Worked example · Easy
Why multiplication happens first
Evaluate \(2+3\times4\).
\[3\times4=12\]\[2+12=\boxed{14}\]
Do not calculate \((2+3)\times4\) unless brackets are actually present.
Chapter 14
Using a calculator
Learn to enter calculations correctly, interpret the display, check whether answers are sensible, and handle standard form, time and fractions efficiently.
By the end of this chapter, you should be able to
- enter multi-step calculations using the correct order of operations;
- use brackets and power/root keys appropriately;
- enter and interpret standard form correctly;
- check calculator answers using estimation, number facts and inverse operations;
- choose a sensible degree of accuracy for a final answer;
- convert calculator time results between decimals and hours/minutes/seconds;
- use a calculator fraction key while still understanding the mathematics.
1
Subtopic 1
Order of operations
A scientific calculator follows the usual order of operations: brackets → powers/roots → multiplication/division → addition/subtraction. When a fraction bar covers a whole expression, use brackets around the numerator and denominator.
1. Brackets
→2. Powers & roots
→3. × and ÷
→4. + and −
(4.65 + 5.72) ÷ (0.651 × 12.7)
(4.65+5.72)÷(0.651×12.7)
Worked example · Medium
Use brackets for a fraction
Evaluate \(\frac{9+3}{4+2}\).
Enter as \((9+3)\div(4+2)\).
\[12\div6=\boxed2\]Worked example · Exam-style
Roots and powers in one calculation
Evaluate \(\sqrt{3.65^2-1.83^2}\), giving the answer to 3 significant figures.
\[\sqrt{3.65^2-1.83^2}=3.158100\ldots\]\[\boxed{3.16}\text{ to 3 s.f.}\]
An estimate using \(\sqrt{4^2-2^2}=\sqrt{12}\approx3.5\) confirms the size is sensible.
Remember for this subtopic
A scientific calculator applies brackets, powers and roots, then multiplication/division, then addition/subtraction. Use brackets around the entire numerator and denominator if entering a written fraction in one line.
Worked example · Exam-style
Carefully enter a multi-stage calculation
Evaluate \(\frac{(9+3)^2-5^2}{7}\).
Step-by-step solution
Bracket the numerator: \((9+3)^2-5^2=144-25=119\). Then \(119\div7=\boxed{17}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Calculate \(7+4\times3\).
Show Answer
Step-by-step solution
Multiply first: \(4\times3=12\), then \(7+12=\boxed{19}\).
2Medium practice [2 marks]
Calculate \((15-3)\div4\).
Show Answer
Step-by-step solution
Brackets: 15−3=12. Then \(12\div4=\boxed3\).
3Medium practice [3 marks]
Calculate \(2^3+\sqrt{81}\).
Show Answer
Step-by-step solution
\(2^3=8\) and \(\sqrt{81}=9\); total \(=\boxed{17}\).
4Exam-style practice [4 marks]
Calculate \(\frac{6^2+4^2}{5^2-12}\).
Show Answer
Step-by-step solution
Numerator \(36+16=52\). Denominator \(25-12=13\). \(52\div13=\boxed4\).
2
Subtopic 2
Standard form on your calculator
Scientific calculators have a key such as ×10x, EXP or EE. This enters “multiply by a power of 10”. If you use that key, do not type an extra ×10.
7 × 107 ÷ (2 × 10−3)
7EXP7÷2EXP−3=
3.5 × 1010
Display formats you may see
The same number may appear as 3.5 × 1010, 3.5E10, or another calculator-style scientific notation. In your written answer, give proper standard form: \(3.5\times10^{10}\).
Worked example · Easy
Standard form calculation
Calculate \((7\times10^7)\div(2\times10^{-3})\).
\[\frac{7}{2}\times10^{7-(-3)}=3.5\times10^{10}\]
Answer: \(\boxed{3.5\times10^{10}}\).
Worked example · Exam-style
Use a formula with standard form
A sphere has radius \(2.48\times10^4\) km. Use \(A=4\pi r^2\) to find its surface area to 2 significant figures.
\[A=4\pi(2.48\times10^4)^2\]\[A=7.72882058\times10^9\]\[\boxed{7.7\times10^9\text{ km}^2}\]
Remember for this subtopic
On many calculators an EXP, EE or ×10x key means “multiply by 10 to the power”. Do not type another ×10 after that key. Re-express the result as \(a\times10^n\), where \(1\leq a<10\).
Worked example · Exam-style
Divide quantities in standard form
Calculate \(\frac{6\times10^7}{3\times10^{-2}}\).
Step-by-step solution
Divide coefficients: \(6\div3=2\). Subtract powers: \(7-(-2)=9\). Answer \(\boxed{2\times10^9}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Write 6 300 000 in standard form.
Show Answer
Step-by-step solution
Move the decimal 6 places: \(\boxed{6.3\times10^6}\).
2Medium practice [2 marks]
Write \(8.4\times10^{-4}\) as an ordinary decimal.
Show Answer
Step-by-step solution
Move four places left: \(\boxed{0.00084}\).
3Medium practice [3 marks]
Calculate \((4\times10^5)\times(2\times10^3)\).
Show Answer
Step-by-step solution
\(4\times2=8\); powers add to 8. \(\boxed{8\times10^8}\).
4Exam-style practice [4 marks]
Calculate \(\frac{9\times10^{-3}}{1.5\times10^2}\) in standard form.
Show Answer
Step-by-step solution
\(9\div1.5=6\); power \(-3-2=-5\). \(\boxed{6\times10^{-5}}\).
3
Subtopic 3
Checking accuracy
Never assume a displayed answer is correct just because the calculator produced it. Use one or more quick checks.
Estimate
Round values, usually to 1 significant figure, and compare the approximate result.
Use number facts
Multiplying by a number below 1 makes a positive number smaller; dividing by a number below 1 makes it larger.
Use inverse operations
Check division with multiplication, subtraction with addition, and square roots by squaring.
Worked example · Easy
Spot an impossible answer
A student writes \(297\div2.8=10.6\). Explain why this must be wrong.
Estimate:
\[300\div3\approx100\]So 10.6 is far too small.
Worked example · Medium
Use a size check
A student writes \(752\div24=18048\). Explain why the answer cannot be correct.
Dividing a positive number by a number greater than 1 must make it smaller. Since 18 048 is much larger than 752, the answer is impossible.
Worked example · Medium
Check by inverse operation
Your calculator gives \(6.9\div75=0.092\). Check it.
\[0.092\times75=6.9\]
The inverse operation returns the starting value, so the answer is consistent.
Remember for this subtopic
Before accepting a calculator result, make an estimate. Check it with number facts or inverse operations. A large answer from dividing by a number smaller than one may be correct; use the operation to decide.
Worked example · Exam-style
Find and explain an incorrect calculator result
A student says \(498\div4.9=10.16\). Explain why it is wrong and give a sensible approximate answer.
Step-by-step solution
Estimate \(500\div5=100\). Since 10.16 is about ten times too small, the entry is incorrect. The actual result is approximately \(101.63\), close to \(100\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Estimate \(403\times19\) to one significant figure.
Show Answer
Step-by-step solution
\(403\approx400\), \(19\approx20\), so \(400\times20=\boxed{8000}\).
2Medium practice [2 marks]
Explain why 5.8 divided by 0.2 is greater than 5.8.
Show Answer
Step-by-step solution
Dividing by \(0.2=\frac15\) multiplies by 5: \(5.8\div0.2=\boxed{29}\).
3Medium practice [3 marks]
Check \(13.6\div0.8=17\) using an inverse operation.
Show Answer
Step-by-step solution
\(17\times0.8=13.6\), so the answer passes the inverse check.
4Exam-style practice [4 marks]
A calculator displays 56000 for \(7.25\times8\). Use an estimate to show the display cannot be right.
Show Answer
Step-by-step solution
\(7.25\approx7\) and \(8\) stays 8, giving about 56. The correct product is \(\boxed{58}\), nowhere near 56 000.
4
Subtopic 4
Interpreting the calculator display
The display may show more digits than should appear in your final answer. Use the context and the question instructions to decide how to present it.
1.8
Money
Write \(\$1.80\), because money normally uses two decimal places.
1.3846153846
Money requiring rounding
Write \(\$1.38\) to the nearest cent.
3.158100695
No stated context
If no accuracy is specified and the answer is not exact, 3 significant figures is often appropriate.
Do not round too early
If an intermediate result is needed in a later calculation, keep the unrounded calculator value and round only the final answer.
Worked example · Easy
Exact money result
$18 is shared equally between 10 people. How much does each receive?
\[18\div10=1.8\]
As money, write \(\boxed{\$1.80}\).
Worked example · Medium
Rounded money result
$18 is shared equally between 13 people.
\[18\div13=1.384615\ldots\]
To the nearest cent, each receives \(\boxed{\$1.38}\).
Remember for this subtopic
Do not copy every display digit automatically. Money is normally given to two decimal places, lengths to the required accuracy, and counts as whole numbers. Keep extra precision in intermediate steps.
Worked example · Exam-style
Interpret a calculator result as a count
A factory needs 14.2 containers to pack all its goods. How many complete containers must it use?
Step-by-step solution
You cannot use 0.2 of a container if all the goods must fit. Round UP to \(\boxed{15\text{ containers}}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
A calculator shows 7.5 for $30 shared equally among four people. Write the money amount.
Show Answer
Step-by-step solution
\(30\div4=7.5\). Write \(\boxed{\$7.50}\).
2Medium practice [2 marks]
Round 16.4873 to 3 significant figures.
Show Answer
Step-by-step solution
The fourth significant digit is 8, so \(\boxed{16.5}\).
3Medium practice [3 marks]
A calculation for a 60 cm length gives 60.006 cm. Round to the nearest centimetre.
Show Answer
Step-by-step solution
The decimal part is below 0.5, so \(\boxed{60\text{ cm}}\).
4Exam-style practice [4 marks]
Tickets cost $3 each. A class has $50. A calculator gives 16.666... tickets. How many complete tickets can be bought?
Show Answer
Step-by-step solution
They can buy \(\lfloor50/3\rfloor=\boxed{16\text{ tickets}}\) because 17 cost $51.
5
Subtopic 5
Calculating with time
Time is not a base-10 measurement: there are 60 minutes in an hour and 60 seconds in a minute. So 1 hour 12 minutes is not 1.12 hours.
Hours + minutes → decimal hoursminutes ÷ 60
1 h 12 min = \(1+\frac{12}{60}=1.2\) h
Decimal hours → hours + minutesdecimal part × 60
6.4 h = 6 h + \(0.4\times60\) = 6 h 24 min
Worked example · Easy
Minutes and seconds to decimal minutes
Convert 8 minutes 36 seconds to minutes.
\[36\div60=0.6\]\[8+0.6=\boxed{8.6\text{ minutes}}\]
Worked example · Easy
Decimal hours to hours and minutes
Write 6.4 hours in hours and minutes.
\[0.4\times60=24\]
So \(\boxed{6\text{ h }24\text{ min}}\).
Worked example · Exam-style
Time in a rate problem
A machine takes 8 minutes to make one item. At the same rate, how long does it take to make 20 items? Give the answer in hours and minutes.
\[8\times20=160\text{ minutes}\]
160 minutes = 2 hours 40 minutes.
Answer: \(\boxed{2\text{ h }40\text{ min}}\).
Remember for this subtopic
Calculator decimal time must be interpreted using 60, not 100: \(0.25\text{ hour}=0.25\times60=15\text{ min}\). A calculator time key may use hours/minutes/seconds notation; check which mode is active.
Worked example · Exam-style
Convert decimal minutes to minutes and seconds
Convert 7.45 minutes into minutes and seconds.
Step-by-step solution
The whole part is 7 minutes. Seconds \(=0.45\times60=27\). Answer: \(\boxed{7\text{ min }27\text{ s}}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Convert 2 hours 30 minutes into decimal hours.
Show Answer
Step-by-step solution
\(30/60=0.5\), so \(\boxed{2.5\text{ hours}}\).
2Medium practice [2 marks]
Convert 0.75 minute into seconds.
Show Answer
Step-by-step solution
\(0.75\times60=\boxed{45\text{ s}}\).
3Medium practice [3 marks]
Convert 3.2 hours to hours and minutes.
Show Answer
Step-by-step solution
\(0.2\times60=12\), so \(\boxed{3\text{ h }12\text{ min}}\).
4Exam-style practice [4 marks]
At 48 km/h, a journey takes 1.375 hours. Write the journey time in hours and minutes and seconds.
Show Answer
Step-by-step solution
\(0.375\times60=22.5\) min, so 22 min + 0.5 min. \(0.5\times60=30\) seconds. \(\boxed{1\text{ h }22\text{ min }30\text{ s}}\).
6
Subtopic 6
Fractions on a calculator
You still need to understand fraction arithmetic without a calculator, but a scientific calculator can enter fractions, mixed numbers and simplify results automatically. The exact key symbol differs between calculator models.
a b/c
2 fraction key 5 + 1 fraction key 2 =
9/10
Calculator models differ
Learn how your calculator switches between fraction and decimal forms and how it enters mixed numbers. The mathematics is the same even if the buttons look different.
Worked example · Easy
Add fractions
Use a calculator to evaluate \(\frac25+\frac12\).
\[\frac25+\frac12=\frac4{10}+\frac5{10}=\boxed{\frac9{10}}\]
Your calculator should simplify the final fraction automatically.
Worked example · Medium
Mixed-number result
Evaluate \(\frac{17}{5}\).
\[\frac{17}{5}=3\frac25\]
Many calculators can toggle between improper and mixed forms.
Remember for this subtopic
A fraction-entry key helps the calculator store a numerator and a denominator correctly. For a mixed number, convert to an improper fraction if needed; simplify and check mentally when possible.
Worked example · Exam-style
Mixed fractions on a calculator
Evaluate \(2\frac13+1\frac56\), giving a mixed-number answer.
Step-by-step solution
Convert: \(2\frac13=\frac73\), \(1\frac56=\frac{11}6\). \(\frac{14}6+\frac{11}6=\frac{25}6=\boxed{4\frac16}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Calculate \(\frac23+\frac16\).
Show Answer
Step-by-step solution
\(\frac46+\frac16=\boxed{\frac56}\).
2Medium practice [2 marks]
Convert \(\frac{9}{4}\) into a mixed number.
Show Answer
Step-by-step solution
\(9\div4=2\) remainder 1, so \(\boxed{2\frac14}\).
3Medium practice [3 marks]
Calculate \(\frac34\times\frac89\).
Show Answer
Step-by-step solution
\(\frac{3\times8}{4\times9}=\frac{24}{36}=\boxed{\frac23}\).
4Exam-style practice [4 marks]
Calculate \(\frac57\div\frac{10}{21}\).
Show Answer
Step-by-step solution
Multiply by the reciprocal: \(\frac57\times\frac{21}{10}=\frac{105}{70}=\boxed{\frac32}\).
Chapter 14 practice
Use your calculator where appropriate, but still write enough working to show the structure of the calculation and check that the result is sensible.
A. Entry and order of operations
1Order of operations[2 marks]
Evaluate \(5+2.8\times(5.2-3.6)\).
Show Answer
\[(5.2-3.6)=1.6\]\[2.8\times1.6=4.48\]\[5+4.48=\boxed{9.48}\]
2Fraction entry[3 marks]
Evaluate \(\frac{9.3+12.3}{8.2-3.4}\).
Show Answer
Enter as \((9.3+12.3)\div(8.2-3.4)\).
\[21.6\div4.8=\boxed{4.5}\]3Powers and roots[3 marks]
Evaluate \(\sqrt{2.7^2+3.6^2}\).
Show Answer
\[\sqrt{7.29+12.96}=\sqrt{20.25}=\boxed{4.5}\]
B. Standard form
4Multiply standard form[3 marks]
Calculate \((6.21\times10^5)(3.78\times10^7)\). Give your answer in standard form to 3 s.f.
Show Answer
\[6.21\times3.78=23.4738\]\[23.4738\times10^{12}=2.34738\times10^{13}\]\[\boxed{2.35\times10^{13}}\]
5Divide standard form[3 marks]
Calculate \((8.34\times10^7)\div(1.78\times10^2)\), giving the answer to 3 s.f.
Show Answer
\[\frac{8.34}{1.78}\times10^5=4.68539\ldots\times10^5\]\[\boxed{4.69\times10^5}\]
C. Check and interpret
6Reasonableness[2 marks]
A calculator display shows \(175\div1.013=177.275\). Explain why this is wrong.
Show Answer
1.013 is greater than 1, so dividing 175 by it must give an answer slightly less than 175. 177.275 is therefore impossible.
7Estimate a check[3 marks]
A student obtains 9.6 for \(39.6\times18.1\). Use an estimate to show this is wrong.
Show Answer
\[39.6\approx40,\qquad18.1\approx20\]\[40\times20=800\]
The answer should be close to 800, not 9.6.
8Money display[2 marks]
$67 is shared equally between 8 people. How much does each person receive?
Show Answer
\[67\div8=8.375\]
Money is rounded to the nearest cent: \(\boxed{\$8.38}\).
D. Time and fractions
9Time to decimal hours[3 marks]
Convert 2 hours 45 minutes to decimal hours.
Show Answer
\[45\div60=0.75\]\[\boxed{2.75\text{ h}}\]
10Decimal hours to time[3 marks]
Write 4.85 hours in hours and minutes.
Show Answer
\[0.85\times60=51\]
\(\boxed{4\text{ h }51\text{ min}}\).
11Decimal minutes[3 marks]
Write 3.75 minutes in minutes and seconds.
Show Answer
\[0.75\times60=45\]
\(\boxed{3\text{ min }45\text{ s}}\).
12Fraction calculation[3 marks]
Evaluate \(\frac34+\frac58\).
Show Answer
\[\frac34=\frac68\]\[\frac68+\frac58=\frac{11}{8}=\boxed{1\frac38}\]
E. Exam-style challenge
13Standard-form context[5 marks]
A planet is \(4.5\times10^9\) km from the Sun. Light travels at \(3\times10^5\) km/s. How many seconds does light take to reach the planet?
Show Answer
\[\text{time}=\frac{4.5\times10^9}{3\times10^5}\]\[=1.5\times10^4\text{ s}\]
\(\boxed{1.5\times10^4\text{ s}}\).
14Accuracy in context[4 marks]
A piece of string 4 m long is cut into 7 equal pieces. Give the length of each piece to a sensible degree of accuracy.
Show Answer
\[4\div7=0.571428\ldots\text{ m}\]
A sensible answer is \(\boxed{0.571\text{ m}}\) to 3 s.f., unless a different unit/accuracy is requested.
15Multi-step calculation[5 marks]
Evaluate \(\frac{6.2\times3.8}{22.7-13.8}\) to 3 significant figures.
Show Answer
Enter as \((6.2\times3.8)\div(22.7-13.8)\).
\[23.56\div8.9=2.64719\ldots\]\[\boxed{2.65}\text{ to 3 s.f.}\]16Time and rate[5 marks]
A worker makes one item every 7.5 minutes. How many hours and minutes are needed for 18 items?
Show Answer
\[7.5\times18=135\text{ minutes}\]
135 minutes = \(\boxed{2\text{ h }15\text{ min}}\).
Chapter 14 high-grade checklist
Enter it correctly
I use brackets when a whole numerator, denominator or root expression must be kept together.
Standard form
I know how my calculator enters powers of 10 and I write the final result in proper standard form.
Estimate
I can check whether a display is sensible using a quick estimate.
Number facts
I know how multiplying/dividing by values above or below 1 should affect the size of an answer.
Interpretation
I choose sensible units and accuracy, especially for money and measurements.
Time and fractions
I can convert time correctly and use my calculator's fraction functions without confusing the underlying mathematics.