Operations with Fractions & Decimals

TipLearning Objectives
  • Add, subtract, multiply, and divide fractions and decimals accurately.
  • Convert between fractions, decimals, and mixed numbers.
  • Perform operations with negative fractions and decimals.
  • Avoid common SAT/ACT mistakes involving improper fractions, reciprocals, and decimal placement.

Key Ideas

Fractions

A fraction represents: \[ \frac{\text{part}}{\text{whole}} \]

To work with fractions efficiently, remember:

  • Common denominators for addition and subtraction
  • Multiply straight across for multiplication
  • Multiply by the reciprocal for division

Decimals

Decimals are another representation of fractions:

  • \(0.25 = \frac{1}{4}\)
  • \(0.6 = \frac{3}{5}\)
  • \(0.875 = \frac{7}{8}\)

Convert by:

  • Decimal → fraction: write over a power of 10
  • Fraction → decimal: divide numerator by denominator
Important

When dividing fractions, do not divide the denominators.
Use the reciprocal: \[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \]

Common Problem Types

1. Adding and Subtracting Fractions

Find a common denominator.

Example:
\[ \frac{3}{4} + \frac{1}{6} \]
LCM of \(4\) and \(6\) is \(12\).
Convert: \[ \frac{9}{12} + \frac{2}{12} = \frac{11}{12} \]

2. Multiplying Fractions

Multiply numerators and denominators.

Example:
\[ \frac{2}{5} \cdot \frac{3}{4} = \frac{6}{20} = \frac{3}{10} \]

3. Dividing Fractions

Multiply by the reciprocal.

Example:
\[ \frac{5}{6} \div \frac{1}{3} = \frac{5}{6} \cdot 3 = \frac{15}{6} = \frac{5}{2} \]

4. Adding & Subtracting Decimals

Line up the decimal points.

Example:
\(3.75 + 0.08 = 3.83\)

5. Multiplying Decimals

Multiply normally, then adjust decimal places.

Example:
\(0.4 \times 0.25 = 0.10\)

6. Dividing Decimals

Move the decimal in the divisor to make it a whole number, then divide.

Example:
\(5.6 \div 0.08\)
Move decimals: \(560 \div 8 = 70\)

7. Converting Between Fractions & Decimals

Fraction → decimal: divide
\[ \frac{7}{20} = 0.35 \]

Decimal → fraction:
\[ 0.28 = \frac{28}{100} = \frac{7}{25} \]

Strategies

  • Reduce fractions to lowest terms when possible.
  • Keep negative signs in front of the whole fraction:
    \(-\frac{3}{4}\) (cleaner than \(\frac{-3}{4}\)).
  • For decimals, estimate to check reasonableness.
  • Use the reciprocal only when dividing fractions.
  • Convert mixed numbers to improper fractions for operations.

Worked Examples

Example 1

Question:
\[ \frac{5}{8} - \frac{1}{3} \]
LCM of 8 and 3 is 24:
\[ \frac{15}{24} - \frac{8}{24} = \frac{7}{24} \]

Example 2

Question:
\(0.6 \times 0.02\)
Numbers: \(6 \times 2 = 12\)
Total decimal places: 3 ⇒ \(0.012\)

Example 3

Question:
\[ \frac{2}{3} \div \frac{4}{9} \]
Reciprocal: \[ \frac{2}{3} \cdot \frac{9}{4} = \frac{18}{12} = \frac{3}{2} \]

Example 4

Question: Convert \(0.875\) to a fraction.
\[ 0.875 = \frac{875}{1000} = \frac{7}{8} \]

Common Mistakes

WarningCommon Mistakes
  • Adding/subtracting fractions without finding a common denominator.
  • Dividing fractions incorrectly instead of using the reciprocal.
  • Misplacing decimal point in multiplication or division.
  • Forgetting to convert mixed numbers into improper fractions before operations.
  • Reducing only numerator or denominator (instead of the entire fraction).

Practice Problems

  1. \(\frac{3}{5} + \frac{2}{15}\)
  2. \(\frac{7}{12} - \frac{1}{8}\)
  3. \(\frac{4}{9} \cdot \frac{3}{5}\)
  4. \(0.32 \div 0.08\)
  5. Convert \(0.56\) to a fraction.

1. \(\frac{3}{5} + \frac{2}{15}\)
Find a common denominator of \(15\).
\(\frac{3}{5} = \frac{9}{15}\)
Add the numerators:
\(\frac{9}{15} + \frac{2}{15} = \frac{11}{15}\)

Answer: \(\frac{11}{15}\)

2. \(\frac{7}{12} - \frac{1}{8}\)
Find a common denominator of \(24\).
\(\frac{7}{12} = \frac{14}{24}\) and \(\frac{1}{8} = \frac{3}{24}\)
Subtract the numerators:
\(\frac{14}{24} - \frac{3}{24} = \frac{11}{24}\)

Answer: \(\frac{11}{24}\)

3. \(\frac{4}{9} \cdot \frac{3}{5}\)
Multiply the numerators and denominators:
\(\frac{4}{9} \cdot \frac{3}{5} = \frac{12}{45}\)
Simplify:
\(\frac{12}{45} = \frac{4}{15}\)

Answer: \(\frac{4}{15}\)

4. \(0.32 \div 0.08\)
Move the decimal point two places to the right in both numbers:
\(0.32 \div 0.08 = 32 \div 8\)
Divide:
\(32 \div 8 = 4\)

Answer: \(4\)

5. Convert \(0.56\) to a fraction.
Write the decimal as a fraction over \(100\):
\(0.56 = \frac{56}{100}\)
Simplify by dividing numerator and denominator by \(4\):
\(\frac{56}{100} = \frac{14}{25}\)

Answer: \(\frac{14}{25}\)

Summary

  • Common denominators for addition/subtraction.
  • Multiply straight across; divide by using the reciprocal.
  • Line up decimals when adding/subtracting.
  • Count decimal places in multiplication/division.
  • Convert between fractions and decimals through division or powers of 10.
  • Reduce fractions to lowest terms to simplify comparisons and answers.
  • Convert mixed numbers to improper fractions before performing operations.
  • For decimal multiplication, count total decimal places from both factors.