Percentages · Eqora guide
Percentage change or percentage points?
Learn which starting value belongs in the denominator, why a rise from 40% to 46% has two valid descriptions, and how to check percentage increases and decreases.
Percentage change answers a comparison question: how large is the change relative to the value you started with? That final phrase controls the entire calculation. If a price rises from $80 to $92, the absolute increase is $12, but the relative increase is 12 divided by the original $80. The result is 0.15, or 15%. Dividing by $92 instead would answer a different question and produce the wrong rate for this change.
Confusion grows when the quantities are already percentages. If a survey result moves from 40% to 46%, it rises by 6 percentage points. Relative to the original 40%, however, it rises by 15%. Both statements can be correct because they measure different things. Percentage points give the direct gap between two percentages. Percentage change compares that gap with the starting percentage.
This guide builds one reliable method for increases, decreases, reverse changes, and repeated changes. You will identify the original and new values, calculate the signed difference, divide by the correct reference, and rebuild the new amount as a check. The examples use market prices and survey rates, but the same reasoning works for grades, attendance, energy use, population, business data, and many homework word problems.
The percentage change formula starts with the original value
Write the values in time order before choosing an operation: original value, then new value. The signed change is new minus original. Divide that change by the original value and multiply by 100. In symbols, percentage change = (new − original) ÷ original × 100%. A positive result describes an increase, a negative result describes a decrease, and zero means there was no change.
Take the price rise from $80 to $92. The difference is 92 − 80 = 12. The original value is 80, so 12 ÷ 80 = 0.15 and 0.15 × 100% = 15%. State the answer in context: the price increased by 15%. OpenStax presents percent increase in the same two stages: find the increase, then compare that increase with the original amount.
The formula is easier to remember when you understand the denominator. The original value represents the whole against which the change is judged. Twelve dollars is a large change from $20 but a small change from $800. The numerator may stay the same while the percentage changes because the reference whole changes. If a problem says ‘compared with last year,’ last year is usually the original value.
Percentage points measure a direct gap between percentages
When both values are percentages, first ask whether the question wants a direct gap or a relative change. A survey rate that rises from 40% to 46% has a direct difference of 46% − 40% = 6 percentage points. The Office for National Statistics defines a percentage point as the difference between percentages. The unit matters because saying ‘up 6%’ would normally describe a relative increase, not this direct subtraction.
To find the relative percentage change, use the same formula as before. The change is 6 and the original percentage is 40, so 6 ÷ 40 × 100% = 15%. A precise report can say, ‘The rate rose by 6 percentage points, from 40% to 46%, which is a 15% increase relative to the original rate.’ Including the start and end values makes the comparison transparent.
Notice that 40% and 46% may themselves describe shares of a larger population. You do not need the population size to calculate the gap in percentage points, but you may need it to interpret how many people the shift represents. If the sample sizes or definitions changed, arithmetic alone cannot make the two rates comparable. Read the labels, period, and denominator behind each published percentage.
A percentage decrease uses the same reference
Suppose a price falls from $92 to $80. The signed change is 80 − 92 = −12. Divide by the original $92: −12 ÷ 92 × 100% ≈ −13.04%. You can report a 13.04% decrease or a percentage change of about −13.04%. The minus sign communicates direction; the word decrease usually uses the positive magnitude after the direction is named.
This result explains why a 15% increase followed by a 15% decrease does not return to the starting value. The first percentage uses the first amount as its reference, while the second uses the larger amount. Starting at $80, a 15% increase produces $92. A 15% decrease from $92 removes $13.80 and leaves $78.20. The same rate acts on a different base.
Reverse questions deserve a fresh calculation. Going from A to B and from B to A uses different original values, so the rates usually differ. Never copy the forward rate and change only its sign. Label the direction, rebuild the formula, and keep enough decimal places until the final step. In financial or measured contexts, follow the rounding rule given in the question.
Check the answer with a multiplier
A good percentage calculation should recreate the new value. Convert the rate to a decimal and form a multiplier. For an increase of 15%, use 1 + 0.15 = 1.15. Then $80 × 1.15 = $92. For a decrease of 13.04%, the approximate multiplier is 1 − 0.1304 = 0.8696, and $92 × 0.8696 is about $80 after rounding.
The multiplier check catches several common mistakes. If you divided by the new value, misplaced the decimal point, or forgot whether the change was positive, the reconstructed amount will not match the stated new value. It also gives you a quick plausibility test: a 15% increase should create a result a little above the original, not ten times larger and not below it.
You can also check by calculating the percentage amount directly. Fifteen percent of $80 is 0.15 × 80 = $12; adding $12 gives $92. This is mathematically equivalent to multiplying by 1.15, but writing both forms makes the meaning visible. When two independent routes agree and both fit the context, your confidence is much stronger than after simply repeating the same division.
Repeated changes multiply instead of simply adding
Successive percentage changes act on successive bases. If a quantity grows by 10% and then by another 10%, its multiplier is 1.10 × 1.10 = 1.21. The total increase is 21%, not 20%. With an original value of 100, the first change gives 110 and the second adds 11, producing 121. The second 10% is calculated from 110.
An increase and equal decrease also do not cancel. A 20% increase followed by a 20% decrease gives 1.20 × 0.80 = 0.96, so the final value is 4% below the original. To reverse a 20% increase exactly, divide by 1.20 or apply a decrease of 1 − 1/1.20 = 16⅔%. The return percentage is smaller because it is taken from the larger amount.
Write one multiplier for each stage and multiply them in order. After finding the combined multiplier, subtract 1 and convert the difference to a percentage. This avoids tracking several changing bases in your head. It also handles tax after a discount, repeated growth, depreciation, and multi-year changes. Keep the story visible so you do not apply a percentage to the wrong intermediate amount.
Know the edge cases and practise the full decision
If the original value is zero, ordinary percentage change is undefined because division by zero is impossible. Report the absolute change and the two values instead. Very small starting values can also create huge rates that are mathematically correct but easy to misinterpret. Give the absolute difference beside the percentage when the scale matters. If two peer measurements have no natural time order, the question may require percent difference, which uses a shared reference such as their mean rather than an original value.
Try a complete example: a weekly pass changes from $24 to $27.60. The difference is $3.60, and $3.60 ÷ $24 × 100% = 15%. Check with $24 × 1.15 = $27.60. Now reverse the direction. The fall from $27.60 to $24 is $3.60 ÷ $27.60 × 100% ≈ 13.04%, not 15%. Explain why the denominator changed.
Finish with a rate example. Attendance rises from 72% to 81%. The direct gap is 9 percentage points. The relative percentage change is 9 ÷ 72 × 100% = 12.5%. Write both statements and name what each measures. If you can choose the reference, preserve the sign, distinguish points from percent, and reconstruct the new amount without looking at this guide, you have learned the method rather than memorized one formula.
Put it into practice now
Choose one problem from your current homework or review set. Attempt it before opening Eqora, then use the app only at the point where your own reasoning stops.
- State what the problem is asking before you solve it
- Identify the first step you cannot justify
- Ask Eqora one focused follow-up about that step
- Finish with a similar problem and no solution in view
The session is complete when the method is clearer, not simply when the worksheet has one more answer.