Fractions can only be added or subtracted directly when they are split into the same size pieces. If the denominators are different, the pieces have different sizes, so the fractions must be rewritten first. This is why the least common denominator is important.
It gives a common unit that lets you combine the numerators correctly.
Key Facts
- To add unlike fractions, first rewrite them with a common denominator.
- The least common denominator is the least common multiple of the denominators.
- Equivalent fractions have the same value, such as 1/2 = 2/4 = 3/6.
- a/b + c/d = (ad + bc)/bd, but using the LCD usually gives smaller numbers.
- a/b - c/d = (ad - bc)/bd, and the denominator stays common after rewriting.
- Always simplify the final answer by dividing the numerator and denominator by their greatest common factor.
Vocabulary
- Denominator
- The bottom number of a fraction that tells how many equal parts make one whole.
- Numerator
- The top number of a fraction that tells how many equal parts are being counted.
- Least Common Denominator
- The smallest denominator that two or more fractions can share after being rewritten as equivalent fractions.
- Equivalent Fraction
- A fraction that has the same value as another fraction even though its numerator and denominator look different.
- Simplify
- To rewrite a fraction in lowest terms by dividing the numerator and denominator by their greatest common factor.
Common Mistakes to Avoid
- Adding the denominators, such as 1/3 + 1/4 = 2/7, is wrong because the denominator names the size of the pieces, not a count to combine.
- Using a common denominator without changing the numerator is wrong because multiplying the denominator changes the size of the pieces unless the numerator is multiplied by the same factor.
- Forgetting to simplify the final answer is incomplete because fractions like 6/8 should be reduced to 3/4.
- Subtracting in the wrong order, such as changing 2/3 - 1/4 into 1/4 - 2/3, is wrong because subtraction depends on order.
Practice Questions
- 1 Find 2/3 + 1/4. Show the common denominator and simplify your answer.
- 2 Find 5/6 - 1/8. Show the equivalent fractions you use before subtracting.
- 3 Explain why 1/2 + 1/3 cannot be found by adding the numerators and denominators to get 2/5.