🔢 Fractions, Decimals, Percents — Topics

Master the core topics and techniques for working with rational numbers in MATHCOUNTS.

Fraction Operations

Adding and Subtracting Fractions

Same denominator: Add/subtract numerators, keep denominator

  • $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$
  • $\frac{5}{8} - \frac{3}{8} = \frac{2}{8} = \frac{1}{4}$

Different denominators: Find common denominator, then add/subtract

  • $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$
  • $\frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}$

Mixed numbers: Convert to improper fractions first

  • $2\frac{1}{3} + 1\frac{1}{4} = \frac{7}{3} + \frac{5}{4} = \frac{28}{12} + \frac{15}{12} = \frac{43}{12} = 3\frac{7}{12}$

Pitfall: Forgetting to find common denominator Fix: Always find LCD before adding/subtracting

Multiplying Fractions

Basic rule: Multiply numerators, multiply denominators

  • $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

Mixed numbers: Convert to improper fractions first

  • $2\frac{1}{3} \times 1\frac{1}{4} = \frac{7}{3} \times \frac{5}{4} = \frac{35}{12} = 2\frac{11}{12}$

Canceling: Cancel common factors before multiplying

  • $\frac{6}{8} \times \frac{4}{9} = \frac{6 \times 4}{8 \times 9} = \frac{24}{72} = \frac{1}{3}$

Pitfall: Not simplifying before multiplying Fix: Cancel common factors first

Dividing Fractions

Basic rule: Multiply by reciprocal

  • $\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$

Mixed numbers: Convert to improper fractions first

  • $2\frac{1}{3} \div 1\frac{1}{4} = \frac{7}{3} \div \frac{5}{4} = \frac{7}{3} \times \frac{4}{5} = \frac{28}{15} = 1\frac{13}{15}$

Pitfall: Forgetting to flip the second fraction Fix: Always multiply by reciprocal

Decimal Operations

Adding and Subtracting Decimals

Method: Line up decimal points, then add/subtract

  • $3.47 + 2.15 = 5.62$
  • $7.89 - 3.24 = 4.65$

Pitfall: Not lining up decimal points Fix: Always align decimal points vertically

Multiplying Decimals

Method: Multiply as whole numbers, then count decimal places

  • $3.4 \times 2.5 = 8.5$ (1 + 1 = 2 decimal places)
  • $0.7 \times 0.3 = 0.21$ (1 + 1 = 2 decimal places)

Pitfall: Misplacing decimal point Fix: Count total decimal places in factors

Dividing Decimals

Method: Move decimal point in divisor and dividend

  • $3.6 \div 0.4 = 36 \div 4 = 9$
  • $0.84 \div 0.07 = 84 \div 7 = 12$

Pitfall: Forgetting to move decimal point Fix: Make divisor a whole number first

Percent Calculations

Finding Percent of a Number

Method: Convert percent to decimal, then multiply

  • $25%$ of 80 = $0.25 \times 80 = 20$
  • $15%$ of 120 = $0.15 \times 120 = 18$

Mental math shortcuts:

  • $10%$ of $n$ = $n \div 10$
  • $5%$ of $n$ = $10%$ of $n \div 2$
  • $1%$ of $n$ = $n \div 100$

Pitfall: Forgetting to convert percent to decimal Fix: Always divide by 100 first

Finding What Percent One Number Is of Another

Method: Divide first number by second, then multiply by 100

  • What percent is 15 of 60? $\frac{15}{60} \times 100% = 25%$
  • What percent is 8 of 32? $\frac{8}{32} \times 100% = 25%$

Pitfall: Dividing in wrong order Fix: Always divide smaller by larger for percents less than 100%

Percent Increase and Decrease

Method: Find change, divide by original, multiply by 100

  • Increase from 80 to 100: $\frac{100-80}{80} \times 100% = 25%$
  • Decrease from 120 to 90: $\frac{120-90}{120} \times 100% = 25%$

Pitfall: Using wrong base for calculation Fix: Always use original amount as base

Conversions

Fraction to Decimal

Method 1: Divide numerator by denominator

  • $\frac{3}{4} = 3 \div 4 = 0.75$

Method 2: Use equivalent fractions

  • $\frac{3}{4} = \frac{75}{100} = 0.75$

Common fractions to memorize:

  • $\frac{1}{2} = 0.5$, $\frac{1}{3} = 0.333…$, $\frac{1}{4} = 0.25$
  • $\frac{1}{5} = 0.2$, $\frac{1}{8} = 0.125$, $\frac{1}{10} = 0.1$

Decimal to Fraction

Method: Write as fraction with appropriate denominator, then simplify

  • $0.75 = \frac{75}{100} = \frac{3}{4}$
  • $0.6 = \frac{6}{10} = \frac{3}{5}$

Fraction to Percent

Method: Convert to decimal, then multiply by 100

  • $\frac{3}{4} = 0.75 = 75%$
  • $\frac{2}{5} = 0.4 = 40%$

Percent to Fraction

Method: Write as fraction with denominator 100, then simplify

  • $75% = \frac{75}{100} = \frac{3}{4}$
  • $40% = \frac{40}{100} = \frac{2}{5}$

Comparing and Ordering

Comparing Fractions

Method 1: Find common denominator

  • $\frac{3}{4}$ vs $\frac{5}{6}$: $\frac{9}{12}$ vs $\frac{10}{12}$, so $\frac{5}{6} > \frac{3}{4}$

Method 2: Convert to decimals

  • $\frac{3}{4} = 0.75$, $\frac{5}{6} = 0.833…$, so $\frac{5}{6} > \frac{3}{4}$

Method 3: Cross multiply

  • $\frac{3}{4}$ vs $\frac{5}{6}$: $3 \times 6 = 18$, $4 \times 5 = 20$, so $\frac{5}{6} > \frac{3}{4}$

Comparing Decimals

Method: Compare digit by digit from left

  • $0.75$ vs $0.8$: $0.8 > 0.75$
  • $0.123$ vs $0.124$: $0.124 > 0.123$

Comparing Percents

Method: Convert to same form, then compare

  • $75%$ vs $0.8$: $75% = 0.75$, so $0.8 > 75%$

Word Problems

Part-Whole Problems

Template: Part = Percent × Whole

  • What is 25% of 80? Part = $0.25 \times 80 = 20$
  • 15 is what percent of 60? $15 = \frac{x}{100} \times 60$, so $x = 25%$

Percent Change Problems

Template: Percent Change = $\frac{\text{Change}}{\text{Original}} \times 100%$

  • Price increased from $50 to $60: $\frac{60-50}{50} \times 100% = 20%$
  • Population decreased from 1000 to 800: $\frac{1000-800}{1000} \times 100% = 20%$

Mixture Problems

Template: Use weighted averages

  • Mix 20% solution with 80% solution to get 50% solution
  • Let $x$ be amount of 20% solution, $y$ be amount of 80% solution
  • $0.20x + 0.80y = 0.50(x + y)$

Common Mistakes

Fraction Mistakes

  • Adding denominators: $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$ (wrong)
  • Not simplifying: $\frac{4}{8} = \frac{4}{8}$ (should be $\frac{1}{2}$)
  • Wrong order in division: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}$ (correct)

Decimal Mistakes

  • Misplacing decimal point: $3.4 \times 2.5 = 85$ (should be 8.5)
  • Not lining up decimals: $3.47 + 2.1 = 3.68$ (should be 5.57)

Percent Mistakes

  • Wrong base: 20% increase from 100 to 120, then 20% decrease = 96 (not 100)
  • Forgetting to convert: 25% of 80 = 25 × 80 = 2000 (should be 20)

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