Operations with Fractions & Decimals
- 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
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
- 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
- \(\frac{3}{5} + \frac{2}{15}\)
- \(\frac{7}{12} - \frac{1}{8}\)
- \(\frac{4}{9} \cdot \frac{3}{5}\)
- \(0.32 \div 0.08\)
- Convert \(0.56\) to a fraction.
- \(\frac{11}{15}\)
- \(\frac{13}{24}\)
- \(\frac{12}{45} = \frac{4}{15}\)
- \(4\)
- \(\frac{56}{100} = \frac{14}{25}\)
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.