Learner Pack
Functional Mathematics
Level 3
Learner Pack
Activity N2: Playing darts
Functional Mathematics
Level 4
FÁS© 2012
Unit 1: Number
1
September 2012
Functional Mathematics Learner Pack
Level 4 Unit 1
Acknowledgements
Acknowledgements
This booklet is part of a pack of resources for Functional Mathematics Level 4
which FÁS commissioned for use in their training programmes. A similar set of
resources has been developed for Functional Mathematics Level 3.
A team from the National Adult Literacy Agency (NALA) and the National Centre
for Excellence in Mathematics and Science Teaching and Learning (NCEMS-TL)
developed and edited the materials.
NALA:
Bláthnaid Ní Chinnéide
Mary Gaynor
Fergus Dolan
John Stewart
Dr Terry Maguire (Institute of Technology, Tallaght)
NCEMS-TL:
Prof. John O’Donoghue
Dr. Mark Prendergast
Dr. Miriam Liston
Dr. Niamh O’Meara
FÁS:
John O’Neill
Louise MacAvin
We are grateful to Kathleen Cramer and her team in Newbridge Youth Training
and Development Centre who gave feedback on extracts from the Level 3
materials.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
Activity
Mountain climbing
N1
(Google images)
This activity links to unit learning outcomes 1.1 and 1.3.
Introduction
In this activity we will introduce the idea of estimation or rounding off by
looking at the heights of different mountain ranges around the world.
What will you learn?
Learning Outcomes
1. Round natural numbers to the closest ten, hundred or thousand.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
Key Learning Points
1. Estimating natural numbers
2. Rounding off natural numbers
3. Exploring examples of mathematics in the world around us
Materials you will need for this activity
FÁS© 2012
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing

Internet access or a printout of the following webpage:
http://en.wikipedia.org/wiki/List_of_highest_mountains

Practice Sheet N1

Solution Sheet N1
What do you need to know before you start?
Maths
When dealing with large numbers it is often easier to give an estimate value
for the number. In order to work out an accurate estimate it is important that
we have a good understanding of rounding off. We need to first understand
place value before deciding the correct value to which the number will be
rounded. Once we have rounded off the number we then have an
approximate value for our original number.
Mountain Heights
One example of where large numbers are used regularly is when discussing
the height of mountains. Mountain climbers often discuss the height of the
next mountain they intend to climb or the difference in height between one
mountain and another. Often, when discussing this they use approximate
numbers rather than the exact height of every mountain.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
Getting Started
As mentioned previously, one of the most important things to understand prior
to learning how to estimate, or approximate or round off numbers is the
concept of place value. Understanding place value will allow us to see what
each digit in a large number represents. For example, if we look at the
number 26,734 we know that we have:
2 Ten Thousands
6 Thousands
7 Hundreds
3 Tens
4 ones
Once we understand place value we can then look at the idea of rounding off.
If I was asked to round the number 26,734 to the nearest ten I would round it
to 26,730. The reason for this is because the next digit to the right, the
number of ones, is less than 5. This makes sense since the number 26,734
is closer to 26,730 than it is to 26,740.
On the other hand, if I was asked to round this number to the nearest
thousand I would round it to 27,000. This is because the next digit to the
right, the number of hundreds, is greater than 5. Again this makes sense
since 26,734 is nearer to 27,000 than it is to 26,000.
In General:
 When the digit to the right of the rounding number is greater than five,
the rounding number increases by 1.
 If the digit to the right of the rounding number is less than five, then the
rounding number remains the same.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
Worked Example 1
Climbing Irish Mountains
The High Climbers Club is a mountaineering club based in Ireland. Each year
its members climb four mountains in Ireland, one in each province. At the
beginning of 2012 they decided that their four mountains for that year would
be Carrantuohill in Co. Kerry, Lugnaquilla in Co. Wicklow, Slieve Gullion in
Co. Armagh and Croagh Patrick in Co. Mayo.
The table below gives the height for these four mountains.
Mountain
Height in metres
Carrantuohill
1 038
Lugnaquilla
926
Slieve Gullion
575
Croagh Patrick
764
a) How many thousands, hundreds, tens and ones are in each of the
heights mentioned above? Complete the following table to show your
answer.
Mountain
Height in metres: Place value
thousands
hundreds
tens
ones
Carrantuohill
Lugnaquilla
Slieve Gullion
Croagh Patrick
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Activity N1: Mountain climbing
b) What approximate height do you get when you round off the height
of Carrantuohill to the nearest ten?
c) Calculate the difference between that approximate height of
Carrantuohill and the actual height of the mountain.
d) Work out an approximate figure, to the nearest hundred, for the total
number of metres the club members planned to climb in 2012.
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Level 4 Unit 1
Activity N1: Mountain climbing
Solution:
a)
Mountain
Height in metres: Place value
thousands
hundreds
tens
ones
Carrantuohill
1
0
3
8
Lugnaquilla
0
9
2
6
Slieve Gullion
0
5
7
5
Croagh Patrick
0
7
6
4
b) We know the height of Carrantuohill is 1,038 metres. In order to
round this height to the nearest ten we must check the value of
the digit to the right of the tens place: that is, the ones.
Since 8 > 5 we know we “round up”.
So if we round off the height of Carrantuohill to the nearest ten the
height would be 1,040 metres.
Note: We can say ‘round off’ or ‘approximate’ to the nearest ten.
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Level 4 Unit 1
Activity N1: Mountain climbing
c) In order to calculate the difference we must take the actual
height from the approximate height.
1040
- 1038
2
So there is only two metres in the difference between the approximate
height and the actual height of Carrantuohill.
d) We can calculate this figure in two different ways.
Option 1: Round each number to the nearest hundred and add
them:
Carrantuohill: 1,038 = 1,000, since 3 < 5
Lugnaquilla: 926 = 900, since 2 < 5
Slieve Gullion: 575 = 600, since 7 > 5
Croagh Patrick: 764 = 800, since 6 > 5
Total metres climbed to the nearest hundred: 1,000 + 900 + 600 + 800
Total metres climbed to the nearest hundred = 3,300 metres.
Option 2: Add up the actual heights of the four mountains and
round off the answer to the nearest hundred:
Total metres climbed = 1038 + 926 + 575 + 764 = 3,303
Now we must round this figure off to the nearest hundred:
3,303 = 3,300 (since 0 < 5)
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
Task 1
The High Climbers Club has contact with another climbing club based in
Central Europe , called On Top of the World. The members of On Top of the
World climb three mountains per year. These mountains are Mount Brocken
in Germany which is 1,142 metres, Torre de Cerrado in Spain which is 2, 648
metres and Mount Ortler in Italy which is 3,902 metres.
Mountain
Height in metres
Mount Brocken
1,142
Torre de Cerrado
2, 648
Mount Ortler
3,902
a) How many thousands, hundreds, tens and ones are in each of the
measurements mentioned above. Complete the table to show your
answers.
Mountain
Height in metres: Place value
thousands
hundreds
tens
ones
Mount Brocken
Torre de Cerrado
Mount Ortler
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
b) The president of On Top of the World club contacted the High Climbers
club to discuss doing a joint climb. He gave approximate
measurements for each of the three mountains that they climb.
He rounded the height of the Spanish mountain to the nearest ten.
He rounded the height of the Italian mountain to the nearest hundred.
He rounded the height of the German mountain to the nearest
thousand.
What height did the president of On Top of the World give for
each mountain? Insert your answers in the following table.
Mountain
Height in metres rounded off
Mount Brocken in Germany
Torre de Cerrado in Spain
Mount Ortler in Italy
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Level 4 Unit 1
Activity N1: Mountain climbing
c) Mount Brocken is the highest peak in Germany and Carrantuohill is the
highest peak in Ireland. Calculate the difference between these two
peaks, to the nearest ten.
d) In 2013 the members of the On Top of the World club will climb their
own usual three mountains as well as climbing Croagh Patrick with
The High Climbers. Calculate to the nearest hundred the total
number of metres that members of the On Top of the World club will
climb in 2013.
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Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing
Task 2
Use the internet to find the heights of the five highest mountains in the world.
Then round the height of each mountain off to the nearest ten and to the
nearest hundred. Use that information to complete the table below.
Mountain Name
Actual Height
Height to the
Height to the
nearest ten
nearest hundred
Practise your skills
FÁS© 2012
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N1: Mountain climbing

Use Practice Sheet N1.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Activity
Calculations
N2
(Google images)
This activity links to unit learning outcomes 1.1 and 1.4.
Introduction
Scientific calculators are used in any situation where we need quick access to
certain mathematical functions. They are very useful for performing difficult
calculations. They can compute square roots, fractions, indices and many
more operations, as well as basic operations such as addition, subtraction,
multiplication and division.
What will you learn?
Learning Outcomes
1. Use a calculator with confidence to perform extended calculations,
requiring functions such as addition, subtraction, multiplication,
division, percent, square-root, pi, 1/x, scientific notation keys, memory
keys and the clear key, while following the conventions of precedence
of operations.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
Key Learning Points
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
1. Performing addition, subtraction, multiplication and division operations
on a calculator
2. Using the calculator to solve problems requiring operations such as
percent, square-root, memory keys and the clear key
3. Becoming familiar with and confidently using specific keys on the
calculator such as the power key, pi, 1/x, scientific notation keys as
well as the use of the second function key
4. Exploring example of mathematics in everyday life
5. Recognising the relevance and usefulness of mathematics in everyday
life
Materials you will need for this activity

Practice Sheet N2

Solution Sheet N2

Scientific calculator
What do you need to know before you start?
You should be familiar with the material from Functional Mathematics Level 3,
particularly working with the different number systems. It will be important to
know how to convert percentages and fractions to decimals and vice versa.
 (Pi) is a mathematical symbol. This symbol is a letter from the Greek
alphabet. In mathematics it represents the fraction
which is the decimal
3.14. There is a  button on the scientific calculator.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Getting started
In the previous activity you learned about rounding off whole numbers.
In general:
When the digit in the next place, to the right of the rounding digit, is
greater than 5, increase the rounding digit by 1.
When the digit in the next place, to the right of the rounding digit, is less
than 5, don’t change the rounding digit.
You can round off decimal numbers in the same way that you round off
whole numbers. We round off decimals to a particular number of decimal
places, depending on how precise we want to be.
For example, we can round off to 2 decimal places or to 3 decimal places.
Another way of saying ‘Round off to 2 decimal places’ , is ‘Give your answer
correct to 2 decimal places’.
If you want your answer correct to 2 decimal places, you look at the digit to
the right of the second decimal place.
If this digit is 5 or bigger you increase the previous digit.
However, if the digit to the right of the second decimal place is less than 5,
you don’t change the previous digit.
When dealing with money, we always give our answer correct to 2 decimal
places.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Worked Example 1
Percentages and fractions
Two sports shops, Lifestyle Sports and Champion Sports, both stock the
latest pair of Puma King football boots. The retail price of the boots is €110.
(Google Images)
Lifestyle Sports is offering a 12% discount. Champion Sports is offering
off the price.
a) How much will the boots cost at Lifestyle Sports?
b) How much will the boots cost at Champion Sports?
c) Which shop would you buy the boots in?
Try this example before looking at the solution.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Solution
a) How much will the boots cost at Lifestyle Sports?
Lifestyle Sports are offering 12% off the boots which cost €110.
12% = 0.12
12 % of 110:
0.12 x 110 = 13.20. Use a calculator.
Therefore Lifestyle Sports are offering €13.20 off the price of the boots.
This means that at Lifestyle Sports the boots will cost: 110 – 13.20 =
€96.80. Use a calculator.
b) How much will the boots cost at Champion Sports?
Champion Sports are offering
off the boots which cost €110.
0.14 x 110 = 15.71. Use a calculator.
Therefore Champion Sports are offering €15.71 off the boots.
This means that at Champion Sports the boots will cost: 110 – 15.71 =
€94.29. Use a calculator.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
c) Which shop would you buy the boots in?
What did you decide? In this case, the same boots cost €96.80 in
Lifestyle Sports and €94.29 in Champion Sports, so Champion Sports
is a little cheaper. However, price is just one thing we think of when
deciding which shop to buy in. What other factors can you think of?
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Task 1
Danny and his brother share a flat. They divide all the bills evenly between
them. This month’s gas bill is €76.50 and the electricity bill is €64.20.
(Google Images)
a) How much does each brother have to pay?
b) Danny pays three quarters of the gas bill and 30% of the electricity bill.
How much does Danny pay for the gas bill and for the electricity bill?
c) How much does Danny’s brother owe him to make sure the total was
divided evenly between the two?
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Worked Example 2
Multiplication and division
You are carpeting your living room, which is a rectangular room. The length
of your room is 6.35m and the width is 4.25m.
Use your calculator to try and work out the following:
a) How much carpet will you need to buy?
b) The carpet costs €16.42 per m². How much will the carpet cost?
Try this example before looking at the solution.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Solution
a) The area of the floor in your living room is (length) x (width)
Area = 6.35 x 4.25
In your calculator press 6.35 x 4.25 =
This will give you an answer of 26.9875m².
Therefore you need 26.9875m² of carpet.
b) You need 26.9875m² of carpet and each m² costs €16.42.
In your calculator press 29.9875 x 16.42 =
This will give you an answer of 443.13475. Therefore it will cost you
€443.13 to carpet your room.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Worked Example 3
Square Roots
You also wish to replace the tiles on your bathroom floor. Your bathroom is a
square room and the area of its floor is 6.25 m².
(Google Images)
What is the length of the bathroom? Try this example before looking at the
solution.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Solution
The area of the floor in the bathroom is (length) x (width).
As it is a square room the length is the same as the width:
So the Area = (length) ².
We know that the area is 6.25m².
6.25m² = (length) ²
So, to find the length we need to find the square root of both sides.
In your calculator press
then press 6.25 =
This will give you an answer of 2.5.
Therefore the length, or width, of your bathroom is 2.5m.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Task 2
You wish to replace the flooring in your room. The room is square and the
area of its floor is 7.29 m². What is the width of the room?
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Worked Example 4
Pi
Your group is doing a gardening project. For your project, the group has been
asked to use a garden that is attached to the centre. You have decided to put
a circular pond in the garden. You can buy plastic fittings to make the outline
of the circular pond. The plastic fittings come in various diameter sizes.
(Google Images)
To make the pond you need to dig a hole in the garden. First, you need to
decide what size you want the pond to be, and what area of ground you
should dig for the pond.
If you want a pond which is 8m in diameter, what should be the ground
surface area of the hole you have to dig?
Give your answer correct to 2 decimal places.
Try this example before looking at the solution.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Solution
The area of a circle is .  r2 (r = radius)
The radius of a circle is half its diameter.
Therefore because the diameter is 8m, the radius is 4m.
Using your calculator:
If you have a Sharp calculator then you use the button
y x to work out powers.
If you have a Casio calculator then you use the button x  to work out powers.
In your calculator press 4 ‘power button’ 2 =
This will give you an answer of 16.
Now you have: A =  16
In your calculator press:  x 16
This will give you an answer of 50.26548.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N2: Calculations
Therefore the area of the circular pond, with a radius of 4m, is 50.27m².
Task 3
In a certain town, some students travel to and from school on the school bus.
The students who live inside a 3 mile radius of the school do not have to pay
a fee for the school bus.
To the nearest square mile, what is the area of the region in which students
do not pay a fee for the school bus?
Give your answer correct to 2 decimal places.
Practise your skills

Use Practice Sheet N2.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Activity
The solar system
N3
(Google images)
This activity links to unit learning outcomes 1.1 and 1.2.
Introduction
When we look at the distance between the planets in the solar system the numbers
we are dealing with are very large. For example, the distance from the Earth to
Mars is approximately 78,300,000 kilometres. Such big figures often make it very
hard to work with distances. This activity will help you to rewrite very large numbers
in order to make them easier to work with.
What will you learn?
Learning Outcomes
1. Convert from scientific notation to standard form and from standard form to
scientific notation.
2. Explain how mathematics can be used to enable an individual to function
more effectively as a person and as a citizen.
Key Learning Points
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
1. Understanding the concept or idea of scientific notation
2. Converting from standard form to scientific notation
3. Converting from scientific notation to standard form
4. Exploring examples of mathematics in everyday life
5. Recognising the importance of mathematics in the world around us
Materials you will need for this activity

Practice Sheet N3

Solution Sheet N3
What do you need to know before you start?
Scientific notation allows us to handle very large or very small numbers easily.
First we need to have a good understanding of decimals. (See the decimals
activities in the Level 3 Learner Pack).
Getting Started
Indices
4 × 4× 4 × 4 × 4 can also be written as 45.
We read this as 4 to the power of 5, as there are 5 fours being multiplied together.
5 is the power or the Index of 4. Indices are the plural of Index.
Other Examples:
3 × 3 = 32
We call this 3 to the power of 2, or more commonly, 3 squared.
2 × 2 × 2= 23
We call this 2 to the power of 3, or more commonly, 2 cubed.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Scientific Notation
In order to write a number in scientific notation we write it as a x 10n where a is
between 1 and 10 (1 ≤ a < 10) and n is an integer.
For Example:
Mars was 78,300,000 kilometres from Earth.
In scientific notation 78,300,000 = 7.83 x 107.
This is because we have to multiply 7.83 by 10 seven times in order to get
78,300,000.
107 means ‘ten to the power of seven’.
Notice that the power to which ten is raised, seven, is equal to the number of
places we have shifted the decimal point in order to show the number in its
longer form:
7.83 x 107 = 78,300,000

We can also write very small numbers using scientific notation.
For Example:
We can express the weight of an object in kilograms or pounds. We know that one
gram is equal to 0.0022 of a pound. We can also write this very small number as
2.2 x 10-3.
Again, the power to which ten is raised shows the number of places we must shift
the decimal point in order to get back to the original number. However, this time
the decimal point is moving in the opposite direction.
Remember:
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Activity N3: The solar system
When the power to which ten is raised is a positive number, the decimal point
moves to the right.
When the power to which ten is raised is a negative number, the decimal point
moves to the left.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Worked Example-Real Life Situation where notations are used
The following table gives the distances between the Earth and other planets in our
solar system:
Planet
Distance to Earth in
Kilometres
Saturn
1.321 x 108
Mercury
7.7 x 107
Pluto
5.913 x 109
Write the distance in words and standard form between the following:
a) Earth and Saturn
b) Earth and Mercury.
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Level 4 Unit 1
Activity N3: The solar system
Solution
a) In scientific notation the distance between Earth and Saturn is 1.321 x 108.
To rewrite this number in standard form we must move the decimal point 8
places to the right.
Therefore, the distance between Earth and Saturn is 132,100,000 kilometres
in standard form.
In words the distance is one hundred and thirty two million, one hundred
thousand kilometres.
b) The distance between Mercury and Earth is written as 7.7 x 107.
This tells us that the decimal point must move 7 places to the right.
7.7 x 107 = 77,000,000 kilometres.
So the distance between Earth and Mercury is 77,000,000 kilometres in
standard form.
In words the distance is seventy seven million kilometres.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Task 1
Planet
Distance to Earth in Kilometres
Saturn
1.321 x 108
Mercury
7.7 x 107
Pluto
5.913 x 109
Using the information from the table, write the distance between Pluto and Earth:
In standard form
_________________________________
In words
_________________________________
_______________________________________________________
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Task 2
The average distance between the Sun and the Earth is 92,900,000 kilometres. The
approximate distance between the Sun and the Moon is 150,000,000 kilometres.
(Google images)
a) The average distance between the Sun and the Earth is 92,900,000
kilometres. Write that distance below using words and scientific notation.
In words this distance is _________________________________________
______________________________________________________.
In scientific notation this distance is ________________________________.
b) The approximate distance between the Sun and the Moon is 150,000,000
kilometres. Write that distance below using words and scientific notation.
In words this distance is _________________________________________
______________________________________________________.
In scientific notation this distance is ________________________________.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N3: The solar system
Task 3
a) The distance between the Earth and the Moon is 384,400 kilometres.
In words this is _________________________________________
______________________________________________________.
In scientific notation this is ________________________________.
b) The distance between the Moon and Mars is 42,100,000 kilometres.
In words this is _________________________________________
______________________________________________________.
In scientific notation this is ________________________________.
Practise your skills

Use Practice Sheet N3.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Activity
Cost of construction
N4
(Google images)
This activity links to unit learning outcome 1.3.
Introduction
During this activity, we will learn about percentage error by looking at
estimates that have been made for different construction projects in Ireland in
the past and comparing those to the actual cost.
What will you learn?
Learning Outcomes
1. Use appropriate strategies including percentage error to compute the
differences between approximations and actual figures.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Key Learning Points
1. Adding natural numbers
2. Understanding the concept of percentage error
3. Calculating percentage error
4. Recognising the relevance and usefulness or mathematics in everyday
life
5. Exploring the use of mathematics in a range of contexts
Materials you will need for this activity

Different items of food or clothing in order to estimate the price

Practice Sheet N4

Solution Sheet N4
What do you need to know before you start?
Maths
Percentage error is the difference between an estimated value and the
accurate value expressed as a percentage.
In Activity N1 we practised estimating the difference between heights of
different mountains. Of course, the estimated height of a mountain would be
different to its actual height. We can represent this difference between the
estimate and the actual height as a percentage. In order to write the
difference as a percentage we would need to calculate the percentage error.
FÁS© 2012
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Construction Projects
Before the Government would commission a company to begin a new
construction project, they would first estimate the total cost of the project.
It is very rare that the actual cost of the project matches the original estimate .
Sometimes governments under – estimate the cost and sometimes they
over – estimate the cost.
When they publish a report on the finished construction project, they often
have to highlight the difference between their original estimate and the
actual cost and they must calculate the percentage error between the two
figures.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Getting Started
Percentage error is the difference between an estimated value and the
accurate value expressed as a percentage.
To see how to calculate percentage error we will first look at a real life
example, from the television show ‘The Apprentice’ in 2010.
One of the teams - Team Elev8 - asked people to guess how many nails
were in a jar. The first person guessed that there were 456 nails in the jar. In
fact, there were 535 nails in the jar.
How close was the person’s guess (or estimate) to the accurate value?
To find out, we subtract 456, the estimated value, from 535, the accurate
value:
535 – 456 = 79
So we know the first person’s estimate was 79 off the correct answer.
To express this as a percentage we calculate:
That gives us the margin of error in fraction form:
To convert a fraction to a percentage we must always multiply by
%.
= 14.77%
So in this example the percentage error is 14.77% :
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Worked Example 1
Aviva Stadium
The then Taoiseach Brian Cowen, TD, officially opened the Aviva Stadium in
Dublin on the 14th May 2010. The construction project was part funded by the
Government and they originally estimated the cost to be €410 million. When
they calculated the actual cost of building the stadium they found that the
Government had over – estimated: the project had cost less than the
Government’s estimate of €410 million: it actually cost €401 million.
Calculate the percentage error, to the nearest hundredth, between the
Government’s estimate and the actual cost of the project.
Solution:
Actual Price = 401,000,000 (401 million)
Estimated Price = 410,000,000 (410 million)
Difference = 410,000,000 – 401,000,000 = 9,000,000
Percentage error =
= 2.244389027%
Therefore, the government believed this project would cost 2.24% more than
it actually did.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Worked Example 2
(Google images)
In 2003, the Government decided to upgrade (make improvements to) the
M50 motorway in Dublin. At the time the National Roads Authority (NRA)
estimated that the total cost of the upgrade would be €318 million.
In 2005, the NRA changed this estimate. They said they now believed the
cost of the project would be €580 million, due to the construction of two
interchanges that they had not originally planned for.
However, in 2010 it was reported that the actual cost of the upgrade would
be €1 billion, which is €1,000 million.
a) Calculate the percentage error between the NRA’s estimate in 2003
and the actual figure.
b) Calculate the percentage error between the NRA’s estimate in 2005
and the actual figure.
Solution
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
a) Calculate the percentage error between the NRA’s estimate in
2003 and the actual figure.
Actual price = 1,000,000,000 (1 billion)
Estimated Price = 318,000,000 (318 million)
Difference = 1,000,000,000 – 318,000,000 = 682,000,000
Percentage error =
= 68.2%
Therefore, in 2003 the NRA estimated that this project would cost 68.2% less
than it actually did.
b) Calculate the percentage error between the NRA’s estimate in
2005 and the actual figure.
Actual price = 1,000,000,000 (1 billion)
Estimated Price = 580,000,000 (318 million)
Difference = 1,000,000,000 – 580,000,000 = 420,000,000
Percentage error =
= 42%
Therefore, in 2005 the NRA estimated that this project would cost 42% less
than it actually did.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Task 1
Underestimating the cost of construction is not confined to Ireland. Even
bigger errors of judgement have been made around the world as the following
examples will show.
a) The Olympic Stadium in London was built for the Olympic Games in
2012. It cost £496 million to complete.
(Google images)
This final cost was more than the original estimate of the London
Organising Committee for the Olympic Games 2012. They had
estimated the cost would be £280 million.
Calculate the percentage error between the actual cost of the
stadium and the original figure estimated by the Olympic Committee.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
b) Construction of the Sydney Opera House began in June 1959. At that
time the Australian government estimated the total cost to be $7
million.
(Google images)
By the time it was finished in January 1973, the total actual cost of
building the Opera House had amounted to $102 million. Calculate
the percentage error between the estimated cost and the actual cost.
c) The Big Dig project in Boston Massachusetts was the most expensive
highway project ever completed in the US. It involved the construction
of a tunnel through the centre of Boston. The original estimate for the
total cost of the tunnel was $2.8 billion ($2,800 million). The actual
cost when it was finished in 2006 was $14.6 billion ($14,600 million).
Calculate the percentage error for this project.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
d) Calculate the total actual cost for all three international projects
mentioned above.
Then calculate the total of the estimates made by the three different
authorities.
Finally, calculate the percentage error between the total actual cost
and total estimated cost for all three projects combined.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Task 2
Estimation Game
Each member of the group brings in a number of items: for example,
food, or items of clothing, or a book or game or ornament. Everyone must
know the price of the items they bring in. This game has a number of
rounds in it. The number of rounds will depend on how many items players
bring in. Before starting the game, appoint a scorekeeper.
Round 1:
Player A shows the item they brought in and asks everyone else in the group
to estimate how much the item cost. Everyone should write down their
estimate. When everyone has written down their estimate Player A tells them
the actual cost of the item.
Everyone then compares the actual cost with their own estimate and must
work out their percentage error.
The person with the estimate closest to the actual cost scores one point.
Round 2 and all other rounds
The game then moves on to the next player, who shows the item they brought
in. Everyone again guesses or estimates how much the item cost and writes
that down. Again, the person with the estimate closest to the actual cost
scores one point. Continue the rounds like that until everybody has shown
the item they brought in. In each round, the person whose estimate is closest
to the actual cost scores one point. The game ends when all the items have
been shown and estimated. The winner is the person with the most points at
the end of the game.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N4: Cost of construction
Practise your skills

Use Practice Sheet N4.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Activity
Interest
N5
(Google images)
This activity links to unit learning outcome 1.1 and 1.5.
Introduction
Whenever you borrow money from the bank it costs you money. This cost is
called interest. It also works the other way around for savings. When you
open a savings account you get paid interest. Interest is the money you are
charged for borrowing money or the money that you are paid for saving
money.
What will you learn?
Learning Outcomes
1. Understand the concept of simple interest.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Key Learning Points
1. Understanding the concept of simple interest
2. Calculating simple interest using the appropriate formulae
3. Applying knowledge to real life questions including savings and credit
options
4. Recognising the importance of mathematics in the world around us
5. Exploring the use of mathematics in a range of contexts
Materials you will need for this activity

Practice Sheet N5

Solution Sheet N5
What do you need to know before you start?
If you wish to open a savings account with a bank you have a number of
choices. Different banks will offer different rates and types of interest. This
interest is added to your savings at the end of the year. You would need to
find out which bank is offering the most amount of money for your savings.
If you need to borrow money from the bank you must pay interest to the
bank. This time you would need to know which bank is charging the least
amount of interest, so that you pay as little extra money as possible.
A sum of money that you borrow or save or invest is called a principal
amount. We calculate how much interest is earned or paid based on that
principal amount. Interest is a percentage of the principal. This
percentage is called a rate of interest.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Getting Started
Simple interest is calculated once a year on the initial amount that you invest
in a savings account or that you borrow.
We can calculate simple interest manually each year or by using the following
formula:
I
R
PT
100
I is the amount of simple interest earned.
R is the rate of interest.
P is the principal amount.
T is the number of years.
See the following example for both these ways of calculating simple interest.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Worked Example 1
You open a savings account and decide to save in it for three years. You
lodge €500 into the account. The rate of simple interest is 4%.
How much interest will you have earned at the end of the three years?
Solution
You are saving €500 @ 4% simple interest for three years.
(4% is 0.04)
0.04 x 500 = €20
At the end of the first year you will earn €20 interest on your principal of
€500.
For the following years you will continue to earn €20 interest on your
principal amount of €500. Any interest you earned and kept in your savings
account will not earn any additional interest.
Therefore, at the end of the three years you will have earned €60.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
We can also use the formula to calculate the simple interest earned in the
example mentioned above.
I = simple interest
R = 4%
P = 500
T=3
I = 0.04 x 1500
I = 60
I represents the amount of simple interest. I = 60, which means that your
savings earned €60 in simple interest over the three year period.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Task 1
How much interest would have to pay the bank when you borrow €2,500 at
3% simple interest over five years?
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Worked Example 2
When you borrow money the total money you pay back is called the amount.
The amount is equal to the principal plus the interest.
This is also the case when you save money: the amount you have at the end
of the period is the principal plus the interest.
(Google Images)
A shop in your town has an offer on plasma TVs. The TVs can be bought for
€900 cash or 12 monthly payments of €89. Calculate how much interest is
charged if you buy the TV using the 12 monthly payments?
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Solution
Amount = Principal + Interest
Total amount paid is €89 x 12 repayments
= €1,068
Principal is the initial amount or the cost of the TV
= € 900
Amount = Principal + Interest
€1,068 = €900 + I
€1,068 - €900 = €900 - €900 + I
€168 = I
Therefore the total interest paid would be €168.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Task 2
A dish washer costing €470 can be bought with 24 monthly repayments of
€22.50.
(Google Images)
How much interest does the shop charge for purchasing the dish washer
using the 24 repayments?
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Worked Example 3
Sofaworld.ie normally charges 12.5% interest for credit. This shop is currently
advertising a one year interest - free repayment plan.
Let’s say you decide to buy a new three piece suite of furniture.
The cost of the suite is €1,440. You will pay for the furniture using 12
monthly repayments meaning that you will have paid for the sofa by the end
of one year and will not have to pay interest.
a) How much will each repayment need to be?
a) How much have you saved on interest by buying while the shop was
offering interest- free shopping for 12 months?
Solution
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
a) You must pay back €1,440 in 12 equal repayments.
€1,440 ÷ 12 = €120
b) I = simple interest
R = 12.5%
P = 1,440
I
12.5
 1,440  1
100
T=1
I  0.125  1,440
I  180
Therefore you have saved €180 by buying while the one year interest free
shopping was offered
Task 3
Diarmuid is buying a new car. It costs €9,000 and the garage will give him
€5,000 for his old car. Diarmuid decides to borrow the remaining amount from
the bank.
How much does Diarmuid need to borrow?
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N5: Interest
Task 3
Diarmuid is buying a new car. It costs €9,000 and the garage will give him
€5,000 for his old car. Diarmuid needs to borrow the remaining amount from
the bank.
a) How much does Diarmuid need to borrow?
b) Diarmuid takes out this loan for 5 years with simple interest and he
pays the bank back a total of €4,800.
How much interest did Diarmuid pay back?
c) What was the rate of simple interest that the bank charged Diarmuid?
Practise your skills

Use Practice Sheet N5.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Activity
Banking options
N6
(Google Images)
This activity links to unit learning outcomes 1.1 and 1.5.
Introduction
Compound interest differs from simple interest. When you save money at
compound interest rates you can earn interest on your interest. The
interest earned at the end of each year is added to the principal amount and
this is reinvested as a new lump sum the next year.
What will you learn?
Learning Outcomes
1. Understand the concept of compound interest.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Key Learning Points
1. Understanding the concept of compound interest
2. Calculating compound interest using the appropriate formulae
3. Differentiating between both types of interest
4. Applying knowledge to real life questions including savings and credit
options
5. Recognising the relevance and use of mathematics in everyday life
Materials you will need for this activity

Practice Sheet N6

Solution Sheet N6
What do you need to know before you start?
Compound interest is calculated once a year on the principal plus any interest
which may have already been earned.
Like simple interest, we pay compound interest on money we borrow and we
earn compound interest on money we save.
The interest rates quoted on investments, credit cards, loans, store credit and
overdrafts are always compound interest unless otherwise stated. When we
hear people on TV or radio talking about interest rates they are talking about
compound interest rates.
Compound interest can be calculated annually, that is, once a year, or more
often.
Per annum (p.a.) means ‘for each year’.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Worked Example 1 - Calculating compound interest annually
You want to invest €6,000 for 3 years at 4.5% per annum (p.a.) compound
interest.
At the end of the first year your €6,000 will earn €270 interest. This interest is
added to the principal and reinvested for year two. Therefore, in year two the
interest earned will be based on your new principal of €6,270.
Complete the following table:
Year
Principal at the
Interest earned
start of the year
1
€6,000
2
€6,270
Amount at the
end of the year
€270
€6,270
3
Note: Use the method of simple interest to calculate the interest earned each
year.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Solution
Year
Principal at the
Interest earned
start of the year
Amount at the
end of the year
1
€6,000
€270
€6,270
2
€6,270
€282.15
€6,552.15
3
€6,552.15
€294.85
€6,847
Note: Use the method of simple interest to calculate the interest earned each
year.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Worked Example 2
Calculating compound interest more often than annually
In reality banks calculate compound interest more than once a year.
It can be calculated
monthly, that is, 12 times a year, once a month
quarterly, that is, four times a year, every three months
bi-annually, that is, twice a year, every six months.
Example
Suppose you want to invest €2,000 for 1½ years at 4% p.a. compound
interest which is compounded bi-annually. Interest is applied every six
months. So that means interest is applied 3 times in the 1½ year period.
Complete the following table:
Period
Principal at
the start of
Interest
I
the period
1
€2,000
2
€2,040
I
Interest
Amount at
R
P T
100
earned
the end of
4
2,000 .5
100
€40
the period
€2,040
3
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Note: T represents time in years and the time in years for each of these
periods is half a year, that is, 0.5.
Solution
Period
Principal at
the start of
Interest
Interest
Amount at
R
P T
100
earned
the end of
I
4
2,000 .5
100
€40
€2,040
I
4
2,040 .5
100
€40.80
€2,080.80
I
€41.62
4
2,040.80 .5
100
€2,122.42
I
the period
1
€2,000
2
€2,040
3
€2,080.80
the period
T represents time in years and the time in years for each of these periods is
half a year, that is, 0.5.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Worked Example 3
Calculating compound interest annually using a formula
Similar to simple interest, there is also a formula for calculating compound
interest directly which is compounded annually or more frequently:
The formula is:
R 

A  P 1 

 100 
n
A is the total amount repaid or the total savings including interest.
R is the rate of interest.
P is the principal amount
n is the number of years or other periods of time, such as months.
Example
Calculate the interest earned on €1000 invested for 4 years at 5% p.a.
compound interest.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Solution
A = amount
R = 5%
P = 1,000
n=4
R 

A  P 1 

 100 
n
5 

A  1,000 1 

 100 
4
A  1,000 1  0.05
4
A  1,000 1.05
4
A  1,000 (1.216)
A  1,216
If Amount = Principal + Interest, then Interest = Amount – Principal.
Therefore the total interest paid is €1,216 - €1,000 = €216
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Worked Example 4 - Calculating compound interest more
often than annually using a formula
If we wish to calculate interest more often than once a year – for example,
monthly, quarterly or bi-annually - then the interest rate p.a. is affected.
Example
A bank is offering 4.2% p.a. compound interest, compounded monthly. If you
deposit €1,000 for two years, how much interest would you earn?
Solution
n
R n

A  P1

 100
A = Amount
P = 1,000
R = 4.2% p.a.
n = 12 periods per year
Remember:
p.a. means ‘per annum’, or once a year, but the interest is now being
compounded monthly, which is 12 times a year.
Therefore the new R value is 4.2% ÷ 12 which is 0.35% per month.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
In this example, n = 12 periods per year.
So for two years n = 12 x 2 = 24
0.35 

A  1,000 1 

100 

24
A  1,000 1  0.0035
24
A  1,000 1.0035
24
A  1,000 (1.088)
A  1,088
If you use a calculator you will get 1,087.469.
Amount = Principal + Interest
Interest = Amount – Principal
Interest = €1,088 - €1,000 = €88
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Task 1
A Building Society pays 4.2% p.a. compound interest on deposits over
€4,000.
If you deposit €5,800 for three years,
a) How much interest would you earn if the interest is calculated
annually?
b) How much interest would you earn if the interest in calculated
quarterly? Quarterly means four times a year.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Worked Example 5
Choosing between simple interest and compound interest
Example
Bank Number One offers 5% simple interest p.a. on any savings accounts
greater than €2,000 and less than €10,000.
Bank Number Two offers 4.8% compound interest p.a., compounded biannually, on any savings accounts greater than €2,000 and less than
€10,000.
You wish to deposit €6,000 for four years. Calculate the interest from both
banks and compare. Which bank has the best offer?
Solution
Bank Number One:
Simple Interest:
I
R
PT
100
I
5
 6,000  4
100
I  0.05  24,000
I  1,200
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Bank Number Two:
Compound Interest
A = amount
R = 4.8% p.a. ÷ 2 = 2.4% bi-annually
P = 6,000
n = twice each year, 4 x 2 = 8 periods
R 

A  P 1 

 100 
n
2.4 

A  6,000 1 

 100 
8
A  6,000 1  0.024
8
A  6,0001.024 
8
A  6,000(1.209)
A  7,254
Interest = Amount – Principal
Interest = €7,254 - €6,000 = €1,254
Bank Number One is offering €1,200 in interest on €6,000 for four years but
Bank Number Two is offering €1,254 for the same amount and time.
Therefore Bank Number Two is giving the better offer.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Task 2
Laura has just turned 28 and wants to save for the next two years so that she
can go on a holiday to South Africa for her 30th birthday. The total cost of the
trip will be €6,000.
(Google Images)
Laura has a lump sum saved already but it is not enough so she has decided
to invest this lump sum into a savings account with Mytown Bank.
Mytown Bank has offered Laura a 3% p.a. compound interest rate, and the
interest is compounded quarterly.
This interest rate over the time period will ensure that her principal lump sum
will amount to €6,000.
How much money did Laura invest in her savings account? Calculate
the principal value (P).
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Task 3
Ronan has just recently opened his hardware store. He purchased stock from
a main supplier; it cost him €8,000. This supplier gave him two years
interest- free credit, which means he does not have to repay the €8,000 for
two years.
Ronan is going to lodge a lump sum with Mytown Bank in order to meet this
payment in two years. Mytown Bank is offering Ronan 4% p.a. compound
interest compounded quarterly.
How much does Ronan need to lodge now in order to have €8,000 in two
years?
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Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N6: Banking options
Task 4
A credit card company charges 3.5% p.a. compound interest which is
compounded monthly on the balance owed each month.
(Google Images)
Jill spent €315.70 on her credit card. How much interest will she have to
pay if she doesn’t clear her credit card before the end of the month?
Practise your skills

Use Practice Sheet N6.
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N7: Pay slips
Activity
Pay slips
N7
(Google Images)
This activity links to unit learning outcomes 1.1 and 1.6.
Introduction
In this activity you will learn how payslips are calculated and how certain
deductions are made using percentages.
What will you learn?
Learning Outcomes
1. Calculate payslips using appropriate statutory deductions.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
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Activity N7: Pay slips
Key Learning Points
1. Becoming familiar with the meaning of terminology on pay slips such
as gross and net pay and how to make such calculations
2. Being able to calculate using a number of real life financial examples
3. Recognising examples of mathematics in everyday life
4. Exploring the use of mathematics in a range of contexts
Materials you will need for this activity

Practice Sheet N7

Solution Sheet N7
What do you need to know before you start?
A salary or wage is the money an employer pays an employee. This could be
a weekly wage or a monthly salary.
All workers get a pay slip. This pay slip shows the pay the worker earned and
any deductions taken off it.
The money that a worker earns is called gross pay. This is always reduced
by deductions and the money that is left is the actual take home pay. This
take home pay is called net pay.
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Activity N7: Pay slips
Getting Started
There are many types of deductions taken from the gross pay. Some of these
are optional, such as a trade union subscription or pension. Others are
compulsory, such as PAYE, PRSI and USC.
PAYE means: Pay As You Earn.
This a tax paid directly from your wages to the Revenue Commissioners. The
Revenue Commissioners collect taxes from citizens on behalf of the Irish
Government.
PRSI means: Pay-related Social Insurance
This is a compulsory deduction from those in employment and is used to fund
social insurance payments.
USC means: Universal Social Charge
It is a tax which is payable if your gross income is more than €10,036 per
year.
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Activity N7: Pay slips
Worked Example 1 -
Calculating Gross and Net Pay
Tom is a painter and decorator. He works a 40-hour week and is paid €9.90
an hour. He gets paid time and a half for overtime.
(Google Images)
This week Tom worked his 40 hours plus 5 hours overtime.
(a) What is Tom’s gross pay?
(b) Tom also paid the 4% USC on his gross income and a further
€108.60 in PAYE and PRSI deductions.
How much USC did Tom pay?
(c) What is Tom’s net pay?
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Activity N7: Pay slips
Solution
a) Tom’s gross pay is:
40 hours at €9.90 per hour
40 x 9.90 = €396
Tom also worked 5 hours overtime.
Overtime is paid at time and a half, that is, €9.90 + half of €9.90
€9.90 ÷ 2 = €4.95
Overtime rate of pay is €9.90 + €4.95 = €14.85
5 hours at €14.85 per hour is:
5 x 14.85 = €74.25
Tom’s gross pay is his basic pay plus his overtime
€396 + €74.25 = €470.25
b)
USC of 4% is calculated from Tom’s gross pay.
Gross pay = 470.25
4% = 0.04
470.25 x 0.04 = 18.81
Therefore €18.81 is deducted from Tom’s salary as USC.
c)
Tom’s net pay is what is left when we work out his gross pay minus
all the deductions:
€470.25 - €18.81 – €108.60 =
Tom’s net pay = €342.84
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Activity N7: Pay slips
Task 1
Siobhán is a cosmetics company sales representative. She earns a basic
salary of €1,100 per month. She also earns 10% commission on any sales
that she makes.
(Google Images)
In December Siobhán sold €2,700 worth of cosmetics.
What is her gross pay for December?
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Activity N7: Pay slips
Worked Example 2 -
Calculating Pay Slips
Sarah has a tax credit of €80 per week. This means that Sarah can earn €80
gross pay each week before certain deductions are made. These deductions
include PAYE, PRSI and Sarah’s payment towards her pension. However,
the USC is payable on gross income.
Complete Sarah’s weekly payslip below.
Name: Sarah Smith
Staff Number: 15364
PRSI Number: 1000001B
Deductions
Date: 25th June 2012
Pay
USC 4%
Basic Pay: 38 x €10.75
PAYE 20%
Overtime: 2 hours x double time
PRSI 4%
Overtime: 3 hours x time and a half
Pension €18.50
Gross Pay =
Total Deductions
Net Pay
Solution
Sarah’s Gross Pay:
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Activity N7: Pay slips
38 hours at €10.75 = €408.50
Double time rate of pay = 2(€10.75) = €21.50
Time and a half rate of pay = (€10.75) ÷ 2 + €10.75 = €16.13
2 hours at €21.50 = €43
3 hours at €16.13 = €48.39
Gross pay = €408.50 + €43 + €48.39 = €499.89
USC:
4% of gross income
0.04 x €499.89 = 19.99
USC = €20
PAYE:
PAYE is calculated on gross income – tax credit
€499.89 - €80 = €419.89
PAYE rate is 20%
0.2 x €419.89
PAYE = €83.98
PRSI:
PRSI is calculated on gross income – tax credit, that is, €419.89
PRSI rate is 4%
0.04 x €419.89 = 16.80
PRSI = €16.80
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Activity N7: Pay slips
Total Deductions:
USC =
€20
PAYE =
€83.98
PRSI =
€16.80
Pension =
€18.50
Sum of deductions = €139.28
Net Pay:
Gross Pay – Total Deductions = €499.89 - €139.28
Net Pay = €360.61
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Task 2
Deirdre works in a busy city centre beauty salon. She earns €10.80 per hour
and works a 39 hour week.
(Google Images)
This week she worked 5 hours overtime paid at time and a half.
Deirdre has a tax credit of €111.
She pays the 4% USC, 20% PAYE and 4% PRSI. She also contributes
€10.30 to her pension fund.
Use that information to complete the following payslip for Deirdre. You
may leave her other employee details blank.
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Activity N7: Pay slips
Name: Deirdre
Staff Number:
PRSI Number:
Deductions
Date:
Pay
USC 4%
Basic Pay:
PAYE 20%
Overtime:
PRSI 4%
Pension
Gross Pay =
Total Deductions
Net Pay
Practise your skills

Use Practice Sheet N7.
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Activity N8: Profit or loss
Activity
Profit or loss
N8
Google Images)
This activity links to unit learning outcomes 1.1 and 1.6.
Introduction
All companies, businesses and trades people must make a profit in order to
survive. They need to be able to keep track of how much money they make
and how much they spend on their business. To do this, they keep a record
called a ‘profit and loss account’. This activity will help you to be able to
calculate profit and loss.
What will you learn?
Learning Outcomes
1. Calculate gross and net profit.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
Key Learning Points
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Level 4 Unit 1
Activity N8: Profit or loss
1. Becoming familiar with the meaning of terminology such as gross and
net profit and how to make such calculations
2. Being able to calculate using a number of real life financial examples
3. Recognising examples of mathematics in everyday life
4. Understanding the importance of mathematics in the world around us
Materials you will need for this activity

Practice Sheet N8

Solution Sheet N8

What do you need to know before you start?
Gross profit is the difference between money coming in, that is, income, and
the cost of making a product or providing a service.
Net profit is money left over after all the expenses are paid.
A loss is when the expenses are greater than the money coming in, that is,
income.
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Activity N8: Profit or loss
Getting Started
Here is an example of a profit and loss account for a self-employed hurley
maker for the financial year 2009/2010.
(Google Images)
Sample profit and loss account
Total Payments, that is, money received
€21,560
Cost of Materials
€ 4,640
Gross profit:
€16,920
Operating Costs
Rent
€4,800
ESB
€1,550
Equipment
€ 620
€6,970
Net profit:
€9,950
The hurley maker’s net profit for that period was €9,950. He will decide how
much to pay himself as a salary and how much to invest back into the
business.
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Activity N8: Profit or loss
Worked Example 1
Gross and Net Profit
Kevin is a self-employed electrician. Last year he received an income of
€65,542 in payments for the work he did.
(Google Images)
He recorded his costs for the year as follows:
Workshop Rent:
€3,200
Van Expenses:
€2,100
Phone Bill:
€385
Advertising on radio:
€590
Insurance:
€678
ESB:
€340
Equipment:
€12,978
a) Create Kevin’s profit and loss account.
b) What was Kevin’s salary last year if he set aside 12% for future
reinvestment?
Solution
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Activity N8: Profit or loss
a) Kevin’s profit and loss account:
Total Income from payments, that is, money received
€65,542
Cost of Materials
€12,978
Gross profit:
€52,564
Net profit:
€45,271
Operating Costs
Rent
€3200
Phone
€385
Insurance
€678
Van
€2100
Advertising
€590
ESB
€340
€ 7,293
Therefore Kevin’s profit is €45,271.
(b)
Kevin’s salary:
Kevin’s profit = €45,271. He reinvested 12% of this profit.
12% = 0.12
0.12 x 45271 = 5432.52
Therefore Kevin reinvested €5,432.52.
Kevin’s salary was then €39,838.48.
Task 1
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Activity N8: Profit or loss
Mrs. O’Brien runs a local grocery store. Her total sales last year were
€42,528 and the total cost of goods last year came to €21,494.57.
(Google Images)
What was Mrs. O’Brien’s gross profit?
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Activity N8: Profit or loss
Task 2
Draw up a profit and loss account for Sean, a self-employed lorry driver:
Total payments:
€42,650
Road Tax & Insurance:
€ 4,500
Maintenance & Diesel:
€ 17,620
Advertising:
€
What is Sean’s salary?
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Assume it is the same as the net profit.
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Activity N8: Profit or loss
Worked Example 2
Cost of a Job
Tony is a gardener. He charges €8.25 an hour for his services.
(Google Images)
At the moment he is trying to get a job re-seeding a lawn for Mr. and Mrs.
Murphy. He has told them that the job will take about 8 hours and that it will
need 6 bags of grass-seed at €24.60 each and 2 bags of fertiliser at €18.75
each.
(a) Tony needs to give Mr. and Mrs. Murphy the price of this job before
he can secure the job. What is the estimated cost of this job?
(b) When Tony is unloading the bags of grass seed he rips three of them
and loses the seed. He must now buy another three bags. It was his
own mistake, so he can’t charge the customers for it. Mr. and Mrs.
Murphy pay Tony €250 for the job. Has Tony made a profit or loss
on this job? How much of a profit or loss?
Solution
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Activity N8: Profit or loss
(a)
Costs:
Materials:
Grass Seed
6 x €24.60
= €147.60
Fertiliser
2 x €18.75
= € 37.50
€185.10
Labour
8 hours at €8.25 an hour
= €66.00
Total Costs: €185.10 + 66 = €251.10
(b)
Profit or loss?
Tony now has to buy another three bags of grass seed.
3 x €24.60 = €73.80
Mr. and Mrs. Murphy have paid Tony €250.
Tony’s actual costs end up being total costs plus the cost of the three extra
bags of seed: €185.10 + 73.80 = €258.90
Therefore Tony makes a loss of €8.90.
However, he does not get any wages for this job now.
So in total Tony makes a loss of €8.90 + €66 = €74.90.
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Activity N8: Profit or loss
Task 3
Niamh is an interior decorator. She charges €9.80 per hour. She is painting a
room for a client. It takes 6.5 hours to paint the room and 3 buckets of paint.
The paint costs €22.60 per bucket. What is the total cost of this job?
Practise your skills

Use Practice Sheet N8.
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Level 4 Unit 1
Activity N9: The transfer window
Activity
The transfer window
N9
(Google Images)
This activity links to unit learning outcomes 1.1 and 1.6.
Introduction
The transfer window is the period during the year in which a football club can
transfer players from other countries onto their playing staff. Each year large
sums of money are exchanged between clubs buying and selling players.
Depending on the form of the player, large profits or losses can be made.
This activity will help to calculate such profit or loss and express it as a
percentage.
What will you learn?
Learning Outcomes
1. Calculate profit and loss on goods sold.
2. Explain how mathematics can be used to enable the individual function
more effectively as a person and as a citizen.
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Activity N9: The transfer window
Key Learning Points
1. Understanding common mathematical terms associated with buying
and selling goods
2. Being able to calculate using a number of real life financial examples
3. Recognising examples of mathematics in everyday life
4. Exploring the use of mathematics in a range of contexts
Materials you will need for this activity

Practice Sheet N9

Solution Sheet N9
What do you need to know before you start?
If a product or service is sold for more than it costs to produce or buy, then the
seller has made a profit.
If a product or service is sold for less than it costs to produce or buy, then the
seller has made a loss.
Getting Started
The percentage profit is the profit expressed as a percentage of the cost
price:
The percentage loss is the loss expressed as a percentage of the cost price:
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Activity N9: The transfer window
Worked Example
Percentage profit or loss
In 2001, Irish international Robbie Keane joined Leeds United from Inter Milan
for a fee of £12 million. Two years later he moved to Tottenham Hotspur for a
fee of £7 million.
(Google Images)
a) Calculate Leeds United’s percentage loss in the sale.
b) In 2008, Tottenham sold Robbie to Liverpool FC for a fee of £19.
Calculate Tottenham’s percentage profit in the sale.
c) Robbie was the sold back to Tottenham for a fee of £12.
Calculate Liverpool’s percentage loss in the sale.
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Solution
a) Leeds United’s percentage loss:
Loss = Cost Price - Selling Price
Loss = 12 million – 7 million = £5 million
Percentage Loss:
500
= 12
= 41.66666%
= 41.67%
Leeds United made a percentage loss of 41.67%
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Activity N9: The transfer window
b) Tottenham’s percentage profit:
Profit = Selling Price - Cost Price
Profit = 19 million – 7 million = £12 million
Percentage Profit:
=
1200
7
= 171.4285%
= 171.43%
Tottenham made a percentage profit of 171.43%
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Activity N9: The transfer window
c) Liverpool’s percentage loss:
Loss = Cost Price - Selling Price
Loss = 19 million – 12 million = £7 million
Percentage Loss:
Liverpool made a percentage loss of 36.84%
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Activity N9: The transfer window
Task 1
In 2007, Spanish striker Fernando Torres joined Liverpool FC for £20 million.
(Google Images)
In 2011, he was sold to Chelsea FC for a British record fee of £50 million.
Calculate Liverpool’s percentage profit in the sale.
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Activity N9: The transfer window
Task 2
In 2003, Irish International Damien Duff joined Chelsea FC for a fee of £17
million. In 2006, he was sold to Newcastle United for a fee of £5 million.
(Google Images)
a) Calculate Chelsea’s percentage loss in the sale.
b) In 2009, Duff joined Fulham FC from Newcastle for a fee of £2.5
million.
Calculate Newcastle’s percentage loss in the sale.
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Activity N9: The transfer window
Task 3
In 1997, Irish International goalkeeper Shay Given joined Newcastle United
for a fee of £1.5 million. In 2009, he was sold to Manchester City for a fee of
£7 million.
(Google Images)
a) Calculate Newcastle’s percentage profit or loss in the sale.
b) In 2011, Manchester City sold Given to Aston Villa making a
percentage loss of 50% on his cost price.
How much of a loss did Manchester City make in the sale?
c) What was their selling price?
Practise your skills

Use Practice Sheet N9.
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Activity N10: Value Added Tax
Activity
Value Added Tax
N10
(Google images)
This activity links to unit learning outcomes 1.1 and 1.6.
Introduction
During this activity we will look at Value Added Tax (VAT).
What will you learn?
Learning Outcomes
:
1. Calculate VAT inclusive and VAT exclusive prices.
2. Explain how mathematics can be used to enable an individual to
function more effectively as a person and as a citizen.
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Activity N10: Value Added Tax
Key Learning Points
1. Calculating percentages
2. Developing an understanding of what VAT is
3. Calculating VAT
4. Exploring the relevance and usefulness of mathematics in the world
around us
Materials you will need for this activity

Receipts from recent shopping trips.

Practice Sheet N10

Solution Sheet N10
What do you need to know before you start?
Maths
The amount of tax we have to pay is usually expressed in percentage form.
For example, if we have to pay a tax of 20% on €200 then we must find 20%
of €200. From studying percentages we know that in order to do this we must
20
first express 20% as a fraction, that is, 100 , and then multiply this fraction by
the sum, €200.
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Activity N10: Value Added Tax
VAT
Value Added Tax (VAT) is a tax the Government charges on consumer
spending. Every individual in Ireland must pay VAT when we purchase goods
or services.
VAT is a tax that is applied at different stages. For example:
Manufacturers must pay VAT when they sell their product to a wholesaler or
distributor.
Then the retailer must pay VAT when they buy from the wholesaler.
Finally the consumer – the shopper - pays VAT to the retailer.
As a result, the rate of VAT has a huge impact on the overall price of goods.
In Ireland there are different rates of VAT depending on the goods or services
you buy.
The following table outlines the rates of VAT as of February 2012.
Standard Rate
Applies to most goods and services.
23%
Reduced Rate
Applies to labour intensive services, for example,
13.5%
cleaning services.
Services Rate
Applies to tourism related activities, for example,
9%
restaurant prices.
Zero Rate
Applies to many food and medicines and to children’s
0%
clothes.
Special Rate
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Applies to the sale of livestock.
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Activity N10: Value Added Tax
Getting Started
The term per cent means ‘for every one hundred’.
A VAT rate of 23% would mean that 23 cent from every 100 cent, or €1,
would go as tax.
Let’s recap on calculating percentages. Here is an example:
In a game of soccer between Manchester United and Liverpool the ball was in
play 55% of the time. The game lasted 95 minutes including injury time.
We can use percentages to calculate the amount of time the ball was in
play.
To do this we need to find 55% of 95.
In mathematics, the word “of” relates to multiplication.
First, write 55% as a decimal: 0.55.
Then, multiply it by 95:.
0.55 x 95 = 52.25
Therefore, during the game between Manchester United and Liverpool the
ball was in play for 52.25 minutes or 52¼ minutes.
“VAT inclusive”
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Activity N10: Value Added Tax
Retailers include the amount of VAT payable in the total price they charge for
the goods or services. That means that the price of most products in Ireland
is VAT inclusive.
For example, if an item in a shop costs €10 then this €10 includes the VAT.
If VAT on this item was charged at a rate of 23% then we know that the total
cost is equal to the original cost + VAT. That is:
Original Cost + VAT = €10
Original Cost + VAT = 123%
(because the Original Cost = 100% and the VAT = 23%).
123% = €10
In order to calculate the amount of VAT included in the price of this item we
must first find 1% and then find 23%:
123% = 10
123
10
%
=
123
123
1% = 0.08
(1 x 23)% = (0.08 x 23)
23% = 1.84

The amount of VAT that we paid on this item was €1.84.

The original cost of the item was €8.16, that is, €10 – €1.84.
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Activity N10: Value Added Tax
Worked Example 1
Royal Rumble Wrestling
The Royal Rumble is a wrestling event held once a year. It is broadcast live
on Sky Sports Box Office. The cost of ordering this event on Sky Box Office
is €17 + VAT, at a rate of 23%.
a) Calculate the total amount of VAT that you must pay if you purchase
this event.
b) Calculate how much change you would get from €30 when you
pay for the event in total.
(Google images)
c) There is a rumour that in 2014 the Royal Rumble will come to Dublin.
They say that the tickets will cost €35 + VAT. The Service Rate of
VAT will be applied: that is, 9%.
Calculate the total cost of a ticket for this show.
d) If the Government were to increase the Service Rate of VAT to 12.5%
in the 2013 budget, how much extra would you have to pay for a
ticket due to this change?
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Activity N10: Value Added Tax
Solution
a) In order to calculate the VAT we must find 23% of 17.
First we must convert 23% to a decimal: 23% = 0.23
Then multiply that by 17:
0.23 x 17 = 3.91
Therefore, the total amount of VAT payable on this purchase is
€3.91.
b) The total cost of this event on Sky Box Office is
€17 + €3.91 = €20.91
In order to calculate the change we would get from €30 we must
subtract the total cost, including VAT, from €30.
Change = 30 – 20.91
Change = €9.09
c) Service rate = 9%
We know 9% = 0.09. Therefore 9% of 35 is 0.09 x 35 = 3.15
VAT = €3.15
Total Cost of Ticket = €35 + €3.15 = €38.15
d) If the service rate was increased to 12.5% then the total amount of
VAT to be paid on the ticket would be 12.5% of €35.
That is 35 x .125 = 4.375. Round that off to €4.38.
VAT = €4.38
Total Cost of Ticket = 35 + 4.38 = €39.38
So, if the rate of VAT went up to 12.5% the ticket would cost €1.23
extra.
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Task 1
Jade runs her family’s convenience store in Athlone. She is responsible for
everything, including pricing all goods in the store.
One day, the first three deliveries that she gets from the wholesaler are

a supply of chocolate cakes

crates of beer

boxes of talcum powder.
Jade knows that the standard rate of VAT applies to the beer and the talcum
powder and that the reduced rate applies to the chocolate cake.
The table below shows the cost price of these items (not including VAT)
and the rate of VAT that Jade must add on to the price.
Item
Cost Price
Rate of VAT
excluding VAT
1 chocolate cake
€ 4.50
Reduced rate
13.5%
1 six pack of beer
€10.00
Standard rate
23%
1 Talcum Powder
€ 3.25
Standard rate
23%
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Activity N10: Value Added Tax
a) Calculate the prices that Jade must sell these products at so as to
include VAT. Round all answers to the nearest cent.
b) If a customer buys 2 chocolate cakes and 2 six packs of beer in
Jade’s shop, how much VAT would they pay in total?
c) If the Government changed the standard rate of VAT to 20% and
changed the reduced rate of VAT to 11%, what would be the
difference in cost for each of the three items?
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Activity N10: Value Added Tax
Worked Example 2
Dining Out
a) Craig treated his friend to a meal in their local restaurant, Bella
Italia. The total cost of the bill came to €90.25. This included VAT
which was charged at a rate of 9%.
Calculate how much VAT Craig paid.
(Google images)
b) On another evening, Craig decided to cook for himself and his
friend. He cooked the exact same food as they had in the Bella
Italia.
The total cost of his food bill was €35.50 including VAT:

€20 worth of the food included VAT at a rate of 13.5%.

The remainder of the food bill included VAT at a rate of 23%.
He also bought wine at a cost of €21. This included VAT at a rate of
23%.
What was the total cost of the meal and drink before VAT?
Solution
FÁS© 2012
118
September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N10: Value Added Tax
(a) Total bill = Original Price + VAT = 90.25
109% = 90.25
109
90.25
%
=
109
109
1% = 0.828
(1 x 9)% = (0.828 x 9)
9% = 7.452
Craig paid €7.45 in VAT for this meal
(b) There are two different rates of VAT to look at here.
Food:
€20 includes VAT at 13.5%
€15.50 (35.50 – 20) includes VAT at 23%
Drink: €21 includes VAT at 23%
We will look at the 13.5% rate first:
Total bill = Original Price + VAT = 20
113.5% = 20
113.5
20
%
=
113.5
113.5
1% = 0.1762
(1 x 100)% = (0.1762 x 100)
Note: The reason we multiply by 100% here is to find 100% since that
is the original price.
100% = €17.62
Price excluding VAT = €17.62
FÁS© 2012
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N10: Value Added Tax
Now we will look at the 23% rate:
The total value of the items which are taxed at this rate is:
15.50 + 21 = 36.50
Total bill = Original Price + VAT = 36.50
123% = 36.50
123
36.5
%
=
123
123
1% = 0.2967
(1 x 100)% = (0.2967 x 100)
100% = €29.67
Price excluding VAT = €29.67
Total cost of the meal, excluding VAT = 17.62 + 29.67 = 47.29
Total cost, before VAT was €47.29.
FÁS© 2012
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N10: Value Added Tax
Task 2
Amanda won a large sum of money in a raffle. She used it to buy herself and
her sister vouchers for a Spa Day and a hotel, and she also paid for the two of
them to have an overnight stay in the hotel.

The Spa Day cost €65 each. This included VAT which was charged
at a rate of 13.5%.

The overnight stay in the hotel cost €180. This included VAT which
was charged at a rate of 9%.
a) Calculate the amount of VAT that Amanda paid for the Spa Day
vouchers for herself and her sister.
b) Calculate the cost of the hotel excluding VAT.
c) Calculate the total cost of the Spa Day and hotel stay, excluding
VAT.
FÁS© 2012
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September 2012
Functional Mathematics
Learner Pack
Level 4 Unit 1
Activity N10: Value Added Tax
Task 3
Find two receipts from recent purchases that you made.
Visit the website (http://www.revenue.ie/en/tax/vat/rates/index.jsp#C) . There
you will find the rate of VAT that was applied to every item on your receipt.
a) Calculate the total amount of VAT you paid for each item.
b) Work out the total amount you would have paid if you did not have to
pay VAT.
Practise your skills

Use Practice Sheet N10.
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September 2012