Biscuit Solution

Back to Maths Grade 9/10

Introduction

This tutorial continues on from this one. In it we're going to follow the steps previously outlined in order to work out how many biscuits you need to purchase for the childcare centre next week (check out the last tutorial if you're not sure what we're talking about).

To avoid having to flip between tutorials, below is the number of children who attended childcare last week, and the number of biscuits they consumed collectively each day.

Day Number of Children Number of biscuits consumed
Monday
15
26.5
Tuesday
14
25
Wednesday
7
14.5
Thursday
20
34
Friday
23
38.5


Step 1: Write out your basic equation and allocate variables

Our basic equation is:

y = mx + c

For the purpose of this example,
~ y is the number of biscuits 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 biscuit consumption, we can make the following two equations:

Equation 1: From Monday's data:         26.5 = 15 m + c
Equation 2: From Tuesday's data:         25 = 14 m + c

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

Part 2: Solve for each unknown variable
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
26.5 = 15 m + c 25 = 14 m + c
26.5 - 15 m = c 25 - 14 m = c
 
Flipped over, these equations look like
c = 26.5 - 15 m c = 25 - 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
26.5 - 15 m = 25 - 14 m
26.5 - 25 - 15 m = -14 m
26.5 - 25 = -14 m + 15 m
1.5 = m
 
or rearranged:
m = 1.5


Now you can solve for c, by substituting m = 1.5 into either Equation 1 or 2. It doesn't matter which equation you use. We've used Equation 1.

c = c
26.5 = 15 m + c
26.5 = (15 x 1.5) + c
26.5 = 22.5 + c
26.5 - 22.5 = c
4 = c
or rearranged:
c = 4


In summary, our final equation is:

y = 1.5 x + 4

Remembering that:
~ y is the number of biscuits 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 with the attendance data recorded from Tuesday through to Friday this week to see if the equation can accurately tell us how many biscuits were consumed.

Tuesday Wednesday Thursday Friday
y = 1.5 x + 4 y = 1.5 x + 4 y = 1.5 x + 4 y = 1.5 x + 4
y = (1.5 x 14) + 4 y = (1.5 x 7.5) + 4 y = (1.5 x 20) + 4 y = (1.5 x 23) + 4
y = 21 + 4 y = 10.5 + 4 y = 30 + 4 y = 34.5 + 4
y = 25 biscuits y = 14.5 biscuits y = 34 biscuits y = 38.5 biscuits



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

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

Once again, here are next week's expected attendance numbers:

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 = 1.5 x + 4
y = (1.5 x 76) + 4
y = 114 + 4
y = 118 biscuits

So in summary, you will purchase 118 biscuits for next week.