This article looks at comparing fractions. Here we will find out which fraction is greater or less, apply the rule, and look at examples of solutions. Let's compare fractions with both equal and different denominators. Let's compare an ordinary fraction with a natural number.
Comparing fractions with the same denominators
When comparing fractions with same denominators, we work only with the numerator, which means we compare the 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 are compared to the numbers 3 and 8.
This follows the rule for comparing fractions with the same denominators: of the existing fractions with the same exponents, the fraction with the larger numerator is considered larger and vice versa.
This suggests that you should pay attention to the numerators. To do this, let's look at an example.
Example 1
Compare the given fractions 65 126 and 87 126.
Solution
Since the denominators of the fractions are the same, we 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 87,126 is greater than 65,126.
Answer: 87 126 > 65 126 .
Comparing fractions with different denominators
Comparison of such fractions can be correlated with comparison of fractions with the same exponents, but there is a difference. Now you need to reduce the fractions to a common denominator.
If there are fractions with different denominators, to compare them you need to:
- find a common denominator;
- compare fractions.
Let's look at these actions using an example.
Example 2
Compare the fractions 5 12 and 9 16.
Solution
First of all, it is necessary to reduce the fractions to a common denominator. This is done this way: find the LCM, that is, the smallest common divisor, 12 and 16 . This number is 48. It is necessary to add 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 the result this way: 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 them to a common denominator, then b · d, that is, the product of these denominators. Then additional factors for fractions will be the denominators of the neighboring fraction. This will be written as a · d b · d and c · b d · b . Using the rule with identical 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 the 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 .
Comparing fractions with the same numerators
If the fractions have the same numerators and different denominators, then the comparison can be made according to the previous point. The result of the comparison is possible by comparing their denominators.
There is a rule for comparing fractions with the same numerators : Of two fractions with the same numerators, the fraction that has the smaller denominator is greater and vice versa.
Let's look at an example.
Example 4
Compare the 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 with a denominator of 31. This is understandable based on the rule.
Answer: 54 19 > 54 31 .
Otherwise, we can look at an example. There are two plates on which there are 1 2 pies, and another 1 16 anna. If you eat 1 2 pies, you will be full faster than just 1 16. Hence the conclusion is that the largest denominator with equal numerators is the smallest when comparing fractions.
Comparing a fraction with a natural number
Comparing an ordinary fraction with a natural number is the same as comparing two fractions with the denominators written in the form 1. For a detailed look, we give an example below.
Example 4
A comparison needs to be made between 63 8 and 9 .
Solution
It is necessary to represent the number 9 as a fraction 9 1. Then we need to compare the fractions 63 8 and 9 1. This is followed by reduction to a common denominator by finding additional factors. After this 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 .
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Comparison rules ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the denominator (same or different) of the fractions being compared.
This section discusses options for comparing fractions that have the same numerators or denominators.
Rule. To compare two fractions with the same denominators, you need to compare their numerators. Greater (less) is a fraction whose numerator is greater (less).
For example, compare fractions:
Rule. To compare proper fractions with like numerators, you need to compare their denominators. Greater (less) is a fraction whose denominator is less (greater).
For example, compare fractions: 
Comparing proper, improper and mixed fractions with each other
Rule. Improper and mixed fractions are always larger than any proper fraction.
A proper fraction is by definition less than 1, so improper and mixed fractions (those containing a number equal to or greater than 1) are greater than a proper fraction.
Rule. Of the two mixed fractions greater (less) is the one whose whole part of the fraction is greater (less). When the whole parts of mixed fractions are equal, the fraction with the larger (smaller) fractional part is greater (smaller).
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 fractions being compared. Rule. To compare two fractions with the same denominators, you need to compare their numerators. Greater (less) is a fraction whose numerator is greater (less). For example, compare fractions:
Comparing proper, improper and mixed fractions with each other.
Rule. Improper and mixed fractions are always larger than any proper fraction. A proper fraction is by definition less than 1, so improper and mixed fractions (those containing a number equal to or greater than 1) are greater than a proper fraction.
Rule. Of two mixed fractions, the one whose whole part of the fraction is greater (less) is greater (smaller). When the whole parts of mixed fractions are equal, the fraction with the larger (smaller) fractional part is greater (smaller).
For example, compare fractions:
Similar to comparing natural numbers on the number line, the larger fraction is to the right of the smaller fraction.
Not only prime numbers You can compare, but so can fractions. After all, a fraction is the same number as, for example, natural numbers. You only need to know the rules by which fractions are compared.
Comparing fractions with the same denominators.
If two fractions have the same denominators, then it is easy to compare such fractions.
To compare fractions with the same denominators, you need to compare their numerators. The fraction that has a larger numerator is larger.
Let's look at an example:
Compare the fractions \(\frac(7)(26)\) and \(\frac(13)(26)\).
The denominators of both fractions are the same and equal to 26, so we compare the numerators. The number 13 is greater than 7. We get:
\(\frac(7)(26)< \frac{13}{26}\)
Comparing fractions with equal numerators.
If a fraction has the same numerators, then the fraction with the smaller denominator is greater.
This rule can be understood by giving an example from life. We have cake. 5 or 11 guests can come to visit us. If 5 guests come, then we will cut the cake into 5 equal pieces, and if 11 guests come, then we will divide it into 11 equal pieces. Now think about in what case would there be a larger piece of cake per guest? Of course, when 5 guests arrive, the piece of cake will be larger.
Or another example. We have 20 candies. We can give the candy equally to 4 friends or divide the candy equally among 10 friends. In what case will each friend have more candies? Of course, when we divide among only 4 friends, the number of candies for each friend will be greater. Let's check this problem mathematically.
\(\frac(20)(4) > \frac(20)(10)\)
If we solve these fractions before, we get the numbers \(\frac(20)(4) = 5\) and \(\frac(20)(10) = 2\). We get that 5 > 2
This is the rule for comparing fractions with the same numerators.
Let's look at another example.
Compare fractions with the same numerator \(\frac(1)(17)\) and \(\frac(1)(15)\) .
Since the numerators are the same, the fraction with the smaller denominator is larger.
\(\frac(1)(17)< \frac{1}{15}\)
Comparing fractions with different denominators and numerators.
To compare fractions with different denominators, you need to reduce the fractions to , and then compare the numerators.
Compare the fractions \(\frac(2)(3)\) and \(\frac(5)(7)\).
First, let's find the common denominator of the fractions. It will be equal to the number 21.
\(\begin(align)&\frac(2)(3) = \frac(2 \times 7)(3 \times 7) = \frac(14)(21)\\\\&\frac(5) (7) = \frac(5 \times 3)(7 \times 3) = \frac(15)(21)\\\\ \end(align)\)
Then we move on to comparing numerators. Rule for comparing fractions with the same denominators.
\(\begin(align)&\frac(14)(21)< \frac{15}{21}\\\\&\frac{2}{3} < \frac{5}{7}\\\\ \end{align}\)
Comparison.
An improper fraction is always larger than a proper fraction. Because an improper fraction is greater than 1, and a proper fraction is less than 1.
Example:
Compare the fractions \(\frac(11)(13)\) and \(\frac(8)(7)\).
The fraction \(\frac(8)(7)\) is improper and is greater than 1.
\(1 < \frac{8}{7}\)
The fraction \(\frac(11)(13)\) is correct and it is less than 1. Let’s compare:
\(1 > \frac(11)(13)\)
We get, \(\frac(11)(13)< \frac{8}{7}\)
Related questions:
How to compare fractions with different denominators?
Answer: you need to bring the fractions to a common denominator and then compare their numerators.
How to compare fractions?
Answer: First you need to decide what category fractions belong to: they have a common denominator, they have a common numerator, they do not have a common denominator and numerator, or you have a proper and improper fraction. After classifying fractions, apply the appropriate comparison rule.
What is comparing fractions with the same numerators?
Answer: If fractions have the same numerators, the fraction with the smaller denominator is larger.
Example #1:
Compare the fractions \(\frac(11)(12)\) and \(\frac(13)(16)\).
Solution:
Since there are no identical numerators or denominators, we apply the rule of comparison with different denominators. We need to find a common denominator. The common denominator will be 96. Let's reduce the fractions to a common denominator. Multiply the first fraction \(\frac(11)(12)\) by an additional factor of 8, and multiply the second fraction \(\frac(13)(16)\) by 6.
\(\begin(align)&\frac(11)(12) = \frac(11 \times 8)(12 \times 8) = \frac(88)(96)\\\\&\frac(13) (16) = \frac(13 \times 6)(16 \times 6) = \frac(78)(96)\\\\ \end(align)\)
We compare fractions with numerators, the fraction with the larger numerator is larger.
\(\begin(align)&\frac(88)(96) > \frac(78)(96)\\\\&\frac(11)(12) > \frac(13)(16)\\\ \\end(align)\)
Example #2:
Compare a proper fraction to one?
Solution:
Any proper fraction is always less than 1.
Task #1:
The son and father were playing football. The son hit the goal 5 times out of 10 approaches. And dad hit the goal 3 times out of 5 approaches. Whose result is better?
Solution:
The son hit 5 times out of 10 possible approaches. Let's write it as a fraction \(\frac(5)(10)\).
Dad hit 3 times out of 5 possible approaches. Let's write it as a fraction \(\frac(3)(5)\).
Let's compare fractions. We have different numerators and denominators, let's reduce them to one denominator. The common denominator will be 10.
\(\begin(align)&\frac(3)(5) = \frac(3 \times 2)(5 \times 2) = \frac(6)(10)\\\\&\frac(5) (10)< \frac{6}{10}\\\\&\frac{5}{10} < \frac{3}{5}\\\\ \end{align}\)
Answer: Dad has a better result.
This article will talk about comparison of mixed numbers. First, we will figure out which mixed numbers are called equal and which are called unequal. Next we will give a rule for comparing unequal mixed numbers, which allows you to find out which number is greater and which is less, and consider examples. Finally, we'll look at how mixed numbers compare to natural numbers and fractions.
Page navigation.
Equal and unequal mixed numbers
First you need to know which mixed numbers are called equal and which are called unequal. Let us give the corresponding definitions.
Definition.
Equal mixed numbers- These are mixed numbers that have equal whole parts and fractional parts.
In other words, two mixed numbers are said to be equal if their entries are exactly the same. If the notation of mixed numbers is different, then such mixed numbers are called unequal.
Definition.
Unequal mixed numbers are mixed numbers whose notations are different.
The stated definitions allow you to determine at a glance whether the given mixed numbers are equal or not. For example, mixed numbers and equal numbers, since their notations are completely the same. These numbers have equal integer parts and equal fractional parts. And mixed numbers and are unequal, since they have unequal integer parts. Other examples of unequal mixed numbers are and , as well as and .
Sometimes it becomes necessary to find out which of two unequal mixed numbers is greater than the other and which is less. We'll look at how this is done in the next paragraph.
Comparison of mixed numbers
Comparing mixed numbers can be reduced to comparing ordinary fractions. To do this, it is enough to convert mixed numbers into improper fractions.
For example, let's compare a mixed number and a mixed number, presenting them in the form improper fractions. We have and . So comparison of original mixed numbers comes down to comparing fractions with different denominators and . Since, then.
Comparing mixed numbers by comparing equal fractions is not the best solution. It is much more convenient to use the following rule for comparing mixed numbers: greater is the mixed number whose integer part is greater, but if the integer parts are equal, then greater is the mixed number whose fractional part is greater.
Let's look at how mixed numbers are compared according to the stated rule. To do this, let's look at the solutions to the examples.
Example.
Which of the mixed numbers and greater?
Solution.
The integer parts of the mixed numbers being compared are equal, so the comparison comes down to comparing the fractional parts and . Since then 
. So a mixed number is greater than a mixed number.
Answer:
Comparison of a mixed number and a natural number
Let's figure out how to compare a mixed number and natural number.
This is fair rule for comparing a mixed number with a natural number: if the integer part of a mixed number is less than a given natural number, then the mixed number is less than a given natural number, and if the integer part of a mixed number is greater than or equal to a given mixed number, then the mixed number is greater than a given natural number.
Let's look at examples of comparing a mixed number and a natural number.
Example.
Compare the numbers 6 and .
Solution.
Whole part mixed number is 9. Since it is greater than the natural number 6, then .
Answer:
Example.
Given a mixed number and a natural number 34, which number is smaller?
Solution.
The whole part of a mixed number is less than 34 (11<34 ), поэтому .
Answer:
A mixed number is less than 34.
Example.
Compare the number 5 and a mixed number.
Solution.
The integer part of this mixed number is equal to the natural number 5, therefore, this mixed number is greater than 5.
Answer:
To conclude this point, we note that any mixed number is greater than one. This statement follows from the rule for comparing a mixed number and a natural number, and also from the fact that the integer part of any mixed number is either greater than 1 or equal to 1.
Comparison of a mixed number and a common fraction
First let's talk about comparison of a mixed number and a proper fraction. Any proper fraction is less than one (see proper and improper fractions), therefore, any proper fraction is less than any mixed number (since any mixed number is greater than 1).
