Maths · 1 Number

1.13 Using a calculator

Before you start

The order of operations: brackets, then powers and roots, then $\times$ and $\div$ , then $+$ and $-$ .
Rounding to a given number of decimal places or significant figures.
There are 60 minutes in an hour and 100 cents in a dollar.
What standard form means: a number written as $a \times 10^{n}$ with $1 \leq a < 10$ .
Speed $=$ distance $\div$ time, and the circle formulas $C = 2 π r$ and $A = π r^{2}$ .

Syllabus objectives covered (3)

Use a calculator efficiently.
Enter values appropriately on a calculator.
Interpret the calculator display appropriately.

Part 1 of 6

The calculator does exactly what you type

A scientific calculator follows the order of operations: brackets first, then powers and roots, then $\times$ and $\div$ , then $+$ and $-$ . It applies that order to the keys you press. It cannot see the fraction line or the layout on your question paper.
Take this calculation. Worked by hand, the top is $16.5$ and the bottom is $2.75$ .
$\frac{12.6 + 3.9}{2.5 \times 1.1} = \frac{16.5}{2.75} = 6$
Now type it straight across with no brackets: $12.6 + 3.9 \div 2.5 \times 1.1$ . The calculator divides first, so it works out $3.9 \div 2.5 \times 1.1 = 1.716$ and then adds $12.6$ . The display shows $14.316$ , which is wrong.
Brackets round the top alone are not enough. $(12.6 + 3.9) \div 2.5 \times 1.1$ divides $16.5$ by $2.5$ and then multiplies by $1.1$ , giving $7.26$ . The $1.1$ was meant to be on the bottom, so the bottom needs its own brackets too.
The rule: a fraction line means one bracket round the whole top and one bracket round the whole bottom.
$(12.6 + 3.9) \div (2.5 \times 1.1) = 6$
If your calculator has a fraction key, it gives you a box for the top and a box for the bottom. That does the same job as the two pairs of brackets. Look at the screen before you press $=$ and check that every number sits in the box it belongs in.
Type the whole calculation in one go when you can. One entry gives one display, so there is nothing to copy down wrongly halfway through and nothing to round too early. That is what using a calculator efficiently means.

Part 2 of 6

Negative numbers, powers and roots

A negative number and a subtraction are different things. If your calculator has a separate negative key, marked $(-)$ , use it to enter a number such as $- 3$ . The subtraction key is for taking one number away from another.
Powers come before the minus sign in the order of operations. So if you type $(-)$ , $3$ , $x^{2}$ , the calculator squares the $3$ and then makes the result negative.
$- 3^{2} = - 9 {(- 3)}^{2} = 9$
To square a negative number, put it in brackets before the power key. The same holds for any even power: ${(- 2)}^{4} = 16$ , but $- 2^{4} = - 16$ .
A root must cover everything it applies to. Here you want the square root of the whole sum, so both numbers must sit under the sign. Use the root template, or type the brackets yourself: $\sqrt{(16 + 9)}$ .
$\sqrt{16 + 9} = \sqrt{25} = 5 but \sqrt{16} + 9 = 13$
A power must hold the whole index. $8^{\frac{2}{3}} = 4$ . If only the $2$ goes into the power, the calculator works out $8^{2} \div 3 = 21.3 \dots$ instead. Check that the screen shows all of $\frac{2}{3}$ raised, or put brackets round the index: 8 ^ (2 ÷ 3).

Part 3 of 6

Keep every digit until the final answer

Round only the final answer. Every value you work out on the way stays in the calculator at full accuracy, because a small rounding early on can change the last figure of the answer.
Here is the problem it causes. A circle has a circumference of $30$ cm and you want its area. First find the radius from $C = 2 π r$ .
$r = \frac{30}{2 π} = 4.774648293 \dots$
Square the full radius and multiply by $π$ . The area is $71.6$ cm² to 3 significant figures.
$A = π \times 4.774648293 \dots^{2} = 71.61972 \dots$
Now suppose you rounded the radius to $4.77$ first. $π \times {4.77}^{2} = 71.48 \dots$ , which rounds to $71.5$ . The radius was out by less than $0.005$ , but squaring it pushed the error into the third figure, and $71.5$ cm² is wrong.
To keep the full value, use the answer key, Ans. It holds the last result exactly. Type $30 \div (2 π)$ and press $=$ , then type $π \times {Ans}^{2}$ and press $=$ . You can also store the value in a memory, or type the whole thing in one line: $π \times {(30 \div (2 π))}^{2}$ .
Still write the value in your working to 4 or 5 figures, such as $r = 4.7746$ , so your method can be followed. Writing a shortened value down is fine. Using the shortened value in the next step is what causes the error.
Use the $π$ key, not $3.14$ . For a circle of radius $4.8$ cm, $π \times {4.8}^{2} = 72.38 \dots$ , which is $72.4$ cm². With $3.14$ you get $72.3456$ , which rounds to $72.3$ cm², a wrong answer. If you have no $π$ key, use $3.142$ , which gives $72.39 \dots$ and the correct $72.4$ .
The same rule runs across the parts of a question. When part (b) uses your answer to part (a), use the unrounded value from (a), not the 3-figure version you wrote down as the answer.

Part 4 of 6

Entering times and standard form

Your calculator counts in tens. Time counts in sixties: there are $60$ minutes in an hour, not $100$ . So $2$ hours $30$ minutes is not $2.30$ hours.
To change minutes into hours, divide by $60$ and add the whole hours. $2$ hours $30$ minutes is $2.5$ hours. In the same way, $45$ minutes is $\frac{45}{60} = 0.75$ hours, and $1$ hour $12$ minutes is $1.2$ hours.
$2 + \frac{30}{60} = 2.5$
Some minutes do not give a short decimal. $3$ hours $20$ minutes is $3 + \frac{20}{60} = 3.333 \dots$ hours. Do not type $3.33$ . Type $3 + 20 \div 60$ , inside brackets when it is part of a bigger calculation, so the calculator keeps every digit.
Here is a car that travels $250$ km in $3$ hours $20$ minutes. The first entry below gives the correct speed, $75$ km/h. Rounding the time to $3.33$ gives $75.1$ km/h. Typing $3.2$ for $3$ h $20$ min gives $78.1$ km/h.
$\frac{250}{(3 + 20 \div 60)} = 75 \frac{250}{3.33} = 75.07 \dots \frac{250}{3.2} = 78.125$
You can also enter a time with the degrees–minutes–seconds key, marked ° ′ ″ (DMS on some calculators). This key works in sixties, so it suits hours and minutes as well as angles. Entering 2° 30′ 0″ gives the calculator the value $2.5$ .
For standard form, use the $\times 10^{x}$ key (marked EXP on some calculators). To enter $3.2 \times 10^{5}$ , type $3.2$ , press that key, then type $5$ . For $4.5 \times 10^{- 3}$ , type $4.5$ , press the key, then $(-)$ and $3$ .
Do not type $\times 10$ before you press the $\times 10^{x}$ key. The key already supplies the $\times 10$ , so typing $3.2 \times 10$ and then the key and $5$ enters $3.2 \times 10^{6}$ , ten times too big.
The key also keeps a standard-form number in one piece. Entered with it, $6 \div 2 \times 10^{3}$ is treated as $6 \div (2 \times 10^{3}) = 0.003$ . If you type $\times 10$ and a power instead, the calculator divides by $2$ and then multiplies by $1000$ , giving $3000$ . Without the key, you need the brackets.

Part 5 of 6

Reading the display: money and time

The display shows a bare number. It does not know whether you were working in dollars, hours or metres. You do, so it is your job to write the number in the form that quantity needs.
In money, the digits after the point are cents, and an amount with cents always takes two decimal places. A display of $4.8$ means $4.80, four dollars and eighty cents. Write $4.80, not $4.8. In the same way $12.5$ means $12.50, and $0.8$ means $0.80, or 80 cents.
When the display has more than two decimal places, round to the nearest cent. $74 \div 7 = 10.571428 \dots$ , so each share is $10.57 to the nearest cent.
In time, a display of $3.25$ hours means $3$ whole hours and $0.25$ of an hour. A quarter of an hour is $15$ minutes, so $3.25$ hours is $3$ hours $15$ minutes, not $3$ hours $25$ minutes.
To change any decimal of an hour into minutes, take off the whole hours and multiply what is left by $60$ . For $1.7$ hours, the $0.7$ gives $42$ minutes, so $1.7$ hours is $1$ hour $42$ minutes.
$0.7 \times 60 = 42$
If your calculator has a ° ′ ″ key, pressing it straight after an answer can switch the display between a decimal and hours, minutes and seconds. If you are not sure how yours behaves, use the multiply-by-60 method, which always works.
For a clock time, add the hours and the minutes separately. A journey of $2.75$ hours lasts $2$ h $45$ min, because $0.75 \times 60 = 45$ . Starting at 09:40, two hours takes it to 11:40, and $45$ more minutes takes it to 12:25. Never add $2.75$ to $9.40$ .

Part 6 of 6

Reading the display: standard form, long decimals and rounding

A very large or very small answer appears in standard form, and calculators show it in different ways. $2.5 \times 10^{- 4}$ might appear as 2.5E-04 or as 2.5×10^-04. Each of these means $2.5 \times 10^{- 4} = 0.00025$ .
Write it on paper in proper standard form: $2.5 \times 10^{- 4}$ . The E form is the calculator's shorthand, not a way of writing the answer. And ${2.5}^{- 4}$ is a different number altogether: it is $0.0256$ .
A long decimal has been cut off to fit the screen. $2 \div 3$ might display as $0.6666666667$ . The final $7$ is the calculator rounding its last digit. The true value is $0. \dot{6}$ , which is $\frac{2}{3}$ .
Some calculators give an exact answer such as $\frac{43}{8}$ or $2 \sqrt{2}$ . When the question wants a decimal, switch the display with the fraction–decimal key (labelled S⇔D on some models), then round.
Non-exact answers are written to 3 significant figures unless the question says otherwise, and angles in degrees to 1 decimal place. Write the full display in your working first, then the rounded answer, so a slip in rounding does not hide a correct method.
An answer that is exact does not need rounding. $(7.2 + 4.9) \div (1.6 \times 2.5) = 3.025$ exactly, so write $3.025$ . Cutting it to $3.03$ throws away accuracy you already had.
When the answer counts whole objects, the context decides which way to round. $196$ people need minibuses that seat $15$ : $196 \div 15 = 13.07$ , so you need $14$ minibuses, because $13$ would leave one person behind. A $25$ m roll of ribbon is cut into $1.8$ m pieces: $25 \div 1.8 = 13.9$ , so you get $13$ pieces, because a $14$ th piece would be too short.
Before you write the answer, ask whether its size makes sense. A rough estimate takes seconds. If the display is far from it, or a walking speed comes out as $450$ km/h, you have mistyped something. Go back and check the entry, not the arithmetic.

Worked examples

Example 1. Calculate $\frac{{4.7}^{2} - \sqrt{18}}{2.3 + 1.65}$ . (a) Write down all the figures on your calculator display. (b) Write your answer correct to 3 significant figures.

Show the working

Estimate first, so you can recognise a wrong display. ${4.7}^{2} \approx 22$ and $\sqrt{18} \approx 4$ , so the top is about $18$ . The bottom is about $4$ . The answer should be near $18 \div 4 = 4.5$ .
(a) The fraction line means brackets round the whole top and the whole bottom. Make sure the root covers only the $18$ .
$({4.7}^{2} - \sqrt{18}) \div (2.3 + 1.65)$
The display shows $4.518318813$ . That is close to the estimate of $4.5$ , so the entry is right.
Without brackets, ${4.7}^{2} - \sqrt{18} \div 2.3 + 1.65$ divides only $\sqrt{18}$ by $2.3$ , and the display shows $21.89537361$ . That is nowhere near $4.5$ , which is how the estimate catches the mistake.
(b) Round your own display. The third significant figure is $1$ and the digit after it is $8$ , so round up to $4.52$ .

Answer: (a) $4.518318813$ (b) $4.52$

Example 2. A cyclist rides $52$ km in $2$ hours $24$ minutes. (a) Calculate her average speed in km/h. (b) At the same average speed, find how long she takes to ride $80$ km. Give your answer in hours and minutes, to the nearest minute.

Show the working

(a) Change the time to hours first. $24$ minutes is $\frac{24}{60} = 0.4$ hours, so the time is $2.4$ hours. Typing $2.24$ would be wrong, because the $24$ is minutes, not hundredths of an hour.
Speed is distance divided by time. The speed is $21.7$ km/h to 3 significant figures.
$52 \div 2.4 = 21.666 \dots$
With $2.24$ you would get $52 \div 2.24 = 23.2$ km/h, an answer that looks reasonable and is wrong. Converting the minutes is the step that protects you.
(b) Time is distance divided by speed. Use the full speed still held by the calculator, not $21.7$ : type $80 \div Ans$ .
$80 \div 21.666 \dots = 3.692307692 \dots$
Read the display as a time: $3$ whole hours and $0.692307 \dots$ of an hour. Multiply that part by $60$ to get the minutes. $41.5 \dots$ minutes is $42$ minutes to the nearest minute.
$0.692307 \dots \times 60 = 41.538 \dots$
If you had used the rounded speed, $80 \div 21.7 = 3.6866 \dots$ hours, which is $3$ hours $41$ minutes. Rounding too early has cost a minute.

Answer: (a) $21.7$ km/h (b) $3$ hours $42$ minutes

Example 3. Calculate $(3.6 \times 10^{- 4}) \div (8 \times 10^{5})$ . Give your answer in standard form.

Show the working

Enter each number with the $\times 10^{x}$ key so it stays in one piece: $3.6$ , the key, $(-)$ , $4$ , then $\div$ , then $8$ , the key, $5$ , then $=$ .
Depending on your calculator, the display shows something like 4.5E-10 or 4.5×10^-10. Both mean $4.5 \times 10^{- 10}$ .
Write it properly in standard form, $4.5 \times 10^{- 10}$ . The first part, $4.5$ , is between $1$ and $10$ , as standard form requires.
Check the size with an estimate. $3.6 \div 8$ is about $0.45$ , and $10^{- 4} \div 10^{5} = 10^{- 9}$ . So the answer is about $0.45 \times 10^{- 9}$ , which is $4.5 \times 10^{- 10}$ .
The trap: if you type the second number as $8 \times 10^{5}$ using a power and no brackets, the calculator divides by $8$ and then multiplies by $10^{5}$ . The display then shows $4.5$ , which is $10^{10}$ times too big. Use the $\times 10^{x}$ key, or put brackets round $8 \times 10^{5}$ .

Answer: $4.5 \times 10^{- 10}$

Example 4. (a) Five friends share a bill of $74 equally. Work out how much each pays. (b) Seven friends share a bill of $74. Each pays the same amount, in whole cents, and together they must pay at least the full bill. Find the smallest amount each can pay.

Show the working

(a) $74 \div 5$ displays $14.8$ . This is money, so the $8$ is tenths of a dollar, which is $80$ cents. Write $14.80.
(b) $74 \div 7$ displays $10.57142857$ . To the nearest cent, that is $10.57.
Now check the context. $7 \times 10.57 = 73.99$ , so seven payments of $10.57 leave the bill $1$ cent short.
Each must pay $1$ cent more: $7 \times 10.58 = 74.06$ , which covers the bill. Here the context means you round up, even though the usual rounding rule says round down.

Answer: (a) $14.80 (b) $10.58

Where people go wrong

Mistake: Typing $2$ hours $30$ minutes as $2.30$ , or $3$ hours $20$ minutes as $3.2$ .

Instead: Divide the minutes by $60$ and add the hours: $2.5$ hours, and $3 + 20 \div 60 = 3.333 \dots$ hours. Or enter the time with the ° ′ ″ key.

The digits after a decimal point count tenths and hundredths, but minutes count sixtieths of an hour. $2.30$ hours is only $2$ hours $18$ minutes.

Mistake: Reading a display of $3.25$ hours as $3$ hours $25$ minutes.

Instead: Multiply the decimal part by $60$ : $0.25 \times 60 = 15$ , so it is $3$ hours $15$ minutes.

$0.25$ is a quarter of an hour. The decimal places on the display are hundredths of an hour, not minutes.

Mistake: Rounding a value halfway through a calculation, then using the rounded value in the next step.

Instead: Keep the full value with the Ans key or a memory, or type the calculation in one line. Round once, at the end.

The small error from early rounding grows in later steps. In the circle example, a radius of $4.77$ instead of $4.7746 \dots$ turned the correct $71.6$ cm² into $71.5$ cm².

Mistake: Typing a fraction straight across, such as $12.6 + 3.9 \div 2.5 \times 1.1$ for $\frac{12.6 + 3.9}{2.5 \times 1.1}$ .

Instead: Bracket the whole top and the whole bottom: $(12.6 + 3.9) \div (2.5 \times 1.1)$ , or use the fraction key.

The calculator follows the order of operations, so it divides before it adds. It cannot see the fraction line on the paper.

Mistake: Copying a display such as 2.5E-04 as the answer, or writing it as ${2.5}^{- 4}$ .

Instead: Write it in standard form: $2.5 \times 10^{- 4}$ .

E is the calculator's shorthand for ' $\times 10$ to the power'. ${2.5}^{- 4}$ means something else entirely: it is $0.0256$ .

Check yourself

Calculate $\frac{8.3 + 2.95}{1.5 \times 2.4}$ . Should you round your answer?
Answer
$(8.3 + 2.95) \div (1.5 \times 2.4) = 11.25 \div 3.6 = 3.125$ . It is exact, so leave it as $3.125$ .
Write $1$ hour $48$ minutes in hours, as a decimal.
Answer
$48 \div 60 = 0.8$ , so $1.8$ hours.
A journey time works out as $2.35$ hours. Write this in hours and minutes.
Answer
$0.35 \times 60 = 21$ , so $2$ hours $21$ minutes.
A money calculation in dollars displays $13.5$ . Write the answer as an amount of money.
Answer
$13.50
Work out ${(- 2.4)}^{2}$ and $- {2.4}^{2}$ . Explain why they are different.
Answer
${(- 2.4)}^{2} = 5.76$ and $- {2.4}^{2} = - 5.76$ . Without brackets, the calculator squares $2.4$ first and then applies the minus sign.
Calculate $(5.2 \times 10^{6}) \div (1.3 \times 10^{- 2})$ . Give your answer in standard form.
Answer
$4 \times 10^{8}$ . Enter each number with the $\times 10^{x}$ key so each one stays in one piece.
$230$ students go on a trip. Each coach seats $52$ students. How many coaches are needed?
Answer
$230 \div 52 = 4.42 \dots$ , so $5$ coaches. Four coaches seat only $208$ .
Find $\sqrt{{3.7}^{2} + {5.2}^{2}}$ correct to 3 significant figures, using a single calculator entry.
Answer
Put the root over the whole sum: $\sqrt{({3.7}^{2} + {5.2}^{2})} = \sqrt{40.73} = 6.382005954 \dots$ , which is $6.38$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Calculate $\frac{\sqrt{{9.8}^{2} - {4.3}^{2}}}{2.7 \times 1.15}$ . Write down all the figures on your calculator display. 2 marks

Mark scheme

M1 77.55 or 8.806… or 3.105 seen, or the correct value from a bracketed entry
A1 2.836150654… (accept 2.83615 or more figures, rounded or truncated)

(b) Write your answer to part (a) correct to 3 significant figures. 1 mark

Mark scheme

B1 2.84 FT their (a) correctly rounded to 3 significant figures

Question 2. A train leaves a station at 08:50. It travels $186$ km at an average speed of $72$ km/h.

(a) Calculate the time the journey takes. Give your answer in hours and minutes. 3 marks

Mark scheme

M1 186 ÷ 72 (= 2.583…)
M1 decimal part of their time × 60, e.g. 0.583… × 60, or 186 ÷ 72 × 60 (= 155 minutes)
A1 2 h 35 min cao

(b) Find the time the train arrives. 1 mark

Mark scheme

B1 11:25 oe, e.g. 11 25 or 11.25 am; FT 08:50 + their (a)

(c) The return journey of $186$ km takes $2$ hours $50$ minutes. Calculate the average speed of the train on the return journey. Give your answer correct to 3 significant figures. 3 marks

Mark scheme

M1 2 h 50 min written as 2.833… h (accept 2.83), 2 5/6 h or 170/60 h, or 170 minutes used
M1 186 ÷ their time in hours, or 186 ÷ 170 × 60
A1 65.6 (65.647…) cao. Using 2.83 gives 65.7 and scores M1 M1 A0; using 2.5 or 2.50 scores M0 M1 A0

Question 3. Paint is sold in tins costing $14.35 each. One tin covers $11.5$ m² of wall. Ana needs to paint $68$ m² of wall.

(a) Calculate the number of tins Ana must buy. 2 marks

Mark scheme

M1 68 ÷ 11.5 (= 5.91…), or 11.5 × 6 = 69 (with 11.5 × 5 = 57.5) seen
A1 6 cao

(b) Calculate the total cost of the tins. 1 mark

Mark scheme

B1 86.10 FT their whole number of tins × 14.35, written with two decimal places

(c) Ana pays with a $100 note. Write down her change. 1 mark

Mark scheme

B1 13.90 FT 100 − their (b), written with two decimal places

Question 4.

(a) Calculate $(4.2 \times 10^{8}) \div (1.5 \times 10^{- 3})$ . Give your answer in standard form. 2 marks

Mark scheme

B2 2.8 × 10¹¹ cao. If B2 is not scored, B1 for 280 000 000 000, or for 2.8 × 10 to any other power

(b) A calculator display shows 7.2E-05. Write this number in standard form. 1 mark

Mark scheme

B1 7.2 × 10⁻⁵ cao

(c) Write the number in part (b) as an ordinary number. 1 mark

Mark scheme

B1 0.000072 cao

How this is examined

Changing between hours-and-minutes and decimal hours in a speed or journey question: A Calculate part in a journey context, where the time is given in hours and minutes or the answer is asked for in hours and minutes, followed by a short part such as an arrival time Going in: minutes $\div 60$ , so $2$ h $50$ min becomes $2.833 \dots$ hours (never $2.50$ ), with the full value used in the division. Coming out: the decimal part $\times 60$ , so $2.583 \dots$ hours becomes $0.583 \dots \times 60 = 35$ minutes, written as 2 h 35 min.
Working out a fraction with powers and roots in one entry, then rounding your own display: A Calculate part that says to write down all the figures on your calculator display, followed by a 1-mark Write your answer correct to 3 significant figures Brackets round the whole numerator and the whole denominator, with the root covering only what it should. The full display written out, such as $2.836150654$ , then that display rounded to 3 significant figures, $2.84$ . An intermediate value such as $77.55$ in the working shows the method.
Using the unrounded value from one step or part in the next: A multi-step Calculate, or a later part that uses the answer to an earlier part The full value carried forward through Ans or a memory, so the final 3-figure answer is correct. A final answer that is wrong because an earlier value was rounded loses the accuracy mark, even when the method mark is earned.
Writing a money answer or a whole-number count in the form the context needs: A cost, change or sharing calculation, or a 'how many tins / coaches / pieces' question Money in dollars written to two decimal places, $86.10 rather than 86.1. A count rounded in the direction the context demands: up when everything must be covered (6 tins, 5 coaches), down when only complete items count (13 pieces), with the unrounded division seen in the working.
Entering standard-form numbers and writing a standard-form display correctly: A Calculate with two numbers in standard form and the answer asked for in standard form, or a Write part that gives a calculator display to rewrite Each number entered with the $\times 10^{x}$ key or in brackets, and the answer written as $a \times 10^{n}$ with $1 \leq a < 10$ , such as $2.8 \times 10^{11}$ . A display copied as 2.8E11 is not standard form.

Words to know

Order of operations: The order a calculator works in: brackets, then powers and roots, then $\times$ and $\div$ , then $+$ and $-$ .
Ans key: Holds the full, unrounded result of the last calculation, so you can use it in the next step.
$\times 10^{x}$ key: Enters a number in standard form as one piece. Labelled EXP on some calculators.
° ′ ″ key: Enters and displays values in sixties: degrees, or hours, with minutes and seconds.
Decimal hours: A time written as a decimal of an hour, e.g. $2.5$ hours $= 2$ hours $30$ minutes and $3.25$ hours $= 3$ hours $15$ minutes.
Significant figures: The digits of a number counted from the first non-zero digit. Non-exact answers are given to 3 significant figures unless the question says otherwise.
Standard form: A number written as $a \times 10^{n}$ , where $1 \leq a < 10$ and $n$ is an integer.
Exact answer: An answer with no rounding, such as $3.025$ . It is written in full, not cut to 3 significant figures.

All Maths lessons Next: 1.14 Time