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NBT Quantitative Literacy and Mathematics: Key Formulas

The formulas and methods most often needed in the National Benchmark Tests: percentages, ratios, rates, interest, measurement, algebra, trigonometry and calculus basics.

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Percentage of an amount

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Percentage of an amount

\frac{p}{100} \times amount

Percentage change

\frac{new - old}{old} \times 100%

Price including VAT (15%)

Price excluding VAT

\times 1, 15

; price excluding VAT = price including VAT

\div 1, 15

Sharing in the ratio a : b

Add the parts, find one part = total

\div (a + b)

, then multiply by each ratio number.

Speed, distance and time

speed = \frac{distance}{time}

; average speed = total distance

\div

total time.

Converting km/h to m/s

Divide by 3,6; to convert m/s to km/h multiply by 3,6.

Mean of a data set

Sum of the values divided by the number of values.

Median of a data set

The middle value after ordering; for an even number of values, the mean of the two middle values.

Simple interest

A = P (1 + in)

Compound interest

A = P (1 + i)^{n}

Area of a rectangle and a triangle

Rectangle: length

\times

breadth. Triangle:

\frac{1}{2} \times

base

\times

height.

Area and circumference of a circle

A = π r^{2}

and

C = 2 π r

Volume of a rectangular box and a cylinder

Box: length

\times

breadth

\times

height. Cylinder:

π r^{2} h

Probability of an event

P = \frac{number of favourable outcomes}{number of possible outcomes}

Best buy comparison

Divide the price by the quantity to find the price per unit, then compare.

1 litre in millilitres and 1 kg in grams

1 litre = 1000 ml; 1 kg = 1000 g.

Fuel consumption per 100 km

Fuel used = (distance

\div

100)

\times

consumption.

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Exponent law for multiplication and division

a^{m} \times a^{n} = a^{m + n}

and

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Exponent law for a power of a power

(a^{m})^{n} = a^{mn}

Negative and fractional exponents

a^{- n} = \frac{1}{a ^{n}}

and

a^{\frac{m}{n}} = (n a)^{m}

Logarithm laws for sums and differences

lo g_{a} x + lo g_{a} y = lo g_{a} (x y)

and

lo g_{a} x - lo g_{a} y = lo g_{a} \frac{x}{y}

Definition of a logarithm

lo g_{a} x = y ⟺ a^{y} = x

Remainder theorem

The remainder when

f (x)

is divided by

(x - k)

f (k)

Sine and cosine rules

\frac{a}{s i n A} = \frac{b}{s i n B}

and

a^{2} = b^{2} + c^{2} - 2 b c cos A

Fundamental trigonometric identity

sin^{2} x + cos^{2} x = 1

Derivative of

x^{n}

n x^{n - 1}

Gradient, midpoint and distance

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

; midpoint

(\frac{x _{1} + x _{2}}{2}; \frac{y _{1} + y _{2}}{2})

;

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Sum of the interior angles of an n-sided polygon

(n - 2) \times 18 0^{\circ}

Areas of similar figures

If the sides are in the ratio

1 : k

, the areas are in the ratio

1 : k^{2}

National Benchmark Tests (NBT) in South Africa

In South Africa, National Benchmark Tests (NBT) is part of: University admission tests.

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