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Comparing Fractions means determining which fractional number is larger or smaller. Fractions consist of two parts, the numerator and the denominator. The number above the fractional bar in a fraction is the numerator, and the bottom number is the denominator. One can compare fractions even if their numerators and denominators are different. Comparing fractions includes a set of rules that is related to the numerator and the denominator and when any two fractions are compared, we can get to know the greater and the smaller fraction. Comparing fractions is crucial, and one must know how to do it. Therefore, one needs to master the following rules described below to compare fractions correctly.
Read Also: Difference Between Fraction and Rational Numbers
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Key Terms: Fractions, Whole Numbers, Numerator, Denominator, Comparing Fractions, Visualization, Cross Multiplication, LCM, Decimals
What is a Fraction?
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A fraction can be defined as a part of a whole that has basically two parts- the numerator and the denominator. The numerator is the number that is on the upper part of the fractional bar and the denominator is the number that is located below the fractional bar.
Check Important Relation Between HCF and LCM
Comparing Fractions With The Same Denominators
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One must follow the steps given below to figure out how to compare fractions with the same denominators.
- Step 1: Check the fraction values given to you. If the denominators are the same, proceed with this step. If the denominators are different, move to rule number 2, described after this rule.
- Step 2: If the denominators are the same, for example, 2/6 and 5/6 have the same denominators, you need to compare the numerator.
- Step 3: In the above example, 5 is greater than 2. Therefore, 5/6 is greater than 2/6.
- Step 4: To write it mathematically, use > or < symbol. For example, 5/6 > 2/6.
Read More: Rationalize the Denominator
Comparing Fractions With Different Denominators
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If the denominators are not the same, follow the steps given below.
- Step 1: Make the denominators equal by taking their LCM. For example, in fractions 2/3 and 4/5, the LCM of denominators of 3 and 5 is 15.
- Step 2: Multiply 2/3 with 5/5 to make the denominator get the value 15. Similarly, multiply 4/5 with 3/3 to get 15.
- Step 3: The first fraction becomes 10/15, and the other becomes 12/15.
- Step 4: Compare the numerators. 10 is smaller than 12. Therefore, 10/15 is less than 12/15.
- Step 5: Hence, 2/3 is less than 4/5, i.e. 2/3 < 4/5.
If the numerators are the same, you can compare fractions with the denominators directly. The fraction with a smaller denominator will be greater. For example, 4/6 and 4/9. 6 is lesser than 9. Hence, it will be greater. Therefore, 4/6 is greater than 4/9.
Also Read:
Decimal Method of Comparing Fractions
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One can compare the fractions directly by dividing and finding their decimal values. Follow the steps to compare fractions by using the decimal method.
- Step 1: Divide the numerator by the denominator. For example, let us compare ½ and ¾. Divide 1 by 2 and 3 by 4.
- Step 2: The decimal values obtained will be 0.5 and 0.75.
- Step 3: You can compare them easily now. 0.75 > 0.5, hence, ¾ is greater than ½.
Read More: Decimal Expansion of Rational Numbers
Comparing Fractions Using Visualization
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Comparing fractions is visual is easier than any other method. However, you must cross-check the values from the rules mentioned above. Make two boxes with the same length and width to check fractions by visuals. Then, divide the box with the given fractions for which you will compare.
For example, represent 4/8 and 4/6 as shown in the diagram below. By observing the diagram, you can determine that 4/6 is greater than 4/8 because it covers more area. Therefore, you can denote every fraction like this and compare them. However, this method gets complicated when comparing bigger values. So, choose the above rules for proper comparison.
Check Important Difference Between Area and Perimeter
Comparing Fractions Using Cross Multiplication
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In order to compare the fractions using cross multiplication, we need to multiply the numerator of one fraction with the denominator of the other fraction. For example, let us compare 1/2 and 3/4.
- Step 1: First we need to cross multiply the given fractions for comparing them. Now we have to multiply the numerator of the first fraction with the denominator of the second fraction, Thus, 1 × 4 = 4.
- Step 2: Similarly, now we will multiply the numerator of the second fraction with the denominator of the first fraction, that is, 3 × 2 = 6.
- Step 3: Now, we have to compare the products received that is 4 and 6. Since 4 < 6, thus, we can easily compare that 1/2 < 3/4.
Also Check:
Things to Remember
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- Comparing fractions involves a set of rules used for determining which fractional number is larger or smaller. Fractions consist of two parts, the numerator and the denominator.
- Compare fractions by checking their denominators first. If they are the same, then compare their numerators.
- If the denominators are not equal, make them equal by taking LCM.
- For smaller fractional values, use the method of visualization.
- One can use the decimal method for large fractional values and divide the numerator by the denominator.
- One can also compare the fraction using the cross-multiplication method in which we need to multiply the numerators of one fraction with the denominators of the other fraction.
Check Also: Cross Multiplication Method of Solving Linear Equation
Sample Questions
Ques. Rahul ordered 2 ½ pizzas, and Rajini ordered 2¾ pizzas. Out of the given two pizzas, who has more pizza? (3 Marks)
Ans. Firstly, you need to convert the mixed fractions into the simple fraction
Pizza ordered by Rahul = 5/2
Pizza ordered by Rajini = 11/4
Since the obtained fractions from the above situation are unlike fractions, we need to convert them into the like fractions.
10/4 and 11/14
Since the numerator in the fraction 11/4 is greater than that in the fraction 10/4
Hence, Rajini ordered more pizza compared to Rahul.
Ques. Is it allowed to compare fractions with the whole like 1? (3 Marks)
Yes. It is allowed to compare fractions with the whole like 1. If a fraction needs to be compared with 1, you can treat one as any other fraction. For example, if you have to compare 3/4 with 1, you can use the decimal method or the same denominator method.
3/4 = 0.75 while
1 = 1.00
In the above example, 1 is greater than 0.75 For the same denominator function, 1 has to be taken as 4/4 and then with 3/4. Thus, numerator 4 is higher than numerator 3, which means, whole number 1 is higher than ¾.
Ques. Which of the following fractions is larger: 8/9 or 5/7 (3 Marks)
Ans. Convert 8/9 and 5/7 into the same denominators by finding out the LCM of the denominators. The LCM, in this case, is 63. So we multiply both 8 and 9 by 7 and get 56/63.
Similarly, multiply 5 and 7 with 9 and get 45/63.
As numerator 56 is higher than numerator 45, 8/9 is higher than 5/7.
Ques. Which of the following fractions is smaller: 90/900 or 80/900 (2 Marks)
Ans. 90/900 = 0.1
80/900 = 0.089
Since 1 is greater than 0.089, we conclude that 80/900 is smaller than 90/900.
Ques. Which of the following fractions is larger: 30/70 or 50/90? (2 Marks)
Ans. 30/70 = 0.42
50/90 = 0.56
As, 0.56 is greater than 0.42, we can say that 5/9 is greater than 3/7.
Ques. Rita was asked to prove that the given fractions are equal: 400/600 and 600/900. Can you prove this using the LCM method? (3 Marks)
Ans. Make the denominators the same by finding the LCM of the denominators of the given fractions.
LCM of 600 and 900 = 1800.
Multiply 400/600 with 3/3, (4/6) × (3/3) = 12/18, and 6/9 with 2/2, (6/9) × (2/2) = 12/18, which will convert them to like fractions with the same denominators.
Thus, here the new fractions with the same denominators will be 12/18 and 12/18.
Hence, we can clearly see that both the fractions are equal: 4/6 = 6/9. Therefore, 4/6 = 6/9.
Ques. Compare the fractions 1/8 and 71/125. (3 Marks)
Ans. For comparing fractions with different denominators, we need to find the LCM of the denominators.
LCM of 8 and 125 = 1000.
So, let us multiply 1/8 with 125/125, i.e., 1/8 × 125/125 = 125/1000.
Now, let us multiply 71/125 with 8/8 = 568/1000.
Now that we have fractions 125/1000 and 568/1000, we can easily compare them.
Since 568 > 125, therefore, 71/125 > 1/8.
Ques. How is 5/11 greater than 4/11? Explain. (3 Marks)
Ans. If the denominators are the same, comparing fractions becomes easier. 5/11 and 4/11 have the same denominators; hence, you can compare the fractions by observing the numerators.
We know that now the fraction with a larger numerator will be the larger fraction. 5 > 4.
Therefore, 5/11 > 4/11.
Ques. Comparing the fractions 1/14 and 1/7, I can say that 1/14 is less than 1/7. Is this statement true or false? (3 Marks)
Ans. To compare the fractions, you must make their denominators equal.
Multiply 1/7 with 2/2, we get 2/14.
1 is less than 2; therefore, 1/14 is less than 2/14, i.e. 1/14 is less than 1/7. Hence, the statement is true.
Ques. If you are given a set of fractions, and if the denominators are the same, you can compare the fractions by checking the numerator. Is this statement true or false? (2 Marks)
Ans. Having the same denominator in the fraction can help you determine which fractional quantity is greater or smaller by checking their numerators. Hence, the statement is true.
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