3. Percentage Mastery for Speed and Accuracy

Why This Chapter Matters

Percentage is the language behind profit and loss, DI, interest, averages, elections, population change, and comparison-based bank-exam questions. This chapter is small in appearance but touches almost everything.

Core Ideas

• Memorise the high-frequency shifts like $50\%=\frac{1}{2}$, $25\%=\frac{1}{4}$, $12.5\%=\frac{1}{8}$, and $62.5\%=\frac{5}{8}$.
• An increase of $x\%$ means multiplying by $\left(1+\frac{x}{100}\right)$.
• A decrease of $x\%$ means multiplying by $\left(1-\frac{x}{100}\right)$.
• If one value is $a\%$ less than another, the reverse comparison is not the same number. Reverse-percentage questions must use the original base carefully.
• Successive changes should be multiplied through factors or handled with the net-change formula, not added blindly.
• Percent of percent questions become easy once the base quantity is clear.

High-Value Formulas

How To Approach Questions

1. Decide the base quantity first.
2. Convert familiar percentages to fractions to cut calculation time.
3. For repeated change, multiply the factors instead of adding the percentages blindly.
4. If the question asks for the reverse comparison, rebuild the base before taking the final percentage.

Worked Examples

Example 1

Prompt: Find $62.5\%$ of $256$.

Approach: Since $62.5\%=\frac{5}{8}$, the answer is $\frac{5}{8}\times 256=160$.

Example 2

Prompt: A quantity increases by $20\%$ and then falls by $10\%$. Find the net multiplier.

Approach: Use $1.2\times 0.9=1.08$. That means a net increase of $8\%$.

Example 3

Prompt: If a price rises by $40\%$, by what percent should consumption fall to keep expenditure unchanged?

Approach: Required reduction $=\frac{40}{140}\times 100=28\frac{4}{7}\%$. This is a standard reverse-base result.

Example 4

Prompt: A salary rises by $10\%$ and again by $10\%$. What is the total increase?

Approach: Do not call it $20\%$. The multiplier is $1.1\times 1.1=1.21$, so the true increase is $21\%$.

Common Mistakes

• Using the wrong base quantity in comparison questions.
• Adding successive percentage changes directly.
• Assuming “20% less” automatically means “20% more” in the reverse direction.
• Treating percent and percentage points as the same idea.

Quick Revision

Percentage questions become manageable once the base is fixed, the fraction form is recognised, and successive changes are handled as multipliers.