Percentages and Fractions - CLAT Quantitative Techniques

Percentages and Fractions: Calculation Techniques

Percentages and fractions are foundational concepts in quantitative aptitude, especially for competitive exams. Mastery of converting between percentages, fractions, and decimals is essential for efficient calculation and problem-solving. This summary focuses on explicit conversions, calculation shortcuts, and ratio-based applications as discussed.

Meaning of Percentage and Basic Conversion

A percentage represents a value out of 100. The percentage sign (%) means "divided by 100." Converting a percentage to a fraction involves dividing the number by 100 and simplifying.

• $x\% = \dfrac{x}{100}$
• Example: $20\% = \dfrac{20}{100} = \dfrac{1}{5}$

Standard Percentage-Fraction Equivalents

Certain…

Percentages and Fractions: Calculation Techniques

Percentages and fractions are foundational concepts in quantitative aptitude, especially for competitive exams. Mastery of converting between percentages, fractions, and decimals is essential for efficient calculation and problem-solving. This summary focuses on explicit conversions, calculation shortcuts, and ratio-based applications as discussed.

Meaning of Percentage and Basic Conversion

A percentage represents a value out of 100. The percentage sign (%) means "divided by 100." Converting a percentage to a fraction involves dividing the number by 100 and simplifying.

• $x\% = \dfrac{x}{100}$
• Example: $20\% = \dfrac{20}{100} = \dfrac{1}{5}$

Standard Percentage-Fraction Equivalents

Certain percentages correspond to standard fractions, which are frequently used for quick calculations. Memorizing these conversions accelerates problem-solving.

• $20\% = \dfrac{1}{5}$
• $40\% = \dfrac{2}{5}$
• $60\% = \dfrac{3}{5}$
• $25\% = \dfrac{1}{4}$
• $50\% = \dfrac{1}{2}$
• $75\% = \dfrac{3}{4}$
• $16\frac{2}{3}\% = \dfrac{1}{6}$
• $83\frac{1}{3}\% = \dfrac{5}{6}$
• $33\frac{1}{3}\% = \dfrac{1}{3}$
• $66\frac{2}{3}\% = \dfrac{2}{3}$
• $14\frac{2}{7}\% = \dfrac{1}{7}$
• $28\frac{4}{7}\% = \dfrac{2}{7}$
• $42\frac{6}{7}\% = \dfrac{3}{7}$
• $12\frac{1}{2}\% = \dfrac{1}{8}$
• $37\frac{1}{2}\% = \dfrac{3}{8}$
• $62\frac{1}{2}\% = \dfrac{5}{8}$
• $87\frac{1}{2}\% = \dfrac{7}{8}$
• $11\frac{1}{9}\% = \dfrac{1}{9}$
• $22\frac{2}{9}\% = \dfrac{2}{9}$
• $8\frac{1}{3}\% = \dfrac{1}{12}$
• $58\frac{1}{3}\% = \dfrac{7}{12}$

Decimal Approximations and Repeating Decimals

Some percentages, when converted to decimals, result in repeating decimals. Recognizing these helps avoid confusion in calculations and interpretation.

• $16\frac{2}{3}\% = 16.66\ldots\% \approx 16.67\%$ (repeating decimal, equivalent to $\dfrac{1}{6}$)
• $33\frac{1}{3}\% = 33.33\ldots\% \approx 33.34\%$ (equivalent to $\dfrac{1}{3}$)
• $66\frac{2}{3}\% = 66.66\ldots\% \approx 66.67\%$ (equivalent to $\dfrac{2}{3}$)
• $14\frac{2}{7}\% = 14.28\%$ or $14.29\%$ (equivalent to $\dfrac{1}{7}$)
• $42.84\%$ or $42.85\%$ (equivalent to $\dfrac{3}{7}$)
• $11.11\%$ (equivalent to $\dfrac{1}{9}$)
• $8.33\%$ (equivalent to $\dfrac{1}{12}$)
• $58.33\%$ (equivalent to $\dfrac{7}{12}$)

Quick Calculation Examples

Applying fraction equivalents simplifies percentage calculations, especially for "of" problems.

• $66.67\%$ of $900$:
  • $66.67\% = \dfrac{2}{3}$
  • $\dfrac{2}{3} \times 900 = 600$
• $42.85\%$ of $2100$:
  • $42.85\% = \dfrac{3}{7}$
  • $\dfrac{3}{7} \times 2100 = 900$
• $37.5\%$ of $640$:
  • $37.5\% = \dfrac{3}{8}$
  • $\dfrac{3}{8} \times 640 = 240$
• $62.5\%$ of $880$:
  • $62.5\% = \dfrac{5}{8}$
  • $\dfrac{5}{8} \times 880 = 550$
• $58.33\%$ of $1728$:
  • $58.33\% = \dfrac{7}{12}$
  • $\dfrac{7}{12} \times 1728 = 1008$

Converting Fractions to Percentages

To convert a fraction to a percentage, multiply by 100.

• $\dfrac{1}{3} \times 100 = 33\frac{1}{3}\%$
• $\dfrac{2}{3} \times 100 = 66\frac{2}{3}\%$
• $\dfrac{1}{7} \times 100 = 14\frac{2}{7}\%$
• $\dfrac{3}{7} \times 100 = 42.85\%$
• $\dfrac{1}{8} \times 100 = 12.5\%$
• $\dfrac{3}{8} \times 100 = 37.5\%$
• $\dfrac{5}{8} \times 100 = 62.5\%$
• $\dfrac{7}{8} \times 100 = 87.5\%$
• $\dfrac{1}{9} \times 100 = 11.11\%$
• $\dfrac{2}{9} \times 100 = 22.22\%$
• $\dfrac{1}{12} \times 100 = 8.33\%$
• $\dfrac{7}{12} \times 100 = 58.33\%$

Ratio Applications in Percentage Increase

When one value is a certain percent more than another, ratios can be established using fraction equivalents.

• If $A$ is $20\%$ more than $B$:
  • $20\% = \dfrac{1}{5}$
  • Ratio $A:B = 6:5$
  • If $B = 200$, then $A = 240$
• General rule: If $A$ is $x\%$ more than $B$, $A:B = (100 + x):100$

Profit and Loss: CP and SP Ratios

Profit percent is always calculated on cost price (CP). The denominator in the fraction represents CP, and the numerator represents profit.

• For $40\%$ profit:
  • $40\% = \dfrac{2}{5}$
  • CP = 5, Profit = 2, SP = 5 + 2 = 7
  • Ratio $CP:SP = 5:7$
• For $16\frac{2}{3}\%$ profit:
  • $16\frac{2}{3}\% = \dfrac{1}{6}$
  • CP = 6, Profit = 1, SP = 6 + 1 = 7
  • Ratio $CP:SP = 6:7$
• If SP is given, CP can be found by unitary method using the ratio.

Example:

• If SP = 2100 and $CP:SP = 6:7$, then $1$ part = $300$, so $CP = 1800$.

Application Example: Selling Price Calculation

When an article is bought at a certain price and sold at a profit percent, use the fraction equivalent to find the selling price.

• Rahul buys a watch for ₹1500 and sells at $20\%$ profit:
  • $20\% = \dfrac{1}{5}$
  • CP = 5, SP = 6 (since profit is 1 part)
  • $5$ parts = ₹1500, so $1$ part = ₹300, $6$ parts = ₹1800
  • Selling price = ₹1800

Common Confusions and Pitfalls

Repeating decimals and close approximations can cause confusion in exams. Always match the decimal or percentage to its fraction equivalent for accuracy.

• $16.66\ldots\%$, $16.67\%$, and $16\frac{2}{3}\%$ all represent $\dfrac{1}{6}$
• $33.33\%$, $33.34\%$, and $33\frac{1}{3}\%$ all represent $\dfrac{1}{3}$
• $66.66\%$, $66.67\%$, and $66\frac{2}{3}\%$ all represent $\dfrac{2}{3}$
• $14.28\%$, $14.29\%$, and $14\frac{2}{7}\%$ all represent $\dfrac{1}{7}$
• $42.84\%$, $42.85\%$ both represent $\dfrac{3}{7}$
• $8.33\%$ and $8\frac{1}{3}\%$ both represent $\dfrac{1}{12}$
• $58.33\%$ and $58\frac{1}{3}\%$ both represent $\dfrac{7}{12}$

Practice and Exam Strategy

Regular practice of these conversions and applications is essential for speed and accuracy. Focus on:

• Memorizing standard percentage-fraction equivalents.
• Applying fraction shortcuts in "of" calculations.
• Using ratios for profit, loss, and comparative percentage problems.
• Recognizing repeating decimals and their fraction forms to avoid confusion in options.

By mastering these explicit conversions and calculation methods, one can efficiently tackle percentage and fraction-based questions in competitive exams.