Tasmania TCE Everyday Adult Standard - Mathematics cheat sheet

Order of operations: evaluate brackets, then indices, then multiplication and division left to right, then addition and subtraction left to right.

Rounding: to round to a place value, look at the next digit; 5 or more rounds up, 4 or less keeps the digit unchanged.

A percentage is a fraction out of 100: 15% = 15/100 = 0.15; to find 15% of $80, compute 0.15 × 80 = $12.

Percentage increase/decrease = (change ÷ original) × 100; a rise from 40 to 50 is (10 ÷ 40) × 100 = 25%.

To increase by a percentage in one step, multiply by (1 + rate); a 20% increase uses a multiplier of 1.20, a 20% decrease uses 0.80.

A ratio a:b shares a total of T into parts of size T ÷ (a + b) per ratio unit; 2:3 on $50 gives $20 and $30.

Scientific notation writes a number as a × 10^n where 1 ≤ a < 10; 4 500 000 = 4.5 × 10^6.

Estimating by rounding each value to one significant figure gives a quick check: 48 × 21 ≈ 50 × 20 = 1000.

Fractions, decimals and percentages are interchangeable; compare quantities by converting to the same form before ordering.

Metric prefixes: kilo = ×1000, centi = ÷100, milli = ÷1000; 1 km = 1000 m, 1 m = 100 cm = 1000 mm.

Area units are squared: 1 m² = 10 000 cm²; volume units are cubed: 1 m³ = 1 000 000 cm³ = 1000 L.

Perimeter of a rectangle = 2(l + w); area of a rectangle = l × w; area of a triangle = ½ × base × height.

Circumference of a circle = πd = 2πr; area of a circle = πr²; use π ≈ 3.14 unless told otherwise.

Volume of a rectangular prism = length × width × height; a container 20 cm × 10 cm × 5 cm holds 1000 cm³ = 1 L.

Speed, distance, time: distance = speed × time; time = distance ÷ speed; keep units consistent (km with hours gives km/h).

Temperature conversion: °F = (°C × 9/5) + 32; °C = (°F − 32) × 5/9.

Scale drawings: a scale of 1:100 means 1 cm on the plan represents 100 cm (1 m) in reality.

When measuring, the smallest marked unit sets precision; a ruler marked in mm is read to the nearest 0.5 mm by estimation.

Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.

In any triangle the interior angles sum to 180°; in any quadrilateral they sum to 360°.

A regular polygon with n sides has interior angle = (n − 2) × 180° ÷ n; a regular hexagon has 120° interior angles.

Pythagoras' theorem: in a right triangle a² + b² = c², where c is the hypotenuse (the side opposite the right angle).

Similar shapes have equal angles and proportional sides; a scale factor k multiplies lengths by k and areas by k².

Congruent shapes are identical in shape and size; similarity preserves shape but allows a change of size.

Nets fold into solids: a cube net has 6 squares; a rectangular prism net has 3 pairs of matching rectangles.

Coordinates are written (x, y): the first number moves horizontally, the second vertically; the origin is (0, 0).

A transformation is a translation (slide), reflection (flip), rotation (turn) or enlargement (scale); only enlargement changes side lengths.

Mean = sum of values ÷ number of values; median = middle value of ordered data; mode = most frequent value.

The range = highest − lowest and measures spread, not centre; it is not affected by how many values lie between.

For an even number of ordered values, the median is the mean of the two middle values.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; values lie from 0 (impossible) to 1 (certain).

Complementary probability: P(not A) = 1 − P(A); the probability of not rolling a 6 is 1 − 1/6 = 5/6.

For independent events, multiply probabilities: two heads in a row = ½ × ½ = ¼.

In a bar or column chart, compare heights; in a line graph, the slope shows the rate of change over time.

A sample is biased if some members of the population are more likely to be selected; a random sample gives each member an equal chance.

Outliers are values far from the rest; they pull the mean strongly but affect the median little.

Simple interest: I = P × r × t, where r is the annual rate as a decimal and t is time in years.

Compound interest grows balance by multiplying by (1 + r)^n; $1000 at 5% for 2 years gives 1000 × 1.05² = $1102.50.

GST in Australia is 10%: GST amount = price × 0.10; price including GST = price × 1.10; to remove GST divide by 1.10.

A discount of 25% means paying 75%; multiply the marked price by 0.75 to get the sale price.

Wages: gross pay = hours × hourly rate; overtime is usually paid at a higher multiple, so compute those hours separately.

Best-buy comparison uses unit price = total cost ÷ quantity; the lower unit price per gram or per litre is better value.

A budget balances total income against total expenses; a surplus occurs when income exceeds expenses, a deficit when expenses exceed income.

A percentage error or percentage change compares the difference to the original value: (new − old) ÷ old × 100.

Identify what is given, what is asked, and which quantity links them before choosing an operation.

Check a result against an estimate: if 19.8 × 4.9 is claimed as 970, the estimate 20 × 5 = 100 rounds to 100, so the claim is wrong.

Work backwards from the answer by reversing operations to verify a solution.

When two rates combine (for example, workers and time), the total work is constant: workers × time stays fixed, so doubling workers halves the time.

Draw a diagram or table to organise multi-step problems and keep track of units.

Read units carefully: a rate of $12 per hour over 90 minutes pays 1.5 × 12 = $18.

Watch for distractors: extra information in a problem may be irrelevant to the quantity requested.

Justify an answer with a short sentence stating the calculation and the unit, so the reasoning can be checked.

Identify what is given, what is asked, and which quantity links them before choosing an operation.

Check a result against an estimate: if 19.8 × 4.9 is claimed as 970, the estimate 20 × 5 = 100 rounds to 100, so the claim is wrong.

Work backwards from the answer by reversing operations to verify a solution.

When two rates combine (for example, workers and time), the total work is constant: workers × time stays fixed, so doubling workers halves the time.

Draw a diagram or table to organise multi-step problems and keep track of units.

Read units carefully: a rate of $12 per hour over 90 minutes pays 1.5 × 12 = $18.

Watch for distractors: extra information in a problem may be irrelevant to the quantity requested.

Justify an answer with a short sentence stating the calculation and the unit, so the reasoning can be checked.