Percentage Increases & Decreases

Percentage increases and decreases describe how an original amount changes by a given proportion of itself. The key idea is that the percentage is always taken from the original amount, which is why increases can be written as finding more than 100% of the original and decreases as finding less than 100%. This topic matters because it provides a consistent method for price changes, growth, discounts, and many real-life comparisons, especially through efficient use of multipliers.

• Percentage increase means making a quantity larger by adding a given percentage of the original amount. If an amount is increased by $p\%$, the new value is the original plus $p\%$ of the original, so the total becomes $(100+p)\%$ of the starting value. This idea is used whenever a value grows, such as in prices, populations, or measurements.
• Percentage decrease means making a quantity smaller by subtracting a given percentage of the original amount. If an amount is decreased by $p\%$, the new value is the original minus $p\%$ of the original, so the total becomes $(100-p)\%$ of the starting value. This applies to discounts, depreciation, shrinkage, and other reductions.
• The original amount is the reference value from which the percentage change is calculated. This matters because the same percentage applied to different starting values gives different numerical changes, so identifying the correct base is essential before doing any calculation.
• A multiplier is the decimal form of the final percentage of the original amount. For an increase by $p\%$, the multiplier is $1+\frac{p}{100}$, and for a decrease by $p\%$, it is $1-\frac{p}{100}$. Multipliers are especially useful because they turn a percentage change problem into a single multiplication.

• The percentage change is always based on the starting amount, not the changed amount. That is why increasing $50$ by $10\%$ adds $5$, because $10\%$ of $50$ is $5$. If the starting amount changes, the numerical size of the percentage change changes as well.
• An increase by $p\%$ can be written as

New amount $= \text{Original amount} \times \left(1 + \frac{p}{100}\right)$

This works because multiplying by $1$ keeps the full original amount, while multiplying by $\frac{p}{100}$ adds the extra proportion. The expression combines both parts into one efficient step.

• A decrease by $p\%$ can be written as

New amount $= \text{Original amount} \times \left(1 - \frac{p}{100}\right)$

This works because the original amount is treated as $100\%$, and a decrease removes $p\%$ from that whole. The remaining proportion is therefore $(100-p)\%$ of the original.

• Thinking in terms of final percentage helps students avoid confusion. For example, an increase of $18\%$ means finding $118\%$ of the original, while a decrease of $18\%$ means finding $82\%$ of the original. This is why multiplier methods are often faster and more reliable than separate add-or-subtract steps.

Non-calculator method

• Find the percentage part first by using easy benchmark percentages such as $10\%$, $5\%$, or $1\%$. This works well when numbers are simple and mental arithmetic is expected, because many percentages can be built from smaller known parts.
• For an increase, calculate the required percentage of the original amount and then add it back to the original. For example, if the change is $p\%$, first find $p\%$ of the amount, then compute

$\text{New amount} = \text{Original} + \text{Change}$

This method is especially helpful when showing working is important.

• For a decrease, calculate the required percentage of the original amount and subtract it from the original. This gives

$\text{New amount} = \text{Original} - \text{Change}$

and is useful when a question emphasizes understanding rather than speed.

Multiplier method

• Convert the percentage change into a multiplier by writing the final percentage as a decimal. Increase by $p\%$ gives multiplier $1+\frac{p}{100}$, while decrease by $p\%$ gives multiplier $1-\frac{p}{100}$. This method is efficient on a calculator because it replaces several steps with one multiplication.
• Use the formula

$\text{New amount} = \text{Original amount} \times \text{multiplier}$

where the multiplier represents the proportion that remains after the change. This is the best method for awkward percentages such as $13\%$, $17.5\%$, or values involving decimals.

• Choose a method based on the context: use building-block percentages for mental or non-calculator work, and use multipliers for accuracy and speed in calculator questions. Both methods are mathematically equivalent, so the main decision is which gives the clearest and fastest route.

• Increase by a percentage is different from finding that percentage of an amount. Finding $15\%$ of a quantity gives only the change part, but increasing by $15\%$ gives the original plus that change. This distinction matters because students often stop after calculating the percentage part and forget to form the new total.

• Decrease by $p\%$ is not the same as multiplying by $p$ as a decimal. A decrease of $20\%$ means keep $80\%$ of the original, so the multiplier is $0.80$, not $0.20$. The decimal $0.20$ represents the amount removed, not the amount left.

• Adding or subtracting percentages directly is different from applying them to values. Two separate percentage changes are not usually combined by simple addition because each change is based on the current amount at that stage. This is why repeated changes are better handled through multiplying successive multipliers.

• | Idea | What it means | Correct interpretation | | --- | --- | --- | | Find $p\%$ of an amount | Calculate the part only | Multiply by $\frac{p}{100}$ | | Increase by $p\%$ | Add the part to the original | Multiply by $1+\frac{p}{100}$ | | Decrease by $p\%$ | Remove the part from the original | Multiply by $1-\frac{p}{100}$ | | New amount after change | Final value after adjustment | Original $\times$ multiplier |

• A percentage increase and the same-sized percentage decrease do not undo each other. If a value goes up by $10\%$ and then down by $10\%$, the second change is based on the larger amount, so the result ends up below the original. This is a key conceptual distinction that prevents reverse-thinking errors.

• Always identify the starting amount first before doing any percentage calculation. The percentage change is taken from the original quantity, so if you choose the wrong base, every later step will be wrong even if the arithmetic is correct.
• Translate the wording into a final percentage before calculating. Phrases such as "increase by" and "decrease by" should immediately suggest $(100+p)\%$ or $(100-p)\%$ of the original, which helps you select the correct multiplier without hesitation.
• Use a reasonableness check after finding the answer. If the value was increased, the result must be larger than the original; if it was decreased, the result must be smaller. This simple check often catches sign errors and incorrect multipliers.
• Watch units and money carefully because exam marks can be lost through presentation as well as method. If the quantity is in grams, litres, or dollars, the final answer should keep the correct unit and, where appropriate, sensible rounding.
• Prefer multipliers for awkward percentages and non-calculator decomposition for friendly percentages. This strategic choice saves time and reduces slips, especially under exam pressure when efficiency matters as much as accuracy.

• Using the percentage itself as the multiplier is a very common mistake. For example, students may use $0.12$ for an increase of $12\%$, but $0.12$ finds $12\%$ of the amount rather than the new total. The correct increase multiplier would be $1.12$.
• Subtracting when the question asks for an increase, or adding when it asks for a decrease, usually comes from reading too quickly. This is why the language of the question should be converted into a clear action word such as "grow" or "reduce" before calculation starts.
• Treating increase and decrease as inverse with the same percentage leads to incorrect conclusions. Increasing by $p\%$ and then decreasing by $p\%$ does not return to the starting value because the second percentage is taken from a changed amount, not the original one.
• Forgetting that percentages greater than 100 represent more than the original can cause confusion. An increase leads to a multiplier above $1$, while a decrease leads to a multiplier below $1$ but still above $0$ unless the entire amount is removed. Recognizing this helps students quickly judge whether a calculated answer is plausible.

• Percentage increases and decreases connect directly to percentage change, where the focus shifts from finding the new amount to describing how much change has occurred. Understanding multipliers here builds the foundation for later topics where the multiplier is inferred from the before-and-after values.
• This topic also links to compound change, where a percentage is applied repeatedly over time. In that setting, the same multiplier idea is used again and again, so a single change becomes repeated multiplication rather than repeated addition.
• In everyday mathematics, percentage increase and decrease appear in discounts, tax, inflation, depreciation, growth rates, and data interpretation. Learning the structure now makes it easier to interpret real-world numerical claims critically and calculate them accurately.
• The topic supports algebraic thinking because percentage changes can be expressed as formulas. Writing relationships such as $N = O\left(1+\frac{p}{100}\right)$ or $N = O\left(1-\frac{p}{100}\right)$ helps students move smoothly between words, arithmetic, and symbolic representation.