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Comparing fractions

How would one compare two fractions? Compared to any other number, a fraction can be greater than, lesser than or equal to it.

1. Comparing two fractions whose numerators are the same

As you know the denominator is a number which represents the number of parts the whole is divided into. If we have two fractions with the same numerator that means we take the same amount of parts, but if the denominator is greater, that means that we take equal amount of smaller parts.

This means that if the numerator of two numbers is the same, but the denominator of the first is greater than the second, then the first fraction is smaller than the other.

Example 1. Compare $\displaystyle{\frac{24}{25}}$ and $\displaystyle{\frac{24}{3}}$.

Numerators are the same, but the denominator of the first is larger than the second which leads us to:

$\displaystyle{\frac{24}{25} < \frac{24}{3}}$.


2. Comparing two fractions whose denominators are the same

Let’s first think about what this means. You have two equal wholes. And you divide them into equal parts. But from the first whole you take more parts than the second. This means that if the denominators in two fractions are equal but the numerator of first is greater than the numerator of second, than the first numerator is greater.

Example 2. Compare $\displaystyle{\frac{22}{26}}$ and $\displaystyle{\frac{2}{26}}$.

$\displaystyle{\frac{22}{26}> \frac{2}{26}}$


3. The general fraction comparison

Let’s compare the fractions $\displaystyle{\frac{a}{b}}$ and $\displaystyle{\frac{c}{d}}$. To compare them we’ll write them down as:

$\displaystyle{\frac{a}{b}}$ ${\square}$ $\displaystyle{\frac{c}{d}}$ and cross multiply them.

$a \cdot d$ $\square$ $b \cdot c$

Since all the numbers $a$, $b$, $c$ and $d$ are integers so are their products so it is easy to compare them.


Example 3. Compare  the fractions $\displaystyle{\frac{8}{5}}$ and $\displaystyle{\frac{4}{7}}$.

$\displaystyle{\frac{8}{5}}$ $\square$ $\displaystyle{\frac{4}{7}}$

$\ 7 \cdot 8$  $\square$  $5 \cdot 4$

$\ 56$ $\square$ $20$

$\ 56 > 20$

$\displaystyle{\frac{8}{5} > \frac{4}{7}}$

One thing to be careful about is where you put the sign. Remember that the numerators will always stay at the sides they are ‘given’.

Why can we cross multiply these numbers?

Remember that we said that if we multiply both the numerator and denominator with the same number their value will remain the same? It may not seem like it but this is exactly what we are doing here. For every two numbers their common multiple is their product. If we have two fractions $\displaystyle{\frac{a}{b}}$ and $\displaystyle{\frac{c}{d}}$ which we have to compare we would multiply them both with the product of their denominator $\ b \cdot d$ and get fractions we can shorten:

$\displaystyle{\frac{abd}{b}}$ $\square$ $\displaystyle{\frac{cbd}{d}}$

$ad$ $\square$ $cb$.


Example 4. Compare – $\displaystyle{\frac{2}{9}}$ and – $\displaystyle{\frac{9}{17}}$.

First compare $\displaystyle{\frac{2}{9}}$ and $\displaystyle{\frac{9}{17}}$.

$\displaystyle{\frac{2}{9}}$ $\square$ $\displaystyle{\frac{9}{17}}$

$\ 34$ $\square$ $81$

$\ 34 > 81$

$\displaystyle{\frac{9}{17} > \frac{2}{9}}$

$\displaystyle{-\frac{9}{17} < – \frac{2}{9}}$


Example 5. Compare $\displaystyle{\frac{48}{7}}$ and $\displaystyle{\frac{96}{14}}$.

$\displaystyle{\frac{48}{7}}$ $\square$ $\displaystyle{\frac{96}{14}}$

$48 \cdot 14$ $\square$ $7\cdot   96$

$672$ $\square$ $672$

$672$ $=$ $672$

$\displaystyle{\frac{48}{7}}$ and $\displaystyle{\frac{96}{14}}$ are equivalent fractions, the same could have been concluded if in the $\displaystyle{\frac{96}{14}}$ both the numerator and the denominator had been divided with $2$, which is equal to one and doesn’t change the value of the fraction.