1 Sets 　　　　　 　　　　 　
　These pictures are of sets of objects.
　See how quickly you can sort out the objects below into different sets,
　and then think of a title for each of the sets you have made.
　　　　　　　　　 　　　　
　Now write down as many sets as you can think of in three
　minutes, starting off with 'a set of tools' and 'a set of picturecards'.
　We can make up a set in any way we choose providing each item
　is distinct from any other; for example:
　a set of kitchen utensils: knife, saucepan, kettle, tin-opener,
　mixing bowl
　a set of four-sided figures:
　a set of square numbers: 1, 4, 9, 16, 25, 36
　It is often useful to talk about sets in mathematics, and so
　special signs are used to write them down. Here are the first five
　things about the language of sets that we need to know.
1 The items in a set are known as the members of the set.
2 A set is enclosed in curly brackets. {1, 3, 5, 7, 9} is the set of odd
　numbers between 0 and 10.
3 We can write a set in two ways:
　(a) by listing the members of the set ____
　　{peaches, pears, plums, pineapples, pomegranates}
　How would you describe this set?
　(b) by writing a rule explaining what the members of the set
　　shallbe　一　{the set of days in the week}
4 Two sets are said to be equal if they have the same members.
　The order of the members does not matter. {a, b, c, d} = {d, b, a, c}
5 If a set has no members it is said to be an empty set and is
　shown by either of these signs: { }  or ¢
　　　　　　Exercises
A  Write down three different sets from the objects pictured below.

B  Write out the members of the following sets. Copy the first one,
　　Which has been done.
　1 {days of the week}
　　{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday,
　　Saturday}
　2 {numbers on a die}
　3 {months which have thirty days}
　4 { even numbers between one and fifteen}
　5 {vowels in the alphabet}
　6 {set of decimal coins}
　7 {months having thirty-two days}
　8 {numbers between eleven and twenty-five which can be divided
　　exactly by three}
　9 {the prime numbers between nought and twenty}
　10 {boys or girls in your form who can jump eight metres high}
　11 {six metric units for measuring weight}
　12 {the numbers between five and thirty-one which can be divided
　　　exactly by both two and three}
C  Make up a rule to explain the members of these sets. You may
　　have to say where a set must start and finish. For example:
　　{seconds, weeks, days, years, minutes, months, fortnights, hours}
　　{units used for measuring time}
　1 {yacht, steamer, canoe, raft, speedboat, rowing boat}
　2 {dog, rabbit, budgerigar, mouse, cat}
　3 {hearts, clubs, diamonds, spades}
　4 {barrel. bottle, vase, bowl, flask}
　5 {binoculars, telescope, microscope, spectacles, magnifying glass}
　6 {blackboard, teacher's desk, pupil's desk, chair, globe}
　7 {Tuesday, Thursday}
　8 {red, orange, yellow, green, blue, indigo and violet}
　9 {4, 8, 12, 16, 20, 24, 28, 32}
　10 {Pacific, Atlantic, Indian, Arctic}
　11 {inches, feet, yards, chains, furlongs, miles}
　１2 {millimeters, centimeters, decimeters, meters, decameters,
　　　hectometers, kilometers }
　13 {77, 88, 99, 110, 121, 132}
　14 {West Riding, East Riding, North Riding}
　15 {Asia, Europe, Africa, America, Australia, Antarctica}
 　　 Sort out these objects into different sets and then make up a ruleto explain each set.
　　　airplane, pencil, tea, fern, car, piece of chalk, coffee, bush, bus,
　　　pen, milk, tree, water, scooter, cactus, flower, bicycle.
E  Say which of the following pairs of sets are equal.
　1 {1, 2, 3, 4, 5} {3, 1, 5, 4, 2}
　2 {January, June, July} {months which begin with the letter J}
　3 {a, b, e, c} {b, d, e, a}
　{2, 4, 6, 8, 10} {even numbers between two and ten}  ( careful ! )
　5 {the three heaviest children in your form}
　　{the three strongest children in your form}
　6 {1, 4, 9, 16} {squares of the first four counting numbers}
　7 {6, 18, 24} {numbers between five and twenty-five which can be
　　divided exactly by six}
　8 {10,  1,000,  1,000,00 } {million, thousand, ten}
F  Make up four sets, and write a rule to explain each of them.

　　　　　　　２　Addition
When we are adding numbers with two or more figures we
cannot usually write the answer down straight away. To make
the working simpler, we add the different parts of the numbers
in turn.
You will remember that  124 = 100 + 20 + 4
and  345 = 300 + 40 + 5
If, therefore, we wished to add 124 and 345, we could set out the
Working like this:
    124 + 345 = 100 + 20 + 4 + 300 + 40 + 5
　　　            = 100 + 300 + 20 + 40 + 4 + 5
　　              = 400 + 60 + 9 = 469
It is easier, however, if we set down the numbers under each
other, remembering to make sure that each digit goes in its
correct column. Now look at the same example again.
　 　124 → 100 + 20 + 4
　+ 345 → 300 + 40 + 5
　　 469    400 + 60 + 9
Normally when we are adding big numbers we set down the sum
as shown on the left, but it is important to understand what we
are really doing.
           Exercises
A  When Leonard and his friends were planning a four-day cycling
　　tour they found they had to do several addition sums in order to
　　work out the journey for each day.
　　The map shows the route they were going to take, and the
　　distances in kilometers.

　　Find the following distances if they started from Newcastle and
　　set off to the North via Morpeth.
　1 Newcastle to Berwick   3 Jedburgh to Corbridge
　2 Berwick to Jedburgh    4 Corbridge to Newcastle
　　Find the following distances if they started from Bishop
　　Auckland and set off to the North via Corbridge.
　5 Bishop Auckland to Otterbum    7 KelsotoMorpeth
　6 Otterburn to Kelso              8 Morpeth to Bishop Auckland
　9 Find the average distance they would have to travel each day
　　by dividing the total distance by four.
　10 If their total cycling time was sixteen hours, find their average
　　speed in km/h by dividing the total distance by sixteen.

　　　　３　Measuring Length
How far is it from the Equator to the North Pole? Make a guess
and then find out how near you were when you have done the
exercises.
When the Metric System was devised in France this distance was
divided into 10,000,000 equal parts, and one of these parts was
taken as a new standard unit of length, the meter.
For small measurements we have fractions of a meter, and for
large measurements we have multiples of a meter.
Here are the metric words for fractions which are used for units
of length, weight and capacity.
　　　　milli-　      one thousandth
　　　　centi-      one hundredth
　　　　deci-       one tennth
From these words we get the following units of length:
　　　　millimeter     centimeter    decimeter
Since decimeters are not used, however, we have the following
Table. The abbreviations are shown in brackets.
　　　　10 millimeters (mm) = 1 centimeter (cm)
　　　100 centimeters (cm) = 1 meter (m)
millimeter ( 1/1000 of a meter )    centimeter ( 1/100 of a meter )
You should examine these units for yourselves on a meter stick.
　　　　　　Exercises
A   millimeter   centimeter   meter
　　　Copy these sentences, filling in the blanks by choosing the
　　　Correct word from those shown above.
　1 One thousand _____ equal one meter.
　2 One hundred _____ equal one meter.
  3 A millimeter is one tenth of a _____ .
　4 A man over two _____ in height is very tall.
　5 A drawing pin is about five _____ long.
　6 A man's palm measures about ten ______ .
　7 A football pitch is about one hundred _____ long.
　8 A pin is about two _____ long.
　9 A pair of school compasses is about ten _____ long.
　10 A pin head is about one _____ wide.B  Change to millimeters:
　　　　　1cm     5cm    30cm     1m     2m37cm
　　　Change to centimeters:
　　　　20mm    151mm     1m     9m     2m900mm
　　　Change to meters:
　　　　200cm    1,000mm    12,000mm     650cm
B  Find all the following measurements in centimeters and
　　millimeters.
　1 My span measures ______
　2 The width of my palm is ______
　3 The length of my foot is ______
　4 The length of my exercise book is ______
　5 The thickness of this book is ______
　6 The length of my pen (pencil) is ______
　7 The width of my thumbnail is ______
　8 The length of my desk is ______
　9 My height is ______
Things to do
If you have a meter rule, or measuring tape, practice measuring
the dimensions of your school and playground. Write down
estimates first and see whether your accuracy improves.
Look in a dictionary and list as many words as you can find
which start with mill- or milli- and are connected with the
number 1,000.

　　　　　４　Geometric Solids
What is the most noticeable difference between the solid things
we see in Nature and those we see in our man-made world?
Perhaps the pictures below will help.

There are many more similarities than we might imagine, but
You will see that the man-made objects are made up of regular
Straight or curved lines, whereas Nature is not so particular.
Which do you find more pleasing to look at?
A solid is a shape which has three dimensions: length, breadth
and height.
Here are some of the common geometric solids you will need
to know.
Prisms
The word 'prism' means 'sawn off', and you can see that if we
cut these prisms, as shown in the diagrams, all the ends have the
same shape. We say a prism has the same cross-section throughout.

Pyramids
A pyramid is a solid which tapers to a point at the top. Its base
may be any shape. Can you think of any unusual pyramids you
have seen?

　　　　　　　Exercises
A  Each of these objects is like one, or more, of the geometric
　　solids we have just met. Say what these solids are in each case.
　　The first one has been done for you.
Eulers Theorem
B  When we describe a geometric solid we refer to the three
　　parts: the faces, the edges and the corners.

　　It has been found that there is an interesting connection between
　　he numbers of these parts.
　　Copy and complete the table and then examine the numbers
　　for each solid. By adding two of the numbers and then
　　subtracting the third number you can obtain the same answer
　　for each one, and so discover Euler's Theorem for yourselves.
　　You may be able to think of other solids to add to your table.

solid	corners	faces	edges	calculation
rectangular prism	8	6	12
triangular prism
square pyramid
tetrahedron
cone

C  Use the following instructions to construct a tetrahedron. You
　　will need thin card, scissors and glue. It will help if you score the
　　folding lines, which have been shown by a broken colored line.
　1 Construct an equilateral triangle with sides l0 cm long.
　2 Bisect each of the sides and join up the mid-points.
　3 Draw in the three flaps.
　4 Cut out the whole figure.
　5 Fold as indicated, and stick down the flaps.

D  Draw a picture of an everyday scene to show examples of the
　　geometric solids you have just met. It would be a good plan to
　　go out into the playground or street first and make a list of the
　　shapes you can see.
E1 Name five games in which a spherical ball is used.
　2 Name one game in which the ball is not like a sphere.
　3 What name is given to a roof which is like half a sphere?
　4 If a piece of a wooden sphere were sawn off, what shape would
　　the flat parts of the two pieces be?
　5 What name should be given to a straight line extending from one
　　side of a sphere to the opposite side, passing through the center?
　6 Explain how spheres are used to make wheels spin more freely.
　　Copy these sentences, putting the correct word in the blanks.
　7 The British Isles are in the Northern _____ , whilst Australia is in
　　the Southern _____ .
　8 A sphere has a _____ surface.
F  Normally we think of a solid object as something completely
　　filled in, but in mathematics anything which has three dimensions
　　( that is length, breadth and height ) is called a solid. You can
　　make solids yourselves, like the two shown below, using straws
　　and pipe cleaners.
　　This illustration shows how to fix two pipe cleaners together
　　so that three straws can be joined at one point.

　　Try some of the other solids you have met in this chapter, or
　　perhaps some of the more difficult ones such as the octahedron,
　　dodecahedron or icosahedron, which you can find out about for
　　yourselves.

　　　　　　５　Averages　　　　　　　　高子数理研
You will perhaps have noticed that some boxes which contain a
Large number of articles display a label which gives the average
Contents for the box.
What do we mean by an average?
If we counted the matches in a large number of boxes, we should
find that some had slightly more, and some slightly less, than 50.
The number is always very close to 50.
Similarly, in the case of the paper clips, some boxes would have
slightly more, and some slightly less, than 100. The number is
always very close to 100.
Sometimes, when we are finding an average, there is a big
difference in the size of the numbers, but usually the average is
round about the middle value.
The table shows a girl's marks in an examination covering six subjects.
___________________________________________________________＿＿＿＿＿＿＿＿＿_
 Maths   English  Science  History   Geography  Housecrft
   8         6        7       10          6           5
-----------------------------------------------------
 Total marks = 8 + 6 + 7 + 10 + 6 + 5 = 42
 Average mark = 42 ÷ 6 = 7
　　　　　　　Exercises
　　　add   multiply   subtract   divide'   total
A1 Copy and complete the following sentence, choosing the correct
　　words from those given above.
　　To find the average of a set of numbers,   the  of all
　　the numbers by the number of members in the set.
　2 Say which of these boys is nearest to the average weight for
　　the set.
　　Albert 32 kg, Bertram 68 kg, Claude 38 kg, Desmond 45 kg, Eustace 50 kg
B  Find, by measuring, which of these lines is the same as the
　　average length for the set. Check your answer by calculating the
　　average in the normal way.
C1 If a motorist averages 9 km on one liter of petrol, how many
　　kilometers can he travel on 25 1itres?
C2 Here are the weights in kilograms of two tug-of-war teams in a
　　school sports.
　Reds: 40, 45, 52, 53, 48, 63, 56
　Yellows: 46, 61, 44, 40, 54, 56, 42
　　Find by how much the average weights of the two teams differ.
D  Try to find out some interesting information about averages
　　Concerning your own school and locality, and make a list of them
　　in your exercise book. Here are a few suggestions.
　1 The average age for your form.
　2 The average weight for your form.
　3 The average size of shoes for your form.
　　For the last five weeks:
　4 The average attendance for your form.
　5 The average attendance for your school.
　6 The average weekly amount collected by your school savings bank.
　　For the last year for which figures are available:
　7 The average daily rainfall for the months of May, June, July and
　　August.
　8 The average daily hours of sunshine for the months of May, June,
　　July and August.
　9 The average monthly accident figures for your town.
10 The average attendance at your nearest football club over the
　　past season.
Things to do
　Make a collection of labels which give the average contents of the packet.
　Find examples of the use of averages and stick them in your book.
　You should be able to find some interesting ones from labels and in
　newspapers and magazines.

　　　　　　　6 Measuring Weight
We have seen that for everyday purposes the most useful weights
are the GRAMME and the KILOGRAMME.
Perhaps, however, like the scientists, doctors and nurses you use
MILLIGRAMMES in your science laboratory for very small,
Accurate measurements.
To understand the metric system, therefore, we need to know
both the fractions and the multiples of a grame.
We have learned the words for fractions: here are the words for
multiples:
　　　　deca-   10 times
　　　　hecto-  100 times
　　　　kilo-    1,000 times
We now have seven units of measurement:
milligrames   centigrames   decigrames
grames
decagrames   hectogrames   kilogrames
Since only three of the units are normally used, however, this is
the table we must remember.
　　　　1,000 milligrames (mg) = 1 grame (g)
　　　　1,000 grames = 1 kilograme (kg)■
As the metric system is based on 10, it is easy to express the
units in decimal form. You will find the following table very
helpful. Copy it into your book and refer to it until you can
remember it.
   　　　1/10 = 0.1       1/100 = 0.01     1/1000 = 0.001
　　1 milligrame equals 1/1000 of a grame   1 mg = 0.001 g
　　10 milligrames equa1 1/100 of a grame  10 mg = 0.01 g
　　100 milligrames equa1 1/10 of a grame  100 mg = 0.1 g
　　1 grame equals 1/1000 of a kilograme   1 g = 0.001 kg
　　10 grames equa1 1/100 of a kilograme  10 g = 0.01 kg
　　100 grames equa1 1/10 of a kilograme  100 g = 0.1 kg
  　　       Exercises
A  Find the total weight of the ingredients for these recipes.
　　　1 Baked jam roll 　　         　　     2 Baked rice pudding
　　　　self raising flour 170 g 　  　　       rice 40 g
　　　　1/2 a level teaspoon of salt 2 g 　  small pat of butter 25 g
　　　　shredded suet 85 g     　　　　       　 sugar 70 g
　　　　150 ml water 150 g  　　　           　1/2 liter of milk 550 g
　　　　nutmeg 25 g
Things to do
　1 Practise weighing in kilogrammes, grammes and milligrammes
　　with any suitable weighing apparatus you have in school.
　2 Make your own spring balance, using a piece of wood, a length
　　of elastic and a tin lid.
　3 Make a collection of labels which give the weights in metric
　　units.

　　　　７　Distance and Direction
We have already seen that an angle is a measurement of
turning and that we can find many different instances of this
kind of measurement. Two examples are: the dials on a cooker
and the hands of a clock.

How many more examples can you think of?
Another important use of angles is concerned with the steering of
A ship. The navigator is constantly checking his position, and
Working out the distance his ship has to travel and the course
she must take. He must be an expert at reading maps and charts,
and drawing and measuring angles.
　Measuring distances
Since a map is a small picture representing a very large area it
must be drawn to a particular scale.
The Scale tells us the actual distance represented by a much
smaller distance on the map.
　　On a map of your town the scale may be l cm to 0.1 km.
　　On a map of the world the scale may be l cm to 500 km.
　Finding a direction
To find a direction we use a compass.
We use the points of the mariner’s compass in our everyday lives,
and you should memorize the dial as soon as possible.
What is the angle between the lines at the center?

To use the 360 degree compass we measure from due north in a
Clockwise direction. The direction is known as the 'bearing', and is
always given in three figures. East, for example, is given as 090
degrees.

       　　 Exercises
A  Make a dial for a mariner's compass as follows:
　1 Cut out a piece of paper 10 cm square. Gummed, colored paper
　　would be very suitable for this purpose.
　2 Fold the paper edge to edge both ways.
11
　3 Fold the paper corner to corner both ways.
　4 Open out the paper and mark in the eight compass points.
B  Change each of the eight points of the mariner's compass to 360°
　　bearings. ( North-east is the same as 045°).
C  The fence on one side of a triangular field is 100 meters long and
　　runs directly West to East. The fence at the western end runs on
　　a bearing of 040°; the fence at the eastern end runs on a bearing
　　of 325°.
　　Draw a diagram to represent the field and, by measuring, estimate
　　the length of the other two fences.
　　Use a scale of l cm to l0 m.
Things to do
　1 At a suitable place in your playground mark out the four cardinal
　　points of the compass. You could use white paint, or perhaps
　　your crafts teacher could suggest a more permanent method.
　2 Find out the difference between True North and Magnetic North
　　　　　　　
　　　　　　　８　Subtraction
　Here are some important points about subtraction, which you
　will find helpful.
1 When we subtract one number from another, or one quantity from
　another, we are finding a difference.
　For, example, the difference in:

2 When a figure in the bottom line is larger than the figure above it,
　we use a method of subtracting known as Equal Addition. It is
　called 'Equal Addition' because we add the same numbers to
　the top and bottom lines.
　Study the example carefully. The figures in color show where a
　number has been added.
　　　　657　　　600 + 50 + 7    　   600 + 150 + 17     　   657
　 　−269　　−200 + 60 + 9　 　−300 +  70 +　 9 　  −269　
    　　　　　　　　　　　　　　             　300 + 　80 +　 8    　　388
3 There are several different ways of expressing subtraction:
　From nine take away four.        　 Take four from nine.
　From nine subtract four.          　 Subtract four from nine.
　By how many is nine more        　 By how many is four less than
　ｔhan four?                    　　　   　 nine?
　How many must be added to  　     How many must be taken from
　four to make nine?             　　   nine to make four?
　Find the difference between   　    Find the difference between
　nine and four.                    　　　　four and nine.
　　　　　　　　Exercises
A  Complete these statements by putting the correct number inthe box.
　　1  29 + 11 = □    4  100−75 = □    7  □ + 77 = 100
　　2  40−11 = □    5  □−25 = 75      8  100−77 = □
　　3  40−29 = □    6  25 + □ = 100    9  100−□ = 77
B  Make up a 'subtraction' problem using the given phrase, and then
　　Find the answer.
　Example: ‘How many more…?'   How many more days are
　　there in July than February in a normal year?
　1 ‘How many more...?'     4 'Which is longer…?'
　2 ‘Which is heavier...?'     5 'How many are left…?'
　3 'Find the difference...'    6 ‘Which is higher...?'
C  Here are the heights of some of the highest mountains and hills
　　in Britain:
　　　Snowdon    1,085m         Kinder Scout    636m
　　　Scafell pike    978m         Ben Macdhui  1,309m
　　　Helvellyn     950m         Whemside       737m
　　　The Cheviot   816m         Devil's Chair     527m
　　　Ben Nevis    1,343m        BrownWilly      419m
　1 How much higher is Whernside than the Devil's Chair?
　2 What is the difference in height between the Cheviot and Scafell
　　pike?
　3 How much lower is Kinder Scout than Whemside?
　4 Find the difference in height between the highest of the
　　mountains and the lowest of the hills.
　5 Which is higher, Ben Macdhui or Ben Nevis, and by how manymeters?
　6 By how many meters is Scafell pike higher than Helvellyn?
　7 If a party climbed Scafell Pike and Helvellyn in one day how
　　many meters would they have climbed altogether?
　8 What is the difference in height between Snowdon, the highest
　　mountain in Wales, and Ben Nevis, the highest mountain in
　　Scotland?
　9 If a party going up Helvellyn had already climbed 480m, how
　　much further had they to climb to the summit?
10 If a mountain is regarded as land rising to a height of 900m or
　　more, how far does the Cheviot fall short of this?

       　　　　　 ９　Volume
If each of the above tanks were filled with water which would
hold most? To find out we should have to measure the space
inside each of the tanks.
In mathematics this is called finding the volume.
The unit of measurement must itself take up space, and so we
use a cube.
　　□ One cubic centimeter
Each edge of a cube has the same measurement and each face
is a square.
It is named according to the measurement of the edges.
See whether yqu can find a quick method of working out the
Number of centimeter cubes □ contained in these rectanglular
Solids.

Now study the following. Perhaps this method is very much like
one you have found out for yourselves.

From the above diagrams we see that:
For a rectangular solid Volume = length × breadth × height.
　　　　　　　　Exercises
A1 Look carefully at this net, or plan, of a one centimeter cube. The
　　colored lines show the flaps that would be needed to stick the
　　cube together.

　　Now draw the net of a three centimeter cube on thin card, using a
　　setsquare to make sure that all the angles are right angles.
　　Cut out your net, fold it along the sides of the squares, and stick
　　it together.
　　How many one centimeter cubes would fit into your three
　　centimeter cube?
A2 Explain why a cube is used to measure volume, and why any
　　other solid shape would not be suitable.
C  Before finding a volume, we must decide what measurements
　　to use. If we are going to use cubic centimeters, the length,
　　breadth and height must be found all in centimeters; for cubic
　　meters, the three measurements must be found all in meters,
　　and so on.
　　Find, in cubic centimeters, the volume of the following
　　rectangular solids. Here is the working for no.1 :
　　　　V =1×b×h              (Notice that the abbreviation for
　　　　　=(4×3×3) cm³                   cubic centimeters is cm³)
　　　　　=36cm³ 
　　　　　length    breadth   height
　　　1    4cm       3cm      3cm
      2    6cm       5cm      4cm
      3    7cm       6cm      5cm
      4    2m       3m50cm   5m

         10　Capacity　　　　　　　　　高子数理研

What have all the above objects got in common? The answer, of
Course, is that they are all vessels for holding some kind of liquid.
When we measure the liquid they contain we say we are finding
their capacity.
The standard unit of capacity in the metric system is the liter
which is about the amount a large bottle will hold.
It is useful to know that when the metric system was worked out
the measurements of length, capacity and weight were all
related to each other. So we find that
the space filled by a liter of water is equal to 1,000 cubic
centimeters, and
the weight of a liter of water is equal to a kilogram (1,000g).
　　1,000 cubic centimeters     1 liter     1 kilogram
We have already met the words milli- (1/1000), centi- (1/100) and
deci- (1/10), but since the centiliter and deciliter are rarely used
we just need to remember that
　　1,000 milliliters (ml) = 1 liter (l)
If you have a liter measure in school make a collection of
containers and find their capacity. You might also find in.
grams the weight of water they will hold and then check
this against the capacity.
　　　　　　　　　Exercises
A  petrol  vinegar  goldfish  saucepan  domestic  thermos
　　　　　Can   bottle    bowl               washing   flask
　　　machine
　　Draw the containers listed above and under each one write
　　down how much you think it might hold. Give your answers in
　　liters.
　　Try to find out for yourselves if your answers were about right.
B1 How many 50 milliliter bottles can be filled from a tank holding
　　1 liter?
B2 How many 5 milliliter doses can be given from a half liter bottle?
B3 How many half liter bottles of milk can be filled from a 250 liter
　　churn?
B4 A family have one liter of milk every day throughout April, May
　　and June. How many liters does this amount to altogether?
B5 How many half liter tins of paint can be filled from a 55 liter tank?
B6 A paraffin lamp can be filled 20 times from a one liter bottle; how
　　many milliliters does the lamp hold? 
B7 If 10 milliliters of flavouring must be added to each of 350
　　bottles, how many liters of flavouring will be required?
B8 If 460 milliliters have been poured from a liter bottle, how many
   milliliters remain?
B9 If a bottle of perfume holds 50 milliliters, how many bottles
　　can be filled from a 4.5 liter tank?
B10 A motorist finds his car uses a liter of oil for every 1,200
　　Kilometers. In 18,000 kilometers how many liters of oil would
　　he use?

C  All the questions in this section refer to the containers shown
　　above.
　1 How many glasses can be filled from 5 liters of milk?
　2 How many times must the kettle be boiled to fill the teapot
　　6 times?
　3 How many liters of orange juice are need to fill 48 glasses?
　4 One liter of milk is used for 28 cups of tea. How many times must
　　the teapot be filled to supply tea for these cups?
　5 How many liters of water would be needed to fill the kettle
　　16 times?
　6 If a bottle of concentrated orange juice holds one liter, how
　　many liters of water must be added to it to fill 36 glasses?
　7 How many times would the teapot have to be filled to supply
　　56 cups?
　8 If 48 children at a party each had a glass of lemonade, how many
　　2 liter bottles were needed?
　9 How many liters of water would be needed to fill 21 cups and
　　21 glasses?
10 At a party 21 adults each had two cups of tea, 15 older children
　　each had a glass of orange juice, and 7 small children each a cup
　　of lemonade. How much liquid does this amount to altogether?
　
　　　　　　　11　Decimal Fractions
　Here are some important facts we have learned about our
　every day numbers:
1 We use the decimal system, which includes both whole numbers
　and fractions.
2 The decimal system is based on the number 10.
3 The place of a figure decides its value.
　This table is similar to one you have seen before. Study it
　Carefully and notice the value of the third decimal place.
　This shows that:
One tenth of a hundredth = one thousandth.

　Multiplying by ten
　　Since  1.2 = 1 + 2/10  then  10×1.2 = 10×1 + 10×2/10
    　　　　　　　　                　　                 = 10 + 2
                    　　　　　　　　   　　              = 12
In each of the examples shown on the left, the top number has
been multiplied by ten, and the answer placed underneath.
What happens to the figures?
Do the noughts make a difference?
In which of these two sets do the noughts help to decide the
　　value of the number?
　Set A  { 0.5,  0.50,   00.5,   00.50 }
　Set B  { 0.05,  50,   0.005,   500 }
In Set A the figures remains in the same place, so each member
of the set has the same value, five tenths.
In Set B the noughts give the figure 5 a different place in each
case, so each member of the set has a different value from the
others.
Addition and Suhtraction
The methods for adding and subtracting decimal fractions are
exactly the same as for whole numbers.
The important thing is to ensure that each figure goes into its
Correct column by putting the decimal points under each other.
　　Examples: 0.57 + 21 + 3.9   0.57   Notice that some nouthts
    　　                      21.00   have been put into complete
        　　                 + 3.90   the columns.
            　　              25.47   This helps to make certain
 that each figure goes into its correct column.
　　　　　　　　　　Exercises
A  Express each of the following as one number, in decimal form.
　Example:  20 + 3 + 7/10 + 6/100 = 23.76
　　10 + 8 + 9/10 
　　50 + 9 + 2/10 + 8/100
　　90 + 6 + 7/10 + 3/100 + 2/1000
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　高子数理研