比较分数

Compare the Numerators

Comparing two or more fractions essentially means figuring out which one is bigger. This can be simple if the fractions have the same denominator (the number on the bottom), but might be a little more complicated if they don't. If the denominators are the same, you can just compare the numerators (the top numbers)!

If numerators and denominators are different, convert each fraction. You'll want either the numerators or the denominators to be the same.

For now though we'll try an example with like denominators.

$\frac{4}{7}$ is bigger! If we imagine the fractions as pictures, we can see the difference:

What if the denominators aren't the same?

If the denominators aren't the same, it can be harder to compare the fractions. For example, even though the numerator in $\frac{3}{5}$ is bigger, $\frac{2}{3}$ is actually a larger fraction.

If you have to compare two fractions with different denominators, you have to make the denominators the same first. You can do that by just multiplying each fraction by the denominator of the other one.

As long as the denominators are the same, it's easy to compare fractions by just comparing their numerators. If the denominators aren't the same, multiply each fraction by the other's denominator and then compare the numerators.

(Video) Comparing Fractions

by Math Antics

Math Antics has an outstanding video on comparing fractions.

$\large{\text{Summary}}$

Comparing fractions just means finding which fraction is bigger. Sadly, this is not always easy.

If you had to compare $\frac13$ and $\frac1{10}$ you might be tempted to say that $\frac{1}{10}$ is bigger since $10$ is bigger than $3$. But, this isn't the case. $\frac13$ is bigger than $\frac1{10}$. The fraction with the bigger denominator is actually smaller! How can this be?

If we were to divide these two fractions, we would see that the decimal value of $\frac13$ is bigger than that of $\frac1{10}$. $$\begin{align} \frac13&=0.333...\\ \frac1{10}&=0.1 \end{align}$$

Even though $0.333\dots$ is a repeating decimal, we can see it is bigger than $0.1.$ This is because the tenths place of $0.333\dots$ has a bigger digit.

If two fractions have the same bottom number, then you can easily compare them. The larger number will have a larger top number.

If you had to compare $\frac14$ and $\frac34$, you would know $\frac34$ is bigger since they have the same bottom number and $3$ is bigger than $1$.

We can use this idea even when fractions don't have the same bottom number by cross multiplying. You multiply the top number of each fraction by the bottom number of the other fraction. The bigger product means that the fraction with that top number is bigger.

Let's use this to compare $\frac2{\color{#29A603}{5}}$ and $\frac3{\color{#8925AE}{4}}$. We can instead rephrase this to compare $2\times\color{#8925AE}{4}$ and $3\times\color{#29A603}{5}$. We get that: $$8<15$$

Since we made $15$ with the numerator of the fraction $\frac34$, we know that it's the bigger fraction.

Checking Your Work

You can find fraction calculators to help check your work. Alternatively, use a pocket calculator. Plug in the fractions you start with and you'll get decimals. If you have equivalent fractions, the decimals will be the same. Otherwise one fraction will be larger or smaller than the other.

Compare the Denominators

Another way to compare positive fractions is to compare denominators when the numerators are the same. With like numerators, whichever fraction has the smaller denominator will be the greater number. This can be easier than comparing numerators when the denominators are the same if finding a least common denominator would be difficult.

For starters let's say you want to compare two fractions with the same numerator. Then the larger fraction has the smaller denominator. For instance,

$$\frac{12}{13} > \frac{12}{17}$$

The numerators are both $12$, but the left fraction has smaller denominator.

This strategy also works for improper fractions. For example,

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

The numerators are both $8$, but the left fraction has smaller denominator.

Let's use an example here:

The correct answer, $\frac{5}{6}$, is easy to see when we compare denominators, because since $6$ is the smallest denominator, $\frac{5}{6}$ is the largest fraction. It would be much harder to find a least common denominator, as the LCD of $6, 7, 8$ and $9$ is $504$.

Finding a common numerator

Finding a common numerator is exactly like finding a common denominator. Simply find the least common multiple of each numerator, and multiply top and bottom of each fraction by the appropriate multiple. Let's do an example:

If we had to find a common denominator, we would have had to multiply each fraction to get an LCD of $220$. Instead, we simply multiply $\frac{4}{7} \times \frac{8}{8} = \frac{32}{56}$, then compare denominators, and see that since $55<56$, $\frac{32}{55}>\frac{32}{56}$

This technique is handy to keep in mind, and can reduce the time it takes to compare certain fractions by quite a bit.