Number & Operations — Fractions

Fraction Operations

Grade 5

✓By the end of this lesson students will be able to add and subtract fractions with unlike denominators by finding equivalent fractions with a common denominator.
✓By the end of this lesson students will be able to multiply a fraction by a whole number or another fraction.
✓By the end of this lesson students will be able to interpret and compute quotients of whole numbers divided by unit fractions, and unit fractions divided by whole numbers.
✓By the end of this lesson students will be able to solve real-world problems involving addition, subtraction, multiplication, and division of fractions.

Key concepts

Adding and Subtracting Unlike Fractions

To add or subtract fractions, they must have the same denominator (a common denominator). If the denominators are different, you need to find a common multiple of the denominators. The least common multiple (LCM) is often the easiest to work with. Once you have a common denominator, rewrite each fraction as an equivalent fraction with that common denominator. Then, add or subtract the numerators and keep the common denominator. Simplify the result if possible.

Multiplying Fractions

To multiply fractions, you multiply the numerators together and multiply the denominators together. The word 'of' often indicates multiplication when working with fractions. For example, '1/2 of 1/4' means 1/2 * 1/4. Always simplify your answer to its simplest form.

(a/b) * (c/d) = (a*c) / (b*d)

Dividing Fractions (Whole Numbers and Unit Fractions)

Division of fractions in Grade 5 focuses on dividing a whole number by a unit fraction (e.g., 3 ÷ 1/4) or dividing a unit fraction by a whole number (e.g., 1/2 ÷ 5). When you divide a whole number by a unit fraction, you are finding how many of those unit fractions are in the whole number. When you divide a unit fraction by a whole number, you are splitting the unit fraction into equal parts. A common method for division is to 'multiply by the reciprocal.' The reciprocal of a fraction is found by flipping the numerator and denominator. For a whole number, write it as a fraction over 1 (e.g., 5 = 5/1), then find its reciprocal.

n ÷ (1/d) = n * d; (1/d) ÷ n = 1 / (d * n)

Key facts to remember

1Fractions must have a common denominator before you can add or subtract them.
2To find an equivalent fraction, multiply both the numerator and denominator by the same non-zero number.
3To multiply fractions, multiply the numerators and multiply the denominators.
4The word 'of' in fraction problems usually means to multiply.
5To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the unit fraction.
6To divide a unit fraction by a whole number, you can multiply the denominator of the unit fraction by the whole number, and keep 1 as the numerator.
7Always simplify your final fraction answer to its simplest form.

Worked examples

Example 1

Add 1/3 + 1/4.

IFind a common denominator for 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

IIRewrite 1/3 as an equivalent fraction with a denominator of 12: (1 * 4) / (3 * 4) = 4/12.

IIIRewrite 1/4 as an equivalent fraction with a denominator of 12: (1 * 3) / (4 * 3) = 3/12.

IVNow add the equivalent fractions: 4/12 + 3/12.

VAdd the numerators and keep the common denominator: (4 + 3) / 12 = 7/12.

Answer

7/12

Example 2

Multiply 2/5 * 3/4.

IMultiply the numerators: 2 * 3 = 6.

IIMultiply the denominators: 5 * 4 = 20.

IIIThe product is 6/20.

IVSimplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 2.

V6 ÷ 2 = 3.

VI20 ÷ 2 = 10.

Answer

3/10

You can sometimes simplify before multiplying to make the numbers smaller.

Example 3

Divide 4 ÷ 1/2.

IUnderstand the problem: How many 1/2-size pieces are in 4 whole units?

IIRewrite the whole number as a fraction: 4 = 4/1.

IIIFind the reciprocal of the divisor (1/2). The reciprocal of 1/2 is 2/1 or 2.

IVChange the division problem to a multiplication problem using the reciprocal: 4/1 * 2/1.

VMultiply the numerators: 4 * 2 = 8.

VIMultiply the denominators: 1 * 1 = 1.

VIIThe result is 8/1, which simplifies to 8.

Answer

Dividing by a unit fraction makes the number larger because you are finding how many small pieces fit into the whole.

Common mistakes

✗Adding or subtracting numerators without first finding a common denominator.
✗Adding or subtracting both numerators and denominators (e.g., 1/2 + 1/3 = 2/5).
✗Forgetting to simplify the final answer to its simplest form.
✗Confusing the rules for multiplying and dividing fractions.
✗Incorrectly finding the reciprocal of a fraction or a whole number.

Exam tips

★Always show your work step-by-step, especially when finding common denominators or simplifying.
★Read word problems carefully to determine whether to add, subtract, multiply, or divide.
★Check if your answer makes sense. For example, multiplying two proper fractions should result in a smaller fraction. Dividing a whole number by a unit fraction should result in a larger whole number.
★Practice finding the least common multiple (LCM) to make adding and subtracting fractions easier and to keep numbers manageable.

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