Maths- Y10 Notes as cards Flashcards
Two events are independent if the probability of one event is unaffected by the probability of another.
e.g. What is the probability of flipping two heads in a row? HH TT HT TH 1/4 = 1/2 x 1/2
Multiplication Rule-
P (a and b) = p(a) x p(b)
*a and b are independent events
Additional Rule-
For mutually exclusive events (events that cannot occur at the same time)
P(A or B) = P(A) + P(B)
Ratio-
Ratio is used to describe a fraction
e.g. 9:6 = 3:2
Always simplify to as low as possible!!
HCF-
Write each number as a product of its primes.
To find the HCF, multiply all factors which appear in both lists
60= 2, 2, 3, 5 72= 2, 2, 2, 3, 3
eg. HCF of 60 and 72 is 2 x 2 x 3= 12
LCM-
The LCM is found by multiplying all the factors which appear in either list.
60= 2, 2, 3, 5 72= 2, 2, 2, 3, 3
2 x 2 x 2 x 3 x 3 x 5 = 360
BIDMAS-
Brackets, Indices, Division, Multiplication, Addition, Subtraction
Percentages-
To change a fraction or a decimal to a percentge you multiply by 100%
e.g. 3/8 = (3/8 x 100/1)%= 37.5%
Working out percentages-
eg 7% of £3200
= 7/100 x 3200/1 = £224
or 0.07 x 3200 = £224
Percentage Increase-
£8400 is increased by 5%= 105% of £8400
= 1.05 (percentage multiplier) x £8400
= 8200
Decrease therefore would be (eg) 0.95 x 8400
Compound Interest-
“A bank pays a fixed interest of 10% on money in accounts. Amanda put £500 in the bank, how much money will she have…”
after:
one year= 500+(10% of 500)= £550
2 years= 550 + (10% of 550)= £605
3 years= 605 + (10% of 605)= £665.50
GENERAL FORMULA…
n
1.10 x 500
1.10= percentage increase
n= years
500= money put in
Reverse Percentages-
Finding the original price
eg cost price = 100% selling price= 100% + 40% £63 = 140% 63/140 = 1% 1%= £0.45 100%= $0.45 x 100 =£45
Fractions-
To add and subtract you must have the same denominator
Multiplying-
Top x top
Bottom x bottom = simplify outcome
When dividing, multiply the reciprocal (upside down) of what you want it to be divided by.
Surds-
a to the power of 0 is always 1
surds are numbers like ⎷3 and ⎷4
It is sometimes helpful to write an expression with an integer denominator- RATIONALISING THE DENOMINATOR
e.g. 6/ route 3
6⎷3 = 6 x ⎷3 = 6 ⎷3 = 2 ⎷3
⎷3 x⎷3 3
A surd is a square root which cannot be reduced to a whole number.
⎷ab = ⎷a x ⎷b
⎷a x ⎷a = a
Standard Form-
n
A x 10
A is always 1-10
eg 15 000 000 = 1.5 x 10^7
Bounds-
12cm long piece of fabric
- 5= lower bound
- 5= upper bound
Averages-
Median- Arrange smallest to largest, this is then the middle number. If there are two middle numbers its the middle of these numbers.
Mean- All data aded and divide by the number of items.
Mode- Number which occurs most frequently
Range- (Largest value) - (smallest value)= range
Moving Averages
To see if there is a trend in the sales figures you can use a moving average
4 point m.a. is most common and involves finding the mean of four quarters
This is then plotted on a graph
Frequency Polygons and scatter graphs-
Vertical axis always= frequency in the frequency polygon the boxes of the bar chart are replaced by a line joining the tops of their mid points
discrete data= counting continuos data =measuring
eg. Weak correlation
Always say if it is weak/strong positive/negative or if there is no correlation
Box and Whisker-
Lowest value
Lower quartile
Median
Upper quartile
Highest value
To find lower, median and upper put numbers in order and the first 1/4, 1/2 and 3/4
interquartile range = upper quartile - lower quartile
Histogram-
The area of each bar shows the frequency of the data.
Bars have varying widths
vertical axis= frequency density
Cumulative frequency=
S shaped curve when on a graph
Data from a frequency table
y = mx + c
Expanding Brackets-
find common factors
Nth term
list of numbers= terms in sequence
Find the common difference and the zero term to find out what you have to add and subtract
Collecting like terms-
4x + 5x -2 -2x +7
if you collect the terms…
7x+5
Notation-
a^2= a x a a^3= a x a x a multiplying = add the indices dividing = subtract the indices only add and subtract ‘ like terms’
Direct Proportion-
If two quantities are in direct proportion, as one increases, the other increases by the same percentage
y is directly proportionate to x .. y=kx
inverse proportion is when one value increases as the other value decreases
y=k/x
Simultaneous equations-
2 equations with 2 unknowns
step1: eliminate the unknowns (add or subtract equations)
Put the answer back into the equation
sometimes equations may need to be multiplied
substitution method
rearrange the so “y=…” for example and put that back into the other equation
In graphs- draw lines and the pint where they meet is the answer
Quadratic Formula
General way to write a quadratic equation is: ax^2 + bx + c= 0
Similarity and Congruence:
Similar shapes/ figures are identical in shape but not in size
e.g. two circles are always similar corresponding sides are in the same ratio and corresponding angles are equal.
Congruent shapes are identical in shape and size- they could be rotated or reflected
Quadratics
Factorising
x^2 + 5x + 6
+ x
2 number which add to make 5 and multiply to make 6
Terms without an x term add to get 0 and mutiny to get the number.
Not all quadratics without an x term can be factorised.
Solving:
Write in the form ax^2 +bx+c=0
Find two numbers which add to get b and multiply to get c
(x )(x )=0 (add in the numbers from 2)
(x )=0
(x )=0
those are the two possibilities and work out and check!
sin= o/h cos= a/h tan= o/a
.
Trial and improvement-
draw a table, each trial write too small, too high, close or yes
cuboid
volume= length x width x height sa= sum of the areas of 6 faces
prism
volume= (area of cross-section) x length sa= sum of sides
cylinder
volume= pi r^2 h sa= 2 pi r h
pyramid
pyramid
volume= 1/3(base area) x height
sa= sum of sides
sphere
volume= 4/3 pi r^2
sa= 4 pi r^2
cone
volume= 1/3 pi r^2h
sa= pi r l
l=slant height
circles
pi r^2= area
pi d= circumference
Pythagorus’ theorem
a^2 + b^2= c^2
Bearings-
always start facing North and move in a clockwise direction. Angles with less than 100 degrees have a 0 in front to make them a 3 figure bearing i.e. 060 degrees
translations-
move up and down side to side but we do not change its shape, size or direction
when an object is translated every (vertex, corner) must be moved in the same way
Circle Theorem
Angle at the centre is double the angle at the circumference
Circle Theorem
Angles in the same segment are equal
Circle Theorem
Angles in a semicircle are 90 degrees
Circle Theorem
Opposite angles in a cyclic quadrilateral add up to 180 degrees
Circle Theorem
The perpendicular from the centre to the chord bisects the chord
Circle Theorem
The angle between the tangent and the radius is 90 degrees
Circle Theorem
Tangent from a point outside the circle are equal in length
Alternative segment theorem
The angle between the tangent and chord at the point of contact is equal to the angle in the alternative segment
Vector-
has both direction and magnitude (size) -scalar has magnitude only
two vectors with the same magnitude and direction are..
equal
a negative vector means…
reversing its direction
A scale factor is…
a number which scales, or multiplies some quantity.
y=cx (c is the scale factor for x)
Scale factors between -1 and 1- the image will be ……… than the object.
smaller
Negative- opposite side of centre of enlargement and is..
upside down
to calculate interior angles= (n-2) x 180 degrees
n is number of sides
