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Percentage Applications
Percentage Fundamentals
A percentage is a way of expressing a number as a fraction of 100. It converts any ratio into a common denominator so that different quantities become directly comparable.
$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$
Converts a part-to-whole ratio into a value out of 100.
$\text{Part}$=The portion or amount being measured(same as whole)
$\text{Whole}$=The total or reference quantity(same as part)
$\text{Part} > \text{Whole}$
→The percentage exceeds 100% (e.g., a 120% increase).
$\text{Whole} = 0$
→Percentage is undefined — you cannot divide by zero.
Quick Fraction Conversions: $\frac{1}{2} = 50\%$, $\frac{1}{3} \approx 33.33\%$, $\frac{1}{4} = 25\%$, $\frac{1}{6} \approx 16.67\%$, $\frac{1}{8} = 12.5\%$. Memorising these speeds up mental math significantly.
Percentage as Multiplier: Any percentage $x\%$ is equivalent to the decimal $\frac{x}{100}$. So $45\%$ of 200 is $0.45 \times 200 = 90$.
Reverse Percentage: If a value is $V$ after a $p\%$ change, the original is $\frac{V}{1 \pm \frac{p}{100}}$ — the denominator accounts for the change direction.
Percentage Increase and Decrease
Percentage change measures how much a value has grown or shrunk relative to its original amount. The original value is always the denominator — this is the single most important rule.
$$\text{\% Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$$
Finds the proportional change expressed as a percentage of the starting value.
$\text{New}$=The final value after change(same as original)
$\text{Original}$=The starting value (always the denominator)(same as new)
$\text{New} > \text{Original}$
→Result is positive — percentage increase.
$\text{New} < \text{Original}$
→Result is negative — percentage decrease.
$\text{New} = \text{Original}$
→No change (0%).
New Value After Increase: $\text{New} = \text{Original} \times \left(1 + \frac{p}{100}\right)$ where $p$ is the increase percentage.
New Value After Decrease: $\text{New} = \text{Original} \times \left(1 - \frac{p}{100}\right)$ where $p$ is the decrease percentage.
Original From Increased Value: $\text{Original} = \frac{\text{New}}{1 + \frac{p}{100}}$ — divide by the growth factor, not multiply by $\left(1 - \frac{p}{100}\right)$.
A percentage change is not symmetric. Increasing a price by $20\%$ and then decreasing by $20\%$ does NOT return to the original value. The decrease operates on a larger base.
$$\text{Net} = V \times \left(1 + \frac{p}{100}\right)\left(1 - \frac{p}{100}\right) = V \times \left(1 - \frac{p^2}{10000}\right)$$
An equal increase and decrease always produces a net loss because $p^2/10000 > 0$.
$V$=Original value(any unit)
$p$=The percentage applied both ways(percent)
Always a Net Loss: Since $1 - \frac{p^2}{10000} < 1$, the final value is always less than the original for any non-zero $p$.
Example: A $\$100$ item increased by $20\%$ becomes $\$120$. Then decreased by $20\%$ gives $120 \times 0.80 = \$96$ — a $\$4$ loss.
Magnitude Grows With $p$: The net loss is $\frac{p^2}{100}$% of the original. At $p = 10\%$ the loss is tiny ($0.01\%$), but at $p = 50\%$ it is $2.5\%$.
Successive Percentage Changes
When multiple percentage changes happen one after another, each change is applied to the result of the previous one. You multiply the successive multipliers to find the overall effect.
$$\text{Net \% Change} = a + b + \frac{a \times b}{100}$$
Shortcut formula for two successive percentage changes applied to the same base.
$a$=First percentage change (positive for increase, negative for decrease)(percent)
$b$=Second percentage change (positive for increase, negative for decrease)(percent)
Both $a$ and $b$ are positive
→Net increase is MORE than $a + b$ (the cross term adds).
One positive, one negative
→Net result depends on which is larger — the cross term reduces the net.
Both negative
→Net decrease is MORE than $|a| + |b|$.
Cross Term $\frac{ab}{100}$: This is the extra change caused by the second percentage acting on the first percentage result. It is what makes successive changes different from simple addition.
Multiplier Method: The most reliable approach — convert each change to a multiplier ($1 + \frac{p}{100}$) and multiply them all. Final value $= V \times \left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)$.
Three or More Changes: Extend by multiplying all multipliers: $V \times (1 + \frac{a}{100})(1 + \frac{b}{100})(1 + \frac{c}{100})$. There is no simple additive shortcut for three.
Percentage of a Percentage
A percentage of a percentage arises when a fraction of a group that is already a subset is being measured. For example, finding $30\%$ of the $60\%$ of students who passed.
$$x\%\text{ of }y\% = \frac{x \times y}{100}\%$$
Multiplying two percentages gives the combined effect as a single percentage.
$x\%$=First percentage (the 'of' part)(percent)
$y\%$=Second percentage (the base)(percent)
$x\%$ or $y\%$ equals $100\%$
→The result equals the other percentage unchanged.
$x\% = y\%$
→Result is $\frac{x^2}{100}\%$ — always smaller than $x$ for $x < 100$.
Decimal Approach: Convert both to decimals and multiply: $30\%$ of $60\% = 0.30 \times 0.60 = 0.18 = 18\%$.
In Word Problems: Read carefully for the nesting structure. '$40\%$ of the $25\%$ who voted' means multiply, not add. The key word is 'of'.
Three-Level Nesting: For $a\%$ of $b\%$ of $c\%$, compute $\frac{a \times b \times c}{10000}\%$ of the total.
Population and Growth Problems
Population growth problems apply percentage increase over multiple time periods — typically years. Each year's growth builds on the previous year's total, making this a compound change.
$$P_{\text{final}} = P_{\text{initial}} \times \left(1 + \frac{r}{100}\right)^n$$
A quantity growing at a fixed annual percentage rate compounds over $n$ periods.
$P_{\text{initial}}$=Starting population or value(people, amount, etc.)
$r$=Growth rate per period(percent per period)
$n$=Number of periods (usually years)(positive integer)
$r < 0$
→Population is declining — use $1 - \frac{|r|}{100}$.
$n = 1$
→Reduces to simple percentage increase.
Growth Factor: The term $\left(1 + \frac{r}{100}\right)$ is the growth factor. For $r = 5\%$, the factor is $1.05$, meaning the population becomes $1.05\times$ each year.
Net Growth Over $n$ Years: The total percentage change is $\left[\left(1 + \frac{r}{100}\right)^n - 1\right] \times 100\%$.
Decline Problems: Replace $+$ with $-$: $P_{\text{final}} = P_{\text{initial}} \times \left(1 - \frac{r}{100}\right)^n$. The population shrinks each period.
Income, Expenditure, and Savings
In income-expenditure problems, the fundamental relationship is that savings equals income minus expenditure. A percentage change in one variable affects the others proportionally.
$$\text{Savings} = \text{Income} - \text{Expenditure}$$
Savings is whatever remains after all spending is subtracted from total income.
$\text{Income}$=Total money earned (salary, wages, revenue)(currency)
$\text{Expenditure}$=Total money spent(currency)
$\text{Savings}$=Money retained after spending(currency)
Constant Income, Changed Expenditure: If income is fixed and expenditure changes by $p\%$, savings change by $\frac{\Delta E}{\text{Income}} \times 100\%$ where $\Delta E$ is the absolute change in expenditure.
Savings as Percentage: $\text{Savings \%} = \frac{\text{Savings}}{\text{Income}} \times 100$. If expenditure is $70\%$ of income, savings are $30\%$.
Both Income and Expenditure Change: New savings $= I_2 - E_2$. Track each separately using their percentage changes, then subtract.
Marks percentage problems involve finding scores, pass thresholds, or comparing performance. The total marks are the base, and each score is expressed as a percentage of that total.
$$\text{Score \%} = \frac{\text{Marks Obtained}}{\text{Total Marks}} \times 100$$
Converts a raw score into a percentage for comparison or grading.
$\text{Marks Obtained}$=The student raw score(marks)
$\text{Total Marks}$=Maximum possible score(marks)
Passing Threshold: If the pass mark is $40\%$ of 500, a student needs $0.40 \times 500 = 200$ marks.
Improvement Calculation: If a student scores $60\%$ and then $78\%$, the improvement is $\frac{78 - 60}{60} \times 100 = 30\%$ improvement — the base is the original score, not the total.
Maximum/Minimum Questions: Given a percentage range (e.g., scored between $65\%$ and $75\%$), convert to absolute marks and then check integer constraints.
Election and Voting Problems
Election percentage problems involve valid votes, invalid votes, and vote shares. The key is to carefully track the base at each stage — the percentage always refers to a specific total.
$$\text{Valid Votes} = \text{Total Votes} \times \left(1 - \frac{\text{Invalid \%}}{100}\right)$$
Subtracts the invalid portion from total votes to get the pool of votes that actually count.
$\text{Total Votes}$=All votes cast including invalid ones(number of votes)
$\text{Invalid \%}$=Percentage of total votes that are rejected/spoiled(percent)
Margin of Victory: If winner gets $a\%$ and runner-up gets $b\%$ of valid votes, the margin is $(a - b)\%$ of valid votes. Convert to absolute numbers using the valid vote count.
Multi-Step Problems: First find valid votes (subtract invalid from total), then find each candidate share, then answer the question — resist the urge to skip steps.
Population vs Voters: In problems involving population, only a fraction are registered voters, and only a fraction of those actually vote. Apply each percentage sequentially.