How to use this sheet: take one topic, cover the formula, and say it before you look. Then answer a past question that needs it, and write the working in full.
Print this page if you want a paper copy for revision at home. The navigation and the questions at the bottom stay off the paper. A printed copy is for home. GLearn does not set the rules of the examination hall.
The revision plan orders the weeks. The how to pass guide covers the timetable, past questions, written answers and exam day. These formulas are standard mathematics. GLearn does not publish a pass rate.
Number and numeration
- Standard form
A × 10n, where A is at least 1 and less than 10, and n is an integer.
- HCF and LCM
For two whole numbers, their product equals their HCF multiplied by their LCM.
- A number in base b
Digits d2, d1 and d0 have value d2 × b2 + d1 × b + d0.
Each digit is a whole number less than the base b. The same pattern continues for more digits: the next digit to the left is multiplied by b³.
Fractions and percentages
- Adding fractions
a/b + c/d = (a × d + b × c) / (b × d), when b and d are not 0.
- Multiplying fractions
a/b × c/d = (a × c) / (b × d), when b and d are not 0.
- Dividing fractions
a/b ÷ c/d = (a × d) / (b × c), when b, c and d are not 0.
- A percent of a quantity
R percent of a quantity Q is (R / 100) × Q.
- One quantity as a percent of another
The percent that A is of B is (A / B) × 100, when B is not 0.
- Increase by a percent
After an increase of R percent, the new amount is the original × (100 + R) / 100.
- Decrease by a percent
After a decrease of R percent, the new amount is the original × (100 − R) / 100.
Ratio
- Sharing in a ratio
When an amount M is shared in the ratio a : b, the share of a is (a / (a + b)) × M.
The share of b is (b / (a + b)) × M. The two shares add to M.
- Equal ratios
If a : b = c : d, then a × d = b × c.
Simple and compound interest
- Simple interest
I = (P × R × T) / 100
P is the principal. R is the rate in percent per year, so 5 percent means R = 5, not a decimal. T is the time in years. Four months is 4/12 of a year. The amount you have at the end is P + I.
- Amount with simple interest
A = P × (1 + (R × T) / 100)
This is the same result as adding the simple interest to the principal. Interest is not added back during the time.
- Compound interest, added once a year
A = P × (1 + R / 100)n
A is the amount after n years. R is the percent per year. The compound interest is A − P. Do not also apply the simple interest formula to the same sum.
Mensuration
- Rectangle
Perimeter = 2 × (length + width). Area = length × width.
- Square
Perimeter = 4 × side. Area = side2.
- Triangle
Perimeter = a + b + c. Area = (1/2) × base × height.
The height is the perpendicular height that belongs to the base you used.
- Parallelogram
Area = base × height. Perimeter = 2 × (a + b).
The height is perpendicular to the base you used. a and b are a pair of adjacent sides.
- Trapezium
Area = (1/2) × (a + b) × height.
a and b are the two parallel sides. The height is perpendicular to those sides.
- Rhombus or kite
Area = (d1 × d2) / 2.
d₁ and d₂ are the diagonals.
- Circle
Circumference = 2 × π × r = π × d. Area = π × r2.
r is the radius and d is the diameter, so d = 2 × r. Use the value of π the question gives. 22/7 and 3.142 are approximations. Leave the answer in terms of π when the question allows it.
- Cube
Volume = side3. Total surface area = 6 × side2.
- Cuboid
Volume = length × width × height. Total surface area = 2 × (length × width + length × height + width × height).
- Prism
Volume = area of the cross section × length.
For a triangular prism, the cross section is the triangular end.
- Cylinder
Volume = π × r2 × h. Curved surface area = 2 × π × r × h. Total surface area = 2 × π × r × (r + h).
r is the radius of the circular end and h is the perpendicular height. The total surface area is for a closed cylinder, including both ends.
Angles and polygons
- Straight line and a point
Angles on a straight line add to 180°. Angles at a point add to 360°. Vertically opposite angles are equal.
- Triangle
The interior angles of a triangle add to 180°.
- Interior angles of a polygon
The interior angles of a polygon with n sides add to (n − 2) × 180°.
A quadrilateral is the case n = 4, so its interior angles add to 360°.
- Exterior angles
The exterior angles of a convex polygon add to 360°. Each exterior angle of a regular polygon with n sides is 360° / n.
- Interior angle of a regular polygon
Each interior angle of a regular polygon with n sides is [(n − 2) × 180°] / n.
- Parallel lines
When a straight line crosses a pair of parallel lines, corresponding angles are equal, alternate angles are equal, and allied angles add to 180°.
Allied angles are the pair of interior angles on the same side of the crossing line.
Pythagoras
- Pythagoras
a2 + b2 = c2
This is for a right angled triangle. c is the side opposite the right angle, the hypotenuse. If the squares of the two shorter sides add to the square of the longest side, the angle opposite the longest side is a right angle.
Statistics
- Mean
Mean = (sum of the values) / (number of values).
- Mean from a frequency table
Mean = Σ(f × x) / Σf.
x is the value, or the midpoint of a class. f is the frequency of that value or class.
- Median
Sort the values. If the count is odd, the median is the middle value. If the count is even, the median is the mean of the two middle values.
- Mode
The mode is the value that occurs most often. A list can have more than one mode.
Probability
- Probability of an event
P(A) = (number of favourable outcomes) / (number of equally likely outcomes).
- The range of a probability
0 ≤ P(A) ≤ 1.
- The complement
P(not A) = 1 − P(A).
- Events that cannot happen together
If A and B cannot happen together, P(A or B) = P(A) + P(B).
Algebra
- Expanding a bracket
a × (b + c) = a × b + a × c.
- Square of a sum
(a + b)2 = a2 + 2 × a × b + b2.
- Square of a difference
(a − b)2 = a2 − 2 × a × b + b2.
- Difference of squares
a2 − b2 = (a − b) × (a + b).
- A linear equation
If a × x + b = c and a is not 0, then x = (c − b) / a.
Sets
- Union of two sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
n(A) is how many elements are in A. The union is everything in A or B or both. The intersection is only what is in both. The subtraction stops the overlap being counted twice.
- Complement
n(A′) = n(U) − n(A).
U is the universal set. A′ is everything in U that is not in A. A and A′ have nothing in common, and together they make U.
- Number of subsets
A set with n elements has 2n subsets.
The count includes the empty set and the set itself.
- The empty set
The empty set has no elements, so n(∅) = 0. It is a subset of every set.
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