Download MONEY MANAGEMENT 12 DIRECTED NUMBERS

Essential Maths Skills
Subject Knowledge and Understanding
for QTS Students
Number
Book C
Contents:
Pages
Chapter 8: Directed Number
1 - 10
Chapter 9: Indices and Standard Form
11 - 22
Chapter 10: Average Speed
23 - 30
Chapter 11: Rational and Irrational Numbers 31 - 32
1
Chapter 8
DIRECTED NUMBERS
This chapter is about positive and negative numbers. Some times the 'signs' are
written in superscript, sometimes not. Both forms are used in this chapter.
POSITIVE NUMBERS
These you know very well. They are numbers such as
3 which can be written as +3
46 which can be written as +46
14.67 which can be written as +14.67
a which can be written as +a
RULE
Any number or letter, which is written without a sign is a POSITIVE NUMBER.
Positive numbers may contain a plus sign, but it is common to see them with no
sign at all.
NEGATIVE NUMBERS
These are numbers (and letters) which have a minus sign in front of them:
NEGATIVE 3 is written -3
NEGATIVE 46 is written -46
NEGATIVE 14.67 is written -14.67
NEGATIVE a is written -a
A negative number, or letter, ALWAYS has a MINUS sign in front of it.
ADDING AND SUBTRACTING DIRECTED NUMBERS
As you can see all numbers have a DIRECTION - positive or negative. This is
best shown, at this stage, by using a number line and doing addition and
subtraction along the number line.
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
2
The number line is INFINITELY LONG, because the set of positive and negative
numbers has no end. The line drawn is just a short part of the number line.
ADDITION using the number line
Eg 1 If you start on 0 and add 3, you move 3 places to the right, and your
answer is +3.
Eg 2 Start at +1 and add 3. Your answer is +4.
Eg 3 Start at -1 and add 3. Your answer is +2.
Eg 4 Start at -2 and add 3. Your answer is +1.
WHEN YOU ADD, YOU MOVE TO THE RIGHT ALONG THE NUMBER LINE.
YOU MOVE TO THE LEFT ALONG THE NUMBER LINE WHEN YOU
SUBTRACT.
SUBTRACTION using the number line
Eg 5 Start at 0 and subtract 1. Your answer is -1.
Eg 6 Start at 0 and subtract 2. Your answer is -2.
Eg 7 Start at -1 and subtract 2. Your answer is -3.
Eg 8 Start at +1 and subtract 2. Your answer is -1.
IS THERE A FASTER WAY OF ADDING AND SUBTRACTING DIRECTED
NUMBERS?
YES! There are certain rules, which when followed, will make these operations
easy. They apply EVERY TIME with numbers and letters.
Eg 1 Another way of writing 0 add 3 is given here. Brackets are placed round
each number and the corresponding sign,
(0) + (+3)
and written out fully is 0 + 3
Answer = 3
Notice that the brackets have been removed in the second line. You are advised
to write the second line for every example to avoid any confusion.
Notice the + sign in front of the second bracket.
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
3
Also, as with a number or a letter, if a BRACKET has no sign in front (like the
first bracket) you must assume that it is positive.
Eg 2
+1 add 3
becomes
(+1) + (+3)
2nd line = +1
+3
= +4
Eg 3
-1 add 3
becomes
(-1) + (+3)
2nd line = -1
+3
= +2
Eg 4
-2 add 3
becomes
(-2) + (+3)
2nd line = -2
+3
= +1
REMEMBER
Look carefully at the signs. You will see that where there is a PLUS sign before
a bracket the sign INSIDE THE BRACKET REMAINS THE SAME. This is an
important rule.
In the second line there are only numbers with their signs (no brackets, no extra
signs).
IF THE SIGNS ARE DIFFERENT, WRITE DOWN IN THE ANSWER THE SIGN
OF THE BIGGER NUMBER AND SUBTRACT THE NUMBERS.
but
IF THE SIGNS ARE THE SAME, WRITE THAT SIGN IN YOUR ANSWER AND
ADD THE NUMBERS.
Look at these
a)
(+5) + (+4)
= +5
+4
SIGNS are the same (like signs). Put that sign down and ADD the numbers
= +9
b)
(+10) + (+7)
= +10
+7
SIGNS are like. Put that sign down and ADD the numbers.
= +17
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
4
c)
(+14) + (-4)
= +14
-4
SIGNS are different (unlike signs). Put the sign of the greater number down and
SUBTRACT the numbers.
= +10
d)
(-20) + (+10)
= -20
+10
= -10
SIGNS are unlike. Put down the sign of the greater number. Subtract the
numbers. Take special note of
e)
(+10) + (-10)
= +10
-10
= 0
f)
(-12) - (-12)
= -12
+ 12
= 0
Go back to Examples 5, 6, 7 and 8 which are subtraction questions.
Eg 5
0 minus 1
becomes (0) - (+1)
= 0 -1
THE MINUS SIGN IN FRONT OF THE SECOND BRACKET CHANGES THE
SIGN INSIDE THE SECOND BRACKET. (- becomes + and + becomes -).
The rules apply now with regard to like and unlike signs so the answer is -1.
Eg 6
0 minus 2
becomes (0) - (+2)
= 0
-2
= -2
Eg 7
-2 subtract 2
= (-2) - (+2)
= -2
-2
= -4
Eg 8
1 take away 2
= (+1) - (+2)
= +1
-2
= -1
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
5
SUMMARY OF RULES
1) Positive numbers, letters or brackets need not have a PLUS SIGN in
front of them, eg,
3 = +3
d = +d
(+16) = +(+16) = +16
2) Negative numbers, letters or brackets always have a minus sign in
front of them, eg,
-d
-(+7) = -7
3) A plus sign in front of a bracket allows the sign INSIDE the bracket to
remain the same.
4) A minus sign in front of a bracket means that the sign INSIDE the
bracket must change (+ to - and - to +).
5) If numbers have different (unlike) signs then write the sign of the
BIGGER number in the answer and SUBTRACT the numbers.
TWO TERMS OR MORE
You may have more than two terms. There are two ways of tackling this type of
question.
(+12) - (+4) + (-10)
Always write out the second line as follows:
+12
-4
-10
+12 -14
= -2
add together numbers with the same sign OR proceed along the line one number
at a time
+12 -4 = +8 (first two terms)
+8 -10 = -2 (answer from first line, then third term)
The rules do not change if there are more than two terms in the question. You
carry on in an orderly way, one step at a time, to solve any problem.
Exercise 1
1) (+3) + (+9)
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
6
2)
3)
4)
5)
6)
7)
8)
9)
10)
(+10) + (-5)
(-15) + (+2)
(-20) + (-20)
(+13) - (+10)
(+24) - (-12)
(-21) - (+21)
(-21) - (-21)
(+12) + (-12) - (-12)
(+100) - (-50) + (+20)
Before going on to multiplication of directed numbers:REMEMBER - when you see a bracket, everything INSIDE the bracket is
multiplied by the number or letter with the sign which is outside the
bracket.
MULTIPLICATION AND DIVISION OF DIRECTED NUMBERS
MULTIPLICATION
POSITIVE multiplied by POSITIVE = +
NEGATIVE multiplied by NEGATIVE = +
__________________________________
POSITIVE multiplied by NEGATIVE = NEGATIVE multiplied by POSITIVE = __________________________________
+ x + = + and - x - = +
_______________________
+ x - = - and - x + = _______________________
So, (+3) x (+4) = +12
(-3) x (-4) = +12
(+3) x (-4) = -12
(-3) x (+4) = -12
REMEMBER (+3)2 and (-3)2 BOTH EQUAL +9
Written out fully, (+3)(+3) and (-3)(-3)
Remember ( )( ) means times
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
7
(+10)2 and (-10)2 = 100
Written out fully, (+10)(+10) and (-10)(-10).
DIVISION
POSITIVE divided by POSITIVE = +
NEGATIVE divided by NEGATIVE = +
_______________________________
POSITIVE divided by NEGATIVE = NEGATIVE divided by POSITIVE = _______________________________
+
=+
+
and
= +
___________________
+
= -
and
= +
__________________
So
and
+10
= +2
+5
+10
= -2
-5
-10
= +2
-5
-10
= -2
+5
SUMMARY OF RULES
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
8
1) When multiplying or dividing LIKE SIGNS, the answer will be positive.
2) When multiplying or dividing UNLIKE SIGNS, the answer will be negative.
3) Multiply or divide numbers as normal.
Exercise 1
1)
3 x 4
2)
-3 x -4
5)
(-4)2
6)
(-12)(+3) 7)
8)
-15 ÷ -5
9)
+15 ÷ -5
3)
(-10)(-4)
(+3)2
+15 ÷ +5
10) +1000 ÷ -10
11) +12 ÷ -6
12) -36 ÷ +6
13) +24 ÷ -6
14) +16 ÷ -3
15) +14 ÷ -3
16) -125 ÷ -5
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4)
SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
9
ANSWERS
EXERCISE 1
(ADDING AND SUBTRACTING)
1)
Becomes +3 + 9 = +12
2)
Becomes +10 - 5 = +5
3)
Becomes -15 + 2 = -13
4)
Becomes -20 - 20 = -40
5)
Becomes +13 -10 = +3
6)
Becomes +24 +12 = +36
7)
Becomes -21 - 21 = -42
8)
Becomes -21 + 21 = 0
9)
Becomes +12 - 12 + 12 = +12
10) Becomes +100 + 50 + 20 = +170
EXERCISE 2
(MULTIPLICATION AND DIVISION)
1)
+12
2)
+12
3)
+40
4)
+9
5)
6)
-36
7)
+3
8)
+3
9)
-3
10) -100
11) -2
12) -6
13) -4
14) -51/3
+16
15) -42/3
16) +25
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Chapter 9
INDICES AND STANDARD FORM
INDICES/INDEX
The index is the power of a number.
23 is said 'two to the power three' OR 'two cubed' and means 2 x 2 x 2 (Ans = 8)
32 is said 'three to the power two' OR 'three squared' and means 3 x 3 (Ans = 9)
Look at these examples with indices
1.
104 - is said 'ten to the power 4'
- means 10 x 10 x 10 x 10 = 10000
2.
53 - is said '5 cubed'
- means 5 x 5 x 5 = 125
3.
a4 - is said 'a to the power 4'
- means a x a x a x a
4.
g6 - is said 'g to the power 6'
- means g x g x g x g x g x g
5.
z7 - is said 'z to the power 7'
- means z x z x z x z x z x z x z
REMEMBER!
X1 = X
The following examples illustrate the rules which apply to indices. You are
advised to learn them.
MULTIPLYING NUMBERS OR LETTERS WITH POWERS
Example 1
23 x 23 = 21 x 21 x 21 x 21 x 21 x 21 = 26 = 64
A QUICK way of doing this is to add the powers when you are multiplying
numbers with powers.
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
12
Example 2
a6 x a3 = a6+3
= a9
b4 x b = b4+1
= b5
Example 3
NOTE
b1 is the same as b
Example 4
3k2 x 4k = 3k2 x 4k1
Written out fully, this is
3 x k 1 x k1 x 4 x k 1
Multiply the numbers
(3 x 4 = 12)
Add the powers
1 + 1 + 1
Answer
= 12 x k3
= 12k3
DIVISION OF NUMBERS OR LETTERS WITH POWERS
Example 1
2
2
3
2
1
=
1
2 x 2 x 2
1
2 x 2
1
1
Now cancel, which gives 21 which is the same as 2
A QUICK way to divide letters or numbers with powers is to subtract the powers
23-2 = 21 or 2
Example 2
a
a
6
3
1
=
1
1
1
1
a x a x a x a x a x a
1
1
a x a xa
1
1
= a6 - 3
= a3
Example 3
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
13
4
1
1
1
b
b x b x b x b
=
1
b
b
1
= b4 - 1
REMEMBER
= b3
b1 is the same as b
Example 4
2
1
12s
12 x s x s
=
1
4s
4 x s
1
12
=3
4
Di v i de t he number s
Subtract the powers
s2 - 1 = s1
3s1 which is the same as 3s
So, the answer is
N.B. Anything to the power zero is 1
2° = 1 6° = 1
147° = 1
a° = 1 b° = 1
2.3° = 1
BUT 6a° = 6 WHY?
Well, 6a° written out fully is
6 x a° = 6 x 1
=6
BRACKETS
Example 1
(a2)3 means a2 x a2 x a2
a2 + 2 + 2 = a 6
A QUICK way of doing this is to MULTIPLY THE POWER INSIDE THE
BRACKET BY THE POWER OUTSIDE THE BRACKET.
a2 x 3 = a 6
You get the same answer, but the latter method is quicker!
Example 2
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
14
(b4)5 = b4 x 5 = b20
Example 3
(2b3)2
= (21 x b3)2
= 21x2 x b3 x 2 = 4b6
Example 4
(42b1)2
= (42 x b1)2
= 42 x 2 x b1 x 2
= 44 x b 2
= 256b2
Example 5
2(a3)4
NOTE - only a3 is inside the bracket
The 2 is not affected by the power outside the bracket!
= 2 x a3 x 4
= 2a12
NEGATIVE INDICES
Example 1
9
-1
means
1
9
1
or
1
9
9 can be w r i t t en as
9
-1
i s w r i t t en as
9
1
1
and i s know n as t he RECI PROCAL of ni ne.
9
WHEN YOU SEE A NEGATIVE POWER THINK 'ONE OVER'.
Example 2
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
15
10
-1
1
=
10
1
=
1
10
-2
1
1
10 =
=
2
100
10
Example 3
a
-1
1
=
a
1
and a
-4
=
1
a
4
BUT
Example 4
-1
4d =
4d
-3
=
4
1
4
x
=
1
d
d
4
1
4
x
=
3
3
1
d
d
In this examples the minus sign APPLIES to the letter d not to the number 4.
Exercise 1
1.
a4 x a 3
3.
3s x 4s5
5.
3-3
7.
8-2
9.
2-5
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2.
4.
6.
8.
10.
b5 x b
33
82
25
4-2
SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
16
STANDARD FORM
Very large and very small numbers must sometimes be expressed in
STANDARD FORM.
A x 10n
where
1 < A < 10
and n is an integer.
Translated this means that A must be a number between 1 and 10 and
n is a positive or negative number.
Here are some examples to clarify this.
Example 1
87000 = 8.7 x 104
The display on the right of the equals sign is 87000 written in standard form.
Example 2
0.000026 = 2.6 x 10-5
The number of places the digits move, gives us the number in the power.
Example 3
146.2 = 1.462 x 102
Digits 'move' to right - sign is +
Digits 'move' 2 places - power is 2.
Example 4
26 = 2.6 x 10
which is
2.6 x 101
Example 5
0.9 = 9.0 x 10-1
Example 6
265 = 2.65 x 102
Example 7
0.0095 = 9.5 x 10-3
Example 8
5 x 10-5 x 3 x 102
Take this step by step as shown below:
First multiply the numbers without powers
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
17
5 x 3 = 15
Secondly, multiply the number with powers
10-5 x 102 = 10-5 + 2 = 10-3
Answer = 15 x 10-3
NOW, this must be converted to standard form as shown:
1.5 x 101 x 10-3
= 1.5 x 10+1 - 3
= 1.5 x 10-2
which is the answer.
Exercise 2
Write these in STANDARD FORM
1.
6500
2.
0.0082
3.
132.3
4.
0.5
5.
43
6.
2660000
7.
0.35
8.
0.7 x 105 x 3 x 104
9.
6 x 103 x 2 x 10-2
10.
9 x 10-1 x 3 x 10-1
FRACTIONAL INDICES
Example 1
16
1
2
=
2
16
1
= 2 16 = 4
4 x 4 = 16
This means the 'square root' of 16, ie, which number multiplied by itself gives
16 - answer is 4.
WHEN YOU SEE FRACTIONAL INDICES THINK 'ROOT SIGN'
Example 2
8
1
3
=
3
8
1
=3 8 = 2
2x2x2 = 8
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
18
This means the 'cube root' of 8, ie, which number multiplied by itself three times
gives 8 - answer is 2.
Example 3
32
1
5
=
5
32
1
= 5 32 = 2
2 x 2 x 2 x 2 x 2 = 32
This means the 'fifth root' of 32, ie, which number multiplied by itself five times
gives 32 - answer is 2.
Example 4
81
1
4
=
4
81
1
= 4 81 = 3
3 x 3 x 3 x 3 = 81
This means the 'fourth root' of 81, ie, which number when multiplied by itself four
times gives 81 - answer is 3.
Example 5
1
3
a =
3
a
1
=3 a
Example 6
1
2
2
d = d
1
=2 d
Example 7
27
2
3
Should be able to do this without a calculator
Fi r st l y, w or k out t he cube r oot of 27 =
3
27 = 3
Then square this ie, 32 = 9
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
19
Example 8
3
4
16
4
=
16
3
3
= 2 =8
Example 9
3
2
1 00
2
=
1 00
3
3
= 10 = 1 000
FRACTIONAL AND NEGATIVE INDICES
Example 1
-
16
1
2
1
=
16
1
=
1
2
=
16
1
4
Example 2
-
16
3
4
1
=
+
16
3
4
1
=
4
=
16
3
1
2
3
=
1
8
Example 3
-
100
3
2
1
=
2
1
=
100
3
10
3
=
1
1000
Example 4
8
1
2
x 8
3
2
=8
1
3
+
2
2
=8
4
2
82 = 64
Example 5
-
8
1
2
x 8
1
2
-
=8
1
1
+
2
2
= 8°
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
20
8° = 1
(Anything to power zero is 1)
Example 6
3
3 x 3
3
2
=
8
3
3
5
8
=3
-3
=
1
3
3
=
1
27
Exercise 3
-
1. 81
-
3. 8
1
4
2.. 8
2
3
-
4. 10 0
5. 12 5
2
3
9. 9
1
2
8. 4
x 9
3
2
6. a°
7. 8 y°
-
2
3
1
2
1
2
-
10. 9
3
2
x 4
3
2
x 9
1
2
2
11. 10° x 10 x 10
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
21
ANSWERS
EXERCISE 1
1.
a7
2.
b6
3.
12s6
5.
1
27
6. 6 4
7.
1
64
8. 3 2
9.
1
32
10.
Exercise 2
1.
6.5 x 103
3.
1.323 x 102
5.
4.3 x 10
7.
3.5 x 10-1
9.
1.2 x 102
4.
27
1
16
2.
4.
6.
8.
10.
8.2 x 10-3
5 x 10-1
2.66 x 106
2.1 x 109
2.7 x 10-1
Exercise 3
1.
1
3
2. 4
3.
1
4
4.
1
100 0
5. 25
6.
1
7. 8
8.
16
9. 1
10.
1
9
11. 1000
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
22
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
23
Chapter 10
AVERAGE SPEED
AVERAGE SPEED = TOTAL DISTANCE
TOTAL TIME
Try to remember the triangle shown below, it will help you to rearrange the
formula.
(i)
to find formula for speed, place a finger over SPEED on the left hand
side and we see DISTANCE .
TIME
DISTANCE
SPEED
(ii)
to find formula for time, cover up TIME and we see
(iii)
to find formula for distance, cover up DISTANCE and we have
SPEED x TIME.
D=ST
S =
D
T
T =
D
S
The most common units that average speed is given in are
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
24
1) Miles per hour - m/h
also written m , mph or mh-1
h
2) Kilometres per hour - k/h
also written km , kmph or kmh-1
h
This may vary, of course, according to which units are used.
Questions are usually straight forward, ie, substituting numbers in a formula BUT
care must be taken with the units. READ the question carefully, and establish
what is being asked for.
If it asks for AVERAGE SPEED, your answer will be given in km/h, or m/h, etc.
If it asks for DISTANCE, your answer will be given in km or miles.
If it asks for the TIME, your answer will be in hours.
THE ABOVE FORMULA CAN BE REARRANGED TO GIVE:
TOTAL DISTANCE = AVERAGE SPEED x TOTAL TIME.
AND
TOTAL TIME =
TOTAL DISTANCE
AVERAGE SPEED
EXAMPLES
1) A car travels 200 km in 4 hours. what is its average speed?
Your answer requires the average speed, so your answer will be in
km/h.
The distance = 200 km
The time = 4 hours
USE THE ORIGINAL FORMULA
Average speed
=
Total distance
Total time
= 200 km
4hrs
=
5 0 km
1 hr
The answer may be written as
50 km/h
OR
2)
50 kmh-1
OR
50km per hr.
A car travels 20 km in 30 mins. What is its average speed in km/h?
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
25
Distance = 20 km
Time = 30 min =
1
hr = 0.5 hr
2
Note: Change the time to hours
Average speed = Total distance
Total time
=
2 0 km
0 .5 hr
= 40 km/h
3) If a car travels at 50 km/h for 2 hours how far has it travelled?
You are being asked to find the DISTANCE which the car has travelled, so use
the following formula.
Distance
4)
= average speed x time
= 50 km x 2hrs
=100 km
A bus travels 300 km at an average speed of 60 km/h. How long does
it take?
You are being asked to find the TIME the bus takes to travel 300 km, so use the
following formula:
Time =
5)
Distance
300
=
= 5 hrs
Average Speed
60
For the first 2 hours of a 260 km journey, the average speed was
30 km/h. If the average speed for the remainder of the journey was
50 km/h, calculate the average speed for the whole journey.
THIS QUESTION MUST BE SPLIT UP INTO THREE PARTS.
Often it helps to make a diagram of the problem, as shown here:
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SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
26
FIRST PART OF JOURNEY
Time = 2 hrs
Average speed = 30 km/h
WE DO NOT KNOW THE DISTANCE FOR THE FIRST PART OF THE
JOURNEY, SO WE MUST FIND IT NOW:
Distance = Speed x Time
= 30 x 2
= 60 km
DISTANCE TRAVELLED IN FIRST 2 HOURS IS 60 km. (You can put this onto
your diagram now).
SECOND PART OF JOURNEY
We have the distance and the average speed for the rest of the journey, BUT we
are not given the TIME this part of the journey took.
Distance = 260 - 60 = 200 km
Average speed = 50 km/h
Time =
Distance
Average speed
= 200 = 4 hrs
50
This can be added to your "picture" of the journey to clarify what you need to do
next.
THIRD PART OF PROBLEM - WHAT IS AVERAGE SPEED OF WHOLE
JOURNEY?
Total distance is 260 km
Total time is 2 hrs + 4 hrs = 6 hrs
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We can now find the average speed
Aver age speed = Tot al Di st ance
Tot al Ti me
= 2 6 0 km
6 hr s
OR = 4 3 .3 km/ h
6) A car travels 40 km at 40 km/h and then 100 km at 50 km/h. What is
the average speed for the whole journey?
Draw a diagram of the information you have:
FIRST PART OF JOURNEY
We know the distance and the average speed. We need to know the TIME of
that part of the journey.
Ti me = Di st ance
Speed
= 40
40
= 1 hr
ON YOUR PICTURE
SECOND PART OF JOURNEY
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We know the distance and the average speed. AGAIN, we need to know how
long this part of the journey took.
Ti me =
=
Di st ance
Speed
100
= 2 hr s
50
We now know the TOTAL TIME (3 hours) and the TOTAL DISTANCE of the
journey (140 km). We need to know the AVERAGE SPEED.
Aver age speed =
=
Tot al di st ance
Tot al t i me
140
3
= 46
2
km/ h
3
= 46.67 km/h
Exercise 1
1) Find the average speed if a car travels a total distance of 400 km in 8
hours.
2) Find the average speed if a car travels 20 km in 20 minutes.
3) A bus travels 500 km at an average speed of 50 km/h. How long does it
take?
4) If a boy cycles for 2.5 hours at an average speed of 20 km/h, how far
has he travelled?
5) A train travels 2 hours at an average speed of 60 km/h, then 3 hours at
an average speed of 70 km/h. What is the average speed of the whole
journey?
6) A bus travels for 9 hours between A and B, which are 270 km apart.
On the return journey from B to A, the average speed is reduced by 3 km/h.
Calculate the time taken for the return journey?
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ANSWERS
Exercise 1
1) Average speed = 400 = 50 km/h
8
2) Average speed = 20
1
3
= 60km/h
(20 mins)
3) Time = Distance = 500 = 10 hrs
Speed
50
4) Distance = 20 x 2.5 = 50 km
5) Distance for 1st part = 60 x 2 = 120 km
Distance for 2nd part = 70 x 3 = 210 km
Total time = 5 hrs
Total distance = 330 km
Average speed = 330 = 66 km/h
5
6)
Speed for 1st part = 270 = 30 km/h
9
Speed for 2nd part = 30 - 3 = 27 km/h
TIME for return journey 270 = 10 hrs
27
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31
Chapter 11
RATIONAL AND IRRATIONAL NUMBERS
All numbers are either rational or irrational.
Rational Numbers
A rational number is a number which can be written as a vulgar fraction in the
form
a
where a and b are integers.
b
eg
4
is a rational number
7
-
1
is a rational number
4
2
1
9
is a rational number because it can be expressed in the form
4
4
1.2 is a rational number because it can be written as 1
2
1
6
= 1 =
10
5
5
22.32 is a rational number because it can be written as 22
32
8
558
= 22
=
100
25
25
5
5 is even a rational number because it can be written as .
1
NB 5 can be classed as a positive integer, a whole number, a natural number or
a rational number.
Any recurring decimal can be written as a fraction and therefore is a rational
number.
Example
Write the decimal 0.121212 ……. As a fraction
If
Then
n
= 0.121212….
100n = 12.121212 …..
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(Equation 1)
(Equation 2)
SUBJECT KNOWLEDGE AND UNDERSTANDING: NUMBER
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Subtracting Equation 1 from Equation 2 gives
100n - n
= 12.121212… - 0.121212….
So
99n
= 12
So
n
= 12/99 = 4/33
Irrational Numbers
Irrational numbers cannot be expressed exactly as fractions. An irrational
number in decimal form does not recur or terminate. Any root that cannot be
simplified to an integer or a fraction is an irrational number.
Some examples of this kind of number that you may have met already are √2, √3
and π. If you ask your calculator for √2 it may give 1.414213562. However, if
we calculate 1.41421356 x 1.414213562 we will obtain 1.999999, ie, not 2
exactly!
There is no rational number which multiplied by itself gives exactly 2.
Similarly the calculator value for π (3.141592654) whilst a very close
approximation is not exactly accurate.
Combining Rational and Irrational Numbers
Two numbers can be added, subtracted, multiplied and divided. The rule for
rationals and irrationals are as shown on these tables.
Add or subtract
Rational
Rational
Rational
Irrational
Irrational
Irrational
Irrational
Either
Mult or divide
Rational
Rational
Rational
Irrational
Irrational
Irrational
Irrational
Either
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