NCEA Numeracy cheat sheet

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.

Order of operations: brackets, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

Adding or subtracting fractions needs a common denominator: a/b ± c/d = (ad ± bc)/(bd).

Dividing by a fraction is multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.

Percentage change = (new − old)/old × 100; a 20% increase then a 20% decrease gives 0.96 of the original, a net 4% fall.

A percentage of an amount: amount × percent/100; to add GST of 15%, multiply by 1.15; to remove it, divide by 1.15.

Ratio sharing: total parts = sum of ratio numbers, one share = amount ÷ total parts, then multiply by each ratio number.

Standard form: a × 10^n with 1 ≤ a < 10; multiplying adds the powers, dividing subtracts them.

Linear patterns: for constant difference d, nth term = d·n + c where c = first term − d; check by substituting n = 1.

Substituting into a formula: replace each variable with its value in brackets, then evaluate using order of operations.

Perimeter is the distance around a shape; area is measured in square units (cm², m²), volume in cubic units (cm³, m³).

Rectangle area = length × width; triangle area = ½ × base × height (height perpendicular to the base); circle area = πr².

Circumference = πd = 2πr; use π ≈ 3.14 unless the question states otherwise.

Prism volume = cross-sectional area × length; cylinder volume = πr²h.

Unit conversions: 1 m = 100 cm = 1000 mm; 1 km = 1000 m; 1 hectare = 10 000 m²; 1 L = 1000 mL = 1000 cm³.

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

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

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of a triangle sum to 180°.

Scale drawings: map distance = real distance × scale factor; 1:100 means 1 cm on the plan represents 100 cm in reality.

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

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

Range = maximum − minimum; interquartile range (IQR) = upper quartile − lower quartile and measures the spread of the middle 50%.

An outlier is usually a value more than 1.5 × IQR below the lower quartile or above the upper quartile.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1 and sum to 1.

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency from data = number of times the event occurred ÷ total number of trials.

A sample is biased if it is not representative of the population; larger random samples give more reliable estimates.

Best-buy comparison: unit price = price ÷ quantity; convert to the same unit (per 100 g, per litre, per item) before comparing.

Wages: pay = hourly rate × hours worked; overtime is usually a stated multiple (e.g. 1.5×) of the normal rate.

Simple interest: I = P × r × t with r as a decimal per year and t in years; total = principal + interest.

Compound growth: amount = principal × (1 + r)^n, where n is the number of compounding periods.

Budget balance: income − fixed costs − variable costs = surplus (positive) or deficit (negative).

Discount then GST: apply the discount first (multiply by 1 − discount rate), then apply GST to the discounted price.

Reading tables and timetables: identify the row and column labels first, and check the units stated in the headings.

Rounding in context: money is usually rounded to 2 decimal places, but keep full values during intermediate steps.

Checking reasonableness: estimate with rounded numbers first, then compare the exact answer to the estimate.