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 $\frac{{5}}{{6}}$ and $\frac{{15}}{{6}}$ .

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

Answer: $\frac{{5}}{{6}}<\frac{{15}}{{6}}$ .

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

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

Make same denominators: $\frac{{2}}{{3}}=\frac{{{2}\cdot{7}}}{{{3}\cdot{7}}}=\frac{{14}}{{21}}$ and $\frac{{5}}{{7}}=\frac{{{5}\cdot{3}}}{{{7}\cdot{3}}}=\frac{{15}}{{21}}$ .

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

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

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

Next example.

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

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

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

Now, do a couple of exercises.

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

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

Next exercise.

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

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

Next exercise.

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

Ignore minuses.

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

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

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

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

Next exercise.

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

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

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

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

Next exercise.

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

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: $\frac{{17}}{{3}}={5}\frac{{2}}{{3}}$ .

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

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

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