PERCENTAGE CRACKER 2.0
SSC CGL MAINS • CAT • IBPS/SBI PO • CDS
Mastering Percentage for Competitive Exams
Eliminate traditional $x$-equations. Learn the exact fractional methods, AB-rules, and weighted shifts used by national exam toppers.
Curriculum & Exam Architecture
The Fraction-Percentage Matrix
Conversion tables, base shifts, reversibility rules ($x\% \text{ of } y = y\% \text{ of } x$), and multiplying factor (MF) method.
Product Constancy & Successive
The famous AB-Rule ($+\frac{a}{b} \rightarrow -\frac{a}{a+b}$), Expenditure-Price-Consumption constraints, and multi-year compound depreciation.
High-Stakes Word Problems
Rigged Elections (Invalid/Unpolled votes), 3D Geometric Volatility, Income-Expense Weighted Averages, and 3-Set Venn Diagrams.
Master Fraction-to-Percentage Equivalence
High-Frequency Derived Fractions (Must Memorize)
Q1. In a standardized quarterly recruitment examination across 4 zonal territories, Candidate B scored $20\%$ more marks than Candidate A, while Candidate A obtained $10\%$ less marks than Candidate C. Furthermore, Candidate D obtained $20\%$ more marks than Candidate C. If the maximum aggregate marks of the exam were $600$ and Candidate C secured $450$ marks, by what percentage are the marks secured by D more than those obtained by A?
Standard Algebraic Method (Slow)
• Given $C = 450$
• $A = 450 - (10\% \text{ of } 450) = 450 - 45 = 405$
• $D = 450 + (20\% \text{ of } 450) = 450 + 90 = 540$
• Difference $(D - A) = 540 - 405 = 135$
• $\%$ more than $A = \frac{135}{405} \times 100 = \frac{1}{3} \times 100 = \mathbf{33.33\%}$
Exam Topper's Base-100 Trick (10 Seconds)
• Ignore the absolute mark ($450$) entirely! Relative $\%$ change does not need actual values.
• Let Base $C = 100$.
• $A = 90$ ($10\%$ less than $C$).
• $D = 120$ ($20\%$ more than $C$).
• Comparison between $D$ and $A$: Base is $A = 90$.
• Excess $= \frac{120 - 90}{90} = \frac{30}{90} = \frac{1}{3} = \mathbf{33\frac{1}{3}\% \text{ or } 33.33\%}$.
Multiplying Factor (MF) & 1-Line Calculation
Integer Percentages
$\text{Final Value} = \text{Initial Value} \times \text{MF}$
Complex Fractional Shifts
Fraction $\frac{a}{b} \implies \text{Inc: } \frac{b+a}{b} \quad \text{Dec: } \frac{b-a}{b}$
Compound Depreciation / Growth Law
If an industrial machine or population changes by $r_1\%, r_2\%, r_3\%$ consecutively over successive years: $$\text{Value after } n \text{ years} = V_0 \times \left(1 \pm \frac{r_1}{100}\right) \times \left(1 \pm \frac{r_2}{100}\right) \times \dots$$ Exam Insight: Always convert $r_n$ into irreducible fractions to enable massive cross-cancellation!
Q2. The market valuation of a specialized industrial robotic unit depreciates at the rate of $16\frac{2}{3}\%$ during the first financial year, depreciates by $14\frac{2}{7}\%$ in the second financial year, and then depreciates further by $12\frac{1}{2}\%$ during the third year. If the total cumulative depreciation suffered by the robotic equipment at the end of $3$ years is $\$52,500$, determine the original purchase cost of the machine when it was brand new.
The Pure Fractional Chain Method (No Complex Equations)
• Therefore, Depreciation $= 8 - 5 = 3 \text{ units}$.
• Given: $3 \text{ units} = \$52,500 \implies 1 \text{ unit} = \frac{52500}{3} = \$17,500$.
• Original Value ($8 \text{ units}$) $= 8 \times 17,500 = \mathbf{\$1,44,000}$.
The Invariant "AB-Rule" & Inverse Dynamics
The Law of Invariant Product ($A \times B = \text{Constant}$)
Whenever two quantities multiply to form a third invariant quantity, any fractional surge in one triggers an automatic reciprocal decrease in the other:
1. $\text{Price} \times \text{Consumption} = \text{Expenditure}$ • 2. $\text{Speed} \times \text{Time} = \text{Distance}$ • 3. $\text{Length} \times \text{Breadth} = \text{Area}$ • 4. $\text{Men} \times \text{Days} = \text{Work}$
Q3. Due to an unexpected disruption in international shipping routes, the domestic price of refined cooking oil shot up abruptly by $32\%$. A restaurant conglomerate resolves to adjust its monthly oil usage such that its overall financial expenditure on cooking oil increases by no more than $10\%$. If the restaurant was consuming $480\text{ litres}$ of cooking oil per month prior to the price hike, how many litres of cooking oil can it now consume monthly?
The "100-Index" Method (No $x$ Variables)
• The consumer has to pull back from the updated price point of $132$ down to their ceiling of $110$.
• Fractional Cut in Consumption:
$$\text{Cut} = \frac{132 - 110}{132} = \frac{22}{132} = \frac{1}{6} \quad (\approx 16\frac{2}{3}\% \text{ reduction})$$
• If original consumption was $480\text{ litres}$, reduction $= \frac{1}{6} \times 480 = 80\text{ litres}$.
• New Allowable Consumption: $480 - 80 = \mathbf{400\text{ Litres}}$.
Q4. The monthly expenditure of an engineering director is $70\%$ of his gross monthly salary. In the subsequent fiscal year, his gross monthly salary is elevated by $24\%$, while his domestic expenditure escalates simultaneously by $15\%$. If his net monthly savings experience an absolute rise of $\$4,500$, find his new revised monthly salary.
Unit Allocation Strategy (Income = Expenditure + Savings)
• $\text{Increase in Savings} = 43.5 - 30 = \mathbf{13.5 \text{ Units}}$.
• Given: $13.5 \text{ units} = \$4,500 \implies 1 \text{ unit} = \frac{4500}{13.5} = \frac{45000}{135} = \frac{1000}{3}$.
• Original Income ($100 \text{ units}$) $= 100 \times \frac{1000}{3} = \$33,333.33$ (or directly find New Income: $124 \times \frac{1000}{3} \approx \$41,333$ or when scaled: $1,24,000$).
Q5. In a mayoral election between two candidates, $10\%$ of enrolled voters did not turn up to vote, and $1,200$ votes cast were declared invalid during scrutiny. The winning candidate secured $54\%$ of the total valid votes and was declared victorious by a margin of $1,920$ votes. Calculate the total count of voters enrolled on the voter registration roll.
The "Reverse Unpacking" Exam Shortcut
• Step 1: Focus on Valid Votes ($V$):
Winner gets $54\% \implies$ Loser gets $(100 - 54)\% = 46\%$.
Winning Margin $= 54\% - 46\% = 8\% \text{ of Valid Votes}$.
• Step 2: Add back the Invalid Ballots:
Total Polled Votes $= 24,000 + 1,200 = \mathbf{25,200 \text{ Votes}}$.
• Step 3: Unpack the Unpolled Percentage:
Since $10\%$ did not turn up, Polled Votes $= 90\%$ of Enrolled ($E$).
$$90\% \times E = 25,200 \implies E = \frac{25200}{0.9} = \mathbf{28,000 \text{ Registered Voters}}.$$
Q6. The radius of the circular base of a right circular cylinder is increased by $16\frac{2}{3}\%$, while its vertical height is simultaneously slashed by $28\frac{4}{7}\%$. Find the exact net percentage variation experienced by the total Volume ($V$) of the cylinder.
The Variable Power-Ratio Method
$$\text{Volume of Cylinder } V = \pi r^2 h \implies V \propto r \times r \times h$$ Constants like $\pi$ do not impact percentage variation. Treat $r$ twice and $h$ once!
$r_2: 6 \rightarrow 7$
Q7. In a premier national defense selection board test comprising $1,500$ officers, $68\%$ of candidates passed the Advanced Physical Endurance Test, $72\%$ cleared the Technical Interview, and $56\%$ cleared both stages. If $60$ candidates who failed in both evaluations decided to file an official re-evaluation petition, how many candidates cleared ONLY ONE of the two stages?
Set Theory Breakdown
68% - 56% = 12%
72% - 56% = 16%
12% + 16% + 56% = 84%
• Total Candidates $= 1,500$.
• Required Number: $28\% \text{ of } 1,500 = 0.28 \times 1,500 = 28 \times 15 = \mathbf{420\text{ Candidates}}$.
Advanced Splitting & Reversibility Drills
Evaluate 57% of 640 mentally:
• 5% = 32
• 1% = 6.4 → 2% = 12.8
• Total = 320 + 32 + 12.8 = 364.8
Evaluate 78% of 375:
• $375\% = 300\% + 75\% = 3 + \frac{3}{4}$
• $3 \times 78 = 234$
• $\frac{3}{4} \times 78 = 58.5$
• Total = 234 + 58.5 = 292.5
Evaluate 87.5% of 720:
• In Fractions: $1 - \frac{1}{8} = \frac{7}{8}$
• $\frac{7}{8} \times 720 = 7 \times 90 = \mathbf{630}$
The 5 Commandments of Quantitative Aptitude
Never Assume $x$: Prefer Base $100$ for simple percentages and irreducible LCM ratios for fractional changes.
Deploy the AB-Rule Instantly: Anytime Product is constant, $+\frac{a}{b}$ triggers $-\frac{a}{a+b}$. Zero scratch work required.
Chain Multiplication: Successive changes compound as ratios: $(1 \pm r_1)(1 \pm r_2)\dots$ Cross-cancel early!
Track the Base: Words following "than", "of", and "as compared to" strictly dictate what goes into the denominator.
