Equivalent Fractions

Equivalent fractions are the fractions having the same value. Same fraction can be represented in many ways. Let us take the following example.

In picture (i) the shaded part is represented by fraction \(\frac{1}{2}\). 

The shaded part in picture (ii) is represented by fraction \(\frac{2}{4}\). In picture (iii) the same part is represented by fraction \(\frac{4}{8}\). SO, the fraction represented by these shaded portions are equal. Such fractions are called equivalent fractions.

We say that \(\frac{1}{2}\) = \(\frac{2}{4}\) = \(\frac{4}{8}\)

Hence, for a given fraction there can be many equivalent fractions.

Making Equivalent Fractions:

We have seen in the above example that \(\frac{1}{2}\), \(\frac{2}{4}\) and \(\frac{4}{8}\) are equivalent fractions.

Therefore, \(\frac{1}{2}\) can be written as \(\frac{1}{2}\) = \(\frac{1 × 2}{2 × 2}\) = \(\frac{1 × 3}{2 × 3}\) = \(\frac{1 × 4}{2 × 4}\) and so on.

Hence, an equivalent fraction of any given fraction can be obtained by multiplying its numerator and denominator by the same number.

Same way, when the numerator and denominator of a fraction are divided by the same number, we get its equivalent fractions.

\(\frac{1}{2}\) = \(\frac{1 ÷ 1}{2 ÷ 1}\) = \(\frac{2}{4}\) = \(\frac{2 ÷ 2}{4 ÷ 2}\) = \(\frac{3}{6}\) = \(\frac{3 ÷ 3}{6 ÷ 3}\) 

We have,

²/₄ = ^{(1 × 2)}/_{(2 × 2)}

³/₆ = ^{(1 × 3)}/_{(2 × 3)}

⁴/₈ = ^{(1 × 4)}/_{(2 × 4)}

We observe that ²/₄, ³/₆ and ⁴/₈ are obtained by multiplying the numerator and denominator of ¹/₂ by 2, 3 and 4 respectively.

Thus, an equivalent fraction of a given fraction can be obtained by multiplying its numerator and denominator by the same number (other than zero).

²/₄ = ^{(2÷ 2)}/_{(4 ÷ 2)} = ¹/₂

³/₆ = ^{(3÷ 3)}/_{(6 ÷ 3)} = ¹/₂

⁴/₈ = ^{(4 ÷ 4)}/_{(8 ÷ 4)} = ¹/₂

We observe that if we divide the numerators and denominators of ²/₄, ³/₆ and ⁴/₈ each by their common factor 2, we get an equivalent fraction ¹/₂.

Thus, an equivalent fraction of a given fraction can be obtained by dividing its numerator and denominator by their common factor (other than 1), if ant.

Note:

(i) Multiplying its numerator (top) and denominator (bottom) by the same number (other than 0).

(ii) Dividing its numerator (top) and denominator (bottom) by their common factor (other than 1).


For Example:

1. Write three equivalent fraction of ³/₅.

Equivalent fractions of ³/₅ are:

^{(3 × 2)}/_{(5× 2)} = ⁶/₁₀,

^{(3 × 3)}/_{(5 × 3)} = ⁹/₁₅,

^{(3 × 4)}/_{(5 × 4)} = ¹²/₂₀

Therefore, equivalent fractions of ³/₅ are ⁶/₁₀, ⁹/₁₅ and ¹²/₂₀.

2. Write next three equivalent fraction of \(\frac{2}{3}\).

We multiply the numerator and the denominator by 2.

We get, \(\frac{2 × 2}{3 × 2}\) = \(\frac{4}{6}\)

Next, we multiply the numerator and the denominator by 3. We get

\(\frac{2 × 3}{3 × 3}\) = \(\frac{6}{9}\).

Next, we multiply the numerator and the denominator by 4. We get

\(\frac{2 × 4}{3 × 4}\) = \(\frac{8}{12}\).

Therefore, equivalent fractions of \(\frac{2}{3}\) are \(\frac{4}{6}\), \(\frac{6}{9}\) and \(\frac{8}{12}\).

3. Write three equivalent fraction of ¹/₄.

Equivalent fractions of ¹/₄ are:

^{(1× 2)}/_{(4× 2)} = ²/₈,

^{(1 × 3)}/_{(4 × 3)} = ³/₁₂,

^{(1× 4)}/_{(4× 4)} = ⁴/₁₆

Therefore, equivalent fractions of ¹/₄ are ²/₈, ³/₁₂ and ⁴/₁₆.



4. Write three equivalent fraction of ²/₁₅.

Equivalent fractions of ²/₁₅ are:

^{(2× 2)}/_{(15 × 2)} = ⁴/₃₀,

^{(2 × 3)}/_{(15 × 3)} = ⁶/₄₅,

^{(2× 4)}/_{(15 × 4)} = ⁸/₆₀

Therefore, equivalent fractions of ²/₁₅ are ⁴/₃₀, ⁶/₄₅ and ⁸/₆₀.



5. Write three equivalent fraction of ³/₁₀.

Equivalent fractions of ³/₁₀ are:

^{(3× 2)}/_{(10× 2)} = ⁶/₂₀,

^{(3 × 3)}/_{(10 × 3)} = ⁹/₃₀,

^{(3× 4)}/_{(10× 4)} = ¹²/₄₀

Therefore, equivalent fractions of ³/₁₀ are ⁶/₂₀, ⁹/₃₀ and ¹²/₄₀.

● Fraction

Representations of Fractions on a Number Line

Fraction as Division

Types of Fractions

Conversion of Mixed Fractions into Improper Fractions

Conversion of Improper Fractions into Mixed Fractions

Equivalent Fractions

Interesting Fact about Equivalent Fractions

Fractions in Lowest Terms

Like and Unlike Fractions

Comparing Like Fractions

Comparing Unlike Fractions

Addition and Subtraction of Like Fractions

Addition and Subtraction of Unlike Fractions

Inserting a Fraction between Two Given Fractions

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