Comparing fractions means determining the larger and the smaller fraction among two parts. In our day-to-day life, we often get stuck with situations where we have to compare two or more fractions. Did you know that you deal with a fraction every time you slice an apple into two parts? We often use fractions without realizing it. Let us learn about fractions and their comparison today!
How to Compare Fractions?
What is a Fraction?
Before exploring the concept of comparing fractions, let us recall fractions. A fraction is nothing but a part of a whole thing. If you break a glass dish into several pieces, can you still say that each part represents a fraction? Yes, reasonably each part can still be called a fraction of the glass dish, but in math, a fraction comes with a rule. The rule is "each of the parts has to be equal." A fraction has two parts; they are called numerator and denominator.
Now, let's discuss what comparing fractions mean. When the two fractions are compared to find out which is greater or which is smaller. The real-time examples of comparing fractions include a number of activities, splitting a bill, following a recipe, checking discounted prices while shopping, comparing sales of a particular product, medical prescriptions by the doctor, scores of tests and exams, etc. Let us go through the different methods of comparing fractions with the help of examples to understand the concept better.
Comparing Fractions with Same Denominators
In this method, we check the denominators to see if they are the same. If the denominators are the same, then the fraction with the bigger numerator is the bigger fraction and the fraction with the smaller numerator is the smaller fraction. If both numerators and denominators are equal, the fractions are also equal. For example, let us compare 6/17 and 16/17
- Step 1: Look for denominators of the given fractions:6/17 and 16/17. Denominators are the same.
- Step 2: Compare numerators. 16>6.
- Step 3: The fraction with a larger numerator would be a larger fraction. Therefore, 6/17 < 16/17.
Comparing Fractions with Unlike Denominators
For comparing fractions with unlike denominators, start by finding the least common denominator(LCM) to make the denominators the same. When the denominators are made the same, the fraction with the larger numerator is the larger fraction. For example 1/2 and 2/5.
- Step 1: Look for denominators of the given fractions: 1/2 and 2/5. Denominators are not the same.
- Step 2: Take LCM of 2 & 5. LCM(2, 5) = 10.
- Step 3: 1/2 = 1/2 × 5/5 and 2/5 = 2/5 × 2/2.
- Step 4: Compare fractions: 5/10 and 4/10. Denominators are the same. Compare numerators, 5>4.
- Step 5: 5/10 > 4/10. The fraction with a larger numerator would be a larger fraction. Thus, 5/10 > 4/10. Therefore, 1/2 > 2/5
Also, if the denominators are different and the numerators are the same, then we can easily compare fractions by looking at their denominators. The fraction with a smaller denominator has a greater value and the fraction with a larger denominator has a smaller value. For example, 2/3 > 2/6.
Decimal Method of Comparing Fractions
In this method, we compare the decimal values of fractions. For this, the numerator is divided by the denominator and the fraction is converted into a decimal. Then, the decimal values are compared. For example, 4/5 and 6/8.
- Step 1: Write 4/5 and 6/8 in decimals. 4/5 = 0.8 and 6/8 = 0.75.
- Step 2: Compare decimal values. 0.8 > 0.75
- Step 3: The fraction with a larger decimal value would be a larger fraction. Therefore, 4/5 > 6/8
Comparing Fractions Using Visualization
We can use various graphical methods and models to visualize larger fractions. Model A and B represent the given respective fractions. 4/8 < 4/6 because 4/6 covers a larger shaded area than 4/8. Note that the smaller fraction occupies a lesser area of the same whole. A point to be taken into consideration here is that the size of models A and B should be exactly the same for the comparison to be valid. Each model is then divided into equal parts equivalent to their respective denominators.
Comparing Fractions Using Cross Multiplication
In this method, we cross multiply the numerator of one fraction with the denominator of the other fraction. The same has been indicated by the arrows in the figure below. In the example given below, when we cross multiply, we get 4 and 6. 4 and 6 are the numerators we would get if we expressed 1/2 and 3/4 with the common denominator 8. The new fractions with the same denominators will be 4/8 and 6/8. Since 6 is a greater numerator, 4/8 < 6/8. Therefore, 1/2 < 3/4
The common methods used to compare fractions are:
- Decimal method
- Cross-multiplication method
- Graphical/Visual method
- LCM method to make the denominators the same
Comparing Fractions Related Topics
Examples on Comparing Fractions
Example 1: Nathan is confused and asks why is 5/11 > 4/11? Can you explain it to him?
5/11 and 4/11 have the same denominators; hence, we can simply compare the fractions by observing the numerators. The fraction with a greater numerator will be a greater fraction. 5 > 4. Therefore, 5/11 > 4/11.
Example 2: Ryan was asked to prove that the given fractions: 4/6 and 6/9 are the same using the LCM method. He is a bit confused. Can you help him?
In this method, we find lcm of the denominators of the given fractions, making the denominator the same. By doing so, we get 18 for both. 12 will be the numerator if we expressed 4/6 and 6/9 with the common denominator 18 (LCM of both the denominators). The new fractions with the same denominators will be 12/18 and 12/18. Hence, both the fractions are equal: 4/6 = 6/9. Therefore, 4/6 = 6/9.
FAQs on Comparing Fractions
What does Comparing Fraction Mean?
Comparing fractions means comparing the given fractions in order to tell if one fraction is less than, greater than, or equal to another. Like whole numbers, we can compare fractions as well using the same symbols: <,> and =.
What is the Rule of Comparing Fractions with the Same Denominator?
When the denominators are the same, the fraction with the lesser numerator is the lesser fraction and the fraction with the greater numerator is the greater fraction. When the numerators are equal, the fractions are considered equivalent.
What is the Rule when Comparing Fractions with the Same Numerator?
When the fractions have the same numerator, the fraction with the smaller denominator is greater. For three or more fractions with the same numerator, arrange them in ascending or descending order first, such as 1/2, 1/4, and 1/6. Now out of these, the fraction with the smaller denominator is 1/2. Thus, 1/2 is the greatest of all the three given fractions.
What are Equivalent Fractions?
The fractions that have different numerators and denominators but are equal in their values are called equivalent fractions. Example: 1/2, 5/10, and 6/12 are equivalent fractions since all of them are equal to half or 0.5.
What is the Easiest Way of Comparing Fractions?
The easiest and the fastest way to compare fractions is to convert them into decimal numbers. Then arrange the decimal numbers in ascending or descending order. The fraction with a greater decimal value would be a greater fraction.
Why do we Need to Compare Fractions?
Comparing fractions is an important component, which helps students develop their number sense about fraction size. This helps them realize that the strategies they use to compare whole numbers do not necessarily compare fractions For example, 1/4 is greater than 1/8 even though the whole number 8 is greater than 4.