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Standard Form

Standard Form

Standard form: word problems and challenge questions

Scenario Based Core (calculator paper) + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Reading the question: which operation?
  4. Units and prefixes
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 1.8: standard form in context
  • 1.8Solve real problems with very large and very small quantities: populations, planets, atoms, cells, computers.
  • 1.8Choose the right operation from the words: how many, per, difference, how many times, total, percentage of.
  • 1.8Convert between units with prefixes (kilo, mega, micro, nano) and square units.
  • 1.9Round sensibly, estimate to check, and give the answer in standard form when asked.

Before you start: calculating in standard form.

Formula bank

Per / density / rateLearn it\(\text{amount}\div\text{the 'per' quantity}\)people per km² = population ÷ area.Use it when the word "per".
How many in / how many timesLearn it\(\text{total}\div\text{one}\); \(\text{bigger}\div\text{smaller}\)Use it when "how many", "how many times bigger".
PrefixesLearn itk \(10^{3}\), M \(10^{6}\), G \(10^{9}\), m \(10^{-3}\), μ \(10^{-6}\), n \(10^{-9}\)Use it when units with prefixes.
Square unitsLearn it1 km² = \(10^{6}\) m²; 1 m² = \(10^{4}\) cm²square the length factor.Use it when area in different units.

Reading the question: which operation?

Word problems are mostly about deciding WHAT to calculate. The standard form part is then just careful calculator work.

The question says…DoExample
how many … in …÷ (whole by one)atoms in 1 kg: 1000 g ÷ mass of one atom
… per … (density, rate)÷ (first by second)people per km²: population ÷ area
difference− (bigger − smaller)Saturn's mass − Venus's mass
how many times bigger÷ (bigger by smaller)Jupiter's diameter ÷ Earth's
total, altogether+total area of four deserts
\(r\%\) of\(\times\tfrac{r}{100}\)70.8% of the Earth's surface
time / distancedistance ÷ speed; speed × timetime for light to reach us

General methodHow to solve any standard form word problem

  1. Underline each quantity and its unit; write each in standard form.
  2. Choose the operation from the words (table above). Write the calculation down BEFORE pressing keys - it earns the method mark.
  3. Make the units match (g and kg, mm and m, km² and m²).
  4. Calculate with the ×10ˣ key.
  5. Round (3 s.f. unless told) and write in standard form if asked. Whole things that must fit: round DOWN.
  6. Estimate with 1 s.f. values to check the size.

Example One titanium atom has a mass of \(7.95\times10^{-23}\) g. How many atoms are in 1 kg? Give your answer in standard form.

How many in → divide; 1 kg = 1000 g first.

  1. \(1000\div(7.95\times10^{-23})=1.2578\ldots\times10^{25}\)

Answer \(1.26\times10^{25}\)

Example A red blood cell is 7 × 10⁻⁶ m wide. How many whole cells fit side by side across 1 cm?

1 cm = 0.01 m; divide; round down.

  1. \(0.01\div(7\times10^{-6})=1428.57\ldots\)

Answer 1428

Try it yourself

1
A country has \(4\times10^{7}\) people and an area of \(2\times10^{5}\) km². Population density?
Show answer

Answer 200 per km²

2
How many times heavier is \(3\times10^{8}\) kg than \(6\times10^{5}\) kg?
Show answer

Answer 500

Units and prefixes

PrefixSymbolMeansPrefixSymbolMeans
kilok\(10^{3}\)millim\(10^{-3}\)
megaM\(10^{6}\)microμ\(10^{-6}\)
gigaG\(10^{9}\)nanon\(10^{-9}\)

1 km = \(10^3\) m, so 1 km² = \((10^3)^2=10^{6}\) m². An area of \(8.82\times10^5\) km² is \(8.82\times10^{11}\) m².

1 km² is NOT 1000 m². Square the conversion factor: \(1000^2=10^6\).

Try it yourself

1
5 micrometres in metres (standard form)?
Show answer

Answer \(5\times10^{-6}\)

2
A virus is 50 nanometres across. How many fit side by side across 1 mm?
Show answer

Answer 20 000

Ways they test it

Type 1 How many in

"How many titanium atoms are in 1 kg?" (2 marks)

Spot it the mass/size of ONE thing.

  1. Convert units.
  2. Total ÷ one (M1).
  3. Standard form (A1).

Type 2 Density / rate

"Calculate the population density." (2 marks)

Spot it "per".

  1. Population ÷ area.

For example \(2.17\times10^{8}\) people, \(8.82\times10^{5}\) km²

  1. \(\tfrac{2.17\times10^8}{8.82\times10^5}=246.0\ldots\)

Answer 246 per km²

Type 3 Difference without a calculator

"Find the difference between \(5.97\times10^{24}\) and \(7.35\times10^{22}\)."

Spot it "difference", Paper 1.

  1. Match the powers.
  2. Subtract the A values.

For example \(5.97\times10^{24}-0.0735\times10^{24}\)

    Answer \(5.8965\times10^{24}\)

    Type 4 Units

    "64 gigabytes … 3.2 megabytes per photo…" (3 marks)

    Spot it prefixes.

    1. Convert both to bytes.
    2. Divide.

    Type 5 Two linked parts

    "(a) How much evaporates? (b) How many families could it supply?"

    Spot it part (b) uses part (a).

    1. Percentage, then divide.

    Mistakes to avoid

    • ✗ area ÷ population✓ population ÷ area"people per km²": people on top.
    • ✗ mixing g and kg✓ convert firstAnswers come out 1000 times wrong.
    • ✗ 1 km² = \(10^3\) m²✓ \(10^6\) m²Square the conversion.
    • ✗ 1428.57 cells → 1429✓ 1428Whole objects that must fit: round down.
    • ✗ 1.26E25✓ \(1.26\times10^{25}\)Proper notation.

    Summary

    • Write the calculation down first.
    • Per / how many in / how many times → divide; difference → subtract; total → add.
    • Make the units match; square the factor for area units.
    • 3 s.f. and standard form if asked; whole objects round down; estimate to check.

    If you see …

    • "per"divide by what follows "per".
    • prefixes (n, μ, k, M, G)convert with powers of 10.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Light travels \(3\times10^{5}\) km/s. How far in \(2\times10^{3}\) s? (standard form)
    Show answer

    Answer \(6\times10^{8}\) km

    2
    A computer does \(2.5\times10^{9}\) operations per second. How many seconds for \(10^{12}\) operations?
    Show answer

    Answer 400

    3
    Write \(8.82\times10^{5}\) km² in m² (standard form).
    Show answer

    Answer \(8.82\times10^{11}\) m²

    Medium

    1
    Pakistan: population \(2.17\times10^{8}\), area \(8.82\times10^{5}\) km². Population density to the nearest whole number?
    Guide me

    What is it asking? people per km²

    What do we know? per → divide

    Plan population ÷ area

    Show working
    1. \(246.0\ldots\)

    Answer 246

    2
    Saturn \(5.7\times10^{26}\) kg, Venus \(4.9\times10^{24}\) kg. Difference in standard form?
    Guide me

    What is it asking? a difference

    What do we know? match the powers

    Plan \(0.049\times10^{26}\)

    Show working
    1. 5.7 − 0.049 = 5.651

    Answer \(5.651\times10^{26}\) kg

    3
    The Sun is \(1.496\times10^{8}\) km from Earth; light travels \(3\times10^{5}\) km/s. Time in minutes, to 1 d.p.?
    Guide me

    What is it asking? time

    What do we know? distance ÷ speed

    Plan then ÷ 60

    Show working
    1. 498.7 s
    2. ÷ 60

    Answer 8.3

    Hard

    1
    A memory card stores 64 gigabytes; a photo needs 3.2 megabytes. How many photos? (1 GB = 10⁹ bytes, 1 MB = 10⁶ bytes)
    Guide me

    What is it asking? a count

    What do we know? convert both to bytes

    Plan divide

    Show working
    1. \(\tfrac{6.4\times10^{10}}{3.2\times10^{6}}=2\times10^4\)

    Answer 20 000

    PartAnswer / workingMarksGuidance
    both in bytesM1
    dividesM1
    20 000A1
    Total: 3 marks
    2
    A bacterium has a mass of \(9.5\times10^{-13}\) g. A colony has a total mass of 0.38 mg. How many bacteria? (standard form)
    Guide me

    What is it asking? a count

    What do we know? 0.38 mg = \(3.8\times10^{-4}\) g

    Plan total ÷ one

    Show working
    1. \(\tfrac{3.8\times10^{-4}}{9.5\times10^{-13}}\)

    Answer \(4\times10^{8}\)

    PartAnswer / workingMarksGuidance
    conversionB1
    divisionM1
    \(4\times10^{8}\)A1
    Total: 3 marks
    3
    \(N=2\times10^{8}\). Smallest positive integer \(M\) such that \(M\times N>10^{10}\)?
    Guide me

    What is it asking? an integer

    What do we know? \(M>10^{10}\div(2\times10^8)\)

    Plan strictly greater

    Show working
    1. M > 50
    2. 50 gives exactly 10¹⁰

    Answer 51

    PartAnswer / workingMarksGuidance
    50 seenM1
    51A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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