Mr F

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Mr Fs Maths Notes

• Number
• 4. Rounding and Approximations

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4. Rounding and Approximations
A Question Imagine you are walking along the
street and someone stops you and asks you to do
this nice little sum in your head in 30 seconds
If you are anything like my students, I can
imagine what you might say, and I cant write it
hereother than say...eek!...
But, with a little knowledge about rounding and
approximations, you should be able to tell that
person that the answer is about 2, and then ask
them to kindly leave you alone
1. Rounding Now, there are lots of degrees of
accuracy you will need to know how to round to,
but the way to tackle any question you could ever
possibly be asked is always the same 1. Circle
the last digit you need what I will call the
Key Digit 2. Look at the unwanted digit to the
right to it if it is 5 or above add one on to
your Key Digit, if it is less than five, leave
your Key Digit alone. 3. Be very careful of the
dreaded number 9
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(a) decimal places The most common degree of
accuracy you are asked to round to is a number of
decimal places. Because mathematicians are lazy,
this is normally shortened down to dp. e.g. 5.96
(2dp) means that the answer was probably really
long, but when rounded to two decimal places, it
was 5.96 The thing you need to remember, and the
thing that sounds really obvious, is that if the
question asks for two decimal places, you must
give two, no more, no less!
Example 2
Example 1
Round 5.639 to 1dp
Round 12.0482 to 2dp
5 . 6 3 9
1 2 . 0 4 8 2
1. This time the Key Digit is in the 2nd decimal
place, which makes it the 4
1. We start by putting a ring around our Key
Digit. Now the question has asked for 1 decimal
place, so our key digit is the 6, as it occupies
the 1st decimal place
2. The unwanted digit to the right of it is an 8,
which is definitely 5 or above, so we must add
one onto our Key Digit
2. Next we look at the digit to the right to it
the unwanted number 3. It is less than 5, so we
leave the key digit alone.
3. So, to two decimal places, our answer is
3. So, to one decimal place, our answer is
12.05
5.6
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Example 4
Example 3
Round 25.72037 to 3dp
Round 3.7952 to 2dp
2 5 . 7 2 0 3 7
3 . 7 9 5 2
1. This time the Key Digit is in the 3rd decimal
place, which makes it the 0
1. This time the Key Digit is in the 2nd decimal
place, which makes it the 9
2. The unwanted digit to the right of it is 3,
which is definitely less than 5, so just leave
our Key Digit alone
2. The unwanted digit to the right of it is a 5,
which is 5 or above, so we must add one onto our
Key Digit
3. So, to three decimal places, our answer is
But if we add one to our key digit, we get 10!
So, we must add one to the next digit as well,
which is the 7
25.720
Be careful Some silly people will put 25.72 down
as the answer thinking that the 0 makes no
difference. But it does! The question has asked
for 3dp, so give them 3dp!
3. So, to two decimal places, our answer is
3.80
(b) nearest whole, 10, 100, 1000 etc These are
the nicest types of rounding questions, and so
long as you have your brain switched on, you
shouldnt get too many of them wrong. But dont
get cocky, as you can easily make
mistakes! Remember the size of your rounded
number should be a similar size to the number in
the question, and you must use zeros to help you
with this
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Example 2
Example 1
Round 3.825 to the nearest whole number
Round 32,825.2 to the nearest hundred
3 . 8 2 5
3 2 8 2 5 . 2
1. We want the nearest hundred, so stick the ring
around the digit in the hundreds column, which is
the 8.
1. Our Key Digit is always the degree of accuracy
the question asks for, which in this case is
whole numbers, so we need the 3.
2. The unwanted digit to the right of it is 8,
which is definitely more than 5, so we add one to
our Key Digit.
2. The unwanted digit to the right of it is a 2,
which is less than 5, so we leave our Key Digit
alone.
3. So, to the nearest whole number, our answer is
3. So, to the nearest hundred, our answer is
4
32,800
Example 4
Example 3
Round 4,365,901 to the nearest thousand
Round 3,999 to the nearest ten
4 3 6 5 9 0 1
3 9 9 9
1. We want the nearest thousand, so our Key Digit
must be the number that represents the thousands
which is the 5
1. We want the nearest ten, so the Key Digit must
be the 9 in the tens column
2. The unwanted digit to the right of it is a 9,
so we add one on, but we then need to add one on
the next 9, and then the 3!
2. The unwanted digit to the right of it is 9,
which is definitely more than 5, so we add one to
our Key Digit.
3. So, to the nearest ten, our answer is
3. So, to the nearest whole number, our answer is
4,000
4,365,000
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(c) significant figures In higher level SATs, and
especially in GCSE and A Level, the nasty
examiners are obsessed with Significant
Figures. Again, there is a lazy way of writing
this, which is sf or sig fig. Crucial The first
significant figure is always the first non-zero
number you come across. The second significant
figure is the number to the right of that, and so
on Remember the size of your rounded number
should be a similar size to the number in the
question, and you must use zeros to help you with
this.
Example 2
Example 1
Round 28.53 to 1 sig fig
Round 5,322 to 2 sig figs
2 8 . 5 3
5 3 2 2
1. The Key Digit has the be the first significant
figure, which must be the 2, as it is the first
non-zero number
1. The Key Digit is in the place of the 2nd
significant figure, which is the 3
2. The unwanted digit to the right of it is 2,
which is definitely less than 5, so we leave our
Key Digit alone
2. Now we carry on as normal looking to the
number to the right, which is an 8, so we add one
on.
3. So, keeping the size of the answer the same as
the question with a zero, to 1 sig fig the answer
must be
3. So, again using zeros to help us, to two sig
figs, our answer is
5300
30
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Example 4
Example 3
Round 0.027 to 1 sig fig
Round 305,216 to 3 sig figs
0 . 0 2 7
3 0 5 2 1 6
1. The 1st sig fig is the 3, the 2nd is the 0 (it
is after the 3, so its significant), so the Key
Digit is the 5
1. Our first significant figure is the first
non-zero number, which means its the 2
2. The unwanted digit to the right of it is 7, so
we add one to our Key Digit.
2. The unwanted digit to the right of it is a 2,
so we leave our Key Digit alone.
3. No need for extra zeros here, so to the 1
significant figure our answer is
3. We need some zeros to make our answer the
correct size, so to 3 sig figs
0.03
305,200
Example 6
Example 5
Round 4.0004 to 2 sig figs
Round 0.089722 to 2 sig figs
4 . 0 0 0 4
0 . 0 8 9 7 2 2
1. The 1st sig fig is the 4, and so the 2nd is
the 0 (it is after the 4, so its significant).
1. Our 1st non zero number is the 8, so the Key
Digit must be the 9.
2. The unwanted digit to the right of it is a 7,
so we add one on, but that gives us 10, so we
must add one to our 8 as well.
2. The unwanted digit to the right of it is 0,
which is definitely less than 5, so we leave our
Key Digit alone
3. Keeping our answer the right size, we have
3. So, to 2 sig figs, our answer is
0.090
4.0
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2. Approximations Right, now that we are experts
at rounding, we can use our skills to find
approximate answers (or estimates) to horrible
looking questions like the one at the start
Now, you would need to be a bit of a freak to do
this in your head, but if you were to round each
number to the nearest whole number, or 1
significant figure then you get
And if you now use our BIDMAS skills, you should
be able to say
The actual answer on the calculator is pretty
close 1.98521838
So, if we want to sound clever, we can say that
Where the funny sign means approximately equal
to
So, always look out for ways to use your rounding
skills to turn tricky looking sums into pretty
easy ones!
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• Good luck with your revision!