Mock revision (number, algebra, shape and space, data handling) Flashcards
How do you carry out long multiplication?
- put one number on top of the other
- multiply each number opposite each other, carrying if necessary
- add these numbers together, adding a 0 in the second row of addition
e.g. 6.3 x 2.1
63 21 x 63 1260 + 1323
! remember to add the . !
so the answer is 13.23
How do you calculate the HCF and LCM?
1) draw a frequency tree for both numbers
2) write down all of the numbers circled multiplied together
3) simplify to include indexes if necessary
HCF is the numbers that appear in both (use lowest index)
LCM is all the numbers (use highest index)
e.g. 150 and 250
150 = 5² x 2 x 3
250 = 5³ x 2
HCF = 5² x2 = 50
LCM = 5³ x 2 x 3 = 750
How do you write an error interval?
- write lower bound
- write ≤
- write the letter
- write >
- write upper bound
e.g. m = 7.6
UB = 7.65
LB = 7.55
so
7.55 ≤ m > 7.65
How do you write an error interval when the number has been truncated?
- write the number
- write ≤
- write the letter
- write >
- write the number but with 1 decimal (lowest) added on
e.g. s = 6.57
6.57 ≤ s > 6.58
How do you convert a recurring decimal to a fraction?
- write x = the number with the recurring digits written out
- multiply x by how many times it takes for the recurring decimals to be subtracted to equal nothing (so the recurring digits line up)
- subtract all of these numbers from each other (including the xs)
- put the remaining number over the number of xs left
e.g.
0.46 recurring
x = 0.46464646 100x = 46.46464646 - 99x = 46
so 46/99
NOTE: if there is a non-recurring number in front of the recurring numbers, carry out the first 2 steps so that the non-recurring number is in front of the decimal, then multiply again so the recurring numbers line up again and continue
How do you find how a total number splits between a ratio?
- add the numbers in the ration to find the total ratio
- divide the total number by the total ratio
- multiply this number by each individual number
e.g. if three people split £1440 between them in the ratio 3:5:7, how much money will each person get?
3+5+7 = 15
1440 / 15 = 96
96 x 3 = 288
96 x 5 = 480
96 x 7 = 672
(add these numbers together to check they add to the total amount)
How do you find the total amount, given the amount of one number in the ratio?
- write the ratio
- write the amount underneath the number you know
- find the conversion factor (how many times the number from the ratio multiplies to give the amount)
- multiply the other numbers in the ratio by this conversion factor
- add all the amounts to find the total amount
e.g. if there are 16 sheep, how many animals are there altogether of sheep, cows and pigs in the ratio 8:1:3?
8 : 1 : 3
x2
16 2 6
16+2+6 = 24 animals
How do you calculate percentage change?
(something increases, how much % did it increase by)
% change = actual change/ original amount x100
How do you calculate percentage increase?
(something increases by x%, what is the new number)
- add the % increase to 100% (the original number)
- divide this number by 100
- multiple original number by this number
e.g. £1552 increase by 20%
100% + 20% = 120%
120 / 100 = 1.2
1552 x 1.2 = £1862.40
How do you calculate percentage decrease?
(something decreases by x%, what is the new number)
- subtract the % decrease from 100% (the original number)
- divide this number by 100
- multiply original number by this number
e.g. £250 decreases by 15%
100% - 15% = 85%
85 / 100 = 0.85
250 x 0.85 = £212.50
How do you calculate reverse percentages?
(finding the original amount after something has increased/decreased e.g. what was the cost before?)
- write out as an equation, with the original number as ?, multiplied by the percentage increase/decrease which equals the amount you are given
- rearrange the equation so that the ? equals the amount divided by the percentage increase/decrease
- solve
e.g. an item on sale for £22.05 reduced by 10%, what was the price before?
percentage decrease = 100% - 10% / 100 = 0.9
? x 0.9 = £22.05
? = 22.05/ 0.9
? = £24.50
How do you calculate compound interest? (x amount in bank for x years, earns x% compound interest)
- total amount in bank = investment x % increase/decrease to the power of the number of years
e.g. £300 in bank for 3 years, earns 2% compound interest
percentage increase = 100% + 2% / 100 = 1.02
total = 300 x 1.02 to the power of 3
total = £3183.62
What are the three rules of indicies?
when x, + the indicies
when /, - the indicies
when there are (), multiply indicies
e.g. (2³)³ = 2 to the power of 9
What is the rule of indicies when it is to the power of 0?
it equals 1
What is the rule of indicies when it is to a negative power?
it becomes 1/x to the power of x
e.g. 2-³ = 1/2³
What is the rule of indicies when the power is a fraction? (fractional indicies)
a number to the power of a fraction is the square root of that number
the denominator is the root
the numerator is the power
e.g. 27 to the 2/3 = (the cube root of 27)² = 3² = 9
What is the rule of indicies when there is a fraction to the power of -1?
you just flip the fraction
e.g. (2/3) to the -1 = 3/2
What is the rule of indicies when there is a fraction to the power of x?
you add the power to both numbers in the fraction
e.g. (2/5)² = 2²/5² = 4/25
remember to simplify if possible!!
What is the rule for negative fractional indicies? (a number to the power of a negative fraction)
the same as fractional indicies but you just add 1/x
e.g. 27 to the -1/3 = 1/ the cube root of 27 = 1/3
How do you simplify surds?
find two factors of the number - one must be a square number
e.g. √75 = √25√3 = 5√3
How do you add or subtract surds?
the surds must be the same
it’s like collecting ‘like terms’
e.g. 3√2 + 5√2 = 8√2
2√27 - 3√3 = 6√3 - 3√3 = 3√3
simplify = 2 x√9√3 = 2x3√3 = 6√3
if the surds are not the same, simplify them so that they have a surd in common and a square number, quick can be square rooted and multiplied with the normal number
e.g. 5√7 + 3√28
5√7 + 3√7 x √4
5√7 + 3√7 x 2
5√7 + 6√7
11√7
What is the rule for multiplying surds that are the same?
they equal the normal number
e.g. √3 x √3 = √9 = √
√5 x √5 = 5
√2 x √2 = 2 etc
How do you expand double brackets with surds?
expand brackets like normal except :
multiplying the same surd equals the normal number
if you are multiplying two different surds you just multiply the two numbers and add the √
e.g. √3 x √3 = 3
√3 x √2 = √6
you then collect the like terms
e.g. (3 + √2) (5 - √2) = 15 - 3√2 + 5√2 - 2 = 13 + 2√2
(3 - √5) (4 - √2) = 12 - 3√2 - 4√5 + √10
How do you rationalise surds?
it is just multiplying out the surd in the denominator - so multiple the numerator and denominator by the denominator
e.g. 7/√2 = 7/√2 x √2/√2 = 7√2/2
remember to simplify fractions if possible!
How do you solve a simultaneous equation (by elimination)?
(4 steps)
1) multiply 1 or both equation so they have the same number of xs or ys (regardless of sign)
2) + or - resulting equations to cancel out the unknowns that are the now the same size
3) solve the resulting equation
4) substitute this into either equation (depending on which looks easiest) and solve resulting equation
eg. 1) 3x + 4y = 29 2) 5x - 2y = 5
2) x2 –> 10x - 4y = 10
1) ——-> 3x + 4y =29 +
————————————
13x = 39
( / 13)
x = 3
sub x into 1)
3x + 4y = 29
3(3) + 4y = 29
9 + 4y = 29
(-9)
4y = 20
( / 4)
y = 5
How do you expand single brackets (1 step)
- multiple term on outside of brackets by each term on the inside of the brackets
eg. 2(a + 5)
2 x a = 2a
2 x 5 = 10
2a + 10
How do you expand double brackets? (2 steps)
1) multiple each term in the left bracket by each term in the right bracket
2) collect the terms
eg. (q + 4) (p + 3)
q x p = pq
q x 3 = 3q
4 x p = 4p
4 x 3 = 12
pq + 3q + 4p + 12
How do you expand triple brackets? (3 steps)
1) multiple the first two brackets as double brackets
2) simplify the resulting expression
3) multiply these terms by the 3rd bracket and simplify
eg. (2x + 1) (x - 3) (3x + 2)
2x x x = 2x²
2x x 3 = -6x
1 x x = x
1 x -3 = -3
(2x² - 6x + x - 3) (3x + 2)
(2x² -5x -3) (3x + 2)
2x² x 3x = 6x³
2x² x 2 = 4x²
-5x x 3x = -15x
-5x x 2 = -10x
-3 x 3x = -9x
-3 x 2 = -6
6x³ + 4x² - 15x - 10x - 9x - 6
6x³ + -11x² - 19x - 6
How do you factorise quadratics? (single x²)
- the reverse of expanding a double bracket
1) there will be an x at the front of each bracket
2) we need to find 2 numbers that multiply to give the number at the end and add to give the number of xs (middle number)
eg. x² + 5x + 6
(x + ) (x + )
3 x 2 = 6
3 + 2 = 5
so
(x + 3) (x + 2)
How do you complete the square?
- we need to rearrange the expression into the form (x + a)² + b
1) half the number of xs to get a
2) subtract a² and combine it with any other number in the original expression
eg. x² + 2x + 6
(x + )² - + 6
2x / 2 = 1 so
(x + 1)² - + 6
1² = 1 so
(x +1)² - 1 + 6
How do you solve a quadratic equation by factorising?
1) factorise equation
eg. x² + 5x = 0
x(x + 5) = 0
2) separate term outside of bracket from terms inside bracket
so x = 0 or x + 5 = 0
3) simplify expression of terms that were inside of bracket
x + 5 = 0
(-5)
x = -5
so x = 0 or x = -5
How do you solve equations by using the quadratic formula?
- quadratic formula is x = -b ± √b² - 4ac / 2a
- a is the number of x²
- b is the number of xs
- c is the number
eg. x² - 5x + 3 = 0
a = 1 b = -5 c = 3
so
x = -(-5) ± √(-5)² - 4(1)(3) / 2(1)
(solve on calculator)
x = 5 + 3.6055 / 2 or x = 5 - 3.6055 / 2
x = 4.18 or x = 0.70
What is the rule for angles in a semi-circle?
they equal 90°
What is the rule for angles in the same segment?
they are equal
What is the rule for angles at a centre?
they are twice the angle on the edge
What is a cyclic quadrilateral?
What is the rule?
all four corners are on the edge of the circle
diagonally opposite angles add to 180°
What is the rule for alternate segments?
diagonally opposite angles are equal
What is the rule for the radius and a tangent?
the radius and tangent make a right angle
What is the rule for a triangle formed by two radii?
it is isosceles
What is the equation for speed?
speed = distance/time
What is the equation for density?
density = mass/volume
How do you find the volume of a cone?
volume = 1/3πr² x h
How do you find the nth term of a linear sequence?
- find the difference between each number (should be the same), this is the number of ns
- plus or minus the amount it takes to get from the start of the sequence to n
e.g. 1, 3, 5, 7
2 2 2
so 2n
1+1 = 2 so
2n + 1
How do you find the nth term of a quadratic sequence?
- find the difference between each number (should be different)
- find the (second) difference between these numbers (should be the same)
- divide this number by 2 to find the amount of n²s
- write out the original sequence, and minus the square numbers from the sequence (if it was 2n², it would be the square numbers multiplied by 2)
- find the nth term of the sequence you are left with (should be linear)
- add the nth term of the linear sequence to the n²s
e.g. 1, 5, 11, 19, 29
4 6 8 10
2 2 2
2/2 = 1n²
-1 5 11 19 29
1 4 9 16 25
——————–
0 1 2 3 4
1 1 1 1
so nth term is n + 1
final nth term would be n² + n + 1
