Percent Increase and Decrease - Misconceptions

Understanding percentages is important in real life, but the concept can be counterintuitive and give rise to persistent student errors. A percentage is a relative measure that depends on context and students must grasp its multiple meanings (for example, part‐whole, fractional or rate interpretations) rather than treating percentages as fixed values (Putrawangsa, Febrian, & Hasanah, 2021). Research confirms that many learners hold misconceptions about percentages, often due to weak conceptual knowledge or incomplete learning (Mula & Hodnik, 2020). The accompanying poster summarizes four common misconceptions about percentages. In the sections below we examine each misconception in turn, explain why it is incorrect, and offer tips for teachers to prevent or resolve it.

1. “20% increase followed by 20% decrease takes me back to the original number.”

A common belief is that a 20% increase immediately followed by a 20% decrease “undoes” the change and returns to the original value. In fact, percentage increases and decreases are multiplicative operations, not additive. As Putrawangsa et al. (2021) note, each percentage is relative to the current whole, so successive changes do not cancel out; similarly, Cambridge researchers note that modelling successive percent increases or decreases is tricky because the “whole” changes after each step (Gould, Rycroft-Smith, & Watson, 2021). For example, if you start with 100 and increase it by 20%, you get 120. A subsequent 20% decrease on 120 removes 24 (since 20% of 120 is 24), leaving 96, not 100. The final value (96) is 4% less than the original. In general, a +20% change followed by –20% yields a net change of (1.20 × 0.80 – 1) × 100% = –4%. In other words, because each percentage is taken from the current value, the two changes do not cancel out.

Students often struggle with this because they think in additive terms, but the percentage change is not symmetric; instead, the percentage change between any two values depends on which is chosen as the starting value.

• Teacher tip: Use step-by-step examples or a number line to model successive percent changes. Ask students to compute each step explicitly (e.g. 100→120→96) and compare to the original. Highlight that a 20% decrease always takes 20% of the new amount, so it will remove more or less absolute quantity depending on the intermediate value. Introducing the idea of a “percentage multiplier” (e.g. × 1.20 for a 20% increase, × 0.80 for a 20% decrease) can help clarify that +20% (multiply by 1.2) and –20% (multiply by 0.8) do not invert each other.

• Teacher tip: Point out real-life contexts. Show that if an item is marked up by 20% and then discounted by 20%, the final price is lower than the original. Concrete examples force students to calculate rather than rely on the myth that “up then down cancel out.”

2. “20% of 30 is 6 so 20% increase is 100% + 6 which gives me 106.”

This error comes from mixing up amounts and percentage points. Saying “20% of 30 is 6” is correct, but treating 6 as a percentage point to add to 100% is wrong. In fact, a 20% increase of 30 means adding 6 to 30, yielding 36 (which is 120% of 30), not “106” of anything. As Putrawangsa et al. (2021) explain, percentages represent relative quantities (operators on a base amount), not standalone. Thus “100% + 6” has no meaning here, because 6 is an amount, not a percent. Instead one should say “30 + 20% of 30 = 30 + 6 = 36”, or see it as 120% of 30.

• Teacher tip: Emphasize the phrase “percent of” vs. “percent more than.” For example, compare “20% of 30” (which is 6) with “20% more than 30” (which is 36). Require students to rewrite “increase by 20%” as “take 20% of the amount and add it to the original.” This language reinforces that we’re adding an amount, not a percentage point.

  
  

• Teacher tip: Use the multiplier approach: increasing 30 by 20% is 30 × 1.20 = 36. Practice converting percent statements to multipliers (e.g. +20% is the same as ×1.2, +100% is the same as ×2). Remind students that “100%” of 30 is 30, so “100% + 6” makes no sense unless 6 is also a percent (which it isn’t in this context).

  
  

3. “You can’t have percentages with decimals, e.g., 14.5% = 14.5 as it is a decimal already.”

Some students mistakenly think that any percentage involving a decimal is somehow invalid. This reveals confusion about what the “%” symbol means. The percent sign simply means “per hundred.” So any real number – even 14.5 – can be a percentage. For example, 14.5% means 14.5 per 100, which equals 0.145 as a decimal. There is no rule forbidding decimal or fractional percentages; we commonly see 0.5%, 12.75%, and so on. Putrawangsa et al. (2021) emphasize that a percentage is a relative value. The value of 20% is not absolute; it is always relative depending on the whole that the 20% refers to. In the same way, 14.5% must be interpreted as 14.5/100 of something, not simply 14.5.

• Teacher tip: Practice conversions between percentages and decimals. Ask students to convert 14.5% to a decimal or fraction (14.5% = 0.145 = 29/200) and to convert 0.725 to a percentage (72.5%). Reinforce that the percent sign is a separate symbol meaning “per 100.”

  
  

• Teacher tip: Use visual aids (like 100-square grids) to show, say, 14.5 out of 100 squares shaded. This makes clear that 14.5% is a valid proportion of 100. Similarly, point out real-world examples with fractional percentages (interest rates like 3.75%, sales tax, etc.) to normalize them.

  
  

• Teacher tip: Highlight that percentages can be greater than 100% or not whole numbers. For example, 125% is 125/100 = 1.25 as a decimal. Reinforce the core idea: “14.5% of X means 0.145 × X.” If a student insists “14.5% means 14.5,” have them test it: e.g. 14.5% of 200 is 29, not 14.5. Such checks quickly show the misunderstanding.

4. “It is impossible to decrease 140 by 150% as the lowest you can get is 0.” (Profit/Loss Context)

Some students assume that it is “impossible” to decrease by more than 100%, arguing that the lowest a quantity you can get is 0. In reality, percentage change can be defined beyond 100%, but the interpretation must be handled with care. A 150% decrease on 140 mathematically means 140 × (1 - 1.50) = 140 × -0.5 = -70. The negative result indicates the new value is 70 below zero. In purely mathematical terms this is allowed, but in practical contexts it usually signals going into debt or deficit. Many contexts (like counting physical objects) do not permit negative results, so a “150% decrease” would not make sense there.

An explanatory note is that 100% decrease brings a quantity to 0 (100% of the original is removed). The key is to teach students that percentages above 100% lead to negative values, and to interpret them in relation to context. For example, “Owing more than you had” or “the deficit is larger than the original value” might be a way to explain a >100% reduction.

• Teacher tip: Clarify that decreasing by 100% means the quantity is gone (zero remains). If a problem states a decrease greater than 100%, ask students: “What does negative quantity mean here?” Discuss what that would imply. In many real-world cases (e.g. population) this is impossible, however, ff it is a conceptual or financial scenario, frame it as going into debt or negative.

Summary

Percent problems require relational thinking, not just arithmetic tricks. In each case above, errors arose from treating percents like fixed numbers instead of relative changes. Teachers should always link 100% to the original whole and reason about parts of that. For example, working with fractions and visual models can ground percentage as “parts per hundred” (Gould, Rycroft-Smith, & Watson, 2021), and then applying a percent is like scaling that whole by a factor. Research suggests that directly addressing these misconceptions (through contrasting examples and guided discussion) can aid conceptual understanding. (Gould, Rycroft-Smith, & Watson, 2021; Mula & Hodnik, 2020). Cambridge experts recommend giving students plenty of practice with multiple representations (bar models, ratio tables, number lines, etc.) to reinforce the underlying ratio concept (Gould, Rycroft-Smith, & Watson, 2021). In short, avoid letting students rely on misremembered rules or intuition; instead, have them convert percents to fractional or decimal form and reason about why the result makes sense (or not). Building these strong conceptual links – and even creating “cognitive conflict” when the shortcut gives a wrong answer – helps students replace myths with true percentage understanding.

References

Gould, T., Rycroft-Smith, L., & Watson, F. (2021). What does research suggest about the teaching and learning of percentage? Cambridge Mathematics, Issue 38. https://www.cambridgemaths.org/Images/espresso_38_teaching_and_learning_of_percentage.pdf

Mula, M., & Hodnik, T. (2020). The PGBE model for building students’ mathematical knowledge about percentages. European Journal of Educational Research, 9(1), 257–276. https://doi.org/10.12973/eu-jer.9.1.257

Putrawangsa, S., Febrian, & Hasanah, U. (2021). Developing students’ understanding of percentage: The role of spatial representation. Mathematics Teaching Research Journal, 13(4), 1–20. https://files.eric.ed.gov/fulltext/EJ1332344.pdf