Ratio. Write \(24 : 60\) in its simplest form.

1. \(2 : 5\)
2. \(4 : 10\)
3. \(12 : 30\)
4. \(5 : 2\)

Answer: \(2 : 5\)

Find the HCF of 24 and 60. It is 12.

\(24 \div 12 = 2\) and \(60 \div 12 = 5\).

Tip: Keep the order the question gives you. \(5:2\) is a different ratio.

Scientific notation. Write \(0{,}000\,62\) in scientific notation.

1. \(6{,}2 \times 10^{-4}\)
2. \(6{,}2 \times 10^{4}\)
3. \(62 \times 10^{-5}\)
4. \(0{,}62 \times 10^{-3}\)

Answer: \(6{,}2 \times 10^{-4}\)

Move the comma to sit after the first non-zero digit: \(6{,}2\).

The comma moved 4 places to the RIGHT, so the exponent is \(-4\).

Tip: A number smaller than 1 always has a negative exponent.

Percentage increase. A jacket costs R850. The price increases by 12%. What is the new price?

1. \(R952{,}00\)
2. \(R862{,}00\)
3. \(R102{,}00\)
4. \(R748{,}00\)

Answer: \(R952{,}00\)

Increase \(= 12\% \times 850 = R102\).

New price \(= 850 + 102\).

Tip: Or in one step: \(850 \times 1{,}12\).

Rational and irrational. Which ONE of the following is an irrational number?

1. \(\sqrt{20}\)
2. \(\sqrt{16}\)
3. \(0{,}25\)
4. \(\tfrac{3}{7}\)

Answer: \(\sqrt{20}\)

\(\sqrt{16} = 4\), a whole number, so it is rational.

\(0{,}25\) and \(\tfrac{3}{7}\) can both be written as common fractions.

\(\sqrt{20}\) does not simplify to a fraction.

Tip: A square root is only rational when the number under it is a perfect square.

Exponent laws. Simplify: \(\left(2x^{2}y^{3}\right)^{3}\)

1. \(8x^{6}y^{9}\)
2. \(2x^{6}y^{9}\)
3. \(6x^{5}y^{6}\)
4. \(8x^{5}y^{6}\)

Answer: \(8x^{6}y^{9}\)

Every factor inside the bracket is cubed, including the 2.

\(2^{3} = 8\), \(\left(x^{2}\right)^{3} = x^{6}\), \(\left(y^{3}\right)^{3} = y^{9}\).

Tip: Forgetting to cube the 2 is the most common error here.