A calculator only does what you type. Use brackets for every numerator, denominator and root, keep every digit until the end, and read the display in the units the question uses.
Put brackets round every top, every bottom and everything under a root. Round only the final answer, and translate the display: \(4.8\) in money means \(\$4.80\), and \(3.25\) hours means 3 hours 15 minutes.
Section 01
Brackets and the order of operations.
A calculator follows the order of operations exactly. Before you trust an answer, estimate it: open the lab on Estimating, round each number to 1 significant figure, and see how close the estimate lands to the exact value. A calculator answer far from your estimate means a keying mistake.
Fractions: a long fraction line acts as brackets. Key \(\dfrac{5.6 + 3.82}{2.4 \times 1.15}\) as (5.6 + 3.82) ÷ (2.4 × 1.15), or use the fraction template.
Roots: the root sign covers everything under it: \(\sqrt{12.5^{2} - 4.8^{2}}\) needs a bracket closed after \(4.8^{2}\).
Negative numbers: use the (−) key, and brackets when squaring: \((-3)^{2} = 9\), but \(-3^{2}\) means \(-(3^{2}) = -9\).
Worked example A fraction
Work out \(\dfrac{5.6 + 3.82}{2.4 \times 1.15}\). Give your answer to 3 significant figures.
Estimate first: \(\dfrac{6 + 4}{2 \times 1} = 5\), so the answer is a small number, roughly 3 to 5.
Key in \((5.6 + 3.82) \div (2.4 \times 1.15)\).
Display: \(3.413\,043...\)
Answer: \(3.41\) (typing \(5.6 + 3.82 \div 2.4 \times 1.15\) without brackets gives \(7.43\), which is wrong)
Worked example A square root
Work out \(\sqrt{12.5^{2} - 4.8^{2}}\) to 3 significant figures.
Key in \(\sqrt{(}12.5^{2} - 4.8^{2})\), closing the bracket after \(4.8^{2}\).
\(12.5^{2} - 4.8^{2} = 156.25 - 23.04 = 133.21\).
\(\sqrt{133.21} = 11.541\,66...\)
Answer: \(11.5\)
Your turn
Work out \(\dfrac{4.2^{3}}{1.9 + 0.75}\). Write down all the figures on your display, then give the answer to 3 significant figures.
\(4.2^{3} = 74.088\) and \(1.9 + 0.75 = 2.65\). Display: \(27.957\,735\,85...\), so \(28.0\) to 3 s.f. (the zero must be written).
Examiner trap
Typing \(5.6 + 3.82 \div 2.4 \times 1.15\) divides only the 3.82. Without brackets the calculator does the division before the addition.
Section 02
Entering times and values.
Time is not decimal: there are 60 minutes in an hour. Either change minutes to a decimal of an hour, or use the degrees-minutes-seconds key (marked \(^{\circ}\ '\ ''\)) which works in base 60.
Time
Decimal hours
2 h 30 min
\(2 + \dfrac{30}{60} = 2.5\)
3 h 15 min
\(3.25\)
1 h 24 min
\(1.4\)
2 h 40 min
\(2.666...\) (keep it as \(2\dfrac{2}{3}\))
Worked example Time from a calculator
A coach travels 170 km at 60 km/h. How long does the journey take, in hours and minutes?
Time \(= 170 \div 60 = 2.8333...\) hours.
The whole number part is 2 hours.
The decimal part is a fraction of an hour: \(0.8333... \times 60 = 50\) minutes.
Rounding the side to \(7.1\) first would give \(4 \times 7.1 = 28.4\), which is wrong to 3 s.f.
Answer: \(28.3\) cm
Examiner trap
Rounding in the middle of a calculation is the commonest way to lose an accuracy mark. Keep the full value on the calculator and round once, at the end.
Stage 2 · Exam
Exam-style questions
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Work out \(\dfrac{(3.7 + 2.15)^{2}}{8.4 - 1.65}\), giving your answer to 3 significant figures.
Exam style+20 XP
A coach travels 170 km at an average speed of 60 km/h. How long does the journey take?
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Use \(v = \sqrt{2gh}\) with \(g = 9.8\) and \(h = 4.5\) to find \(v\), to 3 significant figures.
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A square field has area 2000 m2. Find its perimeter, correct to 3 significant figures.
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