The general rule for comparing fractions. Comparison of ordinary fractions
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.
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Comparing fractions with the same denominators
Compare fractions with 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. Lesson objectives: Lesson steps: 1. Organizational. Let's start the lesson with the words of the French writer A. France: "Learning can be fun .... To digest knowledge, you need to absorb it with appetite." Let's follow this advice, try to be attentive, let's absorb knowledge with great desire, because. they will be useful to us in the future. 2. Actualization of students' knowledge. 1.) Frontal oral work of students. Purpose: to repeat the material covered, which is required when learning a new one: A) regular and improper fractions; (Files are being worked on. Students have them available at every lesson. Answers are written on them with a marker, and then unnecessary information is erased.) Tasks for oral work. 1. Name an extra fraction among the chain: A) 5/6; 1/3; 7/10; 11/3; 4/7. 2. Bring fractions to a new denominator 30: 1/2; 2/3; 4/5; 5/6; 1/10. Find the smallest common denominator of fractions: 1/5 and 2/7; 3/4 and 1/6; 2/9 and 1/2. 2.) Game situation. Guys, our familiar clown (the students met him at the beginning of the school year) asked me to help him solve the problem. But I think you guys can help our friend without me. And the next task. “Compare fractions: a) 1/2 and 1/6; Guys, to help the clown, what should we learn? The purpose of the lesson, tasks (students formulate independently). The teacher helps them by asking questions: a) which of the pairs of fractions can we already compare? b) what tool do we need to compare fractions? 3. Guys in groups (in permanent multilevel). Each group is given a task and instructions for its implementation. First group :
Compare mixed fractions: a) 1 1/2 and 2 5/6; and derive a rule for equating mixed fractions with the same and different integer parts. Instruction: Comparing mixed fractions (using a number beam) Second group: Compare fractions with different denominators and different numerators. (use number beam) a) 6/7 and 9/14; Instruction Third group: Comparison of fractions with one. a) 2/3 and 1; Instruction Consider all cases: (use number ray) a) If the numerator of a fraction is equal to the denominator, ………; Formulate a rule. Fourth group: Compare fractions: a) 5/8 and 3/8; Instruction Use the number beam. Compare the numerators and draw a conclusion, starting with the words: “From two fractions with the same denominators……”. Fifth group: Compare fractions: a) 1/6 and 1/3; 0__.__.__1/6__.__.__1/3__.__.4/9__.__.__.__.__.__.__.__.__.__1__.__.__.__.__.__4/3__.__ Formulate a rule for comparing fractions with the same numerators. Instruction Compare the denominators and draw a conclusion, starting with the words: “From two fractions with the same numerators………..”. Sixth group: Compare fractions: a) 4/3 and 5/6; b) 7/2 and 1/2 using number line 0__.__.__1/2__.__5/6__1__.__4/3__.__.__.__.__.__.__.__.__.__.__.__.__7/2__.__ Formulate a rule for comparing proper and improper fractions. Instruction. Think about which fraction is always greater, right or wrong. 4. Discussion of the conclusions made in groups. Word to each group. Formulation of the rules of students and their comparison with the standards of the corresponding rules. Next, printouts of the rules for comparing various types of ordinary fractions are given to each student. 5. We return to the task set at the beginning of the lesson. (We solve the clown problem together). 6. Work in notebooks. Using the rules for comparing fractions, students, under the guidance of a teacher, compare fractions: a) 8/13 and 8/25; (it is possible to invite a student to the board). 7. Students are invited to perform a test comparing fractions for two options. 1 option. 1) compare fractions: 1/8 and 1/12 a) 1/8 > 1/12; 2) Which is bigger: 5/13 or 7/13? a) 5/13; 3) Which is smaller: 2/3 or 4/6? a) 2/3; 4) Which of the fractions is less than 1: 3/5; 17/9; 7/7? a) 3/5; 5) Which of the fractions is greater than 1: ?; 7/8; 4/3? a) 1/2; 6) Compare fractions: 2 1/5 and 1 7/9 a) 2 1/5<1 7/9; Option 2. 1) compare fractions: 3/5 and 3/10 a) 3/5 > 3/10; 2) Which is bigger: 10/12 or 1/12? a) are equal; 3) Which is smaller: 3/5 or 1/10? a) 3/5; 4) Which of the fractions is less than 1: 4/3; 1/15; 16/16? a) 4/3; 5) Which of the fractions is greater than 1: 2/5; 9/8; 11/12? a) 2/5; 6) Compare fractions: 3 1/4 and 3 2/3 a) 3 1/4 = 3 2/3; Answers to the test: Option 1: 1a, 2b, 3c, 4a, 5b, 6a Option 2: 2a, 2b, 3b, 4b, 5b, 6c 8. Once again we return to the purpose of the lesson. We check the comparison rules and give a differentiated homework: 1,2,3 groups - come up with two examples for each rule and solve them. 4,5,6 groups - No. 83 a, b, c, No. 84 a, b, c (from the textbook). The rules for comparing ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the significant (same or different) of the compared fractions. This section discusses options for comparing fractions that have the same numerator or denominator. 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. To compare proper fractions with the same numerators, you need to compare their denominators. More (less) is the fraction whose denominator is less (greater). 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). Compare two fractions- means to determine which of the fractions is greater, which is less, or to establish that the fractions are equal. Of two fractions with the same denominator, the larger fraction is the one with the larger numerator. Example. A fraction is greater than a fraction, because the fractions in both fractions are the same, but the first fraction has more of them than the second. If we represent the unit as a segment and divide it into 8 shares, then it is easy to see that the fraction is larger: Of two fractions with the same numerator, the fraction with the smaller denominator is larger. Example. A fraction is larger than a fraction, because the number of shares in both fractions is the same, but the shares in the first fraction are larger than in the second. Let's draw two units in the form of circles, divide one into 4 parts, the second into 6 parts. Now you can see that the fraction is larger: To compare fractions that have different numerators and denominators, you need to bring them to a common denominator. After that, they are compared according to the rule for comparing fractions that have the same denominators. Example. Compare fractions: and . Solution: Now we compare them according to the rule for comparing fractions that have the same denominators. Since , it means . Here is another way to compare fractions with different denominators and numerators. Consider first a numerical example. Example. Compare fractions and . Solution: We bring these fractions to a common denominator: Solving this example, you can see that, after reducing the fractions to a common denominator, the comparison problem was actually reduced to comparing the products 2 7 and 4 3. Since 2 7 \u003d 14, and 4 3 \u003d 12, then 2 7 > 4 3. Hence, . Now let's solve the same problem in general view using alphabetic notation. Example. Let fractions and be given, where a And c- zero or natural numbers, b And d- integers. Let's bring the fractions to a common denominator: Hence: Thus, we got the following rule for comparing ordinary fractions: To compare two ordinary fractions, you can multiply the numerator of one fraction by the denominator of the other and compare the resulting products. This rule is called cross rule for comparing fractions. Any proper fraction is less than any natural number. Example. Comparing an improper fraction with a natural number comes down to comparing two fractions. To compare an improper fraction with a natural number, you need to represent the natural number as an improper fraction with a denominator of 1, then they can be compared in one of two ways: using the cross rule, or reduce the fractions to a common denominator. After that, they are compared according to the rule for comparing fractions that have the same denominators. We continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow the beginner to feel like a scientist in a white coat. The essence of comparing fractions is to find out which of the two fractions is greater or less. To answer the question which of the two fractions is greater or less, use such as more (>) or less (<). Mathematicians have already taken care of ready-made rules that allow you 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. The fractions to be compared come across different. The most successful case is when fractions have the same denominators, but different numerators. In this case, the following rule applies: Of two fractions with the same denominator, the larger fraction is the one with the larger numerator. And accordingly, the smaller fraction will be, in which the numerator is smaller. For example, let's compare fractions and and answer which of these fractions is greater. Here the denominators are the same, but the numerators are different. A fraction has a larger numerator than a fraction. So the fraction is greater than . So we answer. Reply using the more icon (>) This example can be easily understood if we think about pizzas that are divided into four parts. more pizzas than pizzas: 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 numerator, the fraction with the smaller denominator is larger. The fraction with the larger denominator is therefore smaller. For example, let's compare fractions and . These fractions have the same numerator. A fraction has a smaller denominator than a fraction. So the fraction is greater than the fraction. So we answer: This example can be easily understood if we think about pizzas that are divided into three and four parts. more pizzas than pizzas: Everyone will agree that the first pizza is bigger than the second one. 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 it will be easy to determine which fraction is greater or less. Let's bring the fractions to the same (common) denominator. Find (LCM) the denominators of both fractions. The LCM of the denominators of the fractions and that number is 6. Now we find additional factors for each fraction. 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 over the first fraction: Now let's find the second additional factor. 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 over the second fraction: 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 denominators, the larger fraction is the one with the larger numerator: The rule is the rule, and we will try to figure out why more than . To do this, select the integer part in the fraction. There is no need to select anything in the fraction, since this fraction is already correct. After selecting the integer part in the fraction, we get the following expression: Now you can easily understand why more than . Let's draw these fractions in the form of pizzas: 2 whole pizzas and pizzas, more than pizzas. When subtracting mixed numbers, sometimes you find that things don't go as smoothly as you'd like. It often happens that when solving some 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 response be received. For example, 10−8=2 10 - reduced 8 - subtracted 2 - difference The minus 10 is greater than the subtracted 8, so we got the normal answer 2. Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2 5 - reduced 7 - subtracted −2 is the difference In this case, we go beyond the usual numbers for us 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, you need the appropriate mathematical background, which we have not received yet. If, when solving examples for subtraction, 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 it will be possible to get a normal answer. And in order to understand whether the reduced fraction is greater than the subtracted one, you need to be able to compare these fractions. For example, let's solve an example. This is a subtraction example. To solve it, you need to check whether the reduced fraction is greater than the subtracted one. more than so we can safely return to the example and solve it: Now let's solve this example Check if the reduced fraction is greater than the subtracted one. We find that it is less: In this case, it is more reasonable to stop and not continue further calculation. We will return to this example when we study negative numbers. It is also desirable to check mixed numbers before subtracting. For example, let's find the value of the expression . First, check whether the reduced mixed number is greater than the subtracted one. To do this, we translate mixed numbers into improper fractions: We got 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're having trouble, be sure to repeat. After reducing the fractions to the same denominator, we get the following expression: Now we need to compare fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the larger fraction is the one with the larger numerator. A fraction has a larger numerator than a fraction. So the fraction is greater than the fraction. This means that the minuend is greater than the subtrahend. So we can go back to our example and boldly solve it: Example 3 Find the value of an expression Check if the minuend is greater than the subtrahend. Convert mixed numbers to improper fractions: We got fractions with different numerators and different denominators. We bring these fractions to the same (common) denominator: Now let's compare the fractions and . A fraction has a numerator less than a fraction, so the fraction is smaller than the fractionComparing fractions with the same numerator
B) bringing fractions to a new denominator;
C) finding the lowest common denominator;
B) 2/6; 6/18; 1/3; 4/5; 4/12.
b) 3/5 and 1/3;
c) 5/6 and 1/6;
d) 12/7 and 4/7;
e) 3 1/7 and 3 1/5;
f) 7 5/6 and 3 1/2;
g) 1/10 and 1;
h) 10/3 and 1;
i) 7/7 and 1.”
b) 3 1/2 and 3 4/5
b) 5/11 and 1/22
b) 8/7 and 1;
c) 10/10 and 1 and formulate a rule.
b) If the numerator of a fraction is less than the denominator,………;
c) If the numerator of a fraction is greater than the denominator,………. .
b) 1/7 and 4/7 and formulate a rule for comparing fractions with the same denominator.
b) 4/9 and 4/3 using the number line:
b) 11/42 and 3/42;
c) 7/5 and 1/5;
d) 18/21 and 7/3;
e) 2 1/2 and 3 1/5;
f) 5 1/2 and 5 4/3;
b) 1/8<1/12;
c) 1/8=1/12
b) 7/13;
c) are equal
b) 4/6;
c) are equal
b) 17/9;
c) 7/7
b) 7/8;
c) 4/3
b) 2 1/5 = 1 7/9;
c) 2 1/5 >1 7/9
b) 3/5<3/10;
c) 3/5=3/10
b) 10/12;
c) 1/12
b) 1/10;
c) are equal
b) 1/15;
c) 16/16
b) 9/8;
c) 11/12
b) 3 1/4 > 3 2/3;
c) 3 1/4< 3 2/3
Comparison of proper, improper and mixed fractions with each other
Comparing fractions with the same denominators
Comparing fractions with the same numerator
Comparing fractions with different denominators and numerators
Comparing a fraction with a natural number
Comparing fractions with the same denominators
Comparing fractions with the same numerator
Comparing fractions with different numerators and different denominators
Subtraction of mixed numbers. Difficult cases.
