Approximation and estimation

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Rounding
We do not always need to know the exact value of a number.
There are about
fourteen hundred
pupils at Eastpark
Secondary School.
There are 1432
pupils at Eastpark
Secondary School.
There are about one
and a half thousand
students at Eastpark
Secondary School.
Why do we sometimes round figures rather
than giving an exact figure?

Rounding
There are four main ways of rounding a number:
The method of rounding used usually depends on what kind
of numbers we are dealing with and how the numbers are
being used.
For example, whole numbers might be rounded to the nearest
power of ten or to a given number of significant figures.
to the nearest 10, 100, 1000, or other power of ten
to the nearest whole number
to a given number of decimal places
to a given number of significant figures.

Round 34871 to
the nearest 100.
Look at the digit in the hundreds position.
The number will be rounded to either 34800 or 34900 as
it is between these two.
Look at the digit in the tens position.
If this digit is 5 or more then we round up to the larger
number (34900). If it was less than 5 we would round
down to 34800.
34871 = 34900 (to the nearest 100)
Rounding to powers of ten
100101102103104
17843

Rounding to powers of ten

Round 2.7524 to
one decimal place.
Look at the digit in the first decimal place.
The number will be rounded to either 2.7 or 2.8 as it is
between these two.
Look at the digit in the second decimal place.
If this digit is 5 or more then we round up to the larger
number (2.8). If it was less than 5 we would round to 2.7.
2.7524 = 2.8 (to 1 decimal place)
Rounding to decimal places
10-410-310-210-1100
42572

Rounding to decimal places

Rounding to significant figures
Numbers can also be rounded to a given number of
significant figures.
The first significant figure of a number is the first digit
which is not a zero.
For example,
4 890 351
and
0.0007506
This is the first significant figure
This is the first significant figure

The second, third and fourth significant figures are the
digits immediately following the first significant figure,
including zeros.
For example,
4 890 351
and
0.0007506
This is the first significant figure.This is the second significant figure.This is the third significant figure.This is the fourth significant figure.
This is the first significant figure.This is the second significant figure.This is the third significant figure.This is the fourth significant figure.

Discrete and continuous quantities
Numerical data can be discrete or continuous.
Discrete data can only take certain values.
Continuous data comes from measuring and
can take any value within a given range. It is
only as accurate as your method of measuring.
For example: boot sizes
the number of children in a class
amounts of money.
For example: the weight of an apple
the time it takes to get to school
heights of 15 year-olds.

The most this could be before
being rounded down is:
Upper and lower bounds for discrete data
The population of the United Kingdom is 59 million to
the nearest million. In what range of values could the
population be in?
The least this could be before
being rounded up is:
58 500 000 59 499 999
We can give the possible range for the population as:
58 500 000 ≤ population ≤ 59 499 999
or 58 500 000 ≤ population < 59 500 000
This value is called
the lower bound…
… and this value is called
the upper bound.

Last year a shopkeeper made a profit of £43 250,
to the nearest £50. What range of values could
this amount be in?
The lower bound is half-way between
£43 200 and £43 250:
£43 225
The upper bound is half-way between
£43 250 and £43 300, minus 1p:
£43 274.99
£43 225 ≤ profit ≤ £43 274.99
The range for this profit is:

Bounds for continuous data
The height of the Eiffel Tower is
324 metres to the nearest metre.
What range of values could its
height be in?
The least this measurement could be
before being rounded up is:
Lower bound = 323.5 m
The most this measurement could
be before being rounded down is
up to but not including:
Upper bound = 324.5 m

Bounds for continuous data
323.5 m ≤ height < 324.5 m
The height could be equal to 323.5 m
so we use a less than or equal to
symbol.
If the length was equal to 324.5 m
however, it would have been rounded
up to 325 m. The length is therefore
“strictly less than” 324.5 m and so we
use the < symbol.
The height of the Eiffel Tower is
324 metres to the nearest metre.
What range of values could its
height be in?

Upper and lower bounds

Village signs

A child’s toy has different shaped pieces that go into the same
shaped holes in a box.
Child’s toy
Another company makes the box with circular holes 4.1cm
diameter to the nearest 0.1cm.
They test the pieces and they do not fit. Explain how this
happened using upper and lower bounds.
One company makes the
pieces 4cm diameter to
the nearest 0.5cm.
One shape is a cylinder.

Carpet fitting
What is the smallest size of carpet that can be cut so
that it fits the room with no gaps round the edge?
The carpet is cut in the shop
and can be cut accurately to
the same accuracy.
The carpet costs £12 per square metre.
What should the shop charge?
A bedroom is measured for a
new carpet.
It is 4.2m by 5.4m measured
to one decimal place.

In the sweet shop
upper bound
lower bound
significant figures
rounded
A sweet shop has some jars of sweets with the number of
sweets on the front, but the numbers have been rounded to
one significant figure.
What is the lowest and highest number of sweets
possible in each jar? Explain your answers using some
or all of the terms:

Approximation and estimation

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