Equivalent Fractions
Two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), are considered equivalent if they represent the same value, even though their numerators and denominators are different. $$ \frac{a}{b} = \frac{c}{d} $$ When two fractions are equivalent, their cross-products are equal: $$ a \cdot d = b \cdot c $$
But what does it really mean for two fractions to be equivalent, and how can you check if they are?
Imagine you have a cake.
If you cut it into two equal parts and take one, you’re eating $ \frac{1}{2} $ of the cake—half of it.
Now, if you cut the same cake into four equal parts and take two slices, you’re eating $ \frac{2}{4} $ of the cake—two slices out of four.
Even though the way you divided the cake is different, the amount you eat remains the same. This is because equivalent fractions represent the same value.
$$ \frac{1}{2} = \frac{2}{4} $$
This simple example illustrates what it means for two fractions to be equivalent.
How can you determine if two fractions are equivalent?
Two fractions are equivalent if they represent the same value.
You can verify this using an important property: the cross-product.
If you have two fractions, $ \frac{a}{b} \quad \text{and} \quad \frac{c}{d} $, they are equivalent if and only if their cross-products are equal: $$ a \cdot d = b \cdot c. $$
To calculate the cross-product, multiply the numerator (a) of the first fraction by the denominator (d) of the second, and the numerator (c) of the second fraction by the denominator (b) of the first.
Example
Consider the fractions 1/2 and 2/4.
$$ \frac{1}{2} \quad \text{and} \quad \frac{2}{4} $$
To check if they are equivalent, compute the cross-products:
$$ 1 \cdot 4 = 2 \cdot 2 $$
$$ 4 = 4 $$
Since the two products are equal, you can conclude that 1/2 and 2/4 are indeed equivalent fractions.
Why is it useful to understand equivalent fractions?
Equivalent fractions are incredibly helpful when it comes to simplifying fractions.
For example, the fraction \(\frac{6}{8}\) can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 2:
$$ \frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} $$
This gives you \(\frac{3}{4}\), which is equivalent to \(\frac{6}{8}\) but uses smaller numbers, making further calculations much easier.
You can also use equivalent fractions to compare fractions.
For instance, to compare \(\frac{2}{3}\) and \(\frac{4}{6}\), you can use the cross-product to check if they are equivalent:
$$ 2 \cdot 6 = 3 \cdot 4 $$
Here, the two products are equal, so the fractions are equivalent:
$$ \frac{2}{3} = \frac{4}{6} $$
Equivalent fractions also play a key role in adding or subtracting fractions, where it’s often necessary to rewrite fractions with a common denominator.
Example. Let’s say you want to add the fractions \(\frac{1}{3}\) and \(\frac{1}{4}\): $$ \frac{1}{3} + \frac{1}{4} $$ To do this, you need a common denominator. The least common multiple (LCM) of 3 and 4 is 12, so you rewrite the fractions with 12 as the denominator:
- For \(\frac{1}{3}\), multiply both numerator and denominator by 4: $$ \frac{1}{3} = \frac{1 \cdot 4}{3 \cdot 4} = \frac{4}{12} $$
- For \(\frac{1}{4}\), multiply both numerator and denominator by 3: $$ \frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12} $$
Now replace \(\frac{1}{3}\) with \( \frac{4}{12} \) and \(\frac{1}{4}\) with \( \frac{3}{12} \):
$$ \frac{1}{3} + \frac{1}{4} $$
$$ \frac{4}{12} + \frac{3}{12} $$
With the same denominator, you can add the numerators:
$$ \frac{4}{12} + \frac{3}{12} = \frac{4+3}{12} = \frac{7}{12} $$
Thanks to equivalent fractions, you’ve successfully added the two fractions.
In summary, understanding equivalent fractions not only simplifies calculations but also provides a strong foundation for tackling more advanced concepts, such as algebraic operations with fractions, fractional equations, and decimal conversions.
