Equivalent Fractions

Two fractions are said to be equivalent $$ \frac{a}{b} \sim \frac{c}{d} $$ if the product of the numerator of the first fraction and the denominator of the second equals the product of the denominator of the first fraction and the numerator of the second. $$ a \cdot d = b \cdot c $$

In other words, two fractions are equivalent if they represent the same rational number.

$$ \frac{a}{b} = \frac{c}{d} $$

The equivalence between two fractions is denoted by the tilde symbol

$$ \frac{a}{b} \sim \frac{c}{d} $$

Example. The following fractions are equivalent $$ \frac{10}{5} \sim \frac{8}{4} $$

How to determine whether two fractions are equivalent

To determine whether two fractions are equivalent, one can proceed in two alternative ways

• Compare their numerical values
  Evaluate each fraction and verify whether they are equal. $$ \frac{a}{b} = \frac{c}{d} $$

  Example. The following fractions are equivalent $$ \frac{10}{5} \sim \frac{8}{4} $$ because they have the same value $$ 10:5 = 8:4 $$ $$ 2 = 2 $$

• Use cross multiplication
  Compute the product of the numerator of the first fraction and the denominator of the second, and compare it with the product of the denominator of the first fraction and the numerator of the second. $$ a \cdot d = b \cdot c $$

  Example. The following fractions are equivalent $$ \frac{10}{5} \sim \frac{8}{4} $$ because the cross products are equal $$ 10 \cdot 4 = 8 \cdot 5 $$ $$ 40 = 40 $$

Which method is preferable?

In general, cross multiplication is preferable because multiplication is typically faster and can often be carried out mentally.

Division, by contrast, is usually more involved and requires additional computational steps.

A practical example

Example 1

Consider the following fractions. We want to determine whether they are equivalent

$$ \frac{9}{2} $$

$$ \frac{27}{6} $$

The value of the first fraction, 9:2, can be computed mentally, 9:2 = 4.5, whereas the value of the second fraction, 27/6, is less immediate.

Therefore, we verify equivalence using cross multiplication

$$ 9 \cdot 6 = 27 \cdot 2 $$

Using basic multiplication facts, we compute both products

$$ 54 = 54 $$

The identity holds.

Therefore, the two fractions are equivalent.

$$ \frac{9}{2} \sim \frac{27}{6} $$

Note. In this simple case, the equivalence can also be recognized directly. Both 27 and 6 are obtained by multiplying 9 and 2 by 3. $$ \frac{27}{6} = \frac{9 \cdot 3}{2 \cdot 3} = \frac{9}{2} $$

Example 2

Now determine whether the following fractions are equivalent

$$ \frac{27}{7} $$

$$ \frac{23}{6} $$

We apply cross multiplication

$$ 27 \cdot 6 = 23 \cdot 7 $$

Note. These products can be computed mentally by decomposing them into sums of simpler products. $$ 27 \cdot 6 = 23 \cdot 7 $$ $$ 20 \cdot 6 + 7 \cdot 6 = 20 \cdot 7 + 3 \cdot 7 $$ $$ 120 + 42 = 140 + 21 $$ $$ 162 = 161 $$

$$ 162 = 161 $$

In this case, the identity does not hold.

Therefore, the two fractions are not equivalent.

Equivalence classes

All fractions that represent the same rational number belong to the same equivalence class.

Each equivalence class is represented by the fraction in lowest terms.

For example, the following equivalent fractions belong to the same equivalence class

$$ \frac{1}{2} = \{ \ \frac{1}{2} \ , \ \frac{2}{4} \ , \ \frac{4}{8} \ , \ \frac{8}{16} \ , \ \dots \} $$

The representative of this class is the fraction in lowest terms.

The following fractions belong to another equivalence class

$$ \frac{1}{3} = \{ \ \frac{1}{3} \ , \ \frac{2}{6} \ , \ \frac{4}{12} \ , \ \frac{8}{24} \ , \ \dots \} $$

The representative of this class is the fraction in lowest terms.

Percentages

An equivalent fraction with denominator equal to 100 is called a percentage.

For example, the following fraction is a percentage because its denominator is 100

$$ \frac{25}{100} $$

Percentages are often written by placing the numerator followed by the % symbol

$$ \frac{25}{100} = 25 \% $$

Note. If the denominator is 1000, the fraction is called a per mille. For example $$ \frac{25}{1000} = 25 ‰ $$ In this case, the symbol ‰ is used.

Example 3

The following fractions belong to the same equivalence class as the previous one, but they are not percentages because their denominator is not 100

$$ \frac{1}{4} \ , \ \frac{2}{8} \ , \ \frac{3}{12} \ , \ \frac{4}{16} \ , \ \frac{5}{20} \ , \ \frac{6}{24} \ , \ \dots $$

And so on.