Mathematics Stage 5
NS5.2.1 Rational numbers
Part 1
Rounding rates and decimals
Number: 43631
Title: NS5.2.1 Rational Numbers
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material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Extract on outcomes from Mathematics Years 7-10 Syllabus © Board of Studies NSW, 2002.
www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf
Overview, Pp
iii-iv
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Contents – Part 1
Introduction – Part 1 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz.................................................................5
Significant figures..............................................................9
Effects of rounding ..........................................................13
To round or not to round? .......................................................15
Estimation .......................................................................17
Applying approximations .................................................21
Recurring decimals to fractions .......................................25
Writing recurring decimals ......................................................27
Fractions and the calculator....................................................28
Changing recurring decimals..................................................30
Converting rates..............................................................35
Further rate conversions.........................................................38
Suggested answers – Part 1 ...........................................43
Exercises – Part 1 ...........................................................49
Part 1
Rounding rates and decimals
1
2
NS5.2.1 Rational numbers
Introduction – Part 1
Rounding numbers was introduced in the stage 4 course. In this course
you need to extend this to rounding decimals to a specified number of
significant figures.
In addition, estimation strategies are further explored. In a calculation
not all digits are meaningful, even though a calculator may generate
them. Students need to develop intuition skills to know what level of
accuracy is needed in an answer. No formal definitions for rounding
procedures in calculations are required at this stage.
Two further areas of rational numbers are also further explored; recurring
decimals and converting rates. Both recurring decimals and rates were
introduced in earlier stages.
Indicators
By the end of Part 1, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
identifying significant figures
•
rounding numbers to a specified number of significant figures
•
using the language of estimation including rounding, approximating
and understanding level of accuracy
•
using symbols for approximation, such as ≈
•
determining the effect of truncating or rounding during calculations
on the accuracy of the results.
•
writing recurring decimals in fraction form using calculator and noncalculator methods
•
Part 1
converting rates from one set of units to another.
Rounding rates and decimals
3
By the end of Part 1, you will have been given the opportunity to work
mathematically by:
•
deciding on an appropriate level of accuracy for the results of
calculations
•
assessing the effect of truncating or rounding during calculations on
the accuracy of the results
•
appreciating the importance of the number of significant figures in a
given measurement
•
using an appropriate level of accuracy for a given situation of
problem solution.
•
recognise that calculators show approximations to recurring decimals
•
justify that 0.9 = 1
•
solve problems involving rates.
Source:
4
Extracts from outcomes of the Mathematics Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_
710_syllabus.pdf > (accessed 04 November 2003).
© Board of Studies NSW, 2002.
NS5.2.1 Rational numbers
Preliminary quiz
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1
2
a
Is 54 closer to 50 or 60? _______________________________
b
Is 11.3 closer to 11 or 12? ______________________________
c
Is 677 closer to 670 or 680? ____________________________
d
Is 677 closer to 600 or 700? ____________________________
Use the number line to help you answer the following.
8.3
8.4
8.5
8.6
8.7
a
Is 8.31 closer to 8.3 or 8.4? _____________________________
b
Is 8.58 closer to 8.5 or 8.6? _____________________________
c
Circle the number closest to 8.4:
8.35
d
8.50
8.44
8.39
Circle the numbers that could be written as 8.6, correct to one
decimal place:
8.54
8.57
8.60
8.63
8.66
Part 1
Rounding rates and decimals
5
3
Calculate the average of 45, 37 and 63 giving your answer to the
nearest integer. _________________________________________
4
An amount of money was rounded to the nearest five cents as $6.45.
What could the amount have been? __________________________
5
Use your calculator to calculate the following, correct to one
decimal place.
6
7
a
364 − 78.2 = ______________________________________
b
56.8 + 89.4
= _______________________________________
π
c
2 3
= ___________________________________________
+
7 10
d
12.8 +14.5
= ________________________________________
14.5 −12.8
Circle the letter beside each exact statement.
A
A table has a length of 145 mm.
B
There are 30 students in the class.
C
452 cars passed this intersection in the last hour.
D
Wilson poured 250 mL into the flask.
Write these fractions as decimals.
a
b
c
d
e
f
6
1
= _______________________________________________
4
3
= _______________________________________________
5
7
= ______________________________________________
10
19
= ______________________________________________
20
23
= ______________________________________________
40
9
= ______________________________________________
16
NS5.2.1 Rational numbers
8
9
Write these decimals as fractions.
a
0.5 = _______________________________________________
b
0.24 = ______________________________________________
c
0.105 = _____________________________________________
d
0.04 = ______________________________________________
a
Write
3
as a decimal. ________________________________
10
b
Write
1
as a decimal. _________________________________
3
c
d
3
1
or ? And by how much?
10
3
(Leave your answer as a fraction.) _______________________
Which is larger:
Comment on the difference between
3
1
and .
10
3
___________________________________________________
___________________________________________________
___________________________________________________
10 One dozen chocolate bars costs $28.20.
a
What is the cost of one chocolate bar? ____________________
b
What is the cost of seven chocolate bars? __________________
11 Roger cycles 54 kilometres in 6 12 hours. What distance does he
cover each hour? (Give your answer correct to one decimal place.)
_______________________________________________________
_______________________________________________________
Write his speed in km/h. __________________________________
Part 1
Rounding rates and decimals
7
12 Phillipa earns $220.85 when she works a seven-hour day.
a
How much does she earn each hour?
___________________________________________________
___________________________________________________
b
How much does she earn in a week when she works 33 hours?
___________________________________________________
___________________________________________________
Check your responses by going to the suggested answers section.
8
NS5.2.1 Rational numbers
Significant figures
All measurements have some degree of uncertainty. How great this
uncertainty is depends on both the accuracy of the measuring device and
the skill of its operator.
30
40 50 60 7
08
0
90
For example, bathroom scales are designed to
measure your mass to the nearest kilogram.
Masses less than this cannot be accurately
detected on this measuring device. So it
would be incorrect to say that you weighed
yourself on such scales and came up with, say,
54.349 kg.
It is important to be honest when reporting a measurement, so that it does
not appear to be more accurate than the equipment allows. You can
achieve this by controlling the number of digits, or significant figures,
used in reporting the measurement.
So the best you could say with 54.349 kg is that you weigh 54 kg.
Only these two digits have any meaning.
Here are the rules for counting significant figures:
•
All non zero digits are significant. Both 3489 and 4.512 contain four
significant figures.
•
Zeros between non-zero digits are always significant. Both 9308 and
40.02 contain four significant figures.
•
Zeros that do nothing but set the decimal point are not significant.
Thus 870 000 has two significant figures. The four zeros are there to
keep spaces between the 87 and the decimal point.
Part 1
Rounding rates and decimals
9
•
Trailing zeros that aren’t needed to hold the decimal point are
significant. For example 5.00 has three significant figures.
This is because the two zeros do not change the size of the number
but tell you that the two decimal places were measured but found to
be zero.
•
Zeros between significant digits are themselves significant.
The number 30.000 has five significant figures. While the zero after
the 3 in 30.000 may at first be seen as a placeholder to set the
decimal point, it is sandwiched between significant digits and so
becomes significant. That zero is no longer a placeholder.
•
Zeros to the left of the first non-zero digits in a number are not
significant; they merely indicate the position of the decimal point.
That is, they set the decimal point. So 0.007 has one significant
figure and 0.026 has two significant figures.
Sometimes confusion arises when a number ends in zeros that are not to
the right of a decimal point. For example, does 9800 have two three or
four significant figures? If the measurement was made to the nearest
hundred, then there are two significant figures. If the number was
measured to the nearest whole then there are four significant figures.
In this course you will avoid this confusion. If you are not sure whether
a zero is significant, assume that it isn’t. So if you read: ‘add the sample
to 200 mL of water’, assume the volume of water is known to one
significant figure.
You will need to refer to these rules for the following activity.
10
NS5.2.1 Rational numbers
Activity – Significant figures
Try these.
1
How many significant figures in each of these?
a
23.4 _______________________________________________
b
0.08582 ____________________________________________
c
6 300 000 __________________________________________
d
20.05 ______________________________________________
e
0.00530 ____________________________________________
f
50.0 _______________________________________________
Check your response by going to the suggested answers section.
In some questions you are asked to give your answer correct to a certain
number of decimal places, or to the nearest multiple of a power of 10.
Now you can round to a certain number of significant figures as well.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Write down the value of π correct to three significant
figures.
b
When built, the Sydney Harbour Bridge required
272 190 litres of paint to give its initial three coats.
Round this to three significant figures.
Part 1
Rounding rates and decimals
11
Solution
a
The first few digits of π are 3.14159… . Writing it correct
to three significant figures gives 3.14.
b
To three significant figures this number rounds to
272 000 litres.
Activity – Significant figures
Try these.
2
3
The distance from Earth to the Sun is 149 597 890 km.
Write this distance correct to:
a
two significant figures ________________________________
b
four significant figures ________________________________
The thickness of a piece of paper was measured as 0.01062 cm.
Write this length correct to two significant figures. _____________
Check your responses by going to the suggested answers section.
You have been practising significant figures. Now check that you can
solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 1.1 – Significant
figures.
12
NS5.2.1 Rational numbers
Effects of rounding
You are already familiar with the effects of rounding when it comes to
money. Items purchased in the supermarket often end in a number of
cents that are not a five or a zero. When the bill is totalled at the cash
register the total is rounded to a value nearest the closest five cents.
The effect is sometimes you pay two cents more than you should but
sometimes you pay two cents les than the total. Since this should even
out it is considered fair.
When counting, your answer is exact unless the count is very large.
For example, the word ‘estimation’ has ten letters. ‘Ten letters’ is not a
measurement; it is a count. The number 10 is exact here.
A measurement always has some uncertainty. Therefore, the result
of any calculation involving measurements has uncertainty.
When measurements are used in calculations, the precision of the
calculated values depends on the precision of the original measurements.
What is the length of a pen?
Emma and Dion measure the same pen. Emma says the pen is 14 cm
while Dion says it is 14.2 cm. Who is correct?
If both measured correctly, Dion’s measurement is just more accurate
than Emma’s. Emma measured to the nearest centimetre.
Dion measured to the nearest millimetre. In this case, a measurement of
14 cm is quite adequate. The extra accuracy given by Dion is often
not required.
The following example shows you that the size of the answer
is important.
Part 1
Rounding rates and decimals
13
Follow through the steps in this example. Do your own working in the
margin if you wish.
Televisions are classified by the diagonal measurement of the
screen in centimetres.
Use Pythagoras’ theorem to calculate the diagonal length of this
television screen. How would this television be classified?
38 cm
x
46 cm
Solution
x 2 = 46 2 + 38 2
= 2116 +1444
= 3560
∴ x = 3560
≈ 59.66573556 ... (using a calculator)
The television would be classified as a 60 cm set.
The exact value of
3560 is unimportant. What is important in
answering this question is that the television would have a
measurement of 60 cm.
•
When using a calculator you may have too many figures for a
reasonable answer.
•
Use the form of the question to decide on a reasonable level
of accuracy.
The answer needs to be at least as accurate as the question.
14
NS5.2.1 Rational numbers
In the above example the dimensions in the question were given to the
nearest centimetre, therefore it would be reasonable to answer to either
the nearest centimetre or the nearest millimetre.
To round or not to round?
Rounding too soon can introduce unnecessary errors, especially as a
calculator has the ability to keep the results of complicated calculations
in its memory.
Consider the sum of 3.45, 7.39, 2.7, 5.24, and 6.104 which will be
rounded to the nearest whole number.
Three students are asked to answer this question.
Round each number then add them
3 + 7 + 3 + 5 + 6 = 24
3.45 + 7.39 + 2.7 + 5.24 + 6.104 = 24.884
Now round the number up to 25.
Who is correct? Sue rounded after finding the sum. Sue is correct.
Ivan rounded each number at the start. He lost accuracy.
As you want the result correct to the nearest whole number, rounding
should normally only occur at the end of the calculation.
If a calculation involves several steps and you want to round as you go,
then keep at least one or two decimal places more, (or significant figures
more) than you are going to finally round to.
This activity will allow you to compare the effect of rounding before and
after calculations on the accuracy of the results.
Part 1
Rounding rates and decimals
15
Activity – Effects of rounding
Try these.
1
The following calculation needs to be carried out:
4.65 ×1.306
2.3
correct to one decimal place.
a
Round off each of these numbers before the calculation, then use
your calculator to calculate the answer.
___________________________________________________
___________________________________________________
b
Perform the calculation first, then round off to one decimal place.
___________________________________________________
___________________________________________________
c
How do the two answers compare? Which is more accurate?
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
The important thing to remember from this activity is that rounding, or
giving your answer to a certain accuracy, should occur after the
calculation is made.
Go to the exercises section and complete Exercise 1.2 – Effects of
rounding.
16
NS5.2.1 Rational numbers
Estimation
Estimation and checking of computations are vital parts of all
mathematics. It is most important that you get into the habit of always
checking that your answers are reasonable. You should decide whether
the answer seems sensible for the context of the question, and whether
the answer is of the appropriate order (size).
It is very easy to press the wrong calculator button and arrive at an
incorrect answer.
Consider the expression shown here. The calculator
answer is 2.064516 … . Correct to two significant
figures this number is 2.1. Did you get this value?
12.8
1.9 + 4.3
A common incorrect answer is 11.036842… ,. This rounds to 11 (given
to two significant figures). How can you tell this answer is incorrect?
13
13
12.8
which gives an answer near 2.
≈
=
1.9 + 4.3 2 + 4 6
•
By rounding:
•
By truncating:
12
12
12.8
which gives an answer near 2.
≈
=
1.9 + 4.3 1+ 4 5
To truncate means to cut off. In this case dropping off digits from one
end of a number causing loss of accuracy or information.
In both cases you can see the answer 2.1 is correct while the erroneous
answer, 11, is incorrect.
Rounding or truncating is a good way to estimate answers.
Of course you lose accuracy, but you do arrive at a quick,
approximate value.
Part 1
Rounding rates and decimals
17
When you are approximating you can
use the symbol or ≈ which means ‘is
approximately equal to’.
12
12.8
For example,
1.9 + 4.3 2 + 4
Follow through the steps in this example. Do your own working in the
margin if you wish.
Calculate the average height of three plants with the following
heights: 30.1 cm, 25.2 cm, 31.3 cm.
Solution
You know that average tells you something about the middle.
So you would expect the answer to lie somewhere between
25 and 31. Or perhaps you could make the following
estimation.
30.1+ 25.2 + 31.3 30 + 30 + 30
≈
3
3
90
=
3
= 30
86.6
= 28.87 .
3
You can round this to 28.9 cm since the three measurements are
When you perform the calculation you obtain
given to one decimal place.
There are three significant figures in each of the heights; even though
you are dividing the sum by a single digit, the three significant figures
should be retained in the answer.
It is important that you round numbers sensibly.
18
NS5.2.1 Rational numbers
Activity – Estimation
Try this.
1
Estimate the result of 4.8 × 56.2 . Show your working, and don’t use
a calculator.
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
In this example you rounded each number to find an estimate.
It is usual to round correct to one significant figure so that calculations
are kept simple.
Of course, a more accurate estimate would have been found if you
round to two significant figures, but then the calculation may become
more difficult.
You have been practising estimation. Now check that you can solve
these kinds of problems by yourself.
Go to the exercises section and complete Exercise 1.3 – Estimation.
An estimate is used to give a reasonable approximation of the
expected result.
Part 1
Rounding rates and decimals
19
20
NS5.2.1 Rational numbers
Applying approximations
Many times an approximate answer will do. For example,
•
this building is some 20 metres tall
•
the garage will need about 10 litres of paint
•
it will take approximately 40 minutes to drive to town.
Sometimes a quick calculation is performed so that the approximate
answer can be given to a fair degree of accuracy. Otherwise you might
only be guessing.
Consider the circle shown. Its radius is 12 cm. Approximate its area,
and then calculate it using your calculator.
The area is given by A = π r 2 .
Substituting r = 12 gives A = π × 12 2 .
12 cm
As a quick approximation the area is 3 ×12 ×12 ,
which is about 3 ×144 or about 3 ×150
(rounding off) to give 450 cm2. (Here π ≈ 3 .)
144 is rounded to 150 as it is easy to mentally multiply 150 by 3.
Using your calculator, the answer given on its screen is 452.3893421.
But should you report all these digits? Are they meaningful?
Part 1
Rounding rates and decimals
21
To answer these questions, consider two other circles.
Both of these have a radius close to 12 cm.
12.2
cm
A = π × r2
11.9
cm
A = π × r2
= π ×12.2 2
= π ×11.9 2
= 467.5946506 cm 2
= 444.8809357 cm 2
Can you see how different these areas are?
The circle that had a radius of 12 cm could have actually been a circle
with radius 12.2 cm or 11.9 cm that had been rounded to two significant
figures. So giving the answer correct to two significant figures is
appropriate. You cannot expect more accuracy than that.
So the area of the circle with r = 12 cm should be reported as 450 cm2.
However, in this course it will also be acceptable to write 452 cm2.
But accuracy beyond this cannot be justified.
The reason the calculator screen fills
with numbers is because π is given on
the calculator as 3.141592654. But we
are not justified in using all those digits.
Calculators just churn out numbers. It
is up to you to determine which of
those are relevant.
Depending on the question, many times an approximate answer will do.
When a more exact answer is required remember that just because a
calculator gives them does not mean you should necessarily report them.
22
NS5.2.1 Rational numbers
Activity – Applying approximations
Try these.
1
For the circle with radius 12 cm, estimate its circumference.
Then use a calculator to give the circumference to an appropriate
number of significant figures.
_______________________________________________________
_______________________________________________________
_______________________________________________________
2
The height of a flagpole needs to be calculated.
a
Use the diagram to estimate the height
of the flagpole.
_______________________________
20
.2
m
_______________________________
11.5 m
b
Use Pythagoras’ theorem to calculate the height of the flagpole,
to an appropriate number of significant figures.
___________________________________________________
___________________________________________________
___________________________________________________
c
Does your estimate agree closely with the calculated value?
Comment.
___________________________________________________
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
23
d
How do you know that your calculated, or estimated value,
looks appropriate as the answer to the question?
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
For all calculations involving measurements it is important to decide on
an appropriate level of accuracy for the results you give.
You have been practicing applying approximations. Now check that you
can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 1.4 – Applying
approximations.
24
NS5.2.1 Rational numbers
Recurring decimals to fractions
All fractions can be written either as terminating decimals or as recurring
(or repeating) decimals.
Examples of terminating decimals:
1
= 0.5
2
3
= 0.75
4
17
= 0.425
40
311
= 0.2488
1250
Some, but not all, fractions can
be written as terminating
decimals. This means the
decimal eventually ends.
A recurring (or repeating) decimal is a decimal number which goes on
forever, but some of the digits are repeated over and over again.
Examples of recurring decimals:
1
= 0.3333...
3
11
= 0.611111...
18
2
= 0.181818...
11
9
= 0.642857142857142857...
14
(The three dots … indicate that the decimal goes on forever.)
Notice how part, or all, of the decimal repeats.
All terminating or repeating decimals can be written as fractions, with a
whole number in the numerator and a whole number in the denominator.
Fractions are part of a group if numbers called rational numbers.
Part 1
Rounding rates and decimals
25
A rational number is a real number that
can be expressed in the form of a fraction
with the numerator and denominator
being whole numbers. (The denominator
cannot equal 0.)
Some numbers, however, are not rational, for example .
It cannot be written as a fraction. The first few digits of are
3.141592653589793238... and it is not rational because the number goes
on forever, and there is no pattern in the digits.
Sometimes we write an approximation
1
for π such as 3 7 . This mixed number is
close; 3.142857142857... and coincides
for the first few decimal places.
So why do some fractions have terminating decimals, while other
fractions have recurring decimals? It is because some fractions can be
written as equivalent fractions with denominators that are powers of
10 (10, 100, 1000, and so on).
3 75
=
4 100
= 0.75
5 625
=
8 1000
= 0.625
1
with a denominator that is a power
3
of 10 (and still keep the numerator as an integer). These fractions just
But you can’t write a fraction like
keep repeating forever.
26
NS5.2.1 Rational numbers
Writing recurring decimals
Because a recurring decimal is one whose digits after the decimal point
do not end but repeat the same sequence forever, a short cut can be used
to write it. These recurring decimals are written using this shortcut:
0.111111… = 0.1
0.121212… = 0.1 2
3
0.74537453… = 0.745
0.5388888… = 0.538
A maximum of two repeater symbols is used. The first dot goes over the
first digit repeating and one over the last digit repeating in the group.
Activity – Recurring decimals to fractions
Try these.
1
Write these using repeater symbols.
a
0.666666666 … = ____________________________________
b
0.3636363636 … = ___________________________________
c
0.0909090909 … = ___________________________________
d
0.142857142857 … = _________________________________
e
0.3456565656 … = ___________________________________
Check your response by going to the suggested answers section.
Remember, only one or two dots are used. Dots are not placed over
every repeating digit.
Part 1
Rounding rates and decimals
27
Fractions and the calculator
Modern calculators are able to change fractions to decimals, and terminating
decimals to fractions. You can do this using the fraction key, ab⁄c .
Try the following on your calculator. If your calculator does not respond
to these instructions refer to the manual that came with the calculator, or
contact your teacher.
(Check your answers at each step before proceeding to the
next question.)
Activity – Recurring decimals to fractions
Try these.
2
9
on your calculator. Press = .
20
Now press the fraction key again. What is shown on the screen?
Enter the fraction
Comment.
_______________________________________________________
_______________________________________________________
3
9
as 9 ÷ 20. Press = . Now press
20
the fraction key. What is shown on the screen? Comment.
Use your calculator to enter
_______________________________________________________
_______________________________________________________
The above two questions show how to change a fraction to a terminating
decimal, or a terminating decimal back to a fraction.
28
NS5.2.1 Rational numbers
4
Use your calculator to change these to fractions. Don’t forget to
press = after each decimal.
a
0.12 = _____________________________________________
b
0.1234 = ___________________________________________
c
0.12345 = __________________________________________
d
0.123456 = _________________________________________
e
0.12345678 = _______________________________________
You have seen that there is a limit to the number of digits that a
calculator can change to a fraction.
Does the calculator recognise repeating decimals and change them to
fraction form? Explore this in the next question.
5
1
to a decimal with your calculator. _______________
3
a
Change
b
Now press the fraction button. Is the decimal expansion
changed back to a fraction? ____________________________
c
Clear the screen and fill it with 0.3333333333 … .
Press
=
then the fraction key. Is this decimal changed to
1
?
3
___________________________________________________
Check your response by going to the suggested answers section.
This activity shows that while fractions can be changed to decimals (and
this works for any fraction you enter on your calculator), not all decimals
can be changed into fraction form. There is a limit to the number of
digits in a decimal number that the calculator can handle.
And certainly, calculators won’t change a screen full of repeating digits
to a fraction. Remember, a screen full of digits is still only an
approximation to the infinite digits in a recurring decimal.
Part 1
Rounding rates and decimals
29
Changing recurring decimals
Here is a method you can use to change recurring decimals to fractions.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Change 0.4 to a fraction.
Solution
Write 0.4 as 0.444444… and let it
equal x.
x = 0.44444...
As there is one repeating digit, multiply 10x = 4.4444...
x = 0.44444...
by 10, and then write the new equation
above the first.
Now perform a subtraction. Notice that 10x = 4.4444...
x = 0.4444...
all the digits to the right of the decimal
point subtract to give zero. Can you see 9x = 4
why?
It is now a simple matter to divide both
sides by 9.
So, 0.4 as a fraction is
x=
4
9
4
.
9
This method uses a little algebra to help you solve the problem.
The purpose of this technique is to eliminate all the repeating digits after
the decimal point.
Use the method to complete this activity.
30
NS5.2.1 Rational numbers
Activity – Recurring decimals to fractions
Try this.
6
Write 0.2 as a fraction.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
What about decimals with more than one repeating digit? The following
examples show two such decimal numbers.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Change 0.5 7 to a fraction.
b
Change 0.93 to a fraction.
Solution
a
Write 0.5 7 as 0.575757… and let
it equal x.
This time there are two repeating
digits so multiply both sides of the
equation by 100 (102) and write it
above the first. Then subtract.
Divide both sides by 99 and
simplify, if possible.
19
Hence x = , and so 0.5 7 as a
33
19
.
fraction is
33
Part 1
Rounding rates and decimals
x = 0.575757...
100x = 57.575757...
x = 0.575757...
99x = 57
57
99
19
=
33
x=
31
57
into the calculator and
99
, whether this fraction reduces to a
You can enter the fraction
check by pressing
=
simpler fraction. If the fraction can be reduced to simpler
terms the calculator will do this. In this case it does.
b
Now 0.93 consists of a repeating and non-repeating part.
As there is only one repeating digit (3), multiply both sides
by ten and proceed as before.
10x = 9.333333...
This time you will notice that not
x = 0.933333...
all digits to the right of the decimal
point disappear.
∴ 9x = 8.4
While the 3s cancel, you are left
with 9.3 – 0.9 = 8.4.
To remove the decimal point,
90x = 84
multiply both sides by 10.
Divide both sides by 90, and use
your calculator to simplify the
fraction, if that is possible.
∴ 0.93 =
84
90
14
=
15
x=
14
15
Once you have obtained your fraction you can always use your calculator
to check it gives the decimal expansion.
The point to remember is that with:
32
•
one repeating digit, multiply both sides by 10,
•
two repeating digits, multiply both sides by 100,
•
three repeating digits, multiply both sides by 1000, and so on.
NS5.2.1 Rational numbers
Activity – Recurring decimals to fractions
Try these.
7
Change these recurring decimals as fractions.
a
0.2 3
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
0.23
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
5
0.34
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Now use your calculator to check whether you are correct.
Check your response by going to the suggested answers section.
Part 1
Rounding rates and decimals
33
You multiply by the appropriate power of 10 (101, 102, 103, and so on),
so that the digits to the right of the decimal point begin to line up at some
early stage for your subtraction.
Be careful when writing recurring decimals. While 0.23 may appear
similar to 0.23 and 0.2 3 , they are different values.
23
, while both 0.23 and 0.2 3 are
100
recurring decimals with the values you calculated above.
0.23 (a terminating decimal) is just
You have been practising recurring decimals. Now check that you can
solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 1.5 – Recurring
decimals to fractions.
34
NS5.2.1 Rational numbers
Converting rates
A rate is a steady or constant relationship between two things.
For example, if you walked 100 metres in a minute your rate (speed, in
this case) is 100 m/min. In two minutes you would walk 200 m, and in
three minutes you would walk 300 m at this rate.
However, it would not take twice as long to boil two eggs, instead of just
one, if they are both in the boiling water at the same time. So, remember
to think carefully when solving problems with rates.
Many times when converting rates it is easier to lay the rate out
side-by-side in sentence form. For example, to convert an interest rate of
6% pa (per annum) to an equivalent interest rate per month, you can write:
1
6% for each year (12 months) = × 6% for each month
12
= 0.5% per month.
It is probably easier to set out a two-step problem as two single steps
rather than trying to make one huge leap. Similarly for multi-step
problems, break them up into smaller steps. For example, change a
speed of 15 m/s to kilometres per hour.
15 metres each second = 60 ×15 metres each minute … step 1
= 900 metres each minute
= 60 × 900 metres each hour … step 2
= 54 000 metres per hour.
From here it is simply a matter of reasoning that 54 000 m/h is equivalent
to 54 km/h.
Many times you are given a rate in one set of units and are asked to
change it to another. The following example shows just such a problem.
Part 1
Rounding rates and decimals
35
Follow through the steps in this example. Do your own working in the
margin if you wish.
Water flows through irrigation pipes at a rate of 12 L/s.
a
Calculate the number of litres that would flow in 1 hour.
b
Calculate the number of kilolitres that would flow in 1 day.
(1 kL = 1000 L ) .
c
Calculate the cost of one day’s water supply at a rate of
45 cents/kL.
Solution
a
12 L flows through in one second.
So in 1 minute 60 ×12 = 720L flows.
And in 1 hour 60 × 720 = 43200 L has passed through the
pipes.
b
In one day 24 × 43200 L = 1036800L passes through
the pipes. This is 1036800L ÷1000 = 1036.8 kL.
c
Cost = 1036.8 × $0.45
= $466.56
In this example you were relying on the following conversions:
•
60 seconds = 1 minute
•
60 minutes = 1 hour
•
24 hours = 1 day
•
100 cents = $1.
When converting from one unit to the other you needed to decide
whether this involves a multiplication or division.
For each of the following decide whether you need to multiply or divide
by the relevant conversion factors. Do this one step at a time.
36
NS5.2.1 Rational numbers
Activity – Converting rates
Try these.
1
A wheel rolls along the ground at 1 metre per second.
Complete the following working to convert 1 m/s to km/h.
a
At the same rate, how far will it travel in 1 hour?
The wheel travels 1 metre in 1 second.
How far will it travel in 1 minute? ______________________ m
How far will it travel in 1 hour? ________________________ m
How far is this in kilometres?_________________________ km
At 1 m/s, the wheel will travel _______ m or _______ km in
1 hour.
b
Express 1 m/s as a speed in km/h. _______________________
c
Peta cycles at 5 m/s What is her speed in km/h?
___________________________________________________
___________________________________________________
d
The speed limit on certain roads is 90 km/h.
Write this limit in m/s.
___________________________________________________
___________________________________________________
___________________________________________________
Check your responses by going to the suggested answers section.
Part 1
Rounding rates and decimals
37
Further rate conversions
In the part above you converted rates by reasoning whether to multiply or
divide by the conversion factor. In this part you will look at an
alternative procedure; that of linking up the conversions.
To start, consider changing a simple distance like 3000 metres to
kilometres. The conversion here is 1 km = 1000 m. So the fraction
1 km ⎛ 1000 m ⎞
⎜ or
⎟ are actually equal to one. This is because the
1 km ⎠
1000 m ⎝
value of the numerator and the value of the denominator are the same.
1 km
3000 m
/
×
= 3 km .
1000 m
1
/
The fraction is written in such a way so that the metres in the numerator
cancel with the metres in the denominator. The remaining units are
kilometres. This method relies on your writing not only the size of the
quantity, but also its units.
For simple conversions like this you would not bother doing it this way.
You would simply argue that as there are 1000 metres to each kilometre,
then 3000 m represents 3 km. However as the units become more
complex this method becomes more convenient.
Here is another simple example.
Follow through the steps in this example. Do your own working in the
margin if you wish.
The distance from Earth to the moon is 240 000 miles.
What is this distance in kilometres?
(Use 1 km = 0.6214 miles.)
38
NS5.2.1 Rational numbers
Solution
240 000 miles ×
1 km
= 386 000 km
0.6214 miles
Notice how ‘miles’ in the numerator in one unit cancels with that in the
denominator of the other. To answer this question you could have also
used the conversion 1 mile = 1.609 km.
Activity – Converting rates
Try this.
2
Use the conversion 1 mile = 1.609 km to change the distance from
Earth to the moon, 240 000 miles, to kilometres. (Answer correct to
the nearest thousand kilometres.)
_______________________________________________________
_______________________________________________________
Did you get the same answer as in the example?
Check your response by going to the suggested answers section.
Linking conversions like this can be extended.
A computer downloads at an average rate of 120 kB/sec.
How long (in minutes and seconds) will it take to download a 30 MB file?
First change this download rate to MB/min. (You will need to know that
1 MB = 1000 kB, and 1 min = 60 s.)
60 sec
120 kB 1 MB
×
×
= 7.2 MB/min .
1 sec 1000 kB 1 min
Part 1
Rounding rates and decimals
39
1 MB
changes kilobytes to megabytes,
1000 kB
60 sec
changes seconds to minutes.
and the
1 min
Go through and cancel the units to check you
The
are left with MB/min.
120 kB
1 MB
60 sec
MB
×
×
= 7.2
1 sec 1000 kB 1 min
min
This means it takes 1 minute to download 7.2 megabytes of data.
So to download a 30 MB file takes:
30 MB ×
1 min
= 4.16 minutes
7.2 MB
= 4 minutes 10 seconds
1
. Or you can use the methods you
6
learned in an earlier unit to change recurring decimals to a fraction.
1
And
of 1 minute (60 seconds) is 10 seconds.)
6
(You might recognise 0.16 as
Did you know? Because time is broken up
into 60 parts (1 minute = 60 seconds) just like
angles you can use the ° ’ ” key on your
calculator.
4.16 SHIFT ° ’ ” gives 4 minutes 10 seconds.
There is no limit to the number of conversions you can link together.
40
NS5.2.1 Rational numbers
Activity – Converting rates
Try this.
3
Speed in suburban streets is now 50 km/h. What is this in m/s?
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Notice how hours are changed to minutes, then minutes to seconds in the
solution provided. There is no need to do a side conversion of 3600
seconds = 1 hour.
(Check that you have linked the conversion fractions in the correct
orientation by going through and crossing out the units that appear both
in the numerator and denominator. You should be left with
metres/second.)
One final example showing how this method works.
Follow through the steps in this example. Do your own working in the
margin if you wish.
A tap drips 8 times per second. Five drops make 1 mL.
How much water, in litres, drips in one hour?
Solution
The question is asking to convert 8 drops/second to litres/hour.
1L
60 sec 60 min
8 drops 1 mL
×
×
×
×
5 drops 1000 mL 1 min
1h
sec
8 ×1×1× 60 × 60 L
=
5 ×1000 ×1×1 h
= 5.76 L / h
So in one hour 5.76 litres drip.
Part 1
Rounding rates and decimals
41
See if you can follow how these conversions are linked together.
Activity – Converting rates
Try these.
4
A fertiliser is made up by dissolving 25 grams for each litre of
solution. A litre of solution covers an area of 20 m2.
a
How many kilograms of fertiliser are needed for each hectare?
___________________________________________________
___________________________________________________
___________________________________________________
b
What weight of fertiliser is needed to cover an area of
245 hectares?
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
With practice you should be able to link up a number of conversions to
change from one set of units to another.
You have been practising converting rates. Now check that you can
solve these kinds of problems by yourself. The exercise also allows you
to practice solving problems with rates.
Go to the exercises section and complete Exercise 1.6 – Converting rates.
42
NS5.2.1 Rational numbers
Suggested answers – Part 1
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
a
50
b
11
c
680
d
700
2
a
8.3
b
8.6
c
8.39
d
8.57, 8.60, 8.63
3
1
45 + 37 + 63
= 48 which rounds to 48.
3
3
4
Any of $6.43, $6.44, $6.45, $6.46, $6.47
5
a
d
16.1
6
a, d are not exact as they are measurements. The others are counts.
7
a
0.25
b
0.6
e
0.575
f
0.5625
8
a
1
2
b
6
25
9
a
0.3
b
0.333333… or as 0.3
c
1
1
1 3
is greater: − =
3
3 10 30
d
10 a
16.9
b
45.5
c
0.6
c
0.7
d
0.95
c
21
200
d
1
25
The two fractions are close, but not the same.
9
1 10
3
=
while = .
3 30
10 30
$28.20 ÷12 = $2.35
b
7 × $2.35 = $16.45
11 54 ÷ 6.5 = 8.3 km. His speed is 8.3 km/h.
12 a
Part 1
$220.85 ÷ 7 = $31.55
Rounding rates and decimals
b
33 × $31.55 = $1041.15
43
Activity – Significant figures
1
a
3
b
4
e
3
f
3
2
a
150 000 000 km
3
0.011 cm
c
2
d
b
149 600 000 km
b
2.64039 … = 2.6
4
Activity – Effects of rounding
1
a
4.7 ×1.3
= 2.7
2.3
c
The second calculation is more accurate as the rounding occurs
after the answer is obtained. While the two answers are close in
this example, that may not always be the case in more complex
calculations.
Activity – Estimation
1
4.8 × 56.2 is approximately 5 × 60 = 300 . (Now when you use a
calculator you expect to get an answer close to this: 269.76.)
Activity – Applying approximations
1
C= 2×π ×r
= 2 × π ×12
This is approximately 25 × 3 (for ease of multiplication) or 75 cm.
(Other approximations are possible.) The calculated answer is
75.4 cm, which can be rounded to 75 cm.
2
a
An estimate: h 2 = 20.2 2 – 11.5 2.
So h2 is about 400 – 140.
(202 = 400, 122 = 144 ) . This gives h2 as about 260 and so h is a
bit more than 15 (15 2 = 225 ) , possibly around 16 m.
b
44
h = 20.2 2 −11.5 2
= 16.6 m (correct to three significant figures)
NS5.2.1 Rational numbers
c
For this example the estimate agrees closely. How did yours
go? An estimate is a quick, approximate value of the actual
answer.
d
The height of the flagpole should be shorter than the hypotenuse
of the triangle. But you also expect its height to be reasonable.
So you would expect that the flagpole would measure
somewhere in the vicinity of 10 m to 20 m.
Activity – Recurring decimals to fractions
1
a
0.6
e
0.345 6
b
0.3 6
c
0.0 9
2
0.45. The fraction is changed to a decimal.
3
9
. The decimal is changed to a fraction.
20
4
a
3
25
b
617
5000
c
2469
20 000
d
0.14285
7
d
1929
15 625
e
No response from the calculator (your calculator may
be different).
5
a
0.33333333
6
10x = 2.222222...
let x = 0.222222...
b
yes
c
no
∴ 9x = 2
2
x=
9
7
Part 1
a
100x = 23.232323...
let
x = 0.232323...
∴ 99x = 23
23
x=
99
Rounding rates and decimals
45
b
10x = 2.3333333...
let x = 0.233333...
∴ 9x = 2.1
So, 90x = 21
21
x=
90
7
=
30
c
let
(multiply both sides by 10)
(let the calculator simplify the fraction for you)
1000x = 345.345345345
x = 0.345345345
999x = 345
345
∴ x=
999
115
=
333
Activity – Converting rates
1
a
Now 1 metres in 1 second is 60 metres in 60 seconds.
60 m in 1 minute = 60 × 60 m in 60 min
= 3600 m in 1 h
= 3600 ÷1000 km in 1 h
= 3.6 km in 1 h
b
So 1 m/s = 3600 m/h = 3.6 km/h
c
Now 1 m/s = 3.6 km/h. So 5 m/s is 3.6 × 5 km/h = 18 km/h.
d
Now 90 km in 1 hour is 90 ×1000 m in 60 min
= 90 ×1000 ÷ 60min1min
= 90 ×1 000 ÷ 60 ÷ 60 m in 1 s
= 25 m in 1 s
= 25m/s.
46
2
1.609 kilometres
240 000 miles
×
= 386 000 km
1 mile
1
3
1h
1 min
50 km 1000 m
×
×
×
= 13.9 m/s
1 km
60 min 60 sec
1h
NS5.2.1 Rational numbers
4
a
1L
10 000 m 2
1 kg
25 g
×
×
×
= 12.5 kg/ha
2
1 ha
1000 g
1 L 20 m
12.5 kg of fertiliser are needed for each hectare.
b
Part 1
The weight is 12.5 × 245 = 3062.5 kg.
Rounding rates and decimals
47
48
NS5.2.1 Rational numbers
Exercises – Part 1
Exercises 1.1 to 1.6
Name
___________________________
Teacher
___________________________
Exercise 1.1 – Significant figures
1
2
How many significant figures in each of the following?
a
434 167 ____________________________________________
b
0.00124 ____________________________________________
c
27.98 ______________________________________________
d
0.01030 ____________________________________________
e
450.0 ______________________________________________
f
75 000 _____________________________________________
The Sun’s atmosphere has a temperature of 5500°C. To how many
significant figures is this given? ____________________________
3
4
Part 1
Write the following correct to two significant figures.
a
234 120 ____________________________________________
b
0.00124 ____________________________________________
c
17.98 ______________________________________________
d
0.01030 ____________________________________________
The largest known dragonfly lived over 270 million years ago.
It had a wingspread of 742 mm. Write this length correct to two
significant figures. _______________________________________
Rounding rates and decimals
49
5
Australia covers an area of 7 682 300 km2. Write this area correct to
three significant figures. __________________________________
6
a
1
of a second.
300
Write this number as a decimal correct to three
A bee can detect events separated by
significant figures.
___________________________________________________
b
Bee wings make 180 beats per second during flight.
How many beats is this in one hour, correct to two
significant figures.
___________________________________________________
7
A student wrote that a typical ant colony contains 40 321 ants.
a
Is the student justified in using this number to this level of
accuracy? Explain.
___________________________________________________
___________________________________________________
b
To how many significant figures should the student have written
this number? ________________________________________
8
Often you will be told to how many significant figures your
answer should be presented. If you are not, the following guides can
be used.
Multiplication and division: round the final result to the least number
of significant figures of any one term. For example:
15.03 × 4.87
= 36.83749371
1.987
= 36.8
The answer, 36.8, is rounded to three significant figures, because the
least number of significant figures was found in the term, 4.87.
The other terms, 15.03 and 1.987, each had four significant figures.
50
NS5.2.1 Rational numbers
Addition and subtraction: round the final result to the least number
of decimal places, regardless of the significant figures of any one
term. For example:
1.003 + 13.45 + 0.0057 = 14.4587
= 14.46
The answer, 14.4587, is rounded to two decimal places, since the
least number of decimal places found in the given terms is two (in
the term 13.45).
Calculate the following giving your answers in an appropriate form.
Part 1
a
(2.023)2 = __________________________________________
b
17.8 = ___________________________________________
c
5.09 ×16.51
= ______________________________________
2.5
d
14.6 + 18.905 + 6.0451 = ______________________________
e
687.45 – 237.255 = ___________________________________
Rounding rates and decimals
51
Exercise 1.2 – Effects of rounding
1
You can write a mass measured as 124.6 g as 0.1246 kg.
Use this example to comment on the fact that the position of the
decimal point has nothing to do with the number of significant digits
in a measurement.
_______________________________________________________
_______________________________________________________
_______________________________________________________
2
Tick the quantities that can be determined exactly.
The number of chairs in the room you are in now.
The number of grams in a kilogram.
The number of stars in the sky.
The number of red blood cells in a litre of blood.
The length of this page.
3
7
To measure the volume in these
measuring instruments look at the
bottom of the curved surface
between air and liquid.
a
5
4
6
3
Which measuring instrument
reads the measurement most
accurately? _________________
2
5
1
P
b
0.6
Q
0.5
0.4
0.3
0.2
0.1
R
Explain why a volume reading of 5.72 mL is reasonable on
instrument P.
___________________________________________________
___________________________________________________
___________________________________________________
52
NS5.2.1 Rational numbers
c
4
What is the volume of liquid in:
i
instrument Q? ___________________________________
ii
instrument R? ____________________________________
A temperature of 18.00 o C was recorded with one of these
thermometers. Which one was it? ___________________________
P
Q
10
8
20
9
10
1
30
2
3
4
R
0
2
Measure the length of this piece of metal using:
the ruler on the right.
6
Part 1
1
CENTIMETRES
___________________
0
the ruler on the left.
1
b
0
___________________
CENTIMETRES
2
a
2
5
1
Calculate the following giving your answers to an appropriate
number of significant figures.
a
π × (23.56)2 = _______________________________________
b
3.45 × 45.772
= _____________________________________
3.45 + 45.772
Rounding rates and decimals
53
7
A restaurant bill is rounded to $98 to the nearest dollar.
What could the bill’s total have been?
_______________________________________________________
8
The crowd at a carnival was given as 3600, correct to the nearest
hundred. In what range of numbers could the exact size of the
crowd be?
_______________________________________________________
54
NS5.2.1 Rational numbers
Exercise 1.3 – Estimation
1
Work out an estimate first, and then do each of these operations on
your calculator. Record your estimate and the calculator’s answer.
a
57 ×17
_______________________________________________________
_______________________________________________________
b
407.2 −153.6
_______________________________________________________
_______________________________________________________
c
58.2 × 31.06
9.35
_______________________________________________________
_______________________________________________________
d
3
2
1
1 ×4 −2
4
5
3
_______________________________________________________
_______________________________________________________
2
Use the level of accuracy suggested to round numbers and estimate
your results. Then use your calculator to check your answers.
a
28 + 30 + 89 – 54 + 76 – 38 (to the nearest 10)
___________________________________________________
___________________________________________________
b
379 + 1280 + 105 (to the nearest 100)
___________________________________________________
___________________________________________________
c
13 692 – 9481 (to the nearest 1000)
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
55
d
7.2 × 5.8 (to the nearest integer)
___________________________________________________
___________________________________________________
e
28.616 ÷ 3.92 (to the nearest integer)
___________________________________________________
___________________________________________________
f
84.5 +16.3
(to the nearest integer)
84.5 −16.3
___________________________________________________
___________________________________________________
3
A number was rounded to 60, correct to one significant figure.
Give three examples of what the number could have been?
_______________________________________________________
4
A number was rounded to 8.5, correct to two significant figures.
Give 3 examples of what the number could have been?
_______________________________________________________
5
Amanda estimated Ivan’s height to be 1.358 metres.
Can her estimate be that exact? Comment.
____________________________________
____________________________________
____________________________________
____________________________________
56
NS5.2.1 Rational numbers
6
a
Explain the difference between rounding and truncating.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
What is the effect of rounding or truncating during calculations
on the accuracy of the results?
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
57
Exercise 1.4 – Applying approximations
1
The floor of a room, 2.900 m by 4.100 m, is to be tiled using large
tiles of area 0.50 m2.
a
Estimate the area of the floor and the number of tiles required.
___________________________________________________
___________________________________________________
b
How many tiles would you order to be sure you had enough?
___________________________________________________
___________________________________________________
2
The walls of the same room are 2.4 m high. Estimate:
a
the perimeter of the room that is, the total distance around the
floor.
___________________________________________________
___________________________________________________
___________________________________________________
b
the total area of the walls
___________________________________________________
___________________________________________________
___________________________________________________
c
(Harder) The amount of paint needed to paint the walls using
two coats if 1 litre covers 16 m2. (Paint can be bought in
500mL, 1 L, 2 L, 4 L, and 10 L cans.)
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
58
NS5.2.1 Rational numbers
3
a
Estimate the number of bricks needed to build this wall.
___________________________________________________
___________________________________________________
b
What is the exact number of bricks (and parts) visible in this
drawing? Describe how you obtained this value.
___________________________________________________
___________________________________________________
___________________________________________________
4
A brick is about 24 cm long and 9 cm high.
a
What is the area of the front face of each brick?
___________________________________________________
___________________________________________________
b
Estimate to the nearest 100 the number of bricks needed for the
outer walls of a rectangular house 15 m long 9 m wide and
3.2 m high. (Show how you got your estimate.)
___________________________________________________
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
59
5
A restaurant bill comes to $130. Seven friends share it equally.
How much to the nearest dollar should each pay? Explain why you
came to this conclusion. (Remember you must pay at least $130
otherwise you aren’t paying enough.)
_______________________________________________________
_______________________________________________________
_______________________________________________________
6
Kyle paddled a 111 km kayak race in 10 h 15 min. Estimate, and
then calculate, his average speed in km/h.
_______________________________________________________
_______________________________________________________
_______________________________________________________
60
NS5.2.1 Rational numbers
Exercise 1.5 – Recurring decimals to fractions
1
2
is displayed as 0.666666667. (Check to see
3
whether your calculator is one of those that do this.) Why do you
On some calculators
think these calculators change the last ‘6’ in the display to ‘7’?
_______________________________________________________
_______________________________________________________
_______________________________________________________
2
a
b
c
Use your calculator to write these numbers as decimals.
i
1
=
9
___________________________________________
ii
2
=
9
___________________________________________
iii
3
=
9
___________________________________________
Can you see a pattern here? If so, use it to write these decimals
without using a calculator.
i
4
=
9
___________________________________________
ii
5
=
9
___________________________________________
iii
8
=
9
___________________________________________
How can you justify that 0.9 = 1 ?
___________________________________________________
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
61
3
a
b
(Harder) Write
7
as a decimal. ________________________
9
7
1 7
can be written as × . Without using a calculator, write
18
2 9
7
.
the decimal equivalent to
18
___________________________________________________
c
Use your calculator to check if you are correct. How did you
go? _______________________________________________
4
5
Write these terminating decimals as fractions in their lowest form.
a
0.7 = ______________________________________________
b
0.65 = ______________________________________________
c
0.175 = ____________________________________________
d
0.032 = ____________________________________________
Write these recurring decimals as fractions in their lowest form.
a
0.1
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
0.1 5
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
62
NS5.2.1 Rational numbers
c
0.15
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
d
3
0.12
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
e
(Harder) 0.12 3
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
f
(Harder) 0.123
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your answers with a calculator.
Part 1
Rounding rates and decimals
63
6
Example: you know that
1
= 0.333333... .
3
1 1
= ÷10 .
30 3
1
So
= 0.03333333... .
30
And
a
(Harder) Explain why 0.63333333… = 0.6 + 0.0333333333… .
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
Explain why 0.63333333… =
1
6
.
+
10 30
___________________________________________________
___________________________________________________
___________________________________________________
1
6
as a fraction. __________
+
10 30
c
Use your calculator to find
d
Use the method outlined in the notes to calculate 0.63333333…
as a fraction.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
e
64
Are your answers to c and d the same? ___________________
NS5.2.1 Rational numbers
7
a
b
7
= 0.777777... , write down (without using a
9
7
calculator) the decimal expansion for
. _________________
90
(Harder) If
What two fractions can you write that will add to 0.2777777…?
___________________________________________________
c
Now write 0.27 as a single fraction. _____________________
Use your calculator to check if you are correct.
Part 1
Rounding rates and decimals
65
Exercise 1.6 – Converting rates
1
A train travels 110 km in an hour and a quarter (1.25 hours).
If it maintains the same average speed, how long will it take to travel
330 km?
_______________________________________________________
_______________________________________________________
2
Rodney earns $192.50 when he works a 7-hour day.
a
Calculate his unit rate of pay. ___________________________
b
How much is he paid when he works 30 hours in a week?
___________________________________________________
3
The paint in a can covers 18 m2 and costs $44.75. How much would
it cost to paint the sides and top of a closed container in the shape of
a rectangular prism, with dimensions 3 m wide, 3 m high and 12 m
long? (Hint: draw a diagram to help you calculate the area to be
painted.)
_______________________________________________________
_______________________________________________________
_______________________________________________________
4
Dennis can type at a rate of 60 words per minute.
a
How long would it take him to type a 3500 word essay at
this rate?
___________________________________________________
___________________________________________________
b
Brendan types at 10 words per minute. How much longer will it
take Brendan to type the same essay?
___________________________________________________
___________________________________________________
___________________________________________________
66
NS5.2.1 Rational numbers
5
A plane leaves the airport at 9:15 am. It reaches its destination, 630
nautical miles away, at 11 am. (A speed of 1 nautical mile per hour
is called a knot).
a
How long did the trip take? _____________________________
b
⎛
distance ⎞
What is the average speed of the aircraft? ⎜speed =
⎟
⎝
time ⎠
___________________________________________________
___________________________________________________
c
If it takes 3 hours to do another trip at the same speed, how far
will the plane travel?
___________________________________________________
___________________________________________________
6
Water flows in a pipe at a rate of 66 L per hour. Change this rate to
mL/min.
_______________________________________________________
_______________________________________________________
_______________________________________________________
7
A leaky tap wastes water at the rate of 1 litre per hour (L/h).
a
How many millilitres are wasted in 1 hour?
___________________________________________________
___________________________________________________
b
At what rate does the tap leak in mL/min.
___________________________________________________
___________________________________________________
Part 1
Rounding rates and decimals
67
8
The interest rate on Jan’s credit card is 18% per annum (per year).
Cash advances attract interest at the daily rate until paid in full.
a
Calculate the daily interest rate, correct to 4 decimal places.
___________________________________________________
b
Find the interest charged on a cash advance of $250 repaid in
full after 11 days.
___________________________________________________
___________________________________________________
9
Fuel consumption of vehicles is measured in L/100 km.
The fuel consumption of Julio’s car is 8 L/100 km.
a
How much fuel is required to travel a distance of 100 km?
___________________________________________________
b
How far can Juan travel on 1 L of fuel?
___________________________________________________
c
What is Julio’s fuel efficiency rate of travelling in km/L?
___________________________________________________
d
How far can Julio travel on a 55 L tank of fuel?
___________________________________________________
10 A grazing property of 15 000 hectares carries 12 sheep/ha.
a
Calculate the total number of sheep. _____________________
b
Shearers are employed at a rate of $189 per 100 sheep shorn.
Find the wages paid to the shearers for the annual clip.
___________________________________________________
___________________________________________________
c
The average clip is 4.54 kg/sheep. Calculate the total weight
of wool.
___________________________________________________
___________________________________________________
68
NS5.2.1 Rational numbers
d
The average price that wool brings is $4/kg. Calculate the
income from the wool.
___________________________________________________
___________________________________________________
11 A car uses 8 L of petrol to travel 104 km. At the same rate of
consumption, how many litres would be needed for a journey of
624 km? (Hint: first find how far the car would go on 1 litre.)
_______________________________________________________
_______________________________________________________
_______________________________________________________
12 Marian is a regular blood donor. It usually takes 8 minutes to give
600 mL of blood. Calculate the rate of flow in litres per hour.
_______________________________________________________
_______________________________________________________
_______________________________________________________
13 The rate of exchange on the Australian dollar to English pounds
sterling (UK£) is UK£0.452 on a certain day.
a
How much would you get for $300 in UK£?
___________________________________________________
b
How much would you get for UK£200 in Australian dollars?
___________________________________________________
14 Calculate the rate of fuel consumption in L/100 km of Karen’s car if
she travels 450 km on 63 L of petrol.
_______________________________________________________
_______________________________________________________
Part 1
Rounding rates and decimals
69
15 Operator-connected phone calls are charged at $2.15 for each three
minutes or part thereof. Susana makes a 10 minute operator
connected call to her brother in Brewarrina. How much did the call
cost her?
_______________________________________________________
_______________________________________________________
16 Eric parks his car in a parking station. It charges $3.50 per half hour
or part thereof, with a maximum daily rate of $35. Quincy arrives at
9:20 am. How much is he charged if he leaves at:
70
a
11:45 am? __________________________________________
b
5:16 pm? ___________________________________________
NS5.2.1 Rational numbers