Common Reasons for Getting Percentage Change Wrong, and the Correct Method (with Compound Change Examples)
The core of percentage change is "using the original quantity as the denominator": the percentage change equals (new value minus old value) divided by the old value, then multiplied by one hundred per cent. When something decreases, the denominator is still the old value, not the new value. Two successive changes cannot simply be added together; you must multiply by (1 plus or minus that percentage) at each step. For example, adding twenty per cent and then adding twenty per cent again gives 1.2 × 1.2 = 1.44 times the original value, which is an increase of forty-four per cent, not forty per cent. This article sets out the common reasons for mistakes, checking steps and practice methods, explains how percentages apply to selling price, original price and profit margin in everyday situations, and provides several examples of calculating percentages and percentage change.
This article was put together automatically from general study experience and is for reference only. For exam and school arrangements, check the latest information from the organisation concerned.
- The denominator in percentage change is always the original quantity, including when it is a decrease.
- First find the amount of change, then divide by the old value, and finally convert to a percentage; this reduces mistakes.
- Successive changes must be multiplied step by step; percentages cannot simply be added.
- Increasing and then decreasing by the same percentage gives a result lower than the original value.
- Percentage points and percentages are two different ways of speaking; check which one the question asks for.
- In questions on selling price, original price and profit margin, likewise first decide the base.
Why percentage change is so easy to get wrong
There is only one main formula for percentage change, but students often go wrong not in the calculation but in their understanding. The most common case is confusing "how much it decreased by" with "what it decreased to": the first is the amount of change, the second is the new value after the change. Which one the question asks for makes a completely different answer.
The second common reason is putting the wrong number in the denominator. When calculating percentage change, the denominator is always the original quantity, that is, the value before the change. In a decrease situation, students easily take the smaller new value as the denominator, and the percentage comes out too large.
The third reason is treating "percentage points" as a "percentage". For example, if a figure rises from twenty per cent to thirty per cent, you can say it rose by ten percentage points, but if the original twenty per cent is the base, the relative change is actually fifty per cent. Both ways of speaking are common in daily life and in the news, so when doing a question, first check which one is being asked.
The fourth reason is confusing "percentage" with "percentage change". A percentage compares a part with the whole: for example, if ten out of forty students in a class pass, the pass rate as a percentage of the class is 25%. Percentage change, on the other hand, compares the same quantity before and after a change. Both are written with a per cent sign, but the base is different, so vague wording in a question easily leads to mistakes.
The percentage change formula and basic examples
Let the old value be a and the new value be b; the percentage change is (b − a) ÷ a × 100%. If b is greater than a, the result is positive, meaning an increase; if b is less than a, the result is negative, meaning a decrease. When calculating, first work out the amount of change, then divide by the old value, and only then convert to a percentage; this makes mistakes less likely. This formula is also often called the percentage formula or the percentage change formula.
Example 1: an item rises in price from $80 to $100. The amount of change is 100 − 80 = $20; dividing by the old value 80 gives 0.25, that is, an increase of 25%.
Example 2: the same item is reduced from $100 to $80. The amount of change is 80 − 100 = −$20; dividing by the old value 100 gives −0.2, that is, a decrease of 20%. The numbers are the same in both questions, but the percentages differ, which shows exactly why the denominator must be the old value.
Another type of question gives the percentage and asks for the new value: the new value equals the old value multiplied by (1 plus the rate of change) or (1 minus the rate of change). For example, $200 increased by 15% gives a new value of 200 × 1.15 = $230; $200 decreased by 15% gives a new value of 200 × 0.85 = $170.
When calculating a percentage, you likewise first need to decide what the whole is. For example, if a class has 40 students and 10 of them pass, the pass rate as a percentage of the class is 10 ÷ 40 × 100% = 25%. This is different from percentage change, because a percentage compares a part with the whole and has no "before" and "after".
Key points for calculating percentage increase and percentage decrease
Percentage increase and percentage decrease are the two directions of percentage change; the method is the same, only the sign of the amount of change differs. Many people ask "how do you calculate percentage increase"; it is simply finding the difference between the new value and the old value, then dividing by the old value. For example, if the old value 50 rises to 65, the increase is 15, and 15 ÷ 50 = 0.3, that is, an increase of 30%.
With percentage decrease, note that a decrease cannot exceed 100%, because the new value cannot fall below zero. If a calculation produces a negative "percentage decrease", it usually means the direction of change has been misjudged, or the new and old values have been swapped. For example, if the old value 80 falls to 60, the decrease is 20, and 20 ÷ 80 = 25%, that is, a decrease of 25%, not 33%.
English questions often use percent or percentage for percentages; in maths questions the two are mostly interchangeable, but percentage usually refers to a percentage value, while percent is more often used in the "how many per cent" form. When students meet them in bilingual questions, they should first confirm whether the question is asking about change or about the relationship between part and whole.
Two successive changes: percentages cannot simply be added
Questions on successive changes are another high-risk spot. When students see "increase by twenty per cent, then increase by twenty per cent again", they easily write it straight down as an increase of forty per cent, but the base is different each time: the second twenty per cent is based on the new value after the first change.
The correct method is to multiply step by step. Let the original value be 100; first increase by twenty per cent gives 100 × 1.2 = 120; then increase by twenty per cent again gives 120 × 1.2 = 144. Compared with the original value, the overall increase is 44%. So two 20% increases together are not 40%.
Successive decreases must likewise be multiplied step by step. An original value of 100 first decreased by twenty per cent gives 80, then decreased by twenty per cent again gives 80 × 0.8 = 64, an overall decrease of 36%, again not 40%.
If there is one increase and one decrease of the same percentage, the result will be lower than the original value. For example, 100 × 1.2 × 0.8 = 96, an overall decrease of 4%. The reason is that the two bases are 100 and 120 respectively, so the twenty per cent decrease is calculated on the larger base. Results like this go against intuition, and students would do well to check them once themselves with simple numbers.
The general form is: after several changes, the new value equals the old value multiplied by (1 plus the rate of change) or (1 minus the rate of change) for each change. If you finally need to report the overall percentage change, you still go back to the original old value as the denominator.
Applying percentages to selling price, original price and profit margin
Percentage change questions often appear in shopping situations, most commonly the relationship between discount, selling price and original price. If the original price and discount rate are known, the selling price is the original price multiplied by (1 minus the discount rate); conversely, if the selling price and discount rate are known, the original price is the selling price divided by (1 minus the discount rate). For example, with an original price of $250, the selling price at 80% of the usual price is 250 × 0.8 = $200; if an item sells for $200 at 80% of the usual price, the original price is 200 ÷ 0.8 = $250.
Profit margin is another common application. Profit is the selling price minus the cost, and the profit margin uses cost as the base, that is, profit ÷ cost × 100%. For example, with a cost of $80 and a selling price of $100, the profit is $20, and the profit margin is 20 ÷ 80 = 25%.
Some questions say "cost plus a certain twenty per cent as the selling price" or "the selling price is a certain per cent higher than the cost"; the wording differs but the base is the same, namely the cost. Students should first write the three names "cost", "selling price" and "profit" on paper, then substitute the numbers. After calculating, check by estimation: if the selling price is higher than the cost, the profit margin must be positive; the selling price after a discount should be lower than the original price.
Learning focus and practice advice from primary school to Form 6
At primary level, first build the concepts of "whole" and "part", and use everyday situations such as discounts and price increases to distinguish "how much was reduced" from "what it was reduced to". In junior secondary, students meet more use of formulas and algebraic expressions; they should be asked to write down the base at every step, not just the answer. Senior secondary students and those preparing for the HKDSE should keep the same principle in more complex situations: first decide which is the original quantity, then decide whether it is a single change or successive changes.
When practising, develop three habits. First, before picking up the pen, write down the "old value" and the "new value". Second, for successive changes, draw a simple flow and mark the multiplier at each step. Third, after calculating, check by estimation: the result after an increase should be greater than the original value, and a decrease cannot exceed one hundred per cent.
When parents help, there is no need to rush to check the answer; first ask "which number is the base in this question?". If the student can point out the base, they can usually correct the mistake themselves. If the student is already secure in using the formula, they can go on to practise questions that work back from the new value and the percentage to the old value; these appear in both junior and senior secondary.
What to look for when choosing tutoring
With questions like percentage change, the problem is usually conceptual rather than speed of calculation, so when choosing support, first check whether the provider will explain the principles of "base" and "successive change" rather than just giving a large number of questions. Common forms of support in Hong Kong include tutoring centres, private tutoring, home tutoring and online tutoring; each differs in arrangements, and fees vary with year level, district and format, generally within a floating range, so parents should find out clearly before deciding.
Whichever form you choose, you can first ask how the provider deals with questions the student gets wrong: do they simply give the answer, or guide the student to find where the base went wrong? In addition, the Consumer Council has always reminded consumers to pay attention to service contracts and fee terms, and to read the relevant documents carefully before enrolling.
Frequently asked questions
Why is the percentage different when something rises from $80 to $100 and when it falls from $100 to $80?
Because percentage change uses the original quantity as the denominator. In the first question the old value is 80, the change is $20, and 20 ÷ 80 = 25%, an increase; in the second question the old value is 100, the change is $20, and 20 ÷ 100 = 20%, a decrease. The amount of change is the same, but the base is different, so the percentage is different.
Why is two successive increases of twenty per cent not an increase of forty per cent?
The second twenty per cent is based on the new value after the first change, not on the original value. Taking an original value of 100, 100 × 1.2 × 1.2 = 144, an overall increase of 44%. Only when the rate of change is very small, or when making a rough estimate, would you approximately add the two percentages together.
What is the difference between percentage points and a percentage?
A percentage point is the difference between two percentages: for example, rising from twenty per cent to thirty per cent is a difference of ten percentage points. Percentage change, by contrast, uses the original percentage as the base: rising from twenty per cent to thirty per cent is a relative increase of fifty per cent. Different wording in the question means a different calculation.
Do primary students need to learn to calculate successive changes?
At primary level, it is better to build a firm foundation in "using the original quantity as the denominator", and to use discounts and price increases to distinguish "how much was reduced" from "what it was reduced to". Step-by-step multiplication for successive changes is generally practised in more depth in junior secondary alongside algebraic expressions.
What is the difference between a percentage and percentage change?
A percentage compares a part with the whole: for example, if 10 out of 40 students in a class pass, the pass rate as a percentage of the class is 25%. Percentage change compares the same quantity before and after a change; the base is different and so is the method.
What is the difference between percent and percentage in English questions?
In maths questions, percent usually refers to the "how many per cent" form, for example 20 percent; percentage more often refers to a percentage value, for example "the percentage increase". The two are sometimes interchangeable, but when you meet a question you should first confirm whether it asks about change or about the relationship between part and whole.