Matematica

Operations With Fractions: Solved Exercises Step by Step

How to add, subtract, multiply and divide fractions using the common denominator and the reciprocal rule, plus five solved exercises of increasing difficulty with verified answers.

Recommended for: Grade 6 · Grade 7 · Grade 8

To handle operations with fractions there are three rules: to add or subtract you need the same denominator (the least common multiple), then you work only on the numerators; to multiply you multiply numerators together and denominators together; to divide you multiply by the reciprocal of the second fraction. At the end you always simplify.

Addition and subtraction: common denominator first

You cannot add the numerators while the denominators are different. First find the least common multiple of the denominators, convert each fraction into an equivalent one with that denominator, and only then add (or subtract) the numerators. The common denominator stays the same and is never added.

Multiplication and division: more direct

Multiplication is the simplest operation: numerator times numerator, denominator times denominator, with no common denominator to look for. Division turns into a multiplication by flipping the second fraction, that is, multiplying by its reciprocal. In both cases it is worth simplifying the result.

The order of operations still applies

When an expression contains several operations, follow the same hierarchy as with whole numbers: multiplication and division first, then addition and subtraction, respecting the brackets. In the exercises below the same scheme is applied at increasing difficulty, and every answer is checked by computing the values.

Solved exercises

1. Compute: 1/4 + 2/3 base

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  1. Find the common denominator: lcm(4, 3) = 12.
  2. Rewrite the fractions: 1/4 = 3/12 and 2/3 = 8/12.
  3. Add the numerators: 3/12 + 8/12 = 11/12.
  4. 11 and 12 share no common factor, so the fraction is already in lowest terms.

Answer: 11/12

2. Compute: 5/6 − 1/4 base

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  1. Common denominator: lcm(6, 4) = 12.
  2. Rewrite: 5/6 = 10/12 and 1/4 = 3/12.
  3. Subtract the numerators: 10/12 − 3/12 = 7/12.
  4. 7 and 12 are coprime, so the result is already reduced.

Answer: 7/12

3. Compute: 3/8 × 4/9 intermedio

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  1. When multiplying, multiply numerators together and denominators together.
  2. Numerators: 3 × 4 = 12 ; denominators: 8 × 9 = 72.
  3. This gives 12/72.
  4. Simplify by dividing by the GCD 12: 12/72 = 1/6.

Answer: 1/6

4. Compute: 5/6 ÷ 10/3 intermedio

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  1. Dividing by a fraction means multiplying by its reciprocal.
  2. 5/6 ÷ 10/3 = 5/6 × 3/10.
  3. Multiply: (5 × 3)/(6 × 10) = 15/60.
  4. Simplify by dividing by the GCD 15: 15/60 = 1/4.

Answer: 1/4

5. Compute: 3/4 ÷ 1/2 − 2/3 avanzato

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  1. Division first: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2.
  2. The expression becomes 3/2 − 2/3.
  3. Common denominator: lcm(2, 3) = 6 ; 3/2 = 9/6 and 2/3 = 4/6.
  4. Subtract: 9/6 − 4/6 = 5/6.

Answer: 5/6

FAQ

How do you add two fractions with different denominators?

Rewrite them over the same denominator, usually the least common multiple of the two denominators. Convert each fraction to an equivalent one with that denominator, then add the numerators while keeping the common denominator unchanged.

Why do you multiply by the reciprocal when dividing fractions?

Dividing by a fraction is the same as multiplying by its inverse: a/b ÷ c/d = a/b × d/c. After multiplying, reduce the result to lowest terms.

When should I simplify a fraction?

At the end of every operation, whenever the numerator and denominator share a common factor. Divide both by their Greatest Common Divisor to get the fraction in lowest terms.