🔢 Fractions, Decimals, Percents — Problem Types

Master the common problem patterns and systematic solution approaches for rational number problems.

Basic Operation Problems

Single Operation Problems

Recognition: One arithmetic operation with fractions, decimals, or percents Template:

1. Identify the operation
2. Set up the calculation
3. Perform the operation
4. Simplify if necessary

Example: What is $\frac{3}{4} + \frac{1}{6}$?

1. Operation: Addition of fractions
2. Setup: $\frac{3}{4} + \frac{1}{6}$
3. Calculate: Find LCD (12), convert: $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$
4. Simplify: Already in simplest form

Common variations:

• Addition: $\frac{a}{b} + \frac{c}{d} = ?$
• Subtraction: $\frac{a}{b} - \frac{c}{d} = ?$
• Multiplication: $\frac{a}{b} \times \frac{c}{d} = ?$
• Division: $\frac{a}{b} \div \frac{c}{d} = ?$

Mixed Operation Problems

Recognition: Multiple operations with rational numbers Template:

1. Identify all operations
2. Apply order of operations
3. Work step by step
4. Check each step

Example: What is $\frac{2}{3} \times \frac{3}{4} + \frac{1}{2}$?

1. Operations: Multiplication, then addition
2. Order: Multiplication first
3. Steps:
   • Multiplication: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$
   • Addition: $\frac{1}{2} + \frac{1}{2} = 1$
4. Answer: 1

Conversion Problems

Fraction to Decimal

Recognition: Convert fraction to decimal form Template:

1. Divide numerator by denominator
2. Round if necessary
3. Check reasonableness

Example: Convert $\frac{5}{8}$ to a decimal

1. Divide: $5 \div 8 = 0.625$
2. Round: Not needed (exact decimal)
3. Check: $0.625 \times 8 = 5$ ✓

Decimal to Fraction

Recognition: Convert decimal to fraction form Template:

1. Write as fraction with appropriate denominator
2. Simplify
3. Check by converting back

Example: Convert 0.75 to a fraction

1. Write: $0.75 = \frac{75}{100}$
2. Simplify: $\frac{75}{100} = \frac{3}{4}$
3. Check: $\frac{3}{4} = 0.75$ ✓

Percent Conversions

Recognition: Convert between percent, decimal, and fraction Template:

1. Identify current and target forms
2. Use conversion formulas
3. Simplify if necessary

Example: Convert 37.5% to a fraction

1. Current: 37.5% (percent)
2. Target: Fraction
3. Convert: $37.5% = 0.375 = \frac{375}{1000} = \frac{3}{8}$

Comparison Problems

Comparing Fractions

Recognition: Questions asking which fraction is larger/smaller Template:

1. Choose comparison method
2. Apply method systematically
3. State conclusion clearly

Example: Which is larger: $\frac{3}{4}$ or $\frac{5}{6}$?

1. Method: Cross multiply
2. Apply: $3 \times 6 = 18$, $4 \times 5 = 20$
3. Conclusion: Since $18 < 20$, we have $\frac{3}{4} < \frac{5}{6}$

Common methods:

• Common denominator
• Cross multiplication
• Decimal conversion
• Benchmark comparison

Ordering Problems

Recognition: Arrange numbers in ascending/descending order Template:

1. Convert all to same form
2. Compare systematically
3. Arrange in requested order

Example: Order from least to greatest: $\frac{2}{3}$, 0.7, 75%

1. Convert: $\frac{2}{3} = 0.666…$, 0.7 = 0.7, 75% = 0.75
2. Compare: 0.666… < 0.7 < 0.75
3. Order: $\frac{2}{3}$, 0.7, 75%

Word Problems

Part-Whole Problems

Recognition: Find part given whole and percent, or vice versa Template:

1. Identify what’s given and what’s asked
2. Use formula: Part = Percent × Whole
3. Solve for unknown
4. Check reasonableness

Example: A store has 120 items. 25% are on sale. How many items are on sale?

1. Given: Whole = 120, Percent = 25%
2. Asked: Part (number on sale)
3. Formula: Part = 0.25 × 120 = 30
4. Answer: 30 items are on sale

Percent Change Problems

Recognition: Find percent increase or decrease Template:

1. Find the change (new - original)
2. Divide by original amount
3. Multiply by 100%
4. Determine if increase or decrease

Example: A price increased from $50 to $65. What is the percent increase?

1. Change: $65 - $50 = $15
2. Divide: $15 ÷ $50 = 0.3
3. Multiply: 0.3 × 100% = 30%
4. Answer: 30% increase

Mixture Problems

Recognition: Combine different concentrations or percents Template:

1. Set up equation with weighted average
2. Solve for unknown
3. Check answer

Example: Mix 20% acid solution with 80% acid solution to get 50% acid solution. If you use 3 liters of 20% solution, how much 80% solution do you need?

1. Setup: $0.20(3) + 0.80(x) = 0.50(3 + x)$
2. Solve: $0.6 + 0.8x = 1.5 + 0.5x$, so $0.3x = 0.9$, so $x = 3$
3. Answer: 3 liters of 80% solution

Estimation Problems

Rounding Problems

Recognition: Round to specified decimal places or significant figures Template:

1. Identify the place to round to
2. Look at the digit to the right
3. Apply rounding rules
4. Check answer

Example: Round 3.14159 to 2 decimal places

1. Place: 2 decimal places (hundredths)
2. Digit to right: 1 (in thousandths place)
3. Rule: 1 < 5, so round down
4. Answer: 3.14

Approximation Problems

Recognition: Find approximate values using estimation Template:

1. Round numbers to make calculation easier
2. Perform calculation
3. Check reasonableness

Example: Estimate $\frac{22}{7} \times 3.14$

1. Round: $\frac{22}{7} \approx 3.14$, so $3.14 \times 3.14 \approx 9.86$
2. Calculate: $3.14^2 = 9.8596$
3. Check: Close to 9.86 ✓

Pattern Problems

Repeating Decimal Patterns

Recognition: Find patterns in repeating decimals Template:

1. Identify the repeating pattern
2. Find the period length
3. Apply pattern rules

Example: What is the 100th digit after the decimal point in $\frac{1}{7}$?

1. Pattern: $\frac{1}{7} = 0.\overline{142857}$ (6-digit period)
2. Period: 6 digits
3. 100th digit: $100 \div 6 = 16$ remainder 4, so 4th digit in period = 8

Fraction Pattern Problems

Recognition: Find patterns in fraction sequences Template:

1. Identify the pattern
2. Find the general term
3. Apply to specific case

Example: What is the sum of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + …$?

1. Pattern: Each term is half the previous
2. Type: Infinite geometric series with $r = \frac{1}{2}$
3. Sum: $S = \frac{a}{1-r} = \frac{\frac{1}{2}}{1-\frac{1}{2}} = 1$

Common Mistakes and Fixes

Fraction Mistakes

Mistake: Adding denominators: $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$ Fix: Find common denominator first

Mistake: Not simplifying: $\frac{4}{8} = \frac{4}{8}$ Fix: Always simplify fractions

Mistake: Wrong order in division: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{4}{5}$ Fix: Multiply by reciprocal

Decimal Mistakes

Mistake: Misplacing decimal point: $3.4 \times 2.5 = 85$ Fix: Count decimal places in factors

Mistake: Not lining up decimals: $3.47 + 2.1 = 3.68$ Fix: Always align decimal points

Percent Mistakes

Mistake: Wrong base: 20% increase then 20% decrease = 0% Fix: Use original amount as base for both calculations

Mistake: Forgetting to convert: 25% of 80 = 25 × 80 Fix: Convert percent to decimal first

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