How to Create a Simple Formula

Back to Maths Grade 9/10

Introduction

In the last two tutorials we learned what equations and formulas are, and why they're useful. In particular, we learned that formulas are useful for describing patterns, and in this tutorial we'll see how to do this with an example.

Childcare Example

You're a new carer at a childcare centre that is open from Monday to Friday. The number of children who attend the childcare centre varies from day to day, and week to week.

Every morning and afternoon the children are offered apples and biscuits to eat as a snack. These snacks are bought every Sunday for the following week. You have been given the task of purchasing the snacks for next week.

 
During your time at the centre, you've noticed that the other carers buy either too much or too little, in part because of the fluctuating attendance. To help decide how much to buy, you write down how many children attended each day this week, as well as the number of apples and biscuits they consumed collectively.

Day Number of Children Number of apples consumed Number of biscuits consumed
Monday
15
33
26.5
Tuesday
14
31
25
Wednesday
7
17
14.5
Thursday
20
43
34
Friday
23
49
38.5



In order to use this information to predict how many apples# to buy for next week you need to create a simple formula.

#NB: The rest of this tutorial will only concern apple purchase. Once you've read through and understood each step, you can consolidate your understanding by repeating the steps to work out how many biscuits to buy. As always, our answer (and working out) is provided for you to check against.


Step 1: Write out your basic equation and allocate variables

The basic equation we'll be using in this algebra series is:

y = mx + c

 
For the purpose of this example, we'll assign the variables in this way:
~ y is the number of apples consumed;
~ x is the number of children attending##; and
~ m and c are apparent or unknown variables that complete the equation.###  

##  It doesn't matter which way around you allocate the x and y variables.
###  You'll notice that once the equation is complete, there will be a swap, and m and c will become bound or known variables, and y and x will become the apparent or unknown variables.


Step 2: Solve equations for apparent (unknown) variables

We need to work out what m and c are; this is called solving the equation for m and c. There are three simple parts to this step.

Part 1: Substitute known variables into equation
Substituting the information you recorded about this week's attendance and apple consumption, we can make the following two equations:
 
Equation 1: From Monday's data:       33 = 15 m + c
Equation 2: From Tuesday's data:       31 = 14 m + c

We've used two days' data to create two equations; this is because we have two unknown variables (m and c). The number of unknown variables dictates how many equations you need to solve them.

Part 2: Solve for each unknown variable
You can only solve for ONE unknown variable at a time. To solve for a variable, you need to create an equation that contains only that one unknown variable. That is, to solve for c, you need to create an equation that only contains c, and not m.

How do you do this?
You created two equations above for this very purpose. First you need to rearrange each equation so that it has the same one variable on one side by itself (the variable you don't want to solve for).

It doesn't matter which variable you choose to solve for first, but for simplicity's sake we'll choose m. This means we need to rearrange equations 1 and 2 so that they both have c on one side by itself.

Equation 1 Equation 2
33 = 15 m + c 31 = 14 m + c
33 - 15 m = c 31 - 14 m = c
           
Flipped over, these equations look like
c = 33 - 15 m c = 31 - 14 m

 
Now that we have two equations describing what c equals, we can combine them to solve for m.

c = c
Equation 1 = Equation 2
33 - 15 m = 31 - 14 m
33 - 31 - 15 m = -14 m
33 - 31 = -14 m + 15 m
2 = m
or rearranged:
m = 2

 
Now we can solve for c by substituting m = 2 into just using one equation. Either one is fine; we've used Equation 1 below.

33 = 15 m + c
33 = (15 x 2) + c
33 = 30 + c
33 - 30 = c
3 = c
or rearranged:
c = 3

 
So, our equation is:

y = 2 x + 3

Remembering that:
~ y is the number of apples consumed, and
~ x is the number of children attending the centre.


Part 3: Check your formula
To confirm this equation is accurate, we use it and Tuesday through to Friday's attendance data to see if the equation can accurately tell us how many apples were consumed from Tuesday through to Friday.

Tuesday Wednesday Thursday Friday
y = 2 x + 3 y = 2 x + 3 y = 2 x + 3 y = 2 x + 3
y = (2 x 14) + 3 y = (2 x 7) + 3 y = (2 x 20) + 3 y = (2 x 23) + 3
y = 28 + 3 y = 14 + 3 y = 40 + 3 y = 46 + 3
y = 31 apples y = 17 apples y = 43 apples y = 49 apples

 
Looking back at the apple consumption data you recorded for this last week, we can confirm that our equation gives accurate predictions.

Step 3: Work out how many apples you need for next week

You check next week's expected attendance numbers; they are as follows:

Day
Number of children
Monday
16
Tuesday
12
Wednesday
5
Thursday
23
Friday
20



Using simple arithmetic, you work out that the total number of children expected to attend the centre next week is seventy-six. Let's put this information into your equation.

y = 2 x + 3
y = (2 x 76) + 3
y = 152 + 3
y = 155 apples

So in summary, you will purchase 155 apples for next week.

Once you've read through and understood each step above, you can consolidate your understanding by repeating the steps to work out how many biscuits to buy. As always, our answer (and working out) is provided for you to check against, in the next tutorial.