In everyday life, we often have to compare fractional quantities. Most often this does not cause any difficulties. Indeed, everyone understands that half an apple is larger than a quarter. But when it comes to writing it down as a mathematical expression, it can get confusing. By applying the following mathematical rules, you can easily solve this problem.
How to compare fractions with the same denominators
Such fractions are most convenient to compare. In this case, use the rule:
Of two fractions with the same denominators but different numerators, the larger is the one whose numerator is larger, and the smaller is the one whose numerator is smaller.
For example, compare the fractions 3/8 and 5/8. The denominators in this example are equal, so we apply this rule. 3<5 и 3/8 меньше, чем 5/8.
Indeed, if you cut two pizzas into 8 slices, then 3/8 of a slice is always less than 5/8.
Comparing fractions with like numerators and unlike denominators
In this case, the sizes of the denominator shares are compared. The rule to be applied is:
If two fractions have equal numerators, then the fraction whose denominator is smaller is greater.
For example, compare the fractions 3/4 and 3/8. In this example, the numerators are equal, which means we use the second rule. The fraction 3/4 has a smaller denominator than the fraction 3/8. Therefore 3/4>3/8
Indeed, if you eat 3 slices of pizza divided into 4 parts, you will be more full than if you ate 3 slices of pizza divided into 8 parts.
Comparing fractions with different numerators and denominators
We apply the third rule:
Comparing fractions with different denominators should lead to comparing fractions with the same denominators. To do this, you need to reduce the fractions to a common denominator and use the first rule.
For example, you need to compare fractions and . To determine the larger fraction, we reduce these two fractions to a common denominator:
- Now let's find the second additional factor: 6:3=2. We write it above the second fraction:
Comparing fractions. In this article we will look at various methods using which you can compare two fractions. I recommend looking at all fractions and studying them sequentially.
Before showing the standard algorithm for comparing fractions, let's look at some cases in which, immediately looking at an example, we can tell which fraction will be larger. There is no particular complexity here, a little analytics and everything is ready. Look at the following fractions:
In line (1) you can immediately determine which fraction is larger, in line (2) this is difficult to do, and here we apply the “standard” (or it can be called the most frequently used) approach for comparison.
The first method is analytical.
1. We have two fractions:
The numerators are equal, the denominators are unequal. Which one is bigger? The answer is obvious! The one with the smaller denominator is larger, that is, three seventeenths. Why? Simple question: What is more - one tenth of something or one thousandth? Of course, one tenth.
It turns out that with equal numerators, the fraction with the smaller denominator is larger. It doesn’t matter whether the numerators are units or other equal numbers, the essence does not change.
Additionally, you can add the following example:
Which of these fractions is greater (x is a positive number)?
Based on the information already presented, it is not difficult to draw a conclusion.
*The denominator of the first fraction is smaller, which means it is larger.
2. Now consider the option when in one of the fractions the numerator is greater than the denominator. Example:
It is clear that the first fraction is greater than one, since the numerator is greater than the denominator. And the second fraction is less than one, so without calculations and transformations we can write:
3. When comparing some ordinary improper fractions, it is clearly visible that one of them has a larger whole part. For example:
In the first fraction the integer part is equal to three, and in the second one, therefore:
4. In some examples it is also clearly visible which fraction is larger, for example:
It can be seen that the first fraction is less than 0.5. Why? To put it in detail:
and the second is more than 0.5:
Therefore, you can put a comparison sign:
Method two. "Standard" comparison algorithm.
Rule! To compare two fractions, the denominators must be equal. Then the comparison is carried out by numerators. The fraction with the larger numerator will be larger.
*This is the main IMPORTANT RULE that is used to compare fractions.
If two fractions with unequal denominators are given, then it is necessary to reduce them to such a form that they are equal. Fractions are used for this.
Let's compare the following fractions (the denominators are unequal):
Let's list them:
How to convert fractions to equal denominators? Very simple! We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first.
More examples:
Please note that it is not necessary to calculate the denominator (it is clear that they are equal); for comparison it is enough to calculate only the numerators.
*All the fractions that we considered above (the first method) can also be compared using this approach.
We could end here... But there is another “win-win” way of comparison.
Method three. Column division.
Look at the example:
Agree that in order to bring to a common denominator and then compare the numerators, it is necessary to perform relatively voluminous calculations. We use the following approach - we perform division by column:
As soon as we detect a difference in the result, the division process can be stopped.
Conclusion: since 0.12 is greater than 0.11, the second fraction will be larger. This way you can do this with all fractions.
That's all.
Sincerely, Alexander.
Let's continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow a beginner to feel like a scientist in a white coat.
The essence of comparing fractions is to find out which of two fractions is greater or less.
To answer the question which of two fractions is greater or less, use such as more (>) or less (<).
Mathematicians have already taken care of ready-made rules that allow them to immediately answer the question of which fraction is larger and which is smaller. These rules can be safely applied.
We will look at all these rules and try to figure out why this happens.
Lesson contentComparing fractions with the same denominators
The fractions that need to be compared are different. The best case is when the fractions have the same denominators, but different numerators. In this case, the following rule applies:
Of two fractions with the same denominator, the fraction with the larger numerator is greater. And accordingly, the fraction with the smaller numerator will be smaller.
For example, let's compare fractions and answer which of these fractions is larger. Here the denominators are the same, but the numerators are different. The fraction has a greater numerator than the fraction. This means the fraction is greater than . So we answer. You must answer using the more icon (>)
This example can be easily understood if we remember about pizzas, which are divided into four parts. There are more pizzas than pizzas:
Everyone will agree that the first pizza is bigger than the second.
Comparing fractions with the same numerators
The next case we can get into is when the numerators of the fractions are the same, but the denominators are different. For such cases, the following rule is provided:
Of two fractions with the same numerators, the fraction with the smaller denominator is greater. And accordingly, the fraction whose denominator is larger is smaller.
For example, let's compare the fractions and . These fractions have the same numerators. A fraction has a smaller denominator than a fraction. This means that the fraction is greater than the fraction. So we answer:
This example can be easily understood if we remember about pizzas, which are divided into three and four parts. There are more pizzas than pizzas:
Everyone will agree that the first pizza is bigger than the second.
Comparing fractions with different numerators and different denominators
It often happens that you have to compare fractions with different numerators and different denominators.
For example, compare fractions and . To answer the question which of these fractions is greater or less, you need to bring them to the same (common) denominator. Then you can easily determine which fraction is greater or less.
Let's bring the fractions to the same (common) denominator. Let's find the LCM of the denominators of both fractions. LCM of the denominators of the fractions and this is the number 6.
Now we find additional factors for each fraction. Let's divide the LCM by the denominator of the first fraction. LCM is the number 6, and the denominator of the first fraction is the number 2. Divide 6 by 2, we get an additional factor of 3. We write it above the first fraction:
Now let's find the second additional factor. Let's divide the LCM by the denominator of the second fraction. LCM is the number 6, and the denominator of the second fraction is the number 3. Divide 6 by 3, we get an additional factor of 2. We write it above the second fraction:
Let's multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to compare such fractions. Of two fractions with the same denominator, the fraction with the larger numerator is greater:
The rule is the rule, and we will try to figure out why it is more than . To do this, select the whole part in the fraction. There is no need to highlight anything in the fraction, since the fraction is already proper.
After isolating the integer part in the fraction, we obtain the following expression:
Now you can easily understand why more than . Let's draw these fractions as pizzas:
2 whole pizzas and pizzas, more than pizzas.
Subtraction of mixed numbers. Difficult cases.
When subtracting mixed numbers, you can sometimes find that things aren't going as smoothly as you'd like. It often happens that when solving an example, the answer is not what it should be.
When subtracting numbers, the minuend must be greater than the subtrahend. Only in this case will a normal answer be received.
For example, 10−8=2
10 - decrementable
8 - subtrahend
2 - difference
The minuend 10 is greater than the subtrahend 8, so we get the normal answer 2.
Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2
5—decreasable
7 - subtrahend
−2 — difference
In this case, we go beyond the limits of the numbers we are accustomed to and find ourselves in the world of negative numbers, where it is too early for us to walk, and even dangerous. To work with negative numbers, we need appropriate mathematical training, which we have not yet received.
If, when solving subtraction examples, you find that the minuend is less than the subtrahend, then you can skip such an example for now. It is permissible to work with negative numbers only after studying them.
The situation is the same with fractions. The minuend must be greater than the subtrahend. Only in this case will it be possible to get a normal answer. And in order to understand whether the fraction being reduced is greater than the fraction being subtracted, you need to be able to compare these fractions.
For example, let's solve the example.
This is an example of subtraction. To solve it, you need to check whether the fraction being reduced is greater than the fraction being subtracted. more than
so we can safely return to the example and solve it:
Now let's solve this example
We check whether the fraction being reduced is greater than the fraction being subtracted. We find that it is less:
In this case, it is wiser to stop and not continue further calculation. Let's return to this example when we study negative numbers.
It is also advisable to check mixed numbers before subtraction. For example, let's find the value of the expression .
First, let's check whether the mixed number being mined is greater than the mixed number being subtracted. To do this, we convert mixed numbers to improper fractions:
We received fractions with different numerators and different denominators. To compare such fractions, you need to bring them to the same (common) denominator. We will not describe in detail how to do this. If you have difficulty, be sure to repeat.
After reducing the fractions to the same denominator, we obtain the following expression:
Now you need to compare fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the fraction with the larger numerator is greater.
The fraction has a greater numerator than the fraction. This means that the fraction is greater than the fraction.
This means that the minuend is greater than the subtrahend
This means we can return to our example and safely solve it:
Example 3. Find the value of an expression
Let's check whether the minuend is greater than the subtrahend.
Let's convert mixed numbers to improper fractions:
We received fractions with different numerators and different denominators. Let us reduce these fractions to the same (common) denominator.
Of two fractions with the same denominators, the one with the larger numerator is greater, and the one with the smaller numerator is smaller.. In fact, the denominator shows how many parts one whole value was divided into, and the numerator shows how many such parts were taken.
It turns out that we divided each whole circle by the same number 5 , but they took different numbers of parts: the more they took, the larger fraction they got.
Of two fractions with the same numerators, the one with the smaller denominator is greater, and the one with the larger denominator is smaller. Well, in fact, if we divide one circle into 8
parts, and the other on 5
parts and take one part from each of the circles. Which part will be larger? 
Of course, from a circle divided by 5 parts! Now imagine that they were dividing not circles, but cakes. Which piece would you prefer, or rather, which share: a fifth or an eighth?
To compare fractions with different numerators and different denominators, you must reduce the fractions to their lowest common denominator and then compare fractions with the same denominators.
Examples. Compare common fractions:
Let's reduce these fractions to their lowest common denominator. NOZ(4 ;
6)=12. We find additional factors for each of the fractions. For the 1st fraction an additional factor 3
(12:
4=3
). For the 2nd fraction an additional factor 2
(12:
6=2
). Now we compare the numerators of the two resulting fractions with the same denominators. Since the numerator of the first fraction is less than the numerator of the second fraction ( 9<10)
, then the first fraction itself is less than the second fraction.
Not only can prime numbers be compared, but fractions too. 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, there will be a larger piece of cake.
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.
