Percentage increase and decrease

A percentage change has two ingredients: the original value (the starting point, sometimes called the base) and the new value (what it became). The change itself is New minus Original, and turning that change into a percentage means dividing by the original value and multiplying by 100.

where: New value is the quantity after the change, Original value is the quantity before the change (always the denominator), and the result is a pure percentage with no units. The same relationship can be run backwards. If you already know the original value and the percentage change $R$, you can jump straight to the new value without a separate subtraction step:

where: the $+$ sign is used for an increase and the $-$ sign for a decrease, and $R$ is the percentage change expressed as a plain number (12, not 0.12). Reading a word problem correctly is the real skill here. If the problem gives you two actual values and asks "by what percentage," use the first pair of formulas. If it gives you one value and a percentage change and asks for the resulting value, use the direct formula instead. The figure below lines the two directions up side by side.

The goal of this derivation is to show, using nothing more than algebra you already know, why the "direct" formula (New value equals Original value times one plus or minus R over 100) is not a separate rule to memorise but is simply the percentage-change formula rearranged to make the new value the subject, so that both formulas describe exactly the same relationship viewed from opposite ends.

Step 1. Start from the percentage increase formula and give the percentage change a name, $R$, so that $R = \dfrac{\text{New} - \text{Original}}{\text{Original}} \times 100$.

Step 2. Divide both sides by 100 to isolate the fraction: $\dfrac{R}{100} = \dfrac{\text{New} - \text{Original}}{\text{Original}}$.

Step 3. Multiply both sides by Original to clear the denominator: $\text{Original} \times \dfrac{R}{100} = \text{New} - \text{Original}$.

Step 4. Add Original to both sides and factor it out on the left: $\text{New} = \text{Original} + \text{Original} \times \dfrac{R}{100} = \text{Original} \times \left(1 + \dfrac{R}{100}\right)$.

Step 5. The same four steps, starting from the percentage decrease formula instead, give $\text{New} = \text{Original} \times \left(1 - \dfrac{R}{100}\right)$. This confirms that the direct formula is just the percentage-change formula solved for the new value, so remembering one relationship is enough to recover the other by simple rearrangement whenever a problem needs it.

A shirt originally priced at ₹800 is marked down by 12%. Find the new price, and separately verify the population example from earlier in the chapter still checks out.

Step 1. Identify the known quantities: Original value = ₹800, percentage decrease $R = 12$. The question asks for the new value directly, so the direct formula applies with the minus sign.

Step 2. Substitute into $\text{New value} = \text{Original value} \times \left(1 - \dfrac{R}{100}\right)$:

Step 3. Compute the product: $800 \times 0.88 = 704$. So the new price of the shirt is ₹704.

Step 4. Verify with the other formula. The actual decrease is $800 - 704 = 96$. Checking, $\dfrac{96}{800} \times 100 = 12$, which matches the given 12%, so the answer is consistent.

Step 5. As a second check on the same core idea, recall the village population that grew from 4,500 to 5,175 (Original = 4500, New = 5175). Using $R = \dfrac{5175-4500}{4500}\times100 = \dfrac{675}{4500}\times100 = 15$, the increase is 15%. Running this $R$ back through the direct formula, $4500 \times \left(1+\dfrac{15}{100}\right) = 4500 \times 1.15 = 5175$, which lands exactly back on the given new value, confirming both formulas describe the same relationship.

• Dividing the change by the new value instead of the original value — the original value is always the base, even when the problem states the new value first in the sentence.
• Mixing up the increase formula ($1 + R/100$) with the decrease formula ($1 - R/100$), especially when a problem uses positive-sounding words for a decrease, such as "marked down by" or "fell by."
• Adding or subtracting successive percentage changes directly (assuming a 10% increase followed by a 10% decrease returns the original value) — each change must be applied to the updated value in turn, not combined arithmetically.
• Forgetting to convert $R$ correctly in the direct formula — $R$ is used as $R/100$ inside the bracket, not as a decimal typed in twice, so a 12% change means using $12/100$, not $0.12/100$.
• Losing track of which quantity a percentage in the question is even attached to — "12% off the marked price" and "12% tax added to the cost price" both use the same formula shape but move the price in opposite directions, so read the wording carefully before choosing a sign.

Ask the tutor: confused about why a 10% rise followed by a 10% fall does not bring you back to the original value — ask your AI tutor to walk through the successive-change calculation with you step by step.