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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.
Application Strategy
Worked Examples
Problem Statement
A shopkeeper increases the price of a jacket by 25%. During a sale, he then offers a 25% discount on the new price. If the original price was Rs. 4,000, what is the final sale price? What is the overall percentage change?
successive percentage changepercentage change
Strategic Toolkit
$$\text{Net} = V \times \left(1 + \frac{25}{100}\right)\left(1 - \frac{25}{100}\right)$$
Step 1
Apply the increase
Increase Rs. 4,000 by 25%.
$$4000 \times 1.25 = 5000$$
Step 2
Apply the decrease
Apply 25% discount on the increased price.
$$5000 \times 0.75 = 3750$$
Step 3
Find net percentage change
Use the successive change formula or compute directly.
$$\text{Net \% Change} = 25 + (-25) + \frac{25 \times (-25)}{100} = -6.25\%$$
Step 4
Verify
Confirm the final value matches.
$$4000 \times (1 - 0.0625) = 4000 \times 0.9375 = 3750$$
Problem Statement
Ahmed's salary was increased by 10% in 2023 and then by 20% in 2024. If his final salary is Rs. 79,200, what was his original salary before any increase?
successive percentage changereverse percentage
Strategic Toolkit
$$S_{\text{final}} = S_0 \times 1.10 \times 1.20$$
Step 1
Find the combined multiplier
Multiply both successive multipliers.
$$1.10 \times 1.20 = 1.32$$
Step 2
Set up the equation
Express the final salary in terms of the original.
$$S_0 \times 1.32 = 79200$$
Step 3
Solve for original salary
Divide to find the starting value.
$$S_0 = \frac{79200}{1.32} = 60000$$
Step 4
Interpret
Ahmed's original salary was Rs. 60,000 — a net increase of 32%.
$$\text{Net \% Change} = 10 + 20 + \frac{10 \times 20}{100} = 32\%$$
Problem Statement
The population of a town was 2,50,000 at the start of 2021. It grows at 8% per year. What is the population at the start of 2024? By what overall percentage has it grown?
population growthcompound change
Strategic Toolkit
$$P = P_0 \times \left(1 + \frac{8}{100}\right)^3$$
Step 1
Identify the periods
From the start of 2021 to the start of 2024 is 3 full years.
$$n = 3$$
Step 2
Apply the compound formula
Compute the population after 3 years at 8% annual growth.
$$P = 250000 \times (1.08)^3$$
Step 3
Calculate
Evaluate the power and multiply.
$$(1.08)^3 = 1.259712$$
Step 4
Final population
Multiply by the original population.
$$P = 250000 \times 1.259712 = 314928$$
Step 5
Overall percentage growth
Convert the growth factor to a percentage.
$$\text{Overall growth} = (1.259712 - 1) \times 100 = 25.97\%$$
Problem Statement
A person spends 75% of his income. If his income increases by 20% but his expenditure remains the same in absolute terms, by what percentage do his savings increase?
income-expendituresavingspercentage change
Strategic Toolkit
$$\text{Savings}_1 = 0.25I, \quad \text{Savings}_2 = 1.20I - 0.75I = 0.45I$$
Step 1
Express original savings
Savings is 25% of income (since 75% is spent).
$$S_1 = I - 0.75I = 0.25I$$
Step 2
Express new income
Income increases by 20%.
$$I_{\text{new}} = 1.20I$$
Step 3
Express new expenditure
Expenditure stays the same in absolute rupees.
$$E_{\text{new}} = 0.75I$$
Step 4
Find new savings
Subtract unchanged expenditure from new income.
$$S_2 = 1.20I - 0.75I = 0.45I$$
Step 5
Calculate percentage increase in savings
The base is the original savings $0.25I$.
$$\text{Increase} = \frac{0.45I - 0.25I}{0.25I} \times 100 = \frac{0.20I}{0.25I} \times 100 = 80\%$$
Problem Statement
In an election between two candidates, 10% of the votes cast were declared invalid. Candidate A received 55% of the valid votes and won by 2,970 votes. How many total votes were cast?
election percentagepercentage of a percentage
Strategic Toolkit
$$0.55V - 0.45V = 0.10V = 2970$$
Step 1
Define valid votes
Let total votes cast be $T$. Valid votes = 90% of $T$.
$$V = 0.90T$$
Step 2
Find each candidate share
Candidate A gets 55% and Candidate B gets 45% of valid votes.
$$A = 0.55V, \quad B = 0.45V$$
Step 3
Use the margin
The difference equals 2,970 votes.
$$0.55V - 0.45V = 0.10V = 2970$$
Step 4
Solve for valid votes
Divide by 0.10 to find valid votes.
$$V = \frac{2970}{0.10} = 29700$$
Step 5
Solve for total votes
Valid votes are 90% of total, so total is valid divided by 0.90.
$$T = \frac{29700}{0.90} = 33000$$
Memorize formulas
Percentage Change
Use when finding how much a value has grown or shrunk relative to the original
Formula
$$\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$$
Successive Percentage Change
Use when two percentage changes are applied one after another
Formula
$$a + b + \frac{ab}{100}$$
Population / Compound Growth
Use when a quantity grows at a fixed rate over multiple periods
Formula
$$P_{\text{final}} = P_{\text{initial}} \times \left(1 + \frac{r}{100}\right)^n$$
Percentage of a Percentage
Use when a percentage is taken of another percentage
Formula
$$x\%\text{ of }y\% = \frac{x \times y}{100}\%$$
New Value After Change
Use when computing the result after a single percentage increase or decrease
Formula
$$\text{New} = \text{Original} \times \left(1 \pm \frac{p}{100}\right)$$
Review
1
A store originally prices a jacket at Rs. 4,500. During a clearance sale, the price is first reduced by 20% and then by an additional 10% on the discounted price. What is the final selling price of the jacket?
Ans
Rs. 3,240
Review
Tap to reveal
Apply the first discount
Reduce the original price by 20%.
$$4500 \times 0.80 = 3600$$
Apply the second discount
Reduce the discounted price by another 10%.
$$3600 \times 0.90 = 3240$$
2
The population of a town was 80,000 at the start of 2020. It grows at 5% per year. What is the approximate population at the start of 2023?
Ans
92,610
Review
Tap to reveal
Set up the compound growth formula
The population grows over 3 years (2020 to 2023) at 5% annually.
$$P_{\text{final}} = 80000 \times \left(1 + \frac{5}{100}\right)^3$$
Calculate the growth factor and final population
Compute $(1.05)^3$ and multiply by the initial population.
$$80000 \times 1.157625 = 92610$$
3
A shopkeeper marks an item at 40% above the cost price. During a festival, he offers a discount of 25% on the marked price. What is his overall profit percentage?
Ans
5%
Review
Tap to reveal
Calculate the marked price
Apply 40% markup to the cost price of 100.
$$\text{MP} = 100 \times 1.40 = 140$$
Calculate the selling price after discount
Apply 25% discount on the marked price.
$$\text{SP} = 140 \times 0.75 = 105$$
Determine profit percentage
Compare selling price to cost price to find the profit percentage.
$$\text{Profit} = \frac{105 - 100}{100} \times 100 = 5\%$$
4
A student scored 72 marks out of 80 in Physics and 84 marks out of 120 in Chemistry. What is her overall percentage across both subjects?
Ans
78%
Review
Tap to reveal
Find total marks obtained and total possible
Add the marks from both subjects separately.
$$\text{Total obtained} = 72 + 84 = 156, \quad \text{Total possible} = 80 + 120 = 200$$
Calculate the overall percentage
Divide total obtained by total possible and multiply by 100.
$$\frac{156}{200} \times 100 = 78\%$$
5
In an election, there were two candidates. 10% of the total votes cast were declared invalid. The winning candidate received 60% of the valid votes and won by 10,800 votes. How many total votes were cast?
Ans
60,000
Review
Tap to reveal
Express valid votes and vote shares
10% of total votes are invalid, so 90% are valid. The winner gets 60% and the loser gets 40% of valid votes.
$$\text{Valid} = 0.9V, \quad \text{Winner} = 0.54V, \quad \text{Loser} = 0.36V$$
Set up the margin equation
The difference between winner and loser votes equals the given margin of 10,800.
$$0.54V - 0.36V = 0.18V = 10800$$
Solve for total votes
Divide the margin by 0.18 to get the total votes cast.
$$V = \frac{10800}{0.18} = 60000$$
6
A man spends 75% of his income. If his income increases by 20% but his expenditure remains the same, by what percentage does his savings increase?
Ans
80%
Review
Tap to reveal
Set up initial income and expenditure
Assume income = 100. With 75% expenditure, savings = 25.
$$\text{Income} = 100, \quad \text{Expenditure} = 75, \quad \text{Savings} = 25$$
Apply the income increase
New income is 20% more, but expenditure stays the same.
$$\text{New income} = 120, \quad \text{New savings} = 120 - 75 = 45$$
Calculate percentage increase in savings
Savings rose from 25 to 45. Use the original savings as the base.
$$\frac{45 - 25}{25} \times 100 = 80\%$$
7
What is 30% of 40% of 500?
Ans
60
Review
Tap to reveal
Apply percentage of a percentage rule
Combine the two percentages first, then apply to the number.
$$30\% \text{ of } 40\% = \frac{30 \times 40}{100}\% = 12\%$$
Apply to the value
Calculate 12% of 500.
$$12\% \text{ of } 500 = \frac{12}{100} \times 500 = 60$$
8
A company's revenue decreased by 15% in the first year and then increased by 20% in the second year. What is the net percentage change in revenue over the two years?
Ans
+2%
Review
Tap to reveal
Apply the successive change formula
Use $a + b + \frac{ab}{100}$ with $a = -15$ and $b = 20$.
$$\text{Net change} = -15 + 20 + \frac{(-15)(20)}{100}$$
Compute the result
Simplify the expression to get the net percentage change.
$$= 5 - 3 = +2\%$$
9
The price of a laptop increased by 25%. By what percentage must the new price be reduced to bring it back to the original price?
Ans
20%
Review
Tap to reveal
Express the new price
With original price of 100, a 25% increase gives 125.
$$\text{New price} = 100 \times 1.25 = 125$$
Find the required reduction percentage
Calculate what percentage of 125 must be subtracted to get back to 100.
$$\frac{125 - 100}{125} \times 100 = 20\%$$
10
A product's price is successively increased by 10%, 15%, and 20%. What is the overall percentage increase?
Ans
51.8%
Review
Tap to reveal
Combine the first two changes
Use the successive change formula with 10% and 15%.
$$10 + 15 + \frac{10 \times 15}{100} = 26.4\%$$
Combine with the third change
Apply the formula again with 26.4% and 20%.
$$26.4 + 20 + \frac{26.4 \times 20}{100} = 51.8\%$$
Verify with successive multipliers
Multiply the individual growth factors to confirm.
$$1.10 \times 1.15 \times 1.20 = 1.518 \implies 51.8\%$$
11
A town's population of 2,00,000 increases by 20% in the first year and by 10% in the second year. In the third year, due to an epidemic, 20% of the population leaves. What is the final population?
Ans
2,11,200
Review
Tap to reveal
Calculate population after each year
Apply each percentage change successively using the new value as the base.
$$200000 \xrightarrow{+20\%} 240000 \xrightarrow{+10\%} 264000 \xrightarrow{-20\%} 211200$$
Verify with successive multipliers
Multiply all individual factors together and apply to the initial population.
$$200000 \times 1.20 \times 1.10 \times 0.80 = 211200$$
12
A salary is first increased by 25% and then the new salary is decreased by 20%. If the final salary is Rs. 50,000, what was the original salary?
Ans
Rs. 50,000
Review
Tap to reveal
Find the net percentage change
Apply the successive change formula with +25% and −20%.
$$25 + (-20) + \frac{25 \times (-20)}{100} = 0\%$$
Conclude the original salary
Since the net change is 0%, the original and final salaries are the same.
$$\text{Original salary} = \text{Final salary} = \text{Rs. } 50{,}000$$
Question 1 of 12
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1
A store originally prices a jacket at Rs. 4,500. During a clearance sale, the price is first reduced by 20% and then by an additional 10% on the discounted price. What is the final selling price of the jacket?