Transcript Document

Higher Tier - Number revision
Contents :
Calculator questions
Long multiplication & division
Best buy questions
Estimation
Units
Speed, Distance and Time
Density, Mass and Volume
Percentages
Products of primes
HCF and LCM
Indices
Standard Form
Ratio
Fractions with the four rules
Upper and lower bounds
Percentage error
Surds
Rational and irrational numbers
Recurring decimals as fractions
Direct proportion
Inverse proportion
Graphical solutions to equations
Calculator questions
Which buttons would you press
to do these on a calculator ?
2.5 + 4.1
3.5
1.7 + 2.8
2.3 – 0.2
1.56
8.5 x 103
3.4 x 10-1
4
1000
1.72 + 5.22
6.31
9.2 6.1
–
7.5
8
3
2.5
1
1
–
3.6
2.3
Long multiplication
Use the method that gives you the
correct answer !!
Question : 78 x 59
70
50
9
Total =
8
3500 400
630
72
3500 + 400 + 630 + 72
Answer : 4602
Now try 84 x 46 and 137 x 23 and check
on your calculator !!
Again use the method that gives
you the correct answer !!
Long division
Question : 2987  23
12 9
23 times
table
23
46
69
92
115
138
161
184
207
230
29
68
23
6 22 20
2987
Answer : 129 r 20
227
Now try 1254  17 and check on
your calculator – Why is the
remainder different?
Best buy questions
Always divide by the price to see
how much 1 pence will buy you
Beans
Large 40087 = 4.598g/p
Small 15034 = 4.412g/p
Large is better value
(more grams for every
penny spent)
32p
0.95L
78p
OR
2.1L
87p
34p
OR
Milk
Large 2.178 = 0.0269L/p
Small 0.9532 = 0.0297L/p
Small is better value
OR (looking at it differently)
{Large 782.1 = 37.14p/L
Small 320.95 = 33.68p/L}
Estimation
If you are asked to estimate an answer to a
calculation – Round all the numbers off to 1 s.f.
and do the calculation in your head.
DO NOT USE A CALCULATOR !!
e.g. Estimate the answer to 4.12 x 5.98
 4 x 6 = 24
Always remember to write down the numbers
you have rounded off
Estimate the answer to these calculations
1.
58 x 21
2.
399 x 31
3.
4.
5.
7.12 x 39.2
0.87
8.
4.89 x 6.01
1.92
47 x 22
6.
377  19
9.
360 x 87
4899  46
7.
1906  44
10. 58 x 21
Units
Learn these rough conversions
between imperial and metric units
1 inch  2.5 cm
1 yard  0.9 m
5 miles  8 km
2.2 lbs  1 kg
1 gallon  4.5 litres
Learn this pattern for converting between the various metric units
Metriclength
weightconversions
conversions
capacity
conversions
Metric
x 1000
km
kg
kl
÷ 1000
x 100
m
gl
x 10
cm
cg
cl
÷ 100
mm
mg
ml
÷ 10
Speed, Distance, Time questions
Speed, Distance and Time are linked by this formula
D
S =
T
To complete questions check that all units are
compatible, substitute your values in and rearrange if necessary.
1.
Speed = 45 m/s
Time = 2 minutes
Distance = ?
45 m/s and 120 secs
S= D
T
45 =
D .
120
45 x 120 = D
D = 5400 m
2.
Distance = 17 miles
Time = 25 minutes
Speed = ?
17 miles and 0.417 hours
S= D
T
S=
17 .
0.417
S = 40.8 mph
3.
Speed = 65 km/h
Distance = 600km
Time = ?
S= D
T
65 = 600 .
T
T = 600 .
65
T = 9.23 hours
Density, Mass, Volume questions
Density, Mass and Volume are linked by this formula
M
D =
V
To complete questions check that all units are
compatible, substitute your values in and rearrange if necessary.
1.
Density = 8 g/cm3
Volume = 6 litres
Mass = ?
8 g/cm3 and 6000 cm3
D= M
V
8=
M .
6000
8 x 6000 = M
M = 48000 g
( or M = 48 kg)
2.
Mass = 5 tonnes
Volume = 800 m3
Density = ?
800 m3 and 5000 kg
D= M
V
D = 5000 .
800
D = 6.25 kg/m3
3.
Density = 12 kg/m3
Mass = 564 kg
Volume = ?
D= M
V
12 = 564 .
V
V = 564 .
12
V = 47 m3
Simple %
Percentage increase and decrease
A woman’s wage increases by 13.7% from
£240 a week. What does she now earn ?
Increase:
13.17% of
New amount:
£240
240 + 31.608 = 271.608
13.17
100
x
Her new wage is
£271.61 a week
240 = 31.608
10% =
Percentages of
amounts
75% =
1% =
25% =
5% =
(Do these without
a calculator)
30% =
£600
45% =
85% =
20% =
50% =
2% =
Simple %
Fractions, decimals and percentages
Copy and complete:
Reverse %
e.g. A woman’s wage increases by 5% to £660 a week. What was her
original wage to the nearest penny?
Original
amount
x 1.05
£660
Original
amount
÷ 1.05
£660
Original amount = 660 ÷ 1.05 = £628.57
e.g. A hippo loses 17% of its weight during a diet. She now weighs 6
tonnes. What was her former weight to 3 sig. figs. ?
Original
weight
x 0.83
6 ton.
Original
weight
Original weight = 6 ÷ 0.83 = 7.23 tonnes
÷ 0.83
6 ton.
Repeated %
e.g. A building society gives 6.5% interest p.a. on all money invested
there. If John pays in £12000, how much will he have in his account
This is not the correct method:
at the end of 5 years.
£12000
12000 x 0.065 = 780
780 x 5x=1.065
3900 x 1.065
x 1.065 x 1.065
12000 + 3900 = £15900
x 1.065
?
He will have = 12000 x (1.065)5 = £16441.04
e.g. A car loses value at a rate of approximately 23% each year.
Estimate how much a $40000 car be worth in four years ?
This is not the correct method:
40000
= 9200x 0.77
£40000 x 0.77
x 0.77 x 0.23
x 0.77
?
9200 x 4 = 36800
40000 – 36800 = $3200
The car’s new value = 40000 x (0.77)4 = $14061 (nearest $)
40
Products of primes
Express 40 as a product of primes
2
20
2
40 = 2 x 2 x 2 x 5 (or 23 x 5)
Express 630 as a product of primes
10
2
630
2
Now do the same for
100 , 30 , 29 , 144
5
315
3
105
3
630 = 2 x 3 x 3 x 5 x 7 (or 2 x 32 x 5 x 7)
35
5
7
HCF
Expressing 2 numbers as a product of primes can help
you calculate their Highest common factor
HCF
e.g. Find the highest
common
factor
of 84 and
Consider
the
numbers
20120.
and 30.
Their Pick
factors
are:
out all the bits that
84 = 2 x 2 x 3 x 7
1, 2, 4, 5, 10, 20 and
2, 3, 5,to6,both.
10, 15, 30
are 1,
common
120 = 2 x 2 xTheir
2 x 3highest
x5
common factor is 10
Highest common factor = 2 x 2 x 3 = 12
LCM
Expressing 2 numbers as a product of primes can also
help you calculate their Lowest common multiple
LCM
e.g. Find the lowest common multiple of 300 and 504.
Consider the numbers 16 and 20.
Their multiples
are:
Pick out
the highest
300 = 22 x 3 x 52
valued
index
for 80,
each100
16,
32,
48,
64,
80,
96
and
20,
40,
60,
2 x 7
504 = 23 x 3Their
prime
factor . is 80
lowest common
multiple
Lowest common multiple = 23 x 32 x 52 x 7 = 12600
15
1
Indices Evaluate:
1/
9
2
-2
3
0
19
3/2
16
3
4
5
2 x
1
2
5
7 
-1
2
1/4
81
-4
10
3
7
-3
2
-1/2
36
Standard form
Write in Standard Form
9.6
0.0001
3 600
0.041
56 x 103
0.2
8 900 000 000
Write as an ordinary number
4.7 x 109
1 x 102
8 x 10-3
5.1 x 104
7 x 10-2 8.6 x 10-1
9.2 x 103
3.5 x 10-3
Calculate 3
x 104 x 7 x 10 -1 without a calculator
Calculate 4.6 x 104 ÷ 2.5 x 108 with a calculator
Calculate 1.5
x 106 ÷ 3 x 10 -2 without a calculator
Ratio
Equivalent
Ratios
? : 10
? : 6
14 : ?
1 : ?
7:2
49 : ?
? : 1
£600 is split between
Anne, Bill and Claire in
the ratio 2:7:3. How
much does each receive?
? : 12
21 : ?
0.5 : ?
Splitting in a given ratio
? : 12
2100 : ?
Total parts = 12
Anne gets 2 of 600 = £100
12
Basil gets 7 of 600 = £350
12
Claire gets 3 of 600 = £150
12
Fractions with the four rules
+–×÷
Learn these steps to complete all fractions questions:
• Always convert mixed fractions into top heavy
fractions before you start
• When adding or subtracting the “bottoms” need to be
made the same
• When multiplying two fractions, multiply the “tops”
together and the “bottoms” together to get your final
fraction
• When dividing one fraction by another, turn the
second fraction on its head and then treat it as a
multiplication
Fractions with the four rules
4⅔  1½
4⅔ + 1½
= 14 + 3
=
=
=
=
=
3
28
6
+
37
6
6
2
9
6
=
1
6
=
14 
3
14 
3
28
9
3
1
9
3
2
2
3
Upper and lower bounds
A journey of 37 km measured to the
nearest km could actually be as long as 37.49999999…. km or as
short as 36.5 km. It could not be 37.5 as this would round up to 38
but the lower and upper bounds for this measurement are 36.5 and
37.5 defined by:
36.5 < Actual distance < 37.5
e.g. Write down the Upper and lower bounds of each of
these values given to the accuracy stated:
9m (1s.f.)
85g (2s.f.)
8.5 to 9.5
84.5 to 85.5
180 weeks (2s.f.)
175 to 185
2.40m (2d.p.)
2.395 to 2.405
4000L (2s.f.)
3950 to 4050
60g (nearest g)
59.5 to 60.5
e.g. A sector of a circle of radius 7cm makes an angle of 320 at the
centre. Find its minimum possible area if all measurements are given
to the nearest unit. ( = 3.14)
320
7cm
Area = (/360) x  x r x r
Minimum area = (31.5/360) x 3.14 x 6.5 x 6.5
Minimum area = 11.61cm2
% error
If a measurement has been rounded off then it is not
accurate. There is a an error between the
measurement stated and the actual measurement.
The exam question that occurs most often is: “Calculate the
maximum percentage error between the rounded off
measurement and the actual measurement”.
e.g. This line has been measured as 9.6cm (to 1d.p.).
Calculate the maximum potential error for this
measurement.
Upper and lower bounds of 9.6 cm (1d.p.) 
9.55 to 9.65
Maximum potential difference (MPD) between
actual and rounded off measurements 
0.05
Max. pot. % error = (MPD/lower bound) x 100
= (0.05/9.55) x 100
= 0.52%
Simplifying roots
Tip: Always look for square numbered
factors (4, 9, 16, 25, 36 etc)
e.g. Simplify the following into the form a  b
20
4 x 5
2 5
8
4 x 2
2 2
45
9 x 5
3 5
72
36 x 2
700
100 x 7
6 2
10 7
Surds A surd is the name given to a number which has been left
in the form of a root. So 5 has been left in surd form.
SIMPLIFYING EXPRESSIONS WITH SURDS IN
A surd or a combination of surds
can be simplified using the rules:
M x N = MN and visa versa
M ÷ N = M/N and visa versa
12
4 x 3
135 ÷ 3
5(5 + 20)
(3 – 1)2
Tips: Deal with a surd as you
would an algebraic term and
always look for square numbers
2 3
45
9 x 5
5 + 100
(3 – 1) (3 – 1)
LEAVING ANSWERS IN SURD FORM
Pythagoras  (14)2 = (6)2 + x2
14 = 6 + x2
x = 8
Answer: x = 22
3 5
15
4 – 23
3 – 3 – 3 + 1
Calculate the length of side x in
surd form (non-calculator
paper):
14
x
6
Rational and irrational numbers
Rational numbers can be expressed in the form a/b. Terminating
decimals (3.17 or 0.022) and recurring decimals (0.3333..or
4.7676..) are rational.
Irrational numbers cannot be made into fractions. Non-terminating
and non-recurring decimals (3.4526473… or  or 2) are irrational.
State whether the following are rational or irrational numbers:
16
52

3
1/5
20
2.3/5.7
What do you need to do to make the following irrational numbers
into rational numbers:

3
53
3
6
20
2
2.7
Recurring decimals as fractions
Learn this technique which changes recurring decimals into
fractions:
Express
0.77777777…..
as a fraction.
Let
n = 0.77777777…..
so 10n = 7.77777777…..
so
9n = 7
so n = 7/9
Express
2.34343434…..
as a fraction.
Let
n=
2.34343434…..
so 100n = 234.34343434…..
so 99n = 232
so n = 232/99
Express
0.413213213…..
as a fraction.
Let
n=
0.4132132132…..
so 10000n = 4132.132132132…..
and
10n =
4.132132132……
so 9990n = 4128
so n = 4128/9990
n = 688/1665
Direct proportion
If one variable is in direct proportion to another (sometimes called
direct variation) their relationship is described by:
pt
Where the “Alpha” can be replaced by an
“Equals” and a constant “k” to give :
p = kt
e.g. y is directly proportional to the square of r. If r is 4 when y is 80,
find the value of r when y is 2.45 .
Write out the variation:
y  r2
Change into a formula:
y = kr2
Sub. to work out k:
So:
And:
Working out r:
Possible direct
variation questions:
80 = k x 42
g  u3
g = ku3
k=5
c  i
c = ki
y = 5r2
s  3v
s = k3v
2.45 = 5r2
t  h2
t = kh2
r = 0.7
xp
x = kp
Inverse proportion
If one variable is inversely proportion to another (sometimes called
inverse variation) their relationship is described by:
p  1/t
Again “Alpha” can be replaced
by a constant “k” to give :
p = k/t
e.g. y is inversely proportional to the square root of r. If r is 9 when y
is 10, find the value of r when y is 7.5 .
Write out the variation:
y  1/r
Change into a formula:
y = k/r
Sub. to work out k:
So:
Possible inverse
variation questions:
10 = k/9
g 1/u3 g = k/u3
k = 30
c  1/i c = k/i
y = 30/r
s  1/3v
s = k/3v
And:
7.5 = 30/r
t  1/h2
t = k/h2
Working out r:
r = 16 (not 2)
x  1/p
x = k/p
Graphical solutions to equations
If an equation equals 0 then its solutions lie
at the points where the graph of the equation
crosses the x-axis.
e.g. Solve the following equation graphically:
x2 + x – 6 = 0
y
-3
y = x2 + x – 6
2
x
All you do is plot the
equation y = x2 + x – 6
and find where it
crosses the x-axis
(the line y=0)
There are two solutions to
x2 + x – 6 = 0
x = - 3 and x =2
Graphical solutions to equations
If the equation does not equal zero :
Draw the graphs for both sides of the equation and
where they cross is where the solutions lie
e.g. Solve the following equation graphically:
x2 – 2x – 11 = 9 – x
y
y=
x2
Plot the following
equations and find
where they cross:
y = x2 – 2x – 20
y=9–x
– 2x – 11
y=9–x
-4
5
x
There are 2 solutions to
x2 – 2x – 11 = 9 – x
x = - 4 and x = 5
If there is already a graph drawn and you are being
asked to solve an equation using it, you must
rearrange the equation until one side is the same as
the equation of the graph. Then plot the other side of
the equation to find the crossing points and solutions.
e.g. Solve the following equation using the graph
that is given: x3 – 4x + 5 = 5x + 5
y
y = x3 – 8x + 7
x
Rearranging the equation x3 – 4x + 5 = 5x + 5 to
get x3 – 8x + 7 :
x3 – 4x + 5 = 5x + 5
Add 2 to both sides
x3 – 4x + 7 = 5x + 7
Take 4x from both sides
x3 – 8x + 7 = x + 7
So we plot the equation y = x + 7 onto the graph to
find the solutions
y
y = x3 – 8x + 7
y=x+7
x
-3
0
3
Solutions lie at –3, 0 and 3
State the graphs you need to plot to solve the
following equations describing how you will find
your solutions:
1.
2.
3.
4.
5.
6.
3x2 + 4x – 2 = 0
7x + 4 = x2 – 4x
x4 + 5 = 0
0 = 8x2 – 5x
2x = 9
6x3 = 2x2 + 5
1.
2.
3.
4.
4x2 + 4x – 6 = 0
4x2 + x - 2 = 7
4x2 – 3x = 2x
3x2 = – 5
If you have got the graph of y= 4x2 + 5x – 6 work
out the other graph you need to draw to solve each
of the following equations:
Solve this equation
graphically:
x3 + 8x2 + 3x = 2x2 – 2x