a < b means a is less than b (so b is greater than a)
a £ b means a is less than or equal to b (so b is greater than or equal to a)
a ³ b means a is greater than or equal to b etc.
a > b means a is greater than b etc.
If you have an inequality, you can add or subtract numbers from each side of the inequality, as with an equation. You can also multiply or divide by a constant. However, if you multiply or divide by a negative number, the inequality sign is reversed.
Solve 3(x + 4) < 5x + 9
3x + 12 < 5x + 9
\ -2x < -3
\x > 3/2 (note: sign reversed because we divided by -2)
Inequalities can be used to describe what range of values a variable can be.
E.g. 4 £ x < 10, means x is greater than or equal to 4 but less than 10.
A man buys 3 fish and 2 chips for £2.80
A woman buys 1 fish and 4 chips for £2.60
Method 1: elimination
First form 2 equations. Let fish be f and chips be c.
We know that:
3f + 2c = 280 (1)
f + 4c = 260 (2)
Doubling (1) gives:
6f + 4c = 560 (3)
(3)-(2) is 5f = 300
\ f = 60
Therefore the price of fish is 60p
Substitute this value into (1):
3(60) + 2c = 280
\ 2c = 100
c = 50
Therefore the price of chips is 50p
Expand (2x + 3)(x - 1):
(2x + 3)(x - 1)
= 2x² - 2x + 3x - 3
= 2x² + x - 3
Factorise 12y² - 20y + 3
12y² - 18y - 2y + 3 [here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].
The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y.
6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y - 3) -1(2y - 3)
The answer is (2y - 3)(6y - 1)
Trial and Improvement
trial and Improvement
Solve t³ + t = 17 by trial and improvement.
Firstly, select a value of t to try in the equation. I have selected t = 2. Put this value into the equation. We are trying to get the answer of 17.
If t = 2, t³ + t = 2³ + 2 = 10 . This is lower than 17, so we try a higher value for t.
If t = 2.5, t³ + t = 18.125 (too high)
If t = 2.4, t³ + t = 16.224 (too low)
If t = 2.45, t³ + t = 17.156 (too high)
If t = 2.44, t³ + t = 16.966 (too low)
If t = 2.445, t³ + t = 17.061 (too high)
So we know that t is between 2.44 and 2.445. So to 2 decimal places, t = 2.44.
LCF,HCM and Nth number
LCM and HCF
The lowest common multiple (LCM) of two or more numbers is the smallest number into which they evenly divide. For example, the LCM of 2, 3, 4, 6 and 9 is 36.
The highest common factor (HCF) of two or more numbers is the highest number which will divide into them both. Therefore the HCF of 6 and 9 is 3.
What is the nth term of the sequence 2, 5, 10, 17, 26... ?
n = 1 2 3 4 5
n² = 1 4 9 16 25
n² + 1 = 2 5 10 17 26
This is the required sequence, so the nth term is n² + 1. There is no easy way of working out the nth term of a sequence, other than to try different possibilities.
Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably be a three in the formula, etc.
In many cases, square numbers will come up, so try squaring n, as above. Also, the triangular numbers formula often comes up. This is n(n + 1)/2 .
A bearing is the angle, measured clockwise from the north direction. Below, the bearing of B from A is 025 degrees (note 3 figures are always given). The bearing of A from B is 205 degrees.
A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB= 8km and AC= 6km, what is the bearing of B from C?
tanC = 8/6, so C = 53.13º
y = 180º - 135º = 45º (interior angles)
x = 360º - 53.13º - 45º (angles round a point)
= 262º (to the nearest whole number)
In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
i.e.: c² = a² + b² in the following diagram:
Find AC in the diagram below.
AB² + AC² = BC²
AC² = BC² - AB²
= 13² - 5²
= 169 - 25 = 144
AC = 12cm
angle A = angle D
angle B = angle E
angle C = angle F
AB/DE = BC/EF = AC/DF = perimeter of ABC/ perimeter of DEF
Two triangles are similar if:
1) 3 angles of 1 triangle are the same as 3 angles of the other
or 2) 3 pairs of corresponding sides are in the same ratio
or 3) An angle of 1 triangle is the same as the angle of the other triangle and the sides containing these angles are in the same ratio.
In the above diagram, the triangles are similar. EF = 6cm and BC = 2cm . What is the length of DE if AB is 3cm?
EF = 3BC, so DE = 3AB = 9cm.
Sin,Cos and Tan
In any right angled triangle, for any angle:
The sine of the angle = the length of the opposite side
the length of the hypotenuse
The cosine of the angle = the length of the adjacent side
the length of the hypotenuse
The tangent of the angle = the length of the opposite side
the length of the adjacent side
The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.
sin = o/h cos = a/h tan = o/a
Often remembered by: soh cah toa
Find the length of side x in the diagram below:
The angle is 60 degrees. We are given the hypotenuse and need to find the adjacent side. This formula which connects these three is:
cos(angle) = adjacent / hypotenuse
therefore, cos60 = x / 13
therefore, x = 13 × cos60 = 6.5
therefore the length of side x is 6.5cm.
The graphs of sin, cos and tan:
The following graphs show the value of sinø, cosø and tanø against ø (ø represents an angle). From the sin graph we can see that sinø = 0 when ø = 0 degrees, 180 degrees and 360 degrees.
Sine and Cosine Formulae:
sin x = sin (180 - x)
e.g. sin 130 = sin (180 - 130) = sin 50
cos x = -cos (180 - x)
The Sine Rule:
This works in any triangle:
a = b = c
sinA sinB sinC
alternatively, sinA = sinB = sinC
a b c
The Cosine Rule:
c² = a² + b² - 2abcosC
can also be written as:
a² = b² + c² - 2bccosA
The area of a triangle
The area of any triangle is ½ absinC (using the above notation)
This formula is useful if you don't know the height of a triangle (since you need to know the height for ½ base × height).
Lines AB and CD are parallel to one another (hence the » on the lines).
a and d are vertically opposite angles. Vertically opposite angles are equal. (b and c, e and h, f and g are also vertically opposite).
g and c are corresponding angles. Corresponding angles are equal. (h and d, f and b, e and a are also corresponding).
d and e are alternate angles. Alternate angles are equal. (c and f are also alternate). Alternate angles form a 'Z' shape and are sometimes called 'Z angles'.
a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d and c, c and a, d and b, f and e, e and g, h and g, h and f are also adjacent).
d and f are interior angles. These add up to 180 degrees (e and c are also interior).
Any two angles that add up to 180 degrees are known as supplementary angles.
The angles around a point add up to 360 degrees.
The angles in a triangle add up to 180 degrees.
The angles in a quadrilateral add up to 360 degrees.
The angles in a polygon (a shape with n sides) add up to 180(n - 2) degrees.
The exterior angles of any polygon add up to 360 degrees.
The area of a triangle = half × base × height
The area of a circle = pr² (r is the radius of the circle)
The area of a parallelogram = base × height
If the radius of the circle is r,
Area of sector = pr² × A/360
Arc length = 2pr × A/360
In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360
Isosceles triangles have two equal angles. The sides of the triangle opposite the equal angles are equal in length to one another.
Equilateral triangles have all of their sides and angles equal. Since there are 180 degrees in a triangle and all the angles are equal, each angle is 60 degrees.
Parallelogram: opposite sides are parallel, opposite angles are equal, the diagonals bisect one another.
Rhombus: (a parallelogram with all four sides of equal length), diagonals bisect one another at right angles.
Trapezium: One pair of opposite sides are parallel.
Square: All sides are equal, all angles are 90 degrees, diagonals bisect one another at 90 degrees.
Rectangle: All angles are 90 degrees, diagonals bisect one another.
Finding the gradient of a straight-line graph
It is often useful or necessary to find out what the gradient of a graph is. For a straight-line graph, pick two points on the graph. The gradient of the line = (change in y-coordinate)/(change in x-coordinate) .
In this graph, the gradient = (change in y-coordinate)/(change in x-coordinate) = (8-6)/(10-6) = 2/4 = 1/2
We can of course use this to find the equation of the line. Since the line crosses the y-axis when y = 2, the equation of this graph is y = ½x + 2 .
Finding the gradient of a curve
To find the gradient of a curve, you must draw an accurate sketch of the curve. At the point where you need to know the gradient, draw a tangent to the curve. A tangent is a straight line which touches the curve at one point only. You then find the gradient of this tangent.
Find the gradient of the curve y = x² at the point (3, 9).