Multiplying and dividing fractions are core skills for working with parts of a whole, rates, scale factors, and measurements. Multiplication of fractions tells you what part of a part you have, such as one half of three fourths. Division of fractions answers how many groups fit or how large each group is when the amounts are fractional.
These skills matter in cooking, construction, science labs, maps, and algebra.
Key Facts
- Multiply fractions straight across: a/b × c/d = ac/bd.
- Divide fractions by multiplying by the reciprocal: a/b ÷ c/d = a/b × d/c.
- The reciprocal of c/d is d/c, as long as c is not 0.
- Simplify before or after multiplying by dividing common factors from the numerator and denominator.
- A mixed number must be changed to an improper fraction before multiplying or dividing.
- A fraction word problem with the word of often means multiplication, such as 2/3 of 12 = 2/3 × 12.
Vocabulary
- Numerator
- The numerator is the top number of a fraction and shows how many parts are being counted.
- Denominator
- The denominator is the bottom number of a fraction and shows how many equal parts make one whole.
- Reciprocal
- A reciprocal is a fraction flipped upside down, so the reciprocal of 3/5 is 5/3.
- Simplify
- To simplify a fraction means to divide the numerator and denominator by common factors until no common factor greater than 1 remains.
- Improper Fraction
- An improper fraction is a fraction whose numerator is greater than or equal to its denominator.
Common Mistakes to Avoid
- Adding across instead of multiplying across is wrong because 2/3 × 1/4 means 2 × 1 over 3 × 4, not 3/7.
- Flipping the first fraction in a division problem is wrong because only the divisor, the fraction after the division sign, becomes its reciprocal.
- Forgetting to convert mixed numbers is wrong because 1 1/2 × 2/3 cannot be multiplied correctly until 1 1/2 becomes 3/2.
- Simplifying only the numerators or only the denominators is wrong because simplification must divide a numerator and a denominator by the same common factor.
Practice Questions
- 1 Compute and simplify: 3/5 × 10/21.
- 2 Compute and simplify: 4/7 ÷ 2/3.
- 3 A recipe uses 3/4 cup of flour for one batch. Explain whether finding the flour needed for 2/3 of a batch uses multiplication or division, then find the amount.