Solving “Calculate,” “Determine,” “Estimate” and “Show That” Questions

Welcome back. In the previous lesson, you practised making comparisons clear by pairing the same feature for both things. Numerical questions need the same clarity, but the examiner must be able to see your method as well as your final number.

In International GCSE Science Double Award, calculations appear in Biology, Chemistry and Physics. The science may change, but the answer structure does not. This lesson gives you a reliable layout for calculate, determine, estimate, and show that questions, including equations, substitutions, unit conversions, and sensible rounding.

By the end, you should be able to make your working easy to mark — and avoid losing marks when a calculator answer is correct but presented badly.

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What the four command words require

The official Pearson definitions are short, but they tell you what the examiner expects.

Here is the practical meaning of each command word.

Command word	What you must do	What your answer should look like
Calculate	Use numerical information to obtain an answer.	Equation, substitution, answer with unit
Determine	Find a numerical value from data, a graph, a table, or a calculation.	A quantitative result, with enough method to show how it was found
Estimate	Find a sensible approximate value.	Clear approximation, often using rounded values or a graph
Show that	Verify the result stated in the question.	Full calculation using the supplied data, ending at the stated value

The command word changes the purpose of your calculation. Your basic layout, however, is nearly always the same.

A key distinction:

Calculate asks “what is the value?”
Show that asks “can you prove that this stated value follows from the data?”

For “show that,” do not simply write the number that is already in the question. You must demonstrate it.

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The calculation spine: a layout that earns method credit

For most two- to four-mark calculations, use this sequence:

1. List useful values, including their units.
2. Convert units if the equation requires it.
3. Write the equation in symbols.
4. Rearrange it first if the unknown is not already the subject.
5. Substitute the numbers, keeping units visible where helpful.
6. Give the final answer with a unit and sensible rounding.

This creates an “audit trail.” If you make one arithmetic mistake at the end, your equation and substitution may still show the examiner that you understood the method.

Worked example: calculate weight

A mass has a mass of $250\text{ g}$. The gravitational field strength is $9.8{\text{ N kg}}^{-1}$.
Calculate its weight.

First, identify that mass must be in kilograms because gravitational field strength is given per kilogram:

$$250\text{ g}=0.250\text{ kg}$$

Write the relationship:

$$W=mg$$

Substitute:

$$W=0.250\times 9.8$$

Give the answer and unit:

$$W=2.45\text{ N}$$

A strong exam answer is therefore:

$$W=mg$$ $$W=0.250\times 9.8$$ $$W=2.45\text{ N}$$

Notice what is visible:

• the correct equation;
• the mass conversion;
• the numerical substitution;
• the final unit.

A bare answer of “2.45” is incomplete. The examiner should not have to guess whether you mean newtons, kilograms, or joules.

The International GCSE specification states that the listed physics formulae are not provided in the examination. Build flashcards that make you recall each relationship in symbols, not merely recognise it in a formula list.

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Use symbols before numbers

Writing the symbolic equation first protects you from several common mistakes:

• choosing the wrong equation;
• putting values in the wrong place;
• forgetting to square a quantity;
• rearranging incorrectly;
• losing a method mark if the arithmetic is wrong.

For example, suppose you need to calculate current from potential difference and resistance.

$$V=IR$$

The question asks for $I$, so rearrange before substituting:

$$I=\frac{V}{R}$$

If $V=12\text{ V}$ and $R=4.0\text{ ohm}$:

$$I=\frac{12}{4.0}$$ $$I=3.0\text{ A}$$

Do not rely only on equation triangles. They can help with simple formulas, but proper rearranging is more reliable when an equation contains squares, roots, fractions, or several stages.

This short Science Shorts video models the useful habits of identifying data, converting units, showing substitution, and rearranging equations. The example is Physics, but the written layout applies across all three sciences.

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“Determine”: make the quantitative step visible

Determine may look less obviously mathematical than “calculate,” but the official definition still requires a quantitative element. You might determine a value from a graph, table, experiment, or formula.

For example:

A reaction produces $5.4{\text{ cm}}^3$ of gas in $180\text{ s}$.
Determine the mean rate of gas production.

A clear approach is:

$$\text{mean rate}=\frac{\text{volume of gas}}{\text{time}}$$ $$\text{mean rate}=\frac{5.4}{180}$$ $$\text{mean rate}=0.030{\text{ cm}}^3{\text{ s}}^{-1}$$

The word “determine” does not mean “write a description of the trend.” You need a number. If the answer comes from a graph, show the two values you read, then show the calculation — particularly if you are finding a gradient.

For a straight-line graph:

$$\text{gradient}=\frac{\text{change in }y}{\text{change in }x}$$

Choose two widely separated points on the best-fit line, not necessarily two plotted data points. Include the gradient’s unit if one is possible from the axes.

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“Estimate”: approximate honestly

An estimate is intentionally approximate. You are being tested on sensible numerical reasoning, not false precision.

You may estimate by:

• reading an approximate value from a graph;
• using an order-of-magnitude calculation;
• rounding awkward values before calculating;
• counting squares or using a line of best fit.

For example:

A rock has a mass of $24.8\text{ g}$ and a volume of $3.1{\text{ cm}}^3$.
Estimate its density.

The exact calculation would be possible, but an estimate can use convenient values:

$$\rho =\frac{m}{V}$$ $$\rho \approx \frac{25}{3}$$ $$\rho \approx 8{\text{ g cm}}^{-3}$$

The approximation sign matters:

$$\approx$$

It tells the examiner that you know the value is not exact.

An estimate such as $8.0000{\text{ g cm}}^{-3}$ would be poor scientific communication. It suggests extreme precision even though you deliberately rounded the input data.

When estimating from a graph, use language such as:

At $25\text{ degrees C}$, the line gives approximately $3.6\text{ g}$.

Do not claim a graph reading is exact unless the value lies exactly on a labelled grid line.

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“Show that”: verify the stated result

A “show that” question supplies the result you need to reach. Your job is to use the information given and demonstrate that the result is correct.

For example:

A runner travels $400\text{ m}$ in $160\text{ s}$.
Show that their average speed is $2.5{\text{ m s}}^{-1}$.

Write the formula:

$$\text{average speed}=\frac{\text{distance moved}}{\text{time taken}}$$

Substitute the data:

$$\text{average speed}=\frac{400}{160}$$

Calculate:

$$\text{average speed}=2.5{\text{ m s}}^{-1}$$

The final line matches the stated value, so you have verified it.

A “show that” checklist

Before moving on, check that you have:

• used the numbers in the question;
• written the relevant equation or relationship;
• shown substitutions;
• included conversions where needed;
• reached the stated answer after sensible rounding.

If you get a different answer, do not change numbers until you get the expected result. Check systematically:

1. Have you copied every value correctly?
2. Are all units compatible?
3. Did you rearrange the equation correctly?
4. Did you square, divide, or multiply correctly?
5. Did you use the correct relationship?

A stated answer can occasionally be reached only after rounding. For instance, a calculator result of $2.47{\text{ m s}}^{-1}$ may support a stated value of $2.5{\text{ m s}}^{-1}$ if the question expects two significant figures.

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Units: use the units required by the equation

Units are not decoration at the end. They tell you whether your answer makes physical and chemical sense.

The safest rule is:

Convert values before substitution, using units that match the equation and the required answer unit.

Some common conversions are:

$$1\text{ kg}=1000\text{ g}$$ $$1\text{ min}=60\text{ s}$$ $$1\text{ kJ}=1000\text{ J}$$ $$1{\text{ dm}}^3=1000{\text{ cm}}^3$$ $$1{\text{ cm}}^3=1.0\times 10^{-6}{\text{ m}}^3$$

Be especially careful with volume, because cubed units change by much more than the length conversion.

Match your answer unit to your input units

Suppose a density question gives mass in grams and volume in cubic centimetres:

$$\rho =\frac{540\text{ g}}{200{\text{ cm}}^3}$$ $$\rho =2.7{\text{ g cm}}^{-3}$$

That is correct if the question asks for density in ${\text{g cm}}^{-3}$.

But if it asks for ${\text{kg m}}^{-3}$, you must convert both values first:

$$540\text{ g}=0.540\text{ kg}$$ $$200{\text{ cm}}^3=2.00\times 10^{-4}{\text{ m}}^3$$ $$\rho =\frac{0.540}{2.00\times 10^{-4}}$$ $$\rho =2700{\text{ kg m}}^{-3}$$

The numerical answer changes because the unit changes. Both density values describe the same material.

Unit checks that catch mistakes

Before accepting a final answer, ask:

• Is the unit the one requested?
• Did I convert minutes to seconds if the equation uses seconds?
• Did I convert grams to kilograms for a Physics equation involving mass?
• If I used a Chemistry concentration equation, have I used the volume unit it requires?
• Is the size of the answer realistic?

For example, a human body mass of $65000\text{ kg}$ should immediately make you suspect that grams have been mistaken for kilograms.

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Rounding: accurate enough, but not falsely precise

The specification requires an appropriate number of significant figures. Follow any instruction in the question first:

• “Give your answer to 2 significant figures.”
• “Give your answer to 3 significant figures.”
• “Give your answer to 1 decimal place.”

If no instruction is given, use a sensible level of precision that matches the data. In many school-science calculations, two or three significant figures is appropriate, but it is not an automatic rule for every question.

Significant figures

Start counting from the first non-zero digit.

• All non-zero digits count.
• Zeros between non-zero digits count.
• Zeros at the start do not count.
• Trailing zeros count if a decimal point makes the precision clear.

For example:

Number	Significant figures	Reason
$846$	3	All digits are non-zero
$0.00470$	3	The leading zeros do not count; the final zero does
$500$	Usually 1	Without a decimal point, the trailing zeros are unclear
$500.0$	4	The decimal point shows the trailing zeros are significant
$4.070$	4	The middle and final zeros are significant

For a quick refresher on recognising significant figures, use this short section of Significant Figures - A Fast Review! by The Organic Chemistry Tutor.

How to round

To round $12.6491$ to three significant figures:

1. Keep the first three digits: $1$, $2$, and $6$.
2. Look at the next digit: $4$.
3. Since it is less than $5$, leave the $6$ unchanged.

$$12.6491\approx 12.6$$

To round $0.003846$ to two significant figures:

$$0.003846\approx 0.0038$$

The zeros before the $3$ are placeholders, not significant figures.

Do not round too early

Keep extra calculator digits during multi-step calculations. Round only on the final answer line unless the question explicitly tells you otherwise.

Early rounding can change a final result enough to lose accuracy. This matters particularly in:

• percentage calculations;
• multi-step energy questions;
• mole calculations;
• graph gradients;
• questions involving squared values.

Write a calculator result in your working if needed, then round only in the final line:

$$I=3.8461538\dots \text{ A}$$ $$I=3.85\text{ A}\text{ (3 significant figures)}$$

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A final five-second calculation check

Before you leave a numerical question, run this quick check:

Check	What to look for
Command word	Did I calculate, determine, estimate, or show that as asked?
Equation	Have I written the correct relationship in symbols?
Rearrangement	Is the quantity asked for isolated correctly?
Units	Did I convert inputs and state the final unit?
Substitution	Are the actual numbers visible in my working?
Rounding	Did I follow the requested precision and round only at the end?
Sense check	Is the magnitude realistic?

The most common avoidable errors are:

• writing only a calculator answer;
• forgetting the unit;
• using grams when kilograms are required;
• converting after doing the calculation;
• giving an exact-looking answer to an estimate;
• writing the answer supplied in a “show that” question without proving it;
• rounding an intermediate answer too soon.

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Turn this into flashcards and past-paper habits

Make these command-word flashcards now:

Flashcard front	Flashcard back
How do I structure a calculation answer?	Values and units; convert if needed; equation; rearrange; substitute; final answer with unit and suitable rounding.
What does “determine” require?	A quantitative answer from supplied information, with a clear route showing how it was obtained.
What does “estimate” require?	A sensible approximate value, often from rounded values or a graph. Use an approximation sign where appropriate.
What does “show that” require?	Verify the stated result using the supplied data and full working.
When should I round?	Follow the question’s instruction. Otherwise keep calculator precision through the working and round sensibly only at the end.

When marking a past paper, add one of these codes beside every calculation error:

• CAL1 — wrong equation selected
• CAL2 — equation not rearranged correctly
• CAL3 — unit conversion error
• CAL4 — substitution or calculator error
• CAL5 — missing or incorrect final unit
• CAL6 — inappropriate rounding
• CAL7 — working not shown
• CAL8 — treated “estimate” or “show that” as an ordinary calculation

Your correction should always be a fully laid-out calculation, not just the correct final number.

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Key takeaways

• Calculate requires a numerical answer with relevant working.
• Determine must include a quantitative result from the supplied information.
• Estimate means approximate honestly; do not give false precision.
• Show that means verify the stated value with equations and substitutions.
• Use the same dependable layout: convert units, write the equation, rearrange, substitute, calculate, round, and state the final unit.
• Keep full calculator precision until the final answer, unless instructed otherwise.
• A correct number without a unit or visible method is an avoidable risk.

Next, you will work on suggest and predict questions: applying known science to unfamiliar situations and data without guessing vaguely.