When we compare whole numbers such as 7 and 5 or even decimals like 3.22 and 3.12, it’s usually pretty easy to tell which is bigger. We can almost instantly tell that 7 > 5, and 3.22 > 3.12.
But with fractions, things can get a little confusing.
Take \( \Large \frac{3}{5}\) and \( \Large \frac{7}{10}\), for example.
It's hard to picture the size of each fraction in your head, right?
A common mistake is thinking smaller numbers in a fraction mean a bigger fraction, which is not always the case. In fact, \( \Large \frac{7}{10}\) is greater than \( \Large \frac{3}{5}\).
We’ve put together a simple guide to comparing fractions with easy-to-follow definitions and instructions, solved examples, practice exercises, and answers to some of the questions our math tutors in Carlsbad often get.
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Fractions are a way to show parts of a whole.
Imagine slicing a pizza into equal pieces—each slice represents a fraction of the entire pizza. A fraction is written using two numbers separated by a line, like this: \( \Large \frac{1}{2}\).
For example, in the fraction \( \Large \frac{3}{4}\), the numerator is 3, meaning we have three parts, and the denominator is 4, meaning the whole is divided into four equal parts.
You Might Also Like: What Are Equivalent Fractions?
There are several ways to compare fractions.
We’ll start with the basics and build up to more advanced cases.
Before we begin doing any math, let’s try an easy way to compare fractions—by looking at them!
Starting with visual comparisons of fractions is great because it helps us see what the numbers mean.
Looking at pictures, like pie charts or bars, helps us see how fractions fit together.
Let’s see this in action and visually compare \( \Large \frac{1}{2}\) and \( \Large \frac{3}{4}\).
The steps are:
1. Pick Two Identical Shapes: For this purpose, we will use squares (circles and rectangles work well too!) that are the same size.
2. Split Each Shape into Equal Parts: Match the number of parts to the denominator of the fraction. Since we’re comparing \( \Large \frac{1}{2}\) and \( \Large \frac{3}{4}\), divide one shape into 2 equal parts and the other into 4 equal parts.
3. Shade the Parts: Color in the number of parts shown by the numerator. For \( \Large \frac{1}{2}\), shade 1 part. \( \Large \frac{3}{4}\), shade 3 parts.
4. Look at the Shaded Areas: Which shape has more of its parts colored in? The one on the right, correct? That’s the bigger fraction!
Easy peasy, right?
Now, let’s do some math!
Check out our guide to comparing fractions with the same numerators:
When we're comparing fractions with the same denominator (e.g. \( \Large \frac{2}{5}\), \( \Large \frac{4}{5}\), \( \Large \frac{1}{5}\)) we're looking at how the pieces are divided into equal parts.
Since the bottom number (the denominator) is the same for these fractions, we have the same number of pieces.
To compare them, just look at the top number (the numerator). The fraction with the bigger top number has more pieces, so it’s the bigger fraction.
For example:
We’ll compare \( \Large \frac{3}{8}\) and \( \Large \frac{5}{8}\).
Check out our video guide to comparing fractions with common denominators:
That was a breeze! Let’s move on to the next step!
To compare fractions with different denominators, we have to make their denominators the same.
How do we do that?
By finding their least common denominator (LCD), or the smallest number that both denominators can divide into evenly.
Once they share the same denominator, we compare their numerators, just like we’ve already learned!
Let’s see this in action. We’ll compare \( \Large \frac{2}{3}\) and \( \Large \frac{3}{4}\) step by step.
1. Find the Least Common Denominator (LCD)
To find the LCD, we ask ourselves: What is the smallest number that both denominators (3 and 4) can divide into?
The smallest number 3 and 4 can divide into is 12. So, the LCD is 12.
2. Convert the Fractions
Now we change the fractions so their denominators match the LCD:
For \( \Large \frac{2}{3}\), multiply both the numerator (top number) and denominator (bottom number) by 4 so the denominator becomes 12:
\( \Large \frac{2}{3}\) x \( \Large \frac{4}{4}\) = \( \Large \frac{2×4}{3×4}\) = \( \Large \frac{8}{12}\)
For \( \Large \frac{3}{4}\), multiply both the numerator (top number) and denominator (bottom number) by 3 so the denominator becomes 12:
\( \Large \frac{3}{4}\) x \( \Large \frac{3}{3}\) = \( \Large \frac{3×3}{4×3}\) = \( \Large \frac{9}{12}\)
3. Compare the Fractions
Now the fractions are \( \Large \frac{8}{12}\) and \( \Large \frac{9}{12}\). Since both have the same denominator, we just compare the numerators:
9>8
This means: \( \Large \frac{9}{12}\)>\( \Large \frac{8}{12}\) or \( \Large \frac{3}{4}\)>\( \Large \frac{2}{3}\)
And that’s how we compare fractions with different denominators—great work!
The more we practice comparing fractions, the easier it gets! Let’s walk through a few examples together.
We’ll compare \( \Large \frac{2}{5}\) and \( \Large \frac{3}{8}\).
1. Find the Least Common Denominator (LCD)
The denominators are 5 and 8. The smallest number that both 5 and 8 divide into is 40. So, the LCD is 40.
2. Convert the Fractions
For \( \Large \frac{2}{5}\), multiply both the numerator (top number) and denominator (bottom number) by 8 so that the denominator becomes 40.
\( \Large \frac{2}{5}\) x \( \Large \frac{8}{8}\) = \( \Large \frac{2×8}{5×8}\) = \( \Large \frac{16}{40}\)
For \( \Large \frac{3}{8}\), multiply both the numerator (top number) and denominator (bottom number) by 5 so that the denominator becomes 40.
\( \Large \frac{3}{8}\) x \( \Large \frac{5}{5}\) = \( \Large \frac{15}{40}\)
3. Compare the Fractions:
Now we compare \( \Large \frac{16}{40}\) and \( \Large \frac{15}{40}\) by looking at their numerators.
\( \Large \frac{16}{40}\)>\( \Large \frac{15}{40}\), or \( \Large \frac{2}{5}\)>\( \Large \frac{3}{8}\)
We’ll use the same steps to compare 47 and 59.
1. Find the Least Common Denominator (LCD):
The denominators are 7 and 9. The smallest number that both 7 and 9 divide into is 63. So, the LCD is 63.
2. Convert the Fractions:
For \( \Large \frac{4}{7}\), multiply both the numerator (top number) and denominator (bottom number) by 9 so that the denominator becomes 63.
\( \Large \frac{4}{7}\) x \( \Large \frac{9}{9}\) = \( \Large \frac{4×9}{7×9}\) = \( \Large \frac{36}{63}\)
For \( \Large \frac{5}{9}\) multiply both the numerator (top number) and denominator (bottom number) by 7 so that the denominator becomes 63.
\( \Large \frac{5}{9}\) x \( \Large \frac{7}{7}\) = \( \Large \frac{5×7}{9×7}\) = \( \Large \frac{35}{63}\)
3. Compare the Fractions:
Now, we compare the fractions by looking at their numerators. Since 36 is greater than 35, we know:
\( \Large \frac{36}{63}\)>\( \Large \frac{35}{63}\), or \( \Large \frac{4}{7}\)>\( \Large \frac{5}{9}\)
Time to practice what you’ve learned. Try working these out by yourself.
Task 1: Compare \( \Large \frac{1}{4}\) and \( \Large \frac{2}{5}\).
Task 2: Compare \( \Large \frac{3}{7}\) and \( \Large \frac{4}{9}\)
Task 3: Compare \( \Large \frac{5}{6}\) and \( \Large \frac{7}{8}\)
Once you’ve solved these, find your answers in the last section. You’re doing great!
Learning to compare fractions can raise some interesting questions. Here are answers to a few common ones that might come up as you practice:
If two fractions have the same numerator, the fraction with the smaller denominator is larger. That’s because the smaller denominator means the whole is divided into fewer (and therefore larger) parts.
We can check this by comparing \( \Large \frac{3}{4}\) and \( \Large \frac{3}{5}\) using the same steps we use for fractions with different denominators.
1. Find the Least Common Denominator (LCD):
The denominators are 4 and 5. The smallest number they both divide into is 20. So, the LCD is 20.
2. Convert the Fractions:
For \( \Large \frac{3}{4}\), multiply both the numerator and denominator by 5:
\( \Large \frac{3}{4}\) x \( \Large \frac{5}{5}\) = \( \Large \frac{15}{20}\)
For \( \Large \frac{3}{5}\), multiply both the numerator and denominator by 4:
\( \Large \frac{3}{5}\) x \( \Large \frac{4}{4}\) = \( \Large \frac{12}{20}\)
3. Compare the Fractions:
Now that the denominators are the same, compare the numerators:
\( \Large \frac{15}{20}\)>\( \Large \frac{12}{20}\), or \( \Large \frac{3}{4}\)>\( \Large \frac{3}{5}\)
This confirms that when two fractions have the same numerator, the one with the smaller denominator is larger!
Yes! Fractions can be equivalent even if they look different. Equivalent fractions represent the same part of a whole.
For example, \( \Large \frac{1}{2}\) and \( \Large \frac{2}{4}\) are equivalent because:
If you simplify \( \Large \frac{2}{4}\) you get \( \Large \frac{1}{2}\).
Or if you multiply \( \Large \frac{1}{2}\) by \( \Large \frac{2}{2}\), you get \( \Large \frac{2}{4}\).
Equivalent fractions are just different ways of showing the same value.
Any fraction with a numerator of 0 is equal to 0, no matter the denominator (as long as the denominator isn’t 0).
For example, \( \Large \frac{0}{5}\) =0, because there are no parts to count.
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If you’ve given our exercises a try, check your answers below to see how you did!
Task 1: \( \Large \frac{1}{4}\) < \( \Large \frac{2}{5}\)
Task 2: \( \Large \frac{3}{7}\) < \( \Large \frac{4}{9}\)
Task 3: \( \Large \frac{5}{6}\) < \( \Large \frac{7}{8}\)
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