Basic Workplace Numeracy Skills – Part 2

Getting to Grips With Multistage Math Problems

For many of us, math sometimes seems abstract and difficult. Strings of numbers and symbols dance across the screen and don't seem to mean much in the "real world."

But, in fact, you can use math to solve a wide range of practical workplace problems – calculating payments, working out measurements, and allocating resources, for example. And, once you understand the basic methods and principles, it's really not as hard as it looks!

This article builds on Basic Workplace Numeracy Skills – Part 1, introducing more complex, multistage calculations, and ways to master them quickly.

How to Apply Math to Workplace Problems

Many workplace math problems may not, at first, look like math problems. To solve them, you must first understand the nature of the problem, identify the data that you're working with, and express the problem using numbers and mathematical operations.

To do this, you can use the following four-step framework:

1. Understand the Problem

First, define the problem that you want to solve, and the outcome that you want to achieve, making a note of all the relevant facts and figures.

For example, say that you want to find out how much fuel you'll use on a long car journey. To define the problem, try writing out each part of it as a sentence. For instance, "I drove 235 miles and used 12 gallons of fuel. How many miles did I drive per gallon? And how much will it cost to drive 586 miles if fuel is about $2.50 per gallon?"

This may seem complicated when you think in terms of gallons, miles and dollars. But you can simplify the problem by leaving out the units and turning the calculation into a series of straightforward numbers, multiplications and divisions (as shown in the worked example, below).

Some problems become less challenging when you visualize them. For example, if you need to calculate the amount of carpet to buy for an irregularly shaped room, sketching a plan of the room and adding the dimensions will make the job much easier.

2. Work Out Which Operations You Need to Perform

Operations are the actions that you perform on numbers to arrive at the answers: addition, subtraction, multiplication, and division. Problems become easier to understand when you know which mathematical operations you need to use to find the solution.

Look out for prompts in what you're being asked to do that could direct you toward the operations that you need to use. The table below shows some key examples.

Operation

Key Words

Addition (+)

Add, sum, total, plus, in all, together, altogether, in all, combined.

Subtraction (−)

Difference, less than, minus, take away, fewer, left over, exceed, remain, lost.

Multiplication (×)

By, of, double, treble, times, factor, multiple.

Division (÷ or ∕ )

Part, percent, share, split, section.

Equals (=)

Is, are, was, were, will be, gives, yields.

For example, if you're asked to combine the total revenues from several sets of sales figures, you'll be dealing with an addition problem. Or, if you're asked to calculate sales of one product as a percentage of total sales, the answer will likely involve division and multiplication. (Calculating percentages is covered in more detail in Part 1.)

3. Carry Out the Calculations

When you've defined your problem and identified the operations that you need to use to solve it, it's time to carry out the calculation.

Break down each step into its simplest form. It's easier to carry out five or six simple steps, each with a single operation, than to tackle a complex, multistage problem all at once.

If you wrote out each part of the problem in words, go through what you've written and strike through each of the parts as you work your way through.

4. Check Your Answer

When you're done, check your answer carefully and make sure that you've covered every step. Repeat the calculation at least once to be sure that you haven't made any errors.

Using PEMDAS in Complex Calculations

Calculations with just a few numbers and a single operation are relatively easy. Sometimes, though, you'll come across problems that involve many numbers and multiple operations.

In circumstances like these, you need to know which operations to perform in which order, or you may end up with incorrect answers. Fortunately, there's a simple tool to help you get this right – PEMDAS.

PEMDAS is an acronym that stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It tells you the order in which to use operations when you want to solve more complex math calculations. Let's look at these terms in more detail.

Parentheses

If any parts of a calculation are inside brackets, or parentheses, tackle these first. Consider this example:

6 × (7 − 3) = ?

Perform the operation inside the brackets first:

7 − 3 = 4

Then, multiply your answer by the other number in the calculation, 6:

6 × 4 = 24

If you had performed the calculation simply by moving from left to right, resolving each operation in turn without the brackets, you would have done this:

6 × 7 − 3

which simplifies to:

42 − 3 = 39

In this case, this is the wrong answer.

Exponents

An exponent is a smaller digit placed to the right and slightly above a number. It shows that you need to multiply the number by itself. The value of the exponent shows how many times to use the number you're multiplying.

For example, 5² is "five squared," or five times five. ("Squares" often show up in calculations of area.) So:

5² is the same as 5 × 5, which equals 25

Similarly, 5³ is called "five cubed." ("Cubes" often appear in calculations of volume.) So:

5³ is the same as 5 × 5 × 5, which equals 125

PEMDAS tells you to work out exponents after you've completed any calculations in parentheses, but before other operations. For example, in…

4² + 7 = ?

…you start by working out 42. So:

4² is the same as 4 × 4, which equals 16

Now that you've calculated the exponent, you have the information you need to complete the calculation:

16 + 7 = 23

Multiplication and Division

When you've worked out any parentheses and exponents in a calculation, you can move on to multiplication and division. The order in which you divide and multiply terms doesn't matter, as long as you do both before you carry out any additions or subtractions. So, it's simplest to move from left to right through the calculation, carrying out each operation as it appears.

For example, in...

9 + 8 × 6 ÷ 4 = ?

...you start by calculating 8 × 6. Then, you do the division, using the result of the multiplication. So:

8 × 6 = 48

48 ÷ 4 = 12

Only then do you move on to the addition:

9 + 12 = 21

Addition and Subtraction

The final step in complex calculations is to calculate additions and subtractions. Again, you can do these in either order, so it's easiest to move from left to right through the calculation.

For example, in…

3² + 5 + 8 − 9 + 6 = ?

...you first deal with the exponent. 3², or 3 × 3, equals 9, so the sum becomes:

9 + 5 + 8 − 9 + 6 = ?

Then, you start at the left and work your way across, working out the additions and subtractions in pairs:

9 + 5 = 14

14 + 8 = 22

22 − 9 = 13

13 + 6 = 19

PEMDAS: Bringing the Elements Together

We've looked at each part of PEMDAS in turn. Now, let's bring all of the elements together in this example:

2³ − 5 × (10 − 6) ÷ 2 + (9 + 5)

Working out the sums in Parentheses (brackets) leads to:

2³ − 5 × (4) ÷ 2 + (14)

Calculating the Exponent simplifies the sum to:

8 − 5 × 4 ÷ 2 + 14

Carrying out Multiplication and Division leads to:

8 − 20 ÷ 2 + 14

8 − 10 + 14

Ending with Addition and Subtraction produces the final sum:

8 − 10 + 14 = 12