Why Two Percentages Can Tell Different Stories: What Students Miss When Comparing Numbers

The mathematics of percentages is often simple. Understanding what those percentages actually describe is a different skill.

By Tymur Levitin | Levitin Language School

Two students improve their test scores by 20%.

Two shops offer a 20% discount.

Two companies report 20% growth.

At first glance, these statements appear to describe equivalent changes.

They do not.

A percentage expresses a relationship between quantities. It does not tell us the absolute size of a change unless we know what the percentage refers to.

Twenty percent of 50 is 10.

Twenty percent of 500 is 100.

Twenty percent of 5,000 is 1,000.

The percentage is identical in all three cases.

The numerical consequences are very different.

This is why learning percentages should involve more than memorising a formula.

Students need to understand the reference quantity, the type of comparison and the conclusion that the available numbers actually support.

1. A percentage is not a quantity by itself

Consider two schools.

School A has 100 students. Twenty percent participate in a science competition.

School B has 1,000 students. Twenty percent participate in the same competition.

Both schools report identical participation rates.

But the numbers of participating students are different.

School A:

[
100\times0.20=20
]

School B:

[
1000\times0.20=200
]

The same percentage represents 20 students in one school and 200 in the other.

Which school has greater participation?

That depends on what we are measuring.

If we mean the proportion of students participating, the schools are equal.

If we mean the total number of participants, School B has more.

Neither interpretation is automatically wrong.

The mistake begins when we answer one question using a number that belongs to another.

A percentage comparison is meaningful only when we understand what is being compared.

2. The missing question: Twenty percent of what?

Imagine a parent reading two advertisements for educational services.

The first offers a 20% discount on a $50 lesson package.

The second offers a 15% discount on a $200 package.

Which discount saves more money?

The first:

[
50\times0.20=10
]

The second:

[
200\times0.15=30
]

The first discount has the larger percentage.

The second produces the larger monetary saving.

But even that does not tell us which package is better value.

The packages may contain different numbers of lessons, different lesson durations or different services.

A larger saving does not automatically mean a lower final price per lesson.

We need to identify the actual decision before selecting the comparison.

Are we comparing discount rates?

Money saved?

Final prices?

Cost per hour?

The answer changes depending on the question.

This is one reason percentages can be persuasive even when they provide only a limited picture.

3. Percentage change depends on the starting value

Suppose two students improve their mathematics scores.

Student A moves from 40 points to 50 points.

Student B moves from 80 points to 90 points.

Both improve by 10 points.

But their relative improvements differ.

For Student A:

[
\frac{50-40}{40}\times100=25%
]

For Student B:

[
\frac{90-80}{80}\times100=12.5%
]

The absolute improvement is identical.

The percentage improvement is not.

Now consider a different example.

Student A moves from 40 to 48 points.

Student B moves from 80 to 96 points.

Both improve by 20% relative to their initial scores.

Yet Student A gains 8 points while Student B gains 16.

This gives us two different ways to describe change:

Absolute change measures the difference between two values.

Relative change measures that difference in relation to a reference value.

Neither measure is universally superior.

They answer different questions.

4. Why a 20% increase followed by a 20% decrease does not return to the starting value

This is one of the most revealing percentage problems.

A product costs $100.

Its price increases by 20%.

The new price is:

[
100\times1.20=120
]

The price then decreases by 20%.

A student might expect the final price to return to $100.

But the second percentage is calculated from $120, not $100.

[
120\times0.80=96
]

The final price is $96.

The two changes do not cancel.

Why?

Because they use different reference values.

The increase is 20% of $100.

The decrease is 20% of $120.

The percentages are equal, but the absolute changes are $20 and $24.

This is not an exception to percentage arithmetic.

It is a direct consequence of how percentages work.

To reverse a 20% increase, the necessary percentage decrease is:

[
\frac{20}{120}\times100\approx16.67%
]

A 16.67% decrease from $120 returns the price approximately to $100.

The key is not remembering a special trick.

It is recognising that every relative change has a reference value.

5. Percentage points are not the same as percent

Suppose a school’s examination pass rate rises from 60% to 75%.

How much has it increased?

There are two legitimate answers, depending on the measure.

The difference between the rates is:

[
75%-60%=15
]

That is an increase of 15 percentage points.

But the relative increase in the pass rate is:

[
\frac{75-60}{60}\times100=25%
]

That is a 25% relative increase.

These statements describe the same change in different ways.

They are not interchangeable.

A report saying that a rate increased by 15% is not equivalent to saying it increased by 15 percentage points.

For students, this distinction matters in mathematics, economics, statistics, geography and the interpretation of public information.

For adults, it matters when reading financial reports, employment statistics, interest rates and educational performance data.

6. The same average can hide different realities

Consider two classes.

Class A has ten students. Eight pass an examination.

Class B has 100 students. Eighty pass.

Both have an 80% pass rate.

But does this mean their educational situations are identical?

Not necessarily.

We know nothing about the difficulty of the examinations, the students’ starting levels, the teaching conditions or the distribution of individual scores.

The pass rate is one summary measure.

It does not describe everything that happened.

Now imagine that Class A consists of students who began with substantial academic difficulties, while Class B consists of students who already had strong preparation.

An identical final pass rate may represent very different learning journeys.

This does not make percentages useless.

It means their interpretation must remain connected to the underlying situation.

A number can be accurate without being a complete description.

7. When the overall percentage hides the behaviour of individual groups

Consider two schools, A and B, each offering a standard course and an advanced course.

Their results are:

School B has a higher pass rate in both courses.

But now calculate the overall results.

School A:

[
\frac{81+2}{90+10}=83%
]

School B:

[
\frac{19+24}{20+80}=43%
]

School A has the higher overall pass rate.

How is this possible?

School A has most of its students in the standard course, where pass rates are high.

School B has most of its students in the advanced course, where pass rates are lower.

The different distributions affect the combined results.

This is an example of Simpson’s paradox: a pattern observed within separate groups can reverse when the groups are combined.

The lesson is not that the overall percentage is mathematically incorrect.

It is that combining groups can change what the result appears to say.

Before comparing institutions, populations or outcomes, we should examine whether the underlying groups are genuinely comparable.

8. Why percentages can mislead in education

Imagine a student moving to another country.

At their previous school, they regularly achieved 85% on tests.

At the new school, they achieve 70%.

Does that automatically mean their knowledge has declined?

No.

The tests may assess different content.

The grading systems may differ.

The language of instruction may be unfamiliar.

The student may be required to explain reasoning rather than select an answer.

The numerical comparison is straightforward:

85% is greater than 70%.

But the educational interpretation is not straightforward.

The percentages may not represent performance under equivalent conditions.

This is especially relevant to internationally mobile families.

In Your Child Was Good at School Before Moving to Switzerland. Why Are Their Grades Lower Now?, we examine how educational expectations and assessment contexts can change after relocation.

The broader principle is that a numerical difference should not automatically be treated as proof of a particular cause.

9. Comparing percentages requires comparing definitions

Suppose two language schools report that 90% of their students achieved their learning goals.

Before comparing the figures, we need to understand what each school means by achieved their goals.

Did students define their own goals?

Were goals established by teachers?

Were results measured by external examinations?

Were only students who completed the programme included?

Were students who stopped attending excluded?

Were the time periods identical?

Without this information, two apparently comparable percentages may describe different outcomes.

A high percentage can be perfectly accurate within its own definition and still be unsuitable for comparison with another institution’s result.

This is not merely a statistical technicality.

It is a question of meaning.

Before comparing numbers, compare the conditions under which those numbers were produced.

10. Percentages in science: a small change may matter enormously

Consider two substances.

Substance A increases in mass from 10 grams to 12 grams.

Substance B increases from 1,000 grams to 1,100 grams.

Substance A increases by 20%.

Substance B increases by 10%.

Which change is greater?

In relative terms, Substance A has the larger increase.

In absolute terms, Substance B gains more mass.

But a scientific interpretation may require further information.

What substances are involved?

What process caused the change?

Are the measurements reliable?

Is the change physically meaningful?

The relevant measure depends on the scientific question.

In chemistry, biology and physics, a percentage may help describe a process, but it does not replace an understanding of that process.

This is why learning physics through English, German or other instructional languages should involve conceptual reasoning alongside calculations.

A student needs to understand what the numbers represent before deciding what they establish.

11. A percentage can be precise while the underlying measurement is uncertain

Suppose a survey estimates that 52% of respondents prefer one option.

Another survey reports 49%.

Can we confidently conclude that the first population has a stronger preference?

Not from those percentages alone.

We would need to examine sample sizes, sampling methods, uncertainty and whether the surveyed populations are comparable.

A percentage displayed with several decimal places may look highly precise.

But numerical formatting does not guarantee that the underlying estimate is equally precise.

This distinction matters when interpreting research, polling, business analytics and educational studies.

Precision in presentation is not the same as certainty in evidence.

12. How students can learn to compare percentages correctly

Instead of treating every percentage problem as an invitation to multiply or divide, students can develop a sequence of questions.

What is the reference quantity?

Twenty percent of what?

Is the reference the original price, the new price, the number of students, the total population or another value?

Are we comparing absolute or relative change?

An increase of ten points and an increase of ten percent may describe different changes.

Are the groups comparable?

Do the percentages refer to populations with similar characteristics, conditions and definitions?

Is the question asking for a mathematical result or an interpretation?

Calculating a percentage may be easy.

Explaining what it means may require more information.

What conclusion can the numbers actually support?

Does the percentage establish a difference, an association, a trend or a cause?

These are not equivalent claims.

Students who learn to ask these questions are developing a transferable skill rather than simply learning another arithmetic procedure.

13. The Tymur Levitin Question: What stays the same when the percentage stays the same?

Suppose two situations both involve a 20% increase.

What do we actually know is identical?

Only the relative rate of increase.

We do not necessarily know that the absolute increases are equal.

We do not know that the final values are equal.

We do not know that the consequences are equally important.

And we do not know that the underlying situations are comparable.

This leads to a broader question:

When two numbers look the same, which properties are genuinely the same — and which have we merely assumed to be the same?

That question applies to far more than percentages.

It applies to grades, prices, salaries, statistics, scientific measurements and everyday decisions.

14. From percentage calculations to mathematical understanding

Students sometimes struggle with mathematics not because they cannot perform an operation, but because they do not know when the operation is meaningful.

A formula may be correct.

The arithmetic may be accurate.

The final number may even look reasonable.

But the interpretation can still be wrong.

This connects to a different problem explored in When a School Question Doesn’t Have Enough Information: Why Students Try to Calculate an Answer Anyway.

In that article, the central question is whether the available conditions permit the requested calculation.

Here, the question is what a completed calculation actually means.

These are complementary skills.

One helps students recognise when a calculation is justified.

The other helps them interpret the result without making unsupported comparisons.

Both are essential to independent mathematical reasoning.

15. How individual teaching can address the problem

A student who repeatedly makes mistakes with percentages may need more than additional exercises of the same type.

First, it is necessary to identify the difficulty.

Does the student understand fractions and ratios?

Can they identify the reference quantity?

Do they distinguish percentage change from percentage points?

Can they interpret a word problem in the language of instruction?

Can they explain why two numerical comparisons produce different conclusions?

Different difficulties require different teaching approaches.

At Levitin Language School, individual online support can combine subject knowledge with the language needed to understand academic tasks.

This includes mathematics and science through English, mathematics through German and broader language learning through school subjects.

The objective is not to replace mathematical practice with abstract discussion.

It is to ensure that students understand what they are practising, why the method works and how to recognise the limits of their conclusions.

Conclusion: The percentage is only the beginning

Twenty percent can represent ten students or two hundred.

A 20% increase and a 20% decrease do not necessarily cancel.

A rise of 15 percentage points is not the same as a 15% relative increase.

And two institutions reporting the same percentage may be measuring different things.

These are not unusual exceptions.

They follow from the basic meaning of percentages as relationships between quantities.

The arithmetic matters.

But so does the reference value, the definition of the measure, the composition of the groups and the question we are trying to answer.

A student who understands these distinctions is doing more than calculating correctly.

They are learning to interpret numerical information.

And that is what makes percentages useful beyond the classroom.

Individual Online Learning at Levitin Language School

Levitin Language School provides individual online education for children, teenagers and adults.

Our educational directions include foreign languages, academic subjects and combined language-and-subject learning. Lessons are arranged according to the student’s needs, instructional language, schedule and the availability of a suitable teacher.

If a student struggles with percentages, mathematical word problems, scientific calculations or academic language, the first step is to clarify the actual learning difficulty rather than assume that every mistake requires the same kind of practice.

Explore Levitin Language School or contact us to discuss individual learning arrangements.

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Tymur Levitin
Founder and Director, Levitin Language School

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