BIRKBECK MATHS SUPPORT
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Calculations 1
In this section we will look at
- practical examples of calculations with x and +
- using brackets and more examples
- an introduction to using a calculator
- examples of calculations using – and ÷ and powers
- formal rules of calculations: BODMAS
- examples of calculations using all rules
Helping you practice
At the end of the sheet there are some questions for you to practice.
Don’t worry if you can’t do these but do try to think about them. This
practice should help you improve. I find I often make mistakes the
first few times I practice, but after a while I understand better.
Videos
All the examples in this worksheet and all the answers to questions
are available as answer sheets or videos.
Good luck and enjoy!
Videos and more worksheets are available in other formats from
www.mathsupport.wordpress.com
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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1. Everyday Calculations with + and x
Most adults will make calculations every day without even thinking about them.
If you have two £5 notes then you know you have £10 and it hardly seems
worth writing this as 2x5 =10. If you are meeting a friend at 8.30 and the walk
takes 20 minutes then you need to leave at 8.10pm and you probably don’t
always need to write down
But in this section we’ll spend time thinking about exactly these sorts of
everyday calculations so we can understand the rules and processes involved
and apply them to more complicated calculations.
Calculation example using multiplication
In a particular cafe, 1 coffee is £1.10, so 3 coffees are £3.30. We can write this as
1 x coffee = 1.10
3 x coffee = 3 x 1.10 = 3.30
Or we could write it in a simpler way where the letter 'C' stands for coffee, so
1 x C = 1.10
3 x C = 3 x 1.10 = 3.30
Here we have started to use algebra (and this just means we are using
symbols, such as the letter C as well as numbers).
Sometimes with algebra we leave out the multiplication sign (x), so the two
following lines mean exactly the same thing:
3 x C = 3 x 1.10 = 3.30
3C = 3 x 1.10 = 3.30
This is just saying that 3 coffees (which we know are £1.10 plus £1.10 plus
£1.10) cost three lots of £1.10 which gives £3.30.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Calculation examples using multiplication and addition
For a similar example coffee still costs £1.10 but now donuts are £1.00. So two
donuts cost 2 x £1.00 = £2.00
However, if we buy 3 coffees and 2 donuts we have
3 x coffees + 2 x donuts = 3 x 1.10 + 2 x 1.00 = 3.30 + 2.00 = 5.30
Using C to mean coffee and D to mean donuts, we can write
3C + 2D = 3x1.10 + 2x1.00 = 5.30
A closer look
Let's now take a careful look at just one part of the calculation above
3 x 1.10 + 2 x 1.00 = 5.30
We know the answer is 5.30, and to get this answer we have to do the
multiplication first and then the adding second.
How to do it wrong!
An important part of learning maths is learning how to get the wrong answer. If
we can see why an answer is wrong, then we can begin to understand when
an answer is right. Notice that if we do the adding first and the multiplication
second we would first add 1.10 and 2 together to get 3.10 in the following way
3 x 1.10 + 2 x 1.00 = 3 x 3.30 x 1.00
giving £9.90 and the wrong answer to the cost of 3 coffees and 2 donuts.
Key point:
The key point of these two examples is that it matters which order you multiply
and add when making any calculation.
If the calculation has adding and multiplying only, then we always need to do
the multiplying first then the adding second.
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Another example with + and x
If I buy three t-shirts and two jumpers I can write the calculation as
3 x t-shirts + 2 x jumpers = 3T + 2J
where we have used T to mean t-shirt and J to mean jumper. Just as I go to buy
them I decide to buy another 2 t-shirts, so now the expression becomes
3 x t-shirts + 2 x jumpers + another 2 x t-shirts = 3T + 2J + 2T
we can buy these T-shirts and jumpers in any order and the cost is the same,
so we can simplify the calculation by putting all the t-shirts together and writing
3 x t-shirts + 2 x t-shirts + 2 x jumpers = 5 x t-shirts + 2 x jumpers
3T + 2T + 2J = 5T + 2J
where both of the two lines above are identical, we have just used words in the
first and algebra in the second. Suppose that the t-shirts are £4.00 and the
jumpers are £12.00 we can then find the total cost by substituting these prices
5T + 2J = 5 x 4.00 + 2 x 12.00
Remembering we have to do the multiplication first, we have
5x4.00 + 2x12.00 = 20.00 + 24.00 = 44.00
So the total is £44.00
The wrong way:
Remember the wrong way would have meant adding first and multiplying
second and this would give
5x4.00 + 2x12.00 = 5x6.00x12.00 = 360.00
So if we carried out the adding and multiplying in the wrong order we would
have to pay £360 instead of the correct amount which was £44.
Key Point
Some of these ideas may seem simple at first, but thinking carefully about how
we make these calculations will help us with more complicated problems.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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2. Brackets
Before we carry on with more examples we will talk a little bit about brackets.
There are two things we can do with brackets, we can multiply out a pair of
brackets or we can factorise using brackets. Multiplying out and factorising are
the opposite of each other and we start with multiplying out.
To do this we will need to introduce two new words: term and expression.
In a calculation there may be many terms added together to make an
expression. For example in the expression 3C + 2D we have two terms, the first
term is 3C (meaning the 3 coffees) and the second term is 2D (meaning the two
donuts) but the whole thing 3C + 2D is the expression.
MULTIPLYING OUT BRACKETS
If you are working (again!) in the cafe selling coffee and cakes, suppose
someone asks for three coffees and two donuts, and then the next person asks
for exactly the same thing (three coffees and two donuts). Then we need to get
2 lots of (three coffees and two donuts)
2 x (three coffees + two donuts)
2 x (3 x coffees + 2 x donuts)
2x (3xC + 2xD)
where each line above represents the same thing and C means coffee and D
means donuts, again we can leave out the multiplication signs and write
2(3C+2D)
We know this means we have to get 3 coffees and 2 donuts and then another 3
coffees and 2 donuts, so in total we have to get 6 coffees and 4 donuts, so the
following three lines represent exactly the same calculation
2 lots of (3 coffees and 2 donuts) = 6 coffees and 4 donuts
2 x (3xC+2xD) = 6xC + 4xD
2(3C + 2D) = 6C + 4D
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and this last line is important for simplifying calculations as we notice that there
is a 2 outside the bracket, meaning two lots of each thing listed inside the
bracket, so this 2 applies to every term inside the bracket (where the first term
is 3C and the second term is 2D).
We say we have 'multiplied out' the bracket if we remove the brackets by
multiplying each term by the number outside the bracket, as follows
2(3C + 2D) = 2x3C + 2x 2D = 6C + 4D
but remember this just means that 2 lots of “3 coffees and 2 donuts” is the same
as 6 coffees and 4 donuts.
How to do it wrong!
As it always it helps to do things wrong a few times (because if we don't make
mistakes then we can't understand the mistakes) we’ll look at the calculation
again, and notice that if we had not included the brackets then we would write
2 lots of three coffees and 2 donuts
2 x 3 coffees and 2 donuts
2 x 3C + 2D
6C + 2D
which represents 6 coffees and 2 donuts, which gives us the wrong answer
because we forgot to use brackets to double the number of donuts. By not
using brackets, we couldn't multiply out to get the right answer which is
2(3C + 2D) = 2x3C + 2x2D = 6C + 4D
Key Point
The key point is that before we multiply or add anything, we must first check
the brackets in a calculation. When we multiply out a bracket we must make
sure that the thing outside the bracket multiplies each term inside the bracket.
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Another calculation example with brackets
A washing machine in a launderette needs 3 one pound coins (each pound is
written as 1.00) and the dryers need 5 twenty pence pieces (each 20 pence is
written as 0.20). So the total amount of money we need is:
3 x 1.00 + 5 x 0.20
But I want to do three sets of washes including the drying so I need three lots of
the whole amount above and so I write
3 lots of (three one pound coins and five twenty pence pieces)
3 x (three one pound coins + five twenty pence pieces)
3 x (3 x 1.00 + 5 x 0.20)
3 (3 x 1.00 + 5 x 0.20)
and we can now finish this calculation in two ways. As said before we always
need to consider the brackets first in any calculation, but we can either simplify
the terms inside the bracket or multiply out the bracket. Both ways should give
us the same answer in the end so we will try them both.
FIRST WAY
So we will start by simplifying the bracket, this means simplifying the
calculation inside the bracket. Remembering the rule about multiplying before
adding: first we multiply then second we add the things inside the bracket
3 ( 3 x 1.00 + 5 x 0.20) = 3 (3.00 + 1.00) = 3 (4.00)
Then multiply the terms inside the bracket by the number outside the bracket.
3 (4.00) = 3 x 4.00 = 12.00
SECOND WAY
This time we first multiply out, so we multiply each term inside the bracket by
the number outside the bracket. Remember there are two terms, the first is
3x1.00 and the second is 5x0.20 and each has to be multiplied by 3 to give us:
3 (3 x 1.00 + 5 x 0.20) = 3 x 3 x 1.00 + 3 x 5 x 0.20
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Now that we have removed the brackets, we only have multiplication and
addition left, so we will do the multiplications first and the additions second
3 x 3 x 1.00 + 3 x 5 x 0.20 = 9.00 + 3.00 = 12.00
and we get exactly the same answer as before so we can be sure that the
answer is 12.00 and that we understand the maths of brackets.
Key points
There are two important points about this.
- when we have brackets in a calculation, we must look at the brackets first. If
we can simplify the terms inside the bracket we should do this straight away,
we must also make sure that whatever happens to the whole bracket (in this
case being multiplied by three) must happen to every term inside the bracket
- if there are two ways to make a calculation then they should give the same
answer. This is a really good way for us to check the answer we get is correct.
Remember too that we must still obey the rule that says multiplication happens
before addition. But from now on we will always check the brackets first.
FACTORISING WITH BRACKETS
So now we will introduce the opposite of multiplying out which is called
factorising. Again an example: if you are going to Alton Towers for the day and
you are going with 4 adults and 8 children then we can use a for adults and c
for children and write this as
4 adults + 8 children = 4a + 8c
To find out if an expression can be factorised we look at each term (there are
two terms here the 4a and the 8c) and look for a number (or an algebraic
symbol – more of this in 04 Algebra) that goes into each term. We can see that
4 goes into each term since the first term can be written 4 x a and the second
term is the same as 2 x 4 x a, so we can write the whole expression as
4a + 8c = 4 x a + 8 x c = 4 x a + 2 x 4 x a
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this means we can factorise using the 4 to give us
4 adults and 8 children is the same as 4 lots of “1 adult and 2 children”
4 x a + 8 x c = 4 x (1 x a + 2 x a)
4a + 8c = 4 (a + 2c)
Notice that we can write 1a or just a to mean one adult. Also we can check our
answer is correct by multiplying out the bracket to get the original expression.
4(a+ 2c) = 4xa + 4x2xc =4a + 8c.
Remember this factorisation just tells us that 4 adults and 8 children is the same
as 4 lots of “1 adult plus 2 children”. Writing the '4 lots of' is the factorising.
4. Using calculators
As our calculations get more complicated we will need to use a calculator and
whether you are buying a new calculator or already have one you will need to
check the type of calculator and be sure you know how to use it.
If you want to practice every day calculations most calculators are fine as long
as they have + , - , x , ÷ , = as well as a button that allows you to include the
brackets ( ) and to calculate powers, usually this button looks like
.
Sometimes the button for = is written as Enter or Ans and sometimes the divide
is written as / and the multiply as *. But you can use the exercises below to
check the type of calculator and that you are using the right buttons.
Calculator types and Scientific Calculators
One of the best ways to understand your calculator is to read the instructions (if
you’ve lost them you may be able to get copies from the manufacturers
website) but it is important to know whether your calculator is algebraic or nonalgebraic. To find this out type in the following
This expression is 5+3x2 and could relate to having a £5 note plus three £2
coins (which is £11 in total) and if you calculator is an algebraic calculator you
will get the answer 11 as we expect.
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However if your calculator is non-algebraic you will get the answer 16 because
your calculator makes the calculations as the numbers and symbols are typed.
It has calculated 5+3 first to get 8 then multiplied this by 2 to get 16.
Which type of calculator you use depends on your preference, although most
maths courses will recommend a scientific calculator as these are algebraic.
From now on we will assume that you are using a scientific calculator
If you want to buy a new calculator generally any calculator from the Casio FX
series are recommended (these are scientific algebraic calculators and are not
too expensive). If you are studying a specific course such as nursing,
healthcare, economics, business or science you should check the course
documents to see which calculator you need but the usual recommended
calculators include the Casio FXs. It is worth noting that if you are using one of
these calculator the answers may display as a fraction and to convert to
decimal you should use the button S⇔D.
Practice using your calculator
Try these calculations and to check you get the right answers
1. 3 x 14.2 = 42.6
6. 0.3 (18x2 + 3x9.2) = 35.64
2. 15 x 0.002 x 17 = 0.51
7. 2.1 (10 +2x3) + 14 = 47.6
3. 18 + 19x5 = 113
8. 32x0.03 + 5(1.2 + 0.6 x 3) = 15.96
4. 18 x 240 = 4320
9. 3 x(8.2 + 1) + 2 x (5.9+ 3.8) = 24.6
5. (18 x2 + 3)x14 = 546
10. 2 x ( 9x2 + 4 (3+ 7) + 5) = 126
Notice that in some of these calculations we have used two sets of brackets,
and it doesn't matter if there are 2 or 3 or more brackets, the same rules apply
with multiplying out and simplifying and your calculator should do this
automatically.
We look at more examples with calculators in the second sheet on calculations
and if you want to practice fractions and scientific notation go to the next
worksheet 02 Calculations 2.
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5. Everyday calculations with – and ÷
We have looked at calculations with multiplications, additions and brackets and
we have practiced these on a calculator. Now we will look at subtracting and
dividing with brackets, later we will look at powers or indices
Calculation example with subtraction
Remember when multiplying or dividing numbers: a negative (–) multiplied or
divided by a positive (+) makes a negative (–). But a negative (–) multiplied or
divided by a negative (–) makes positive (+).
Example: A cinema is offering a deal. Each time you buy 5 tickets you get 5
pound returned. So if you buy five tickets we can write the total cost as
five tickets reduced by £5
5 x tickets - £5
Now, if 4 groups do this then we have 4 times the whole thing
4 x (5 x tickets – £5) = 4 (5T - 5)
where T represents tickets. If each ticket costs £2.00, then we can write
4 x (5 x 2.00 – 5)
And we can finish the calculation in more than one way (by simplifying the
brackets or by multiplying out the brackets), but we always look at the
brackets first.
FIRST WAY
We can simplify the bracket, which means calculating the things inside the
bracket. Remembering multiplication happens before addition so we can write
4 x (5 x 2.00 – 5) = 4 x (10.00 – 5) = 4 x (5.00) = 4 x 5.00 = 20.00
Giving us the answer 20.00
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SECOND WAY
We can multiply out the bracket. So first we multiply each separate term
inside the bracket by the thing outside the bracket. Then simplify the result,
remembering to multiply before adding
4 x (5 x 2.00 – 5) = 4 x 5 x 2.00 – 4 x 5 = 40.00 – 20 = 20.00
Remember that a positive number multiplied by a negative number is a
negative number, so 4 x –5 is the same as –4 x 5 which is the same as –20.
THIRD WAY
Try typing the whole expression into your calculator, including the brackets. You
should get 20.00 or 20 as the answer. So all three calculation methods agree
and if you get a different answer double check to try to find the problem.
Calculation example with division:
If I am cooking and I have eight potatoes and four fish fingers and I want to
divide them between two children I write
Eight potatoes plus four fish fingers all divided between two
8 potatoes + 4 fish fingers all divided between 2
( 8 potatoes + 4 fish fingers ) divided by 2
we can use P for potatoes and F for fish fingers
the thing outside the bracket which is ÷2 applies to each term inside the
bracket which is 8P+4F giving
We can then write the two terms in the bracket separately as
So each child has 4 potatoes and 2 fish fingers.
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6. Powers or indices in calculations
Here is a harder example with powers.
You need to remember that
and if you want to revise these types
of calculations look through the indices section of worksheet 01 Numbers 3.
Here is the example: Let’s suppose the Interest rate on a credit card is 20% per
year, so if we spend £200 but don't pay the balance, then after a year we owe
the original 200 plus 20% of 200, and we can write this as
by remembering the work on percentages and decimals, we can write
from the factorising with brackets section we can see that both terms have 200
in, so we can write this as
using a calculator we see that after one year we owe £240 so combining all this
information into one line we get:
The next year though, we still don't pay off the money we owe and the bank
adds on 20% to the new amount £240 which (just as before) gives
but 240 is the same as 1.20 x 200, so the total amount we owe after two years
can also be written as 200 x1.20 x 1.20. So after 2 years the amount we owe is
And what happens if we leave it for five years? Well we can see the same
pattern for each year, so for five years adding on 20% each year we can write
and now using the calculator (with the button
) we can see that we get
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Notice that the power of 5 only applies to the 1.20 and not to the 200, in this way
we get the correct answer that after 5 years we owe nearly £500.
If we then remember that we had spent £200 on 3 different credit cards, each
with the same 20% interest, then the amount we owe after five years would be
So we first check the brackets to see if they can be removed. They can since
there is only one term in the brackets, which means we can simply write
Notice again that the power 5 only applies to the 1.20. So having dealt with the
brackets we calculate the indices part next and we could write it as
and carry out the multiplication on a calculator, which gives us 1492.992
And to check our workings we enter the original expression into the calculator
using the
button and all being well we get the same answer = 1492.992.
So we now have several ways to check that if we borrow £200 on three credit
cards, with 20% interest rate per year, then after five years we owe £1492.99.
7. Rules for calculations
In general for any calculation we should always use the following rules to make
sure we get the right answer.
Sometimes the rules for calculations are called BODMAS, which stands for
B = Brackets (get rid of them or only consider the things inside them)
O = Orders (this is another word for indices or powers)
DM = Division and Multiplication
AS = Addition and Subtraction
This tells you what to do first, second, third and fourth when you need to work
on a calculation to simplify an expression or find the answer.
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Example 1:
So first we look at the brackets. We simplify the terms inside the brackets so we
focus on the expression inside the brackets
. Now looking only
inside the brackets we next focus on the powers and we simplify
the same as
, which is
, so we can re-write our whole expression as
There aren't any more powers inside the brackets, so we next look at any
multiplications or divisions in the brackets and we carry out the multiplications
inside the brackets, well
, which gives us
Finally we look at the additions or subtractions inside the bracket and get
Now it's easy we can just multiply the 4 and the 50 to give 200, so in full:
There is another way to carry out this calculation, we could first multiply out the
brackets and then simplify it. Also we could type the whole expression into our
calculator. Both ways should give you the answer 200 and these are great ways
to check your understanding of how to multiply out and use your calculator.
Below are the workings for these other ways:
Calculating by multiplying out the brackets
First we multiply each term by in the brackets by four to give
Next we look at the powers. The power of 3 only applies to the two and we can
leave this as it is or write it as
. However it is fine to move on to the
multiplication if we only carry out the multiplications not involving powers:
This is all find and correct but it would now be wrong to write
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Instead we need to first calculate the powers before multiplying
,so
Calculating with brackets and powers on the calculator
If you are using a scientific algebraic calculator (for example any of the Casio
FX series) we can enter the full equation using the buttons:
If you get an error message it may be because you need to move the cursor
forward to the main line after using the
button. But when entered correctly
this will give you the answer 200.
Another example using a calculator
Suppose we want to calculate the expression
Again we start with the terms inside brackets, then consider powers, then
multiplication or division, then addition and subtraction and it would be good
practice to try this.
However, to type the whole think into a scientific algebraic calculator we enter
If you get Error or get a different answer try breaking each section down bit by
bit to find the problem. The answer should be -0.875
Key point
To calculate any expression correctly you need to use the BODMAS rules. Start
with the brackets then look a powers then multiplication and division and finally
addition and subtraction. If you use a scientific calculator it should use the
BODMAS rules automatically and you can double check your answers by
breaking down small sections of a calculation to be sure you are right.
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8. Now your turn
Generally the more maths you practice the easier it gets. If you make mistakes
don’t worry. I generally find that if I make lots of mistakes I understand the
subject better when I have finished. If you want to see videos explaining these
ideas and showing the answers visit www.mathsupport.wordpress.com
A) Calculations with + and x. Try these without a calculator.
1)
4)
2)
5)
3)
6)
when you finish these calculations check your answers using a calculator, if
any answers disagree it’s really good practice to find the problem.
B) Calculations with -, +, ÷, x, brackets and powers
1)
–
2)
–
3)
– –
–
9)
10)
4) –
11)
5) –
6)
7)
8)
–
–
12)
13)
C) Calculations to try with a calculator
1)
5)
2)
6)
3)
7)
4)
8)
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D) Calculations with word problems
1) I get three twenty pound notes out of the bank. I need to buy presents for my
three nieces. I decide to get the same for all of them, and I buy them each three
packs of pens, and 4 colouring books. Use the letter P to mean pens and B to
mean books and write an expression for the total money you have left after
buying the presents.
2) Using the answer to the first question calculate how much money you have
left by substituting the cost of pens which are £1.00 per pack and the books are
£2.00 each. Try to show each line of your workings and check your answer
using your calculator.
3) A family hire a holiday villa for two weeks at £100 per night. Then they hire
bikes for five people. The bikes are £10 a day and they hire them for three days.
They then hire two jet skis for a week which costs £80 each but they get a 15%
discount. Write the calculation and then calculate how much they spent.
4) A holiday caravan is available to rent throughout the year. It is booked for 13
weeks during the summer and 12 weeks during the winter season. Between
these dates is booked for 7 short breaks of three nights. Write the calculation
showing what percentage of the 365 days of the year the caravan is occupied.
5) A patient is to receive 180 mg of a drug over four days. He is to take the
medication as tablets three times a day. Show how you would calculate the
number of mg the patient should take each time he takes his medication.
6) In the above calculation the tablets are only available as 5mg tablets. How
many tablets should the patient take each time?
7) There are 1,000 millilitres in a litre and 284 millilitres in a pint. If there are 18.2
grams of fat in one litre of milk, show how you would calculate the numbers of
grams of fat in a pint of milk?
8) The gear ratio in a car relates the number of teeth on the smaller gear to the
number of teeth on the larger gear. The ratio is 1:5 means the larger gear has 5
times more teeth than the smaller gear and so the smaller gear rotates five
times each time the larger one rotates once. The gear ratio on the 6 gear of an
Enzo Ferrari is 0.76:1 show how to calculate the number of times the smaller
gear rotates if the larger one rotates 286 times?
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