Calculations, time, and ratio

Most practical problems in Essential Mathematics come down to adding, subtracting, multiplying and dividing whole numbers, fractions and decimals. You are expected to do the core calculations by hand first, because a tradesperson quoting a job, a nurse checking a dose or a shop assistant counting change often has no device in hand.

This includes square roots of perfect squares (√49 = 7, √144 = 12) and moving between index form and ordinary numerals (2⁵ = 32, 10⁴ = 10 000, 81 = 3⁴).

Order of operations. Evaluate brackets first, then indices and roots, then multiplication and division working left to right, and finally addition and subtraction working left to right. For example 20 − 2 × 3² = 20 − 2 × 9 = 20 − 18 = 2. Doing the subtraction first would give 18 × 9 = 162, which is wrong.

Place value after the decimal point. In 4.375 the 3 means three tenths, the 7 means seven hundredths and the 5 means five thousandths. Place value lets you compare decimals (0.45 is larger than 0.405 because both have 4 tenths, but 0.45 has 5 hundredths and 0.405 has 0 hundredths) and turn fractions into decimals by dividing the numerator by the denominator: 5/8 = 5 ÷ 8 = 0.625, and 2/3 = 0.666…, which is 0.67 to two decimal places.

Multiples of ten. Multiplying by 10, 100 or 1000 moves every digit one, two or three places to the left; dividing moves them to the right. To multiply by 300, multiply by 3 and then by 100: 2.4 × 300 = 7.2 × 100 = 720. To divide by 400, divide by 4 and then by 100: 1800 ÷ 400 = 450 ÷ 100 = 4.5.

Using technology discerningly. A calculator is the sensible tool for longer multi-step work with fractions, decimals, roots and indices, but you still need to enter it correctly. Use brackets and the fraction key, and write down the calculation you entered so a reader can follow your method. For example (3½ + 2.75) × √81 ÷ 2³ = 6.25 × 9 ÷ 8 = 7.03125. Always check a calculator result by re-entering it or by an inverse operation.