaluating the subtraction, since it is further left $=$= $49$49 Evaluating the addition last

##### Example 3

Evaluate the expression $\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32

Think: Using the order of operations, we start by working inside the grouping symbols. Inside, we should perform the divisions (from left to right) before the addition. After that, we should evaluate the power, and then finally the multiplication.

Do:

 $\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32 $=$= $\left(4+35\div5\right)\times3^2$(4+35÷​5)×32 Evaluating division inside the grouping symbols first $=$= $\left(4+7\right)\times3^2$(4+7)×32 Evaluating the other division inside the grouping symbols $=$= $11\times3^2$11×32 Evaluating the last operation (addition) inside the grouping symbols $=$= $11\times9$11×9 Evaluating the power next $=$= $99$99 Evaluating the multiplication last

Reflect: In this expression it is possible to evaluate the power first, or at the same time as the division, since it doesn't affect the calculations going on inside the grouping symbols. In this way, we can often perform more than one operation at a time once we are comfortable with the order of operations:

 $\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32 $=$= $\left(4+7\right)\times9$(4+7)×9 $=$= $11\times9$11×9 $=$= $99$99

#### Practice questions

##### Question 1

Evaluate $2\times12+30-28$2×12+3028

##### Question 2

Evaluate $-18+21\div7$18+21÷​7

## Calculators

When using a calculator it is important to understand the calculation that is being performed and the order of the operations involved. In particular, it is a good idea to have a reasonable estimate about what the output of the calculation should be.

Try the following calculation on your calculator. Type it in exactly from left to right.

$6+3\times7$6+3×7

Some calculators will display $63$63, and others will display the correct answer of $27$27. This is because some calculators can correctly identify that, following the order of operations, the multiplication needs to be completed before the addition of $6$6. Other calculators will evaluate $6+3$6+3 once it is typed in, without waiting for the $\times7$×7.

When faced with a numeric calculation, you should first look at it and decide whether it is something that you can do in your head (mental computation), something that can be done quickly by hand, or if you need a calculator. This decision will be different for everyone, so you'll just have to try some questions out and see.

Remember!

The more you perform calculations by hand or in your head, the faster you will get. You don't always need to resort to a calculator!

### Using a calculator

Most high school students will have a scientific calculator that looks something like one of these: #### Exploration

Suppose that in the process of answering a question we need to calculate the product $351\times996$351×996, and decide that it will be easier to use a calculator rather than writing it out by hand.

Before putting the values into the calculator, we can quickly estimate the answer mentally by rounding. $351$351 is close to $350$350, and $996$996 is close to $1000$1000, so we can expect the answer to be roughly $350\times1000=350000$350×1000=350000.

Now we can enter the product into the calculator by pressing each button individually in the exact order:

 $\editable{3}$3 $\editable{5}$5 $\editable{1}$1 $\editable{\times}$× $\editable{9}$9 $\editable{9}$9 $\editable{6}$6 $\editable{=}$=

The display screen on the calculator shows that the answer to this calculation is $349596$349596. This is close to our earlier estimate of $350000$350000, so we can be confident that we have pressed the correct buttons and gotten the right answer.

#### Practice questions

##### Question 3

Use a calculator to evaluate $214+443$214+443.

##### Question 4

Using a calculator, evaluate $47-\left(14+\left(14\div7\right)\right)$47(14+(14÷​7))

##### Question 5

Using a calculator evaluate $2^3\times3+81$23×3+81

## Negative numbers on a calculator

Most calculators have a special button to make negative numbers, here are some examples on some common calculator types. For example, to evaluate $-351\times996$351×996 using a calculator we would press the same buttons as before, but we will also need to press a button to make the first number negative.

Some calculators require pressing the negative sign before putting in the rest of the number, like this:

 $\editable{\left(-\right)}$(−) $\editable{3}$3 $\editable{5}$5 $\editable{1}$1 $\editable{\times}$× $\editable{9}$9 $\editable{9}$9 $\editable{6}$6 $\editable{=}$=

Some calculators require pressing the negative sign after putting in the number, like this:

 $\editable{3}$3 $\editable{5}$5 $\editable{1}$1 $\editable{\left[+/-\right]}$[+/−] $\editable{\times}$× $\editable{9}$9 $\editable{9}$9 $\editable{6}$6 $\editable{=}$=

Some calculators will accept both ways.

Try this calculation on your calculator (note that the correct answer is $-349596$349596). Which way do you need to enter the negative on your calculator to get the correct answer?

#### Practice question

##### Question 6

We want to use a calculator to find the value of:

$-6.3\times9\times\left(-14.08\right)$6.3×9×(14.08)

1. Before entering the calculation in to your calculator, it is useful to have an estimation in mind. Choose the correct statement:

The product of two negatives and a positive could be positive or negative.

A

The product of two negatives and a positive is a positive.

B

The product of two negatives and a positive is a negative.

C
2. Now, use your calculator to find the value of:

$-6.3\times9\times\left(-14.08\right)$6.3×9×(14.08)

3. Which of the following expressions also have the same value as the product in the previous part? Select all of the correct answers.

$6.3\times\left(-9\right)\times14.08$6.3×(9)×14.08

A

$6.3\times9\times14.08$6.3×9×14.08

B

$-14.08\times9\times\left(-6.3\right)$14.08×9×(6.3)

C

$-6.3\times9\times14.08$6.3×9×14.08

D

Consider the expression $9989-\left(-9885\right)$9989(9885). To enter this into a calculator, some calculators need the negative number inside parentheses, while other calculators will accept the subtraction of a negative number without parentheses. Try the two methods below, and see what your calculator does. Make sure you know which method gives the correct answer (which is $19874$19874 for this example).

Method 1:

 $\editable{9}$9 $\editable{9}$9 $\editable{8}$8 $\editable{9}$9 $\editable{-}$− $\editable{\left(-\right)}$(−) $\editable{9}$9 $\editable{8}$8 $\editable{8}$8 $\editable{5}$5 $\editable{=}$=

Method 2: Some calculators (particularly older ones) require grouping symbols when negatives are involved in a calculation.

 $\editable{9}$9 $\editable{9}$9 $\editable{8}$8 $\editable{9}$9 $\editable{-}$− $\editable{(}$( $\editable{9}$9 $\editable{8}$8 $\editable{8}$8 $\editable{5}$5 $\editable{\left(-\right)}$(−) $\editable{)}$) $\editable{=}$=

Finally, remember that when using a calculator to perform calculations, it is as much about learning how to read the mathematical questions as it is learning how to use your tool effectively!

#### Practice question

##### Question 7

We want to use a calculator to find the value of:

$\frac{26-64}{-58+53}$266458+53

1. Before entering the calculation in to your calculator, it is useful to have an estimation in mind. Choose the correct statement:

The answer will be negative as we are dividing a positive number by a negative number.

A

The answer will be positive as we are dividing a negative number by a negative number.

B

The answer will be negative as we are dividing a negative number by a positive number.

C

The answer will be positive as we are dividing a positive number by a negative number.

D
2. Now, use your calculator to find the value of $\frac{26-64}{-58+53}$266458+53, giving your answer as an exact decimal.

3. Which two of the following expressions have the same value?

$\frac{26+64}{-58-53}$26+645853

A

$\frac{64-26}{58-53}$64265853

B

$\frac{64-26}{53-58}$64265358

C

$\frac{-26+64}{-53+58}$26+6453+58

D

### Outcomes

#### 1.1.6

choose and use addition, subtraction, multiplication and division, or combinations of these operations, to solve practical problems

#### 1.1.7

apply arithmetic operations according to their correct order

#### 1.1.12

use a calculator appropriately and efficiently for multi-step calculations