Comparison of Fractions

Introduction

A fraction is a subset of a whole with two parts: the numerator and the denominator. The numerator is the number on the fractional bar’s upper half, while the denominator is the number on the fractional bar’s lower half. When comparing fractions, a set of rules relating to the numerator and denominator must be followed. When two fractions are compared, we can determine greater and smaller. Comparison of fractions has various uses in our daily life, such as comparing the marks in an exam, the ingredients involved in a recipe, etc.

Various Methods Involving Comparison of Fractions

The various methods by which we can compare fractions are listed as follows:

• Comparing Fractions with like denominator: When comparing fractions with the same denominators, determining the greater or smaller fraction becomes easy. We can just look for the fraction with the larger numerator after confirming if the denominators are the same. The fractions are equal if the numerators and denominators are both equal.

For example, comparing 8/15 and 10/15. Since both of these fractions have the same denominator, we compare the numerator and determine the greater number. Here, ten is greater than 8, so 10/15 is the larger fraction.

• Comparing Fractions with unlike denominators: We must convert unlike denominators to like denominators before comparing fractions. To do so, we must find the denominators’ Least Common Multiple (LCM). We can simply compare fractions when the denominators are the same.

For Example, comparing 3/4 and 2/5. The denominators are unlike, so we take the LCM of (4,5) 20. Now we convert the fractions that the denominators become the same. So now the fractions become 15/20 and 8/20. Denominators are the same. We compare in the same way. Here, 15 is greater than eight, so 3/4 is the larger fraction here.

• Decimal Method of Comparing Fractions: We think about the decimal upsides of parts in this method. The denominator partitions the numerator, changing the division into a decimal. 

For Example, we compare 1/2 and 3/4. We convert both the fractions into decimals and get 1/2 = 0.5 and 3/4 = 0.8. We compare the decimals now and determine that 0.5>0.8, therefore 1/2 is the larger fraction. 

• Comparing Fractions using Cross Multiplication: For comparing fractions utilizing cross augmentation, we duplicate one division’s numerator with the other part’s denominator. 

For Example, Comparing 2/3 and 4/5. Cross multiplying the given fractions 2×5=10 and 3×4=12. Since 12>10, hence 4/5 is the larger fraction.  

• Comparing Fractions using Visualizations: We can utilize various graphical tools and models to depict greater fractions.

For Example, we take two models, A and B, representing two fractions. The model that we took has eight blocks with four colored in Model A, and in Model B, we have six blocks and four colored blocks. Because 4/6 covers a bigger shaded area than 4/8, we can simply establish 4/8< 4/6. It’s worth noting that the smaller percentage takes up less space within the same whole. One thing to keep in mind here is that for the comparison to be legitimate, the sizes of models A and B must be identical. The denominators of each model are then divided into equal halves for each model.

• Equivalent Fractions: Equivalent fractions represent the same portion of a larger whole. We multiply the numerator and denominator by the same number to get an equivalent fraction.

For Example, We compare 2/3 and ⅘, and both should have the same denominator of 15. We multiply 2/3 by 5/5 and 4/5 by 3/3 to find equivalent fractions with denominators of 5. 2/3 x 5/5 =10/15 and 4/5 x 3/3 = 12/15. Now both have the same denominator, and on comparing, we get that 12/15 is larger, which implies that 4/5 is the larger fraction.

Solved Example

Ryan was given to prove that the fractions 4/6 and 6/9 are equal. Is it possible to demonstrate this using the LCM method?

By determining the LCM of the denominators of the given fractions, we can make the denominators the same. The LCM of the numbers 6 and 9 is 18. As a result, we’ll multiply 4/6 by 3/3, (4/6) x (3/3) = 12/18, and 6/9 by 2/2, (6/9) x (2/2) = 12/18, converting them to like fractions with the same denominators. 12/18 and 12/18 will be the new fractions with the same denominators. As a result, both fractions are equal: 6/9 = 4/6. As a result, 4/6 = 6/9.

Conclusion

We learned how to compare fractions. We have some rules to compare the fractions, mainly including comparing by decimal, comparing like fractions, comparing unlike fractions, and defining equivalent fractions. The numerators of two fractions with similar denominators can be compared. Because it comprises more whole pieces, the fraction with the larger numerator is the greater fraction. Because it comprises fewer whole pieces, the fraction with the smaller numerator is smaller. Equal numerators indicate equivalent fractions when the denominator of two fractions is the same.