Percentage

1. Core Concepts & Theoretical Blueprint

Percentage is a standardized fraction with denominator 100, used as the universal language connecting Profit & Loss, Simple/Compound Interest, and Data Interpretation.

Absolute Core Definition:

Fraction-to-Percentage Conversion (memorize, do not compute live):

Fraction	%	Fraction	%
1/2	50%	1/8	12.5%
1/3	33.33%	1/9	11.11%
1/4	25%	1/11	9.09%
1/5	20%	1/12	8.33%
1/6	16.66%	1/16	6.25%
1/7	14.28%	1/20	5%

Percentage Change Formula:

Successive Percentage Change Formula:

(negative sign for a decrease)

Product-Constancy Formula (if A × B = constant, and A changes by x%):

The Universal Trap: The most exploited trap is "percentage of what?" — failing to correctly identify the base. "A is 20% more than B" does NOT mean "B is 20% less than A." A common secondary trap: applying percentage increase/decrease on the already-changed value instead of the original base.

2. Exhaustive Question Typology

                              PERCENTAGE
                                  |
      -----------------------------------------------------------
      |            |               |               |            |
  Type 1:      Type 2:        Type 3:         Type 4:       Type 5:
  Basic %      % Change /     Successive %    Comparison    Reverse %
  of a Number  Increase-      Change          ("A is x%     (Find original
               Decrease                       more/less     value from
                                               than B")      final value)
      |            |               |               |            |
  Type 6:      Type 7:        Type 8:         Type 9:
  Product-     Population/    Percentage      Error/
  Constancy    Depreciation   in terms of     Approximation
  (Ratio       (Repeated %    Fractions       problems in
  Fixing)      change)        (DI-linked)     DI sets

Type 1 — Basic percentage of a number:

• Core Scenario: "Find 35% of 480."
• Governing Equation: $\text{Result} = \frac{35}{100} \times 480$

Type 2 — Percentage increase/decrease:

• Core Scenario: "A number is increased/decreased by x%. Find the new value."
• Governing Equation: $\text{New Value} = \text{Original} \times \left(1 \pm \frac{x}{100}\right)$

Type 3 — Successive percentage changes:

• Core Scenario: "A quantity is increased by a%, then decreased by b%." Find net effect.
• Governing Equation: $\text{Net \%} = a + b + \frac{ab}{100}$

Type 4 — Comparison type ("A is x% more/less than B"):

• Core Scenario: "A's salary is 25% more than B's. By what % is B's salary less than A's?"
• Governing Equation: B is less than A by $\frac{x}{100+x} \times 100\%$

Type 5 — Reverse percentage (find original value):

• Core Scenario: "After a 20% increase, the value becomes 480. Find the original value."
• Governing Equation: $\text{Original} = \frac{\text{Final Value}}{1 \pm \frac{x}{100}}$

Type 6 — Product-constancy/ratio-fixing:

• Core Scenario: "Price of sugar increases by 25%. By what % must consumption be reduced to keep expenditure constant?"
• Governing Equation: $\text{Reduction \%} = \frac{x}{100+x} \times 100$

Type 7 — Population growth/depreciation:

• Core Scenario: "Population grows by r% every year for n years."
• Governing Equation:
  $$P_n = P_0\left(1 \pm \frac{r}{100}\right)^n$$

Type 8 — Percentage via fractions (rapid DI-style calculation):

• Core Scenario: Data-interpretation values requiring quick % computation.
• Governing Equation: Convert % to nearest standard fraction and multiply.

Type 9 — Error/approximation percentage:

• Core Scenario: "While measuring, an error of x% excess/deficit occurs. Find % error in the computed area."
• Governing Equation: Use successive change formula with each dimension's % error.

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Basic Percentage of a Number

MCQ 1. Find 35% of 480. (A) 168 (B) 160 (C) 172 (D) 156

Correct Answer: (A) Solution: $\dfrac{35}{100}\times480=168$.

MCQ 2. Find 12.5% of 640. (A) 80 (B) 75 (C) 85 (D) 70

Correct Answer: (A) Solution: $12.5\%=\dfrac18$. $\dfrac18\times640=80$.

MCQ 3. What percentage of 250 is 45? (A) 18% (B) 20% (C) 16% (D) 22%

Correct Answer: (A) Solution: $\dfrac{45}{250}\times100=18\%$.

Type 2 — Percentage Increase/Decrease

MCQ 1. A number 250 is increased by 24%. Find the new value. (A) 310 (B) 300 (C) 315 (D) 305

Correct Answer: (A) Solution: $250\times1.24=310$.

MCQ 2. A number 480 is decreased by 15%. Find the new value. (A) 408 (B) 400 (C) 412 (D) 415

Correct Answer: (A) Solution: $480\times0.85=408$.

MCQ 3. If a number is increased by 40%, it becomes 700. Find the increase in absolute terms. (A) 200 (B) 210 (C) 190 (D) 220

Correct Answer: (A) Solution: Original$=700/1.4=500$. Increase$=700-500=200$.

Type 3 — Successive Percentage Changes

MCQ 1. A quantity is increased by 20%, then decreased by 10%. Find the net percentage change. (A) +8% (B) +10% (C) +6% (D) +12%

Correct Answer: (A) Solution: Net$=20-10+\dfrac{20\times(-10)}{100}=10-2=8\%$.

MCQ 2. A price is increased by 25% and then decreased by 20%. Find the net effect. (A) 0% (no change) (B) +5% (C) -5% (D) +10%

Correct Answer: (A) Solution: Net$=25-20+\dfrac{25\times(-20)}{100}=5-5=0\%$.

MCQ 3. A number is increased by 30%, then again increased by 20%. Find the overall percentage increase. (A) 56% (B) 50% (C) 54% (D) 58%

Correct Answer: (A) Solution: Net$=30+20+\dfrac{30\times20}{100}=50+6=56\%$.

Type 4 — Comparison Type

MCQ 1. A's salary is 25% more than B's. By what % is B's salary less than A's? (A) 20% (B) 25% (C) 18% (D) 22%

Correct Answer: (A) Solution: $\dfrac{25}{125}\times100=20\%$.

MCQ 2. X's income is 60% more than Y's. By what percent is Y's income less than X's? (A) 37.5% (B) 40% (C) 35% (D) 42%

Correct Answer: (A) Solution: $\dfrac{60}{160}\times100=37.5\%$.

MCQ 3. P's weight is 20% less than Q's. By what percent is Q's weight more than P's? (A) 25% (B) 20% (C) 22% (D) 28%

Correct Answer: (A) Solution: $\dfrac{20}{100-20}\times100=\dfrac{20}{80}\times100=25\%$.

Type 5 — Reverse Percentage

MCQ 1. After a 20% increase, a value becomes 480. Find the original value. (A) 400 (B) 384 (C) 420 (D) 410

Correct Answer: (A) Solution: Original$=480/1.2=400$.

MCQ 2. After a 15% decrease, a quantity becomes 340. Find the original quantity. (A) 400 (B) 390 (C) 410 (D) 395

Correct Answer: (A) Solution: Original$=340/0.85=400$.

MCQ 3. A shopkeeper's sales increased by 25% to reach ₹15,000. Find the original sales. (A) ₹12,000 (B) ₹11,500 (C) ₹12,500 (D) ₹11,000

Correct Answer: (A) Solution: Original$=15000/1.25=12000$.

Type 6 — Product-Constancy/Ratio-Fixing

MCQ 1. The price of sugar increases by 25%. By what % must consumption be reduced to keep expenditure constant? (A) 20% (B) 25% (C) 18% (D) 22%

Correct Answer: (A) Solution: $\dfrac{25}{125}\times100=20\%$.

MCQ 2. The price of oil falls by 20%. By what % should consumption increase to keep expenditure the same? (A) 25% (B) 20% (C) 22% (D) 28%

Correct Answer: (A) Solution: $\dfrac{20}{80}\times100=25\%$.

MCQ 3. If the length of a rectangle is increased by 50%, by what % should the breadth be decreased to keep the area unchanged? (A) 33.33% (B) 30% (C) 35% (D) 40%

Correct Answer: (A) Solution: $\dfrac{50}{150}\times100=33.33\%$.

Type 7 — Population Growth/Depreciation

MCQ 1. A town's population grows by 10% each year. If the current population is 20,000, find the population after 2 years. (A) 24,200 (B) 24,000 (C) 24,400 (D) 23,800

Correct Answer: (A) Solution: $20000\times1.1^2=20000\times1.21=24200$.

MCQ 2. A machine depreciates by 10% each year. If its current value is ₹50,000, find its value after 3 years. (A) ₹36,450 (B) ₹35,000 (C) ₹37,000 (D) ₹36,000

Correct Answer: (A) Solution: $50000\times0.9^3=50000\times0.729=36450$.

MCQ 3. The population of a village increases by 10% in the first year and decreases by 10% in the second year. If the initial population was 22,000, find the population after 2 years. (A) 21,780 (B) 22,000 (C) 21,500 (D) 21,900

Correct Answer: (A) Solution: Net%$=10-10-1=-1\%$. Population$=22000\times0.99=21780$.

Type 8 — Percentage via Fractions

MCQ 1. Find 16.66% of 720. (A) 120 (B) 110 (C) 130 (D) 100

Correct Answer: (A) Solution: $16.66\%=1/6$. $720/6=120$.

MCQ 2. Find 6.25% of 960. (A) 60 (B) 55 (C) 65 (D) 50

Correct Answer: (A) Solution: $6.25\%=1/16$. $960/16=60$.

MCQ 3. Find 9.09% of 1100. (A) 100 (B) 95 (C) 105 (D) 90

Correct Answer: (A) Solution: $9.09\%=1/11$. $1100/11=100$.

Type 9 — Error/Approximation Problems

MCQ 1. While measuring the side of a square, an error of 4% excess is made. Find the % error in the calculated area. (A) 8.16% (B) 8% (C) 8.5% (D) 7.8%

Correct Answer: (A) Solution: Since area involves squaring, use successive change with a=b=4: $4+4+\dfrac{16}{100}=8.16\%$.

MCQ 2. In measuring the sides of a rectangle, one side is taken 5% in excess and the other 4% in deficit. Find the error % in the calculated area. (A) 0.8% excess (B) 1% excess (C) 0.5% excess (D) 1.2% excess

Correct Answer: (A) Solution: Net$=5-4+\dfrac{5\times(-4)}{100}=1-0.2=0.8\%$ excess.

MCQ 3. The side of a cube is measured with an error of 2% excess. Find the % error in the calculated volume. (A) 6.12% (B) 6% (C) 6.5% (D) 5.8%

Correct Answer: (A) Solution: Apply successive change three times for volume (side³): first two: $2+2+\dfrac4{100}=4.04\%$; then combine with third 2%: $4.04+2+\dfrac{4.04\times2}{100}=6.04+0.08=6.12\%$.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1 — Fraction Substitution Method

• Application: Any "% of a number" calculation, especially with numbers like 12.5%, 16.66%, 33.33%, 6.25%.
• Mental Model: Never multiply decimals. Instantly convert the percentage to its memorized fraction and divide.

Shortcut 2 — The $\frac{x}{100 \pm x}$ Reciprocal Trick

• Application: All Type 4 and Type 6 problems (comparison and product-constancy).
• Mental Model: Skip full algebra. Directly plug x into $\frac{x}{100+x} \times 100$ (increase-based) or $\frac{x}{100-x} \times 100$ (decrease-based).

5. Deep-Dive: Most Frequently Asked Questions

Problem 1 (SSC/RRB Standard): The population of a town increases by 10% in the first year and decreases by 10% in the second year. If the initial population was 22,000, find the population after 2 years.

Traditional Method (Slow): Year 1: 22000 × 1.10 = 24200. Year 2: 24200 × 0.90 = 21780. (~35-40 seconds.)

Exam Shortcut (Fast): Net % = $10 - 10 + \frac{10 \times (-10)}{100} = -1\%$. New Population = 22000 × 0.99 = 21,780. (Under 15 seconds.)

Problem 2 (UPSC/Banking Advanced): In an examination, a candidate secures 34% marks and fails by 30 marks. Another candidate scores 42% marks and gets 42 marks more than the passing marks. Find the maximum marks and the passing marks.

Step-by-Step Breakdown:

1. Let maximum marks = M, passing marks = P.
2. $0.34M = P - 30$
3. $0.42M = P + 42$
4. Subtract: $0.08M = 72 \Rightarrow M = 900$
5. $0.34 \times 900 = P - 30 \Rightarrow 306 = P - 30 \Rightarrow P = 336$
6. Answer: Maximum marks = 900, Passing marks = 336.

6. Chapter Checklist for Students

• I have memorized the fraction-equivalent table for 1/3 through 1/20.
• I never assume "x% more than B" and "x% less than A" are symmetric — I apply the $\frac{x}{100\pm x}$ correction.
• I apply the successive percentage change formula directly instead of two sequential multiplications.
• I always identify the correct base value before computing any percentage change.
• I can set up and solve "passing marks / maximum marks" simultaneous percentage equations within 60 seconds.