Which fraction is bigger online. Comparison of fractions: rules, examples, solutions
Two unequal fractions are subject to further comparison to find out which fraction is larger and which fraction is smaller. To compare two fractions, there is a rule for comparing fractions, which we will formulate below, and we will also analyze examples of the application of this rule when comparing fractions with the same and different denominators. In conclusion, we will show how to compare fractions with the same numerators without reducing them to a common denominator, and also consider how to compare an ordinary fraction with a natural number.
Page navigation.
Comparing fractions with the same denominators
Comparing fractions with the same denominators is essentially a comparison of the number of equal shares. For example, the common fraction 3/7 determines 3 parts 1/7, and the fraction 8/7 corresponds to 8 parts 1/7, so comparing fractions with the same denominators 3/7 and 8/7 comes down to comparing the numbers 3 and 8, that is , to comparing numerators.
From these considerations it follows rule for comparing fractions with the same denominator: Of two fractions with the same denominator, the larger fraction is the one whose numerator is larger, and the smaller is the fraction whose numerator is smaller.
The stated rule explains how to compare fractions with the same denominators. Consider an example of applying the rule for comparing fractions with the same denominators.
Example.
Which fraction is larger: 65/126 or 87/126?
Solution.
The denominators of the compared ordinary fractions are equal, and the numerator 87 of the fraction 87/126 is greater than the numerator 65 of the fraction 65/126 (if necessary, see the comparison of natural numbers). Therefore, according to the rule for comparing fractions with the same denominators, the fraction 87/126 is greater than the fraction 65/126.
Answer:
Comparing fractions with different denominators
Comparing fractions with different denominators can be reduced to comparing fractions with the same denominators. To do this, you just need to bring the compared ordinary fractions to a common denominator.
So, to compare two fractions with different denominators, you need
- bring fractions to a common denominator;
- compare the resulting fractions with the same denominators.
Let's take a look at an example solution.
Example.
Compare the fraction 5/12 with the fraction 9/16.
Solution.
First, we bring these fractions with different denominators to a common denominator (see the rule and examples of reducing fractions to a common denominator). As a common denominator, take the lowest common denominator equal to LCM(12, 16)=48 . Then the additional factor of the fraction 5/12 will be the number 48:12=4 , and the additional factor of the fraction 9/16 will be the number 48:16=3 . We get 
And 
.
Comparing the resulting fractions, we have . Therefore, the fraction 5/12 is smaller than the fraction 9/16. This completes the comparison of fractions with different denominators.
Answer:
Let's get another way to compare fractions with different denominators, which will allow you to compare fractions without reducing them to a common denominator and all the difficulties associated with this process.
To compare fractions a / b and c / d, they can be reduced to a common denominator b d, equal to the product of the denominators of the compared fractions. In this case, the additional factors of the fractions a/b and c/d are the numbers d and b, respectively, and the original fractions are reduced to fractions and with a common denominator b d . Recalling the rule for comparing fractions with the same denominators, we conclude that the comparison of the original fractions a / b and c / d is reduced to comparing the products of a d and c b .
From this follows the following rule for comparing fractions with different denominators: if a d>b c , then , and if a d Consider comparing fractions with different denominators in this way. Example. Compare the common fractions 5/18 and 23/86. Solution. In this example, a=5 , b=18 , c=23 and d=86 . Let's calculate the products a d and b c . We have a d=5 86=430 and b c=18 23=414 . Since 430>414 , the fraction 5/18 is greater than the fraction 23/86 . Answer: Fractions with the same numerators and different denominators can certainly be compared using the rules discussed in the previous paragraph. However, the result of comparing such fractions is easy to obtain by comparing the denominators of these fractions. There is such rule for comparing fractions with the same numerator: Of two fractions with the same numerator, the one with the smaller denominator is the larger, and the one with the larger denominator is the smaller. Let's consider an example solution. Example. Compare the fractions 54/19 and 54/31. Solution. Since the numerators of the compared fractions are equal, and the denominator 19 of the fraction 54/19 is less than the denominator 31 of the fraction 54/31, then 54/19 is greater than 54/31. The rules for comparing ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the denominators (same or different) of the compared fractions. rule. To compare two fractions with the same denominators, you need to compare their numerators. More (less) is the fraction whose numerator is greater (less). For example, compare fractions: rule. Improper and mixed fractions are always greater than any proper fraction. A proper fraction is, by definition, less than 1, so improper and mixed fractions (having a number equal to or greater than 1) are greater than a proper fraction. rule. Of two mixed fractions, the larger (less) one is the one in which the integer part of the fraction is larger (less). When the integer parts of mixed fractions are equal, the fraction with the larger (less) fractional part is greater (less). For example, compare fractions: Similar to the comparison of natural numbers on the number axis, a large fraction is to the right of a smaller fraction. Fractions are usually compared to find out which is larger and which is smaller. To compare fractions, you need to bring them to the same denominator, then the fraction with a large numerator is large, and with a smaller one, it is smaller. The hardest part is figuring out how to make fractions have the same denominator, but it's not as hard as it sounds. We'll show you how to do it all. Read on! Find out if the denominators of fractions are the same or not. The denominator is the number below the fraction line, at the bottom, and the numerator is at the top. For example, the fractions 5/7 and 9/13 do not have the same denominator. You need to bring them to the same denominator. Find a common denominator. To compare fractions, you first need to find a common denominator. This is necessary for comparison, as well as for performing mathematical operations with fractions, addition, subtraction, and so on. In the case of addition or subtraction, it is necessary to look for the lowest common denominator. However, in this case (comparison of fractions), you can only multiply the denominators of both fractions, and the resulting number will be a common denominator. Remember, this way of finding the common denominator ONLY works when comparing fractions (not addition, subtraction, etc.) Change the numerators of fractions. Once you find the common denominator, in this case 91, you will need to change the numerators so that the value of the fraction stays the same. To do this, you need to multiply the numerators of one fraction by the denominator of the second, and the numerator of the second by the denominator of the first. Like this: This article deals with the comparison of fractions. Here we will find out which of the fractions is greater or less, apply the rule, and analyze examples of the solution. Compare fractions with the same and different denominators. Let's compare an ordinary fraction with a natural number. When comparing fractions with the same denominators, we work only with the numerator, which means we compare fractions of a number. If there is a fraction 3 7 , then it has 3 parts 1 7 , then the fraction 8 7 has 8 such parts. In other words, if the denominator is the same, the numerators of these fractions are compared, that is, 3 7 and 8 7 the numbers 3 and 8 are compared. This implies the rule for comparing fractions with the same denominators: of the available fractions with the same indicators, the larger one is considered to be the one whose numerator is larger and vice versa. This suggests that you should pay attention to the numerators. To do this, consider an example. Example 1 Compare the given fractions 65 126 and 87 126 . Solution
Since the denominators of the fractions are the same, let's move on to the numerators. From the numbers 87 and 65 it is obvious that 65 is less. Based on the rule for comparing fractions with the same denominators, we have that 87126 is greater than 65126. Answer: 87 126 > 65 126 . The comparison of such fractions can be compared with the comparison of fractions with the same exponents, but there is a difference. Now we need to reduce the fractions to a common denominator. If there are fractions with different denominators, to compare them you need: Let's take a look at these steps with an example. Example 2 Compare fractions 5 12 and 9 16 . Solution
The first step is to bring the fractions to a common denominator. This is done in this way: the LCM is found, that is, the least common divisor, 12 and 16. This number is 48. It is necessary to inscribe additional factors to the first fraction 5 12, this number is found from the quotient 48: 12 = 4, for the second fraction 9 16 - 48: 16 = 3. Let's write it down like this: 5 12 = 5 4 12 4 = 20 48 and 9 16 = 9 3 16 3 = 27 48. After comparing the fractions, we get that 20 48< 27 48 . Значит, 5 12 меньше 9 16 . Answer: 5 12 < 9 16 .
There is another way to compare fractions with different denominators. It is performed without reduction to a common denominator. Let's look at an example. To compare fractions a b and c d, we reduce to a common denominator, then b · d, that is, the product of these denominators. Then the additional factors for fractions will be the denominators of the neighboring fraction. This is written as a · d b · d and c · b d · b . Using the rule with the same denominators, we have that the comparison of fractions has been reduced to comparisons of the products a · d and c · b. From here we get the rule for comparing fractions with different denominators: if a d > b c, then a b > c d, but if a d< b · c , тогда a b < c d . Рассмотрим сравнение с разными знаменателями. Example 3 Compare fractions 5 18 and 23 86. Solution
This example has a = 5 , b = 18 , c = 23 and d = 86 . Then it is necessary to calculate a · d and b · c . It follows that a d = 5 86 = 430 and b c = 18 23 = 414 . But 430 > 414 , then the given fraction 5 18 is greater than 23 86 . Answer: 5 18 > 23 86 .
If the fractions have the same numerator and different denominators, then you can perform the comparison according to the previous paragraph. The result of the comparison is possible when comparing their denominators. There is a rule for comparing fractions with the same numerators :
Of two fractions with the same numerator, the larger fraction is the one with the smaller denominator, and vice versa. Let's look at an example. Example 4 Compare fractions 54 19 and 54 31. Solution
We have that the numerators are the same, which means that a fraction with a denominator of 19 is greater than a fraction that has a denominator of 31. This is clear from the rule. Answer: 54 19 > 54 31 . Otherwise, you can consider an example. There are two plates on which 1 2 pies, anna another 1 16 . If you eat 1 2 pies, you will get full faster than just 1 16. Hence the conclusion that largest denominator with the same numerators is the smallest when comparing fractions. A comparison of an ordinary fraction with a natural number is the same as a comparison of two fractions with the denominators written in the form 1. Let's take a look at an example below for more details. Example 4 It is necessary to perform a comparison 63 8 and 9 . Solution
It is necessary to represent the number 9 as a fraction 9 1 . Then we have the need to compare fractions 63 8 and 9 1 . This is followed by reduction to a common denominator by finding additional factors. After that, we see that we need to compare fractions with the same denominators 63 8 and 72 8 . Based on the comparison rule, 63< 72 , тогда получаем 63 8 < 72 8 . Значит, заданная дробь меньше целого числа 9 , то есть имеем 63 8 < 9 . Answer: 63 8 < 9 . If you notice a mistake in the text, please highlight it and press Ctrl+EnterComparing fractions with the same numerator
Comparison of proper, improper and mixed fractions with each other.
Steps
Comparing fractions with the same denominators
Comparing fractions with different denominators
Comparing fractions with the same numerator
Comparing a fraction with a natural number
