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Quantitative Aptitude

Percentage

Mixture & Alligation10 / 51Quadrilaterals & Polygons
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Concepts (2)

Mastering percentages is crucial for SSC CGL. Focus on quick conversions, increase/decrease, and base value identification for speed and accuracy in Arithmetic and Data Interpretation.

Core Formula

Percentage means 'per hundred' or 'out of every hundred'. It's a way to express a part of a whole as a fraction of 100.

1. Basic Percentage Calculation: Percentage = (Value / Base Value) * 100%

2. Converting Fraction to Percentage: (Numerator / Denominator) * 100%

3. Converting Decimal to Percentage: Decimal Value * 100%

4. Converting Percentage to Fraction/Decimal: X% = X / 100

5. Percentage Increase/Decrease: Percentage Change = [(New Value - Original Value) / Original Value] * 100%

Worked Example 1

Q: What is 15% of 200? Convert 3/8 to a percentage.

Solution:

  • 15% of 200:
    • Using the formula: (15 / 100) * 200 = 0.15 * 200 = 30
  • Convert 3/8 to a percentage:
    • Using the formula: (3 / 8) * 100% = 0.375 * 100% = 37.5%

Worked Example 2

Q: A product's price increased by 20% to ₹240. What was its original price?

Solution:

  • Let the original price be P.
  • An increase of 20% means the new price is P + 0.20P = 1.20P.
  • We are given the new price is ₹240.
  • So, 1.20P = 240
  • P = 240 / 1.20
  • P = 2400 / 12 = 200
  • The original price was ₹200.

Shortcuts & Tricks

  1. Common Fraction-to-Percent Conversions (Memorize!):
    • 1/2 = 50%, 1/3 = 33.33% (or 33 1/3%), 1/4 = 25%, 1/5 = 20%, 1/6 = 16.66% (or 16 2/3%), 1/7 = 14.28%, 1/8 = 12.5%, 1/9 = 11.11% (or 11 1/9%), 1/10 = 10%, 1/11 = 9.09%, 1/12 = 8.33% (or 8 1/3%).
  2. Multiplier Method for Changes:
    • Increase by X%: Multiply by (1 + X/100). E.g., 20% increase -> multiply by 1.20.
    • Decrease by X%: Multiply by (1 - X/100). E.g., 20% decrease -> multiply by 0.80.
  3. Successive Percentage Changes (Net Change):
    • If a value changes by A% and then by B%, the net percentage change is A + B + (AB/100). (Use positive for increase, negative for decrease).

Common Mistakes

  1. Incorrect Base Value: Always identify the correct 'original' or 'base' value for percentage calculations. E.g., Profit % is always on Cost Price, not Selling Price, unless specified.
  2. Direct Addition/Subtraction of Percentages: You cannot directly add or subtract percentages of different base values or successive changes. E.g., a 20% increase followed by a 10% decrease is NOT a net 10% increase. Use the multiplier method or successive change formula.
  3. Calculation Errors with Decimals: Be careful when converting percentages to decimals (e.g., 12.5% is 0.125, not 1.25).

Derivation (brief)

The term 'percent' originates from the Latin 'per centum', meaning 'by the hundred'. It's a convenient way to standardize comparisons of quantities by expressing them as parts of a common whole (100). For example, saying '20 out of 50' is less intuitive than '40 out of 100' or '40%'. This standardization simplifies understanding and comparison across different total quantities.

Advanced Examples

Q: A's salary is 25% more than B's. By what percentage is B's salary less than A's?

Solution (Ratio Method):

  • 25% = 1/4. A's salary is 1/4 more than B's.
  • If B's salary is 4 units, A's salary is 4 + 1 = 5 units.
  • B's salary is less than A's by 1 unit.
  • Percentage less = (Difference / A's Salary) * 100%
  • = (1 / 5) * 100% = 20%

Q: The population of a town increased by 10% in the first year and decreased by 20% in the second year. If the current population is 44,000, what was the population 2 years ago?

Solution (Multiplier Method):

  • Let the original population be P.
  • After 1st year: P * (1 + 10/100) = P * 1.10
  • After 2nd year: (P * 1.10) * (1 - 20/100) = P * 1.10 * 0.80
  • P * 0.88 = 44,000
  • P = 44,000 / 0.88
  • P = 4400000 / 88 = 50,000
  • The population 2 years ago was 50,000.

Variation Types

Percentages are foundational to many SSC CGL topics:

  • Profit & Loss: Calculating profit/loss percentage, discount percentage.
  • Simple & Compound Interest: Interest rates are expressed as percentages.
  • Data Interpretation (DI): Analyzing data presented in pie charts, bar graphs, tables often requires calculating percentages of totals or percentage changes.
  • Ratio & Proportion: Percentages can often be converted to ratios for easier calculation.

Time-Saving Methods

  1. Fraction Equivalents for Changes: Instead of (1 + X/100), use fraction equivalents. E.g., 20% increase is 1/5 increase. So, Original : New = 5 : (5+1) = 5:6. This is faster for complex problems.
  2. Mental Math for Common Percentages: Practice calculating 10%, 1%, 50%, 25% mentally. E.g., to find 35% of 800: (10% of 800) * 3 + (5% of 800) = (80 * 3) + (40) = 240 + 40 = 280.
  3. Cross-Multiplication for 'X% of Y is Z' problems: If X% of Y = Z, then (X/100) * Y = Z. This can be rearranged quickly to find any unknown.
  4. Percentage Point vs. Percentage Change: Understand the difference. If a rate goes from 10% to 12%, it's a 2 percentage point increase, but a (2/10)*100 = 20% percentage change.
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Successive percentage changes aren't additive. Use `x + y + xy/100` for two changes or the multiplier method for multiple changes to find the overall net percentage effect quickly and accurately.

Core Formula

For two successive percentage changes, x% and y%, the net percentage change is given by: Net Change = (x + y + xy/100)%

Remember to use positive values for increases and negative values for decreases. This formula is a cornerstone for speed in SSC CGL exams.

Alternatively, the Multiplier Method is highly versatile, especially for multiple changes or when the initial value is known: Final Value = Original Value * (1 ± P1/100) * (1 ± P2/100) * ... Here, P1, P2, etc., are the successive percentages. Use '+' for an increase and '-' for a decrease.

Worked Example 1 (Using x + y + xy/100)

Question: The price of an article is increased by 20% and then decreased by 20%. The net change in price is:

Solution: Here, x = +20% (increase) and y = -20% (decrease). Using the formula: Net Change = (x + y + xy/100)% = (20 + (-20) + (20 * -20)/100)% = (0 - 400/100)% = -4%

So, there is a net decrease of 4% in the price of the article.

Worked Example 2 (Using Multiplier Method)

Question: The population of a town increases by 5% in the first year and decreases by 5% in the second year. If the present population is 20,000, what will it be after 2 years?

Solution:

  1. First year increase: 5%. The multiplier is (1 + 5/100) = 1.05.
  2. Second year decrease: 5%. The multiplier is (1 - 5/100) = 0.95.

Using the Multiplier Method: Population after 2 years = Present Population * Multiplier1 * Multiplier2 = 20,000 * 1.05 * 0.95 = 20,000 * (105/100) * (95/100) = 20,000 * (21/20) * (19/20) = 20,000 * (399/400) = 50 * 399 = 19,950

Alternatively, using the x + y + xy/100 formula for net percentage change: x = +5, y = -5 Net Change = (5 - 5 + (5 * -5)/100)% = (-25/100)% = -0.25% Final population = 20,000 * (1 - 0.25/100) = 20,000 * 0.9975 = 19,950.

Shortcuts & Tricks

  1. For two changes of x% increase and x% decrease: The net change is always a decrease of (x^2)/100 %. For example, +20% and -20% results in a (20^2)/100 = 400/100 = 4% decrease. This is a crucial time-saver.
  2. Fractional Equivalents: Convert common percentages to fractions for quicker calculations. For example, 20% = 1/5. An increase of 20% means the new value is (1 + 1/5) = 6/5 of the original. A decrease of 20% means it's (1 - 1/5) = 4/5 of the original. This is especially useful for three or more changes.
    • Example: Price increases by 20% (5->6) then decreases by 25% (4->3).
    • Overall ratio: (5 * 4) -> (6 * 3) = 20 -> 18. Net change = (18-20)/20 * 100% = -10%.
  3. Multiplier Method for 3+ changes: When dealing with three or more successive changes, the multiplier method is generally more straightforward and less prone to errors than chaining the x + y + xy/100 formula.

Common Mistakes

  1. Simple Addition/Subtraction: The most common error is assuming that a +20% and -20% change cancels out to 0%. This is incorrect because the second percentage is applied to the new base, not the original.
  2. Sign Errors: Forgetting to use negative signs for decreases (discounts, reductions) in the x + y + xy/100 formula leads to incorrect results.
  3. Calculation Precision: Rounding off intermediate steps, especially with decimals or fractions, can lead to slightly off answers, which are often provided as distractors in options.

Derivation (brief)

Let the original value be V. If V increases by x%, the new value V1 = V * (1 + x/100). If V1 then changes by y% (increase or decrease), the final value V2 = V1 * (1 + y/100) (using + for increase, - for decrease). Substituting V1 into the second equation: V2 = V * (1 + x/100) * (1 + y/100). To find the net percentage change, we calculate ((V2 - V) / V) * 100%. = ((V * (1 + x/100) * (1 + y/100) - V) / V) * 100% = ((1 + x/100) * (1 + y/100) - 1) * 100% = (1 + y/100 + x/100 + xy/10000 - 1) * 100% = (x/100 + y/100 + xy/10000) * 100% = (x + y + xy/100)%. This derivation validates the formula.

Advanced Examples

Question: A shopkeeper first increases the price of an item by 10%, then gives a discount of 20%, and finally offers an additional festive discount of 5%. What is the overall percentage change in the price?

Solution (Multiplier Method - Recommended for 3+ changes): Let the original price be P.

  1. Increase by 10%: Multiplier = (1 + 10/100) = 1.1
  2. Discount of 20%: Multiplier = (1 - 20/100) = 0.8
  3. Additional discount of 5%: Multiplier = (1 - 5/100) = 0.95

Final Price = P * 1.1 * 0.8 * 0.95 = P * 0.88 * 0.95 = P * 0.836

Net change = (Final Price - Original Price) / Original Price * 100% = (P * 0.836 - P) / P * 100% = (0.836 - 1) * 100% = -0.164 * 100% = -16.4%

Thus, there is an overall decrease of 16.4% in the price.

Variation Types

Successive percentage change concepts appear in various forms:

  • Successive Discounts: Often phrased as "two successive discounts of 10% and 20%", which means -10% and -20% changes.
  • Population Growth/Decline: As seen in examples, population can increase or decrease over consecutive periods.
  • Area/Volume Changes: If the sides of a rectangle change by x% and y%, its area changes by x + y + xy/100. For a square, if the side changes by x%, the area changes by 2x + x^2/100. For 3D objects (e.g., cuboid), if length, breadth, and height change by x, y, and z percent respectively, the volume change is best found using the multiplier method: (1+x/100)(1+y/100)(1+z/100) - 1.
  • Profit/Loss on successive transactions: A product sold at a profit, then resold at a loss, etc.

Time-Saving Methods

  • Master Fractional Equivalents: For percentages like 10% (1/10), 20% (1/5), 25% (1/4), 33.33% (1/3), 16.66% (1/6), etc., converting to fractions allows for quick multiplication of ratios. This is often the fastest method for complex problems.
  • Prioritize x + y + xy/100 for two changes: For any two successive changes, this formula is unbeatable for speed. Practice it until it's second nature.
  • Systematic Multiplier Method for 3+ changes: While you can chain the x + y + xy/100 formula, the multiplier method is less error-prone and often faster for three or more changes, especially with non-integer percentages.
  • Look for x, -x pattern: Instantly identify (x^2)/100 decrease for x% increase followed by x% decrease.
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