Comparing Fractions

To compare two fractions we first need to make same denominators using equivalence of fractions. After this denominators are already equal, so we compare numerators just like we compared integers.

If we compare mixed numbers then we compare integer parts. If integer parts are equal then we need to compare fractional parts.

Example 1. Compare 56\frac{{5}}{{6}} and 156\frac{{15}}{{6}}.

Denominators are already equal, so we compare numerators: since 5<15{5}<{15} then 56<156\frac{{5}}{{6}}<\frac{{15}}{{6}}.

Answer: 56<156\frac{{5}}{{6}}<\frac{{15}}{{6}}.

Now, let's do an example with unlike denominators.

Example 2. Compare 23\frac{{2}}{{3}} and 57\frac{{5}}{{7}}.

Make same denominators: 23=2737=1421\frac{{2}}{{3}}=\frac{{{2}\cdot{7}}}{{{3}\cdot{7}}}=\frac{{14}}{{21}} and 57=5373=1521\frac{{5}}{{7}}=\frac{{{5}\cdot{3}}}{{{7}\cdot{3}}}=\frac{{15}}{{21}}.

So, we compare 1421\frac{{14}}{{21}} and 1521\frac{{15}}{{21}}.

Denominators are equal, so we compare numerators: since 14<15{14}<{15} then 23<57\frac{{2}}{{3}}<\frac{{5}}{{7}}.

Answer: 23<57\frac{{2}}{{3}}<\frac{{5}}{{7}}.

Next example.

Example 3. Compare 911-\frac{{9}}{{11}} and 13\frac{{1}}{{3}}.

We don't need to make denominators equal, because negative number is always less than positive.

Answer: 911<13-\frac{{9}}{{11}}<\frac{{1}}{{3}}.

Now, do a couple of exercises.

Exercise 1. Compare 73\frac{{7}}{{3}} and 95\frac{{9}}{{5}}.

Answer: 73>95\frac{{7}}{{3}}>\frac{{9}}{{5}}.

Next exercise.

Exercise 2. Compare 257{2}\frac{{5}}{{7}} and 578{5}\frac{{7}}{{8}}.

Answer: 257<578{2}\frac{{5}}{{7}}<{5}\frac{{7}}{{8}}. Hint: compare integer parts: 2<5{2}<{5}.

Next exercise.

Exercise 3. Compare 223-{2}\frac{{2}}{{3}} and 215-{2}\frac{{1}}{{5}}.

Ignore minuses.

Integer parts are equal, so we compare fractional parts: 23\frac{{2}}{{3}} and 15\frac{{1}}{{5}}.

Since 23>15\frac{{2}}{{3}}>\frac{{1}}{{5}} then 223>215{2}\frac{{2}}{{3}}>{2}\frac{{1}}{{5}}.

Add minus sign and change direction of inequality: 223<215-{2}\frac{{2}}{{3}}<-{2}\frac{{1}}{{5}}.

Answer: 223<215-{2}\frac{{2}}{{3}}<-{2}\frac{{1}}{{5}}.

Next exercise.

Exercise 4. Compare 13\frac{{1}}{{3}} and 215{2}\frac{{1}}{{5}}.

Notice that 13\frac{{1}}{{3}} is proper fraction so it can be treated as mixed number with integer part 0.

Now, compare integer parts: since 0<2{0}<{2} then 13<215\frac{{1}}{{3}}<{2}\frac{{1}}{{5}}.

Answer: 13<215\frac{{1}}{{3}}<{2}\frac{{1}}{{5}}.

Next exercise.

Exercise 5. Compare 173\frac{{17}}{{3}} and 215{2}\frac{{1}}{{5}}.

First fraction is improper, so this is not a mixed number.

We have two ways: either to convert improper fraction to mixed number or convert mixed number to improper fraction.

Let's choose first way: 173=523\frac{{17}}{{3}}={5}\frac{{2}}{{3}}.

Now, we compare 523{5}\frac{{2}}{{3}} and 215{2}\frac{{1}}{{5}}.

Since 5>2{5}>{2} then 523>215{5}\frac{{2}}{{3}}>{2}\frac{{1}}{{5}}.

Answer: 173>215\frac{{17}}{{3}}>{2}\frac{{1}}{{5}}.