gcse maths revison

8/6/2019 GCSE Maths Revison

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Number

Rounding and approximating

Multiplying and dividing by powers of 10

y When you multiplyby a power of 10, the decimal point moves to the right.e.g. 45.9 X102 = 4590

y When you divide by a power of 10, the decimal point moves to the left.e.g. 45.9 102 = 0.459

y The number of places the decimalpointmoves depends on the number of zeros or thepower of 10.

Prime factors

y When a number is written as a product of prime factors, it is written as a multiplicationconsisting only of prime numbers.

e.g. 30 = 2 X 3 X 5

y To find the prime factors of a number, divide by a prime numbers until the answer is aprime number.

y Products of prime factors can be expressed in index form.e.g. 2 X 2 X 19 = 22 X 19

Lowest common multiple and highest common factor

y Thelowestcommonmultiple (LCM) of two numbers is the smallest number in the timestables of both of the numbers.e.g. The LCM of 6 and 7 is 42. _ The LCM of 8 and 20 is 40.

y Thehighestcommon factor (HCF) of two numbers is the biggest number that dividesexactly into the two numbers.

e.g. The HCF of 24 is 18 and 6. _ The HCF of 45 and 36 is 9.

y To find the LCM and HCF, write out the multiples and factors of each number.


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Fractions

One quantity as a fraction of another

y To write one quantity as a fraction of another, write the first quantityas thenumerator(brojnik)and the second quantity as the denominator(nazivnik).e.g. what is 8 as fraction of 20?

Write as

then cancel to

.

Percentage

The percentage multiplier

y Using the percentage multiplier is the best way to solve percentage problems.y The percentage multiplier is the percentage expressed as a decimal.

e.g. 72% gives a multiplier of 0.72. _ 20% gives a multiplier of 0.20 or 0.2.

y The multiplier for a percentage increase or decrease is the percentage multiplieradded toor subtracted from 1.

e.g. An 8% increase is a multiplier of 1.08 (1 + 0.08). _ A 5% decrease is a multiplier of 0.95

(1- 0.05).

Calculating a percentage increase or decrease

y To calculate the new value after a quantity is increased ordecreased by a percentage,simply multiply the original quantity by the percentage multiplier for the increase or

decrease.

e.g. What is the new cost after a price of 56 is decreased by 15% ? ~ Work it out as 0.85 X

56 = 47.60 .

Expressing one quantity as a percentage of another

y To calculate one quantityas apercentage of another, divide the first quantity by thesecond. This will give a decimal, which can be converted to a percentage.

e.g. A plant grows from 30 cm to 39 cm in a week. What is the percentage growth? ~ The

increase is 9 cm. 9 30 = 0.3 and this is 30%.

Compound interest

y When money is invested in, for example, a savings account, it can earn a certain rate ofinterest each year.

y This interest is then added to the original amount and the newtotalamount then earnsinterest at the same rate in the following year.

y The process whereby interest is accumulated each year is called compound interest.y Rememberthe formula as total amount = P X (1 + x)n, where P is the original amount of

(money) invested, x is the interest rate expressed as a decimal or a fraction and n is the

number of years for which money is invested.

y This type of problem can also be about increasing or decreasing populations, salaries,weights, etc.


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Reverse percentage

y After an amount has been increased or decreased by a certain percentage, the originalamount can be found from the new amount.

y There are two methods to do this: the unitarymethod and the multipliermethod.e.g. After a 22% increase, the population of a village is 1464. What was the population

originally? ~ The multiplier for a 22% increase is 1.22. The calculation is: 1464 1.22 = 1200 .So the original population was 1200. (the multiplier method - simpler)

Ratio

Ratios

y Aratiois a way of comparing the sizes of two or more quantities.y A colon (:) is used to show ratios. e.g. 3 : 4 and 6 : 20 are ratios.y A quantity can be divided into portions

that are in a givenratio.

y The process has three steps: Step 1: Add the separate parts of

the ratio

Step 2:Divide this number into theoriginal quantity

Step 3: Multiply this answer by theoriginal parts of the ratio

e.g. Share $40 in the ratio 2 : 3. ~ Add 2 and 3 to find the total number of parts: 5. Divide

40 by 5 to find the value of each part: 8. multiply each term in the ratio by 8 : 2X8 = 16,

3X8=24. So $40 divided in the ratio 2 : 3 gives shares of $16 and $24 .

NOTE: There are other methods to solve this problem. This is just one of thepossibilities.

y When one part of the ratio is known, it is possible to calculate other values.Direct proportion problems

y When solving directproportion problems,work out the costof one item. This is called

theunitarymethod.

e.g. If eight cans of cola cost $3.60, how

much do five cans cost? ~ The cost of one

can is 360 8 = 45p, so five cans cost 5 X0.45 = $2.25.


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Powers and reciprocals

Powers

y Powers are convinient way of writing repetitive multiplications.y Powers are also called indices (singular: index)y Thepower "2" has a special name: "squared".y Thepower "3" has a special name: "cubed".

Negative indices

y A negative index is a convenient way of writing the reciprocal of a number or term.Standard form

y Standard form (also called standard index form) is a way of writing very large or verysmall numbers, using powers of 10.

y Every standard form number is of the form a X 10n, where and n is a positiveor negative whole number.

y Standard form numbers can be combined, using the rules of powers.y Count how many places you have to move the decimal point to get the numerical value of

the power. Moving rightisnegative, leftispositive.

Rational numbers

y A rationalnumber is any number that can beexpressed as a fraction.

y Some fractions result in terminatingdecimalsand some fractions result in recurring

decimals.

e.g.

this is a terminating decimal.

= 0.3333... this is a reccuring decimal.

Rules of indices

=


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Variation

Direct variation

y Directvariation means the same as directproportion.y

There is a direct variation between two variables when one variable is a simple multiple ofthe other.

y When two variables are proportional to each other, you can set up an equation with aconstant of proportionality.

y When a power or root of one variable is directly proportional to another varia ble, theprocess is the same as for a linear direct variation.

e.g. y is directly proportional to the square of x. ~ y x2 => y = kx2, where k is the constant

of proportionality.

y In an examination question, some numerical information will be given to enable theconstant of proportionality to be found.

e.g. y is directly proportional to the square root of x. When y = 6, x = 4. ~ y =>

y = k. 6 = k => 6 = 2k => k = 3

y A question will usually give the value of one variable and ask for the value of the othervariable.

e.g. y is directly proportional to the cube of x. If y = 20 when x = 2, what is the value of y

when x = 4? ~ When x =4, y =2.5 X 43 =2.5 X 64

= 160.

Inverse variation

y There is an inverse variation between two variables when one variable decreases as theother increases such that the product of the two variables is constant.

y This is represented mathematically by the reciprocal.e.g. y is inversely proportional to x. ~

y Inverse variation problems involving powers, roots and given numerical information aresolved using the same method as for direct variation problems.

Limits

Limits of accuracy

y The range of values between the limits of accuracy is called the rounding error.y e.g. the length of a piece of paper is 24.7 cm accurate to one decimal place. What are the

limits of accuracy? ~ 24.65 lenght 24.75 (continuous data)

y Data can be continuous ordiscrete (integer).


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Calculating with limits of accuracy

y When we combine two or more linear values,the errors in the linear measures will be

compounded producing a larger error in the

calculated value.

y It is important to combine the values togetherin the correct way. The table below shows the

combinations to give the maximum and

minimum values for the four rules of arithmetic.

Operation Maximum Minimum

Addition (a + b) amax + bmax amin + bminSubtraction (a - b) amax - bmin amin - bmaxMultiplication (a X b) amax X bmax amin X bminDivision (a b) amax bmin amin bmax

Basic algebra

Expansion and simplification

y When you are asked to expand and simplify an expression, it means expand any bracketsand then simplify by collecting like terms.

Factorisation

y To factorise an expression, look for the highestcommon factor of both terms.Linear equations

Solving linear equations

y Solving an equation means finding thevalue of the variable that makes it true.

y There are different ways of solvingequations. Rearrangement is the most

efficient method.

y When an equation contains brackets youshould multiply out the brackets first andthen solve the equation in the normal

way.

y When a letter appears onboth sides of an equation, use the "change sides, change signs"rule.


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Simultaneous equations

Simultaneous equations

y Simultaneous equations can be solved by one of two methods: the eliminationmethod &the substitutionmethod.

y Follow these steps to use the elimination method to solve simultaneous equations:Step 1: Balance the coefficients of one of the variables.Step 2: Add or subtract the equations to eliminate this variable.Step 3: Solve the resulting linear equation to find the value of the other variable.Step 4: Substitute the value back into one of these previous equations.Step 5: Solve this equation to find the value of the second variable.

y The substitutionmethod works by substituting one equation into the other.Algebra 2

Quadratic expansion

y A quadratic expressionis one in which the highest power of any term is 2.y There are three methods for quadratic expansion.y Splittingthe brackets - The terms inside the first brackets are split and used to multiply

the terms in the second brackets.

y FOIL - FOIL stands for First, Outer, Inner and Last. This is the order in which terms aremultiplied.

y Box method - This is similar to the box method used for long multiplication.

Factorising quadratic expressionsFactorising a quadratic with a unit coefficient of x2

y When an expression such as x2 + ax + b is factorised, it is always of the form:x2 + ax + b = (x+p)(x + q) where p + q = a and pq = b.

y When an expression such as ax2 + bx + c is factorised, it is always of the form:ax2 + bx + c = (mx + p)(nx + q), where mn = a and pq = c.

y Once a quadratic expression has been factorised, it can be used to solve the equivalentquadratic equation, but remember, the equation must be in form as ax2 + bx + c = 0 before

it can be solved.

Solving the general quadratic by the quadratic

formula

y One way to solve this type of equation is to use the quadratic formula. This formula cansolve any soluble quadratic equation.

y The solutions of the equation ax2 + bx + c = 0 are given by:


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Quadratic equations with no solution

y The quantity b2 - 4ac in the quadratic formula is known as the discriminant.y When b2 - 4ac is positive, the equation has two roots or solution.y When b2 - 4ac is zero, the equation has one repeated root or one solution.y When b2 - 4ac is negative, the equation has no roots and is not soluble.

NOTE : You will not be asked about this in GCSE examinations so, if this happens, you have made a

mistake and should check your working.

Linear graphs

y You only need twopoints to draw a straight-line graph. However, it is better to use threepoints, because the third point acts as a check.

y Plot the points you have found and jointhem up to draw the line.Gradients

y Thegradient of a line is a measure of its slope.y It is calculated by dividingthe verticaldistance between two points on the line by the

horizontal distance between the same two points.

y Lines that slope from left to right have negative gradients.y Note the right-angled triangles drawn along grid lines. These are used to find gradients.y To draw a line with a certain gradient, for every unit moved horizontally, move upwards

(or downwards if the gradient is negative) by the number of units of the gradient.

Equations of lines

y When a graph is expressed in the form y = mx + c, the coefficient of x, m, is the gradient,and the constant term, c, is theintercept on the y-axis.

Linear graphs and equations

Uses of graphs - solving simultaneous equations.

y You can find the solution of two simultaneous equations by drawing their graphs on thesame pair of axis. The graphs intersect is the solution.


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Parallel and perpendicular lines

y If two lines are parallel their gradients are the same.y If two lines are perpendicular their gradients are negative reciprocals of each other. For

example, lines with a gradient of 2 and -1/2 are perpendicular to each other.

Quadratic graphs

y Quadratic graphs always have the samecharacteristic shape, which is called a parabola.

Using graphs to solve quadratic equations

y To solve a quadratic equation from its graph, read the values where the curve crosses thex-axis.

The significant points of a quadratic graph

y A quadratic graph has fourpoints that are of interest to a mathematician.y The points where the graph crosses the x-axis are called the roots.y The point where the graph crosses the y-axis is called the intercept.y The point which is the maximum or minimum point of the graph is called the vertex.

Other graphs

Reciprocal graphs

y A reciprocal equation has the form y = a/x.y All reciprocal graphs have a similar shape and some

symmetry properties.

y The lines y =x and y = -x are lines of symmetry.y The closer x gets to zero, the nearer the graph gets

to y-axis.

y As x increases, the graph gets closer to the x-axis.y The graph never actually touches the axes, but gets closer and closer to them. The axes

are called asymptotes.


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The nth term

nth term of a sequence

y A linear sequence has the same difference between consecutive terms.y

The nth term of a linear sequence is always of the formA

n b.y To find the coefficient of n, A, find the difference between consecutive terms.y To find the value of b, work out the difference between A and the first term of the

sequence.

Special sequances

y There are many special sequences that you should be able to recognise.y The even numbers 2, 4, 6, 8, 10 ,12 ... The nth term is 2n.y The odd numbers 1, 3, 5, 7, 9, 11 ... The nth term is 2n - 1.y The square numbers 1, 4, 9, 16, 25, 36 ... The nth term is n2.y The triangular numbers 1, 3, 6, 10, 15, 21 ... The nth term is

1

/2n(n + 1).y The powers of 2 2, 4, 8, 16, 32, 64 ... the nth term is 2 n.y The powers of 10 10, 100, 1000, 10 000, 100 000 ... The nth term is 10n.y The prime numbers 2, 3, 5, 7, 11, 13, 17, 19 ... There is no nth term as there is no pattern to

the prime numbers.

NOTE:The only even prime number is 2.

Inequalities

Solving inequalities

y An inequality is an algebraic expression that uses the signs (less than), (greater than), (less than or equal to) and (greater than or equal to).

y The solutions to inequalities are a range of values.y Linearinequalities can be solved using the same rules that you use to solve equations.y The solutions to a linear inequality can be shown on a numberline by using the

convection than an open circle is a strictinequality and a filled-in circle is an inclusive

inequality.

Graphical inequalities (linear programming)

y A linearinequality can be plotted on a graph.y The result is region that lies on one side or the other of a straight boundaryline.y The first step is to draw the boundary line. This is found by replacing the inequality sign

with an equals sign. If the inequality is strict ( ) the boundary line should be

dashed. If the inequality is inclusive ( ) the boundary line should be solid.

y Once the boundary line is drawn, the required region is shaded.


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y To confirm on which side of the line the region lays, choose any point that is not on theboundary line and test it in the inequality. If it satisfies the inequality, that is the side

required. If it doesn't, the other side is required.

y You will be required to show more than one inequality on the same graph.y In this case, it is clearer to shade the regions that are not required, so that the required

region is left unshaded.

gcse maths revison

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