Then we act according to the rule: we multiply the first fraction by the fraction inverse to the second (that is, by an inverted fraction, in which the numerator and denominator are reversed). When multiplying fractions, multiply the numerator by the numerator, and the denominator by the denominator.
Consider examples for dividing mixed numbers.
We begin the division of mixed numbers by converting them to improper fractions. Then we divide the resulting fractions. To do this, multiply the first fraction by the inverted second. 20 and 25 by 5, 3 and 9 by 3. We got the wrong fraction, so it is necessary.
Convert mixed numbers to improper fractions. Further, according to the rule of dividing fractions, we leave the first number and multiply it by the reciprocal of the second. We reduce 15 and 25 by 5, 8 and 16 - by 2. From the resulting improper fraction, select the whole part.
We replace mixed numbers with improper fractions and divide them. To do this, we rewrite the first fraction without changes and multiply by the inverted second. We reduce 18 and 36 by 18, 35 and 7 by 7. The result is an improper fraction. We select from it the whole part.
In this article, we will analyze multiplication of mixed numbers. First, we will voice the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next, we will talk about the multiplication of a mixed number and a natural number. Finally, we will learn how to perform multiplication of a mixed number and common fraction.
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Multiplication of mixed numbers.
Multiplication of mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.
Let's write down multiplication rule for mixed numbers:
- First, the mixed numbers to be multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule of multiplying a fraction by a fraction.
Consider examples of applying this rule when multiplying a mixed number by a mixed number.
Example.
Perform mixed number multiplication and .
Solution.
First, we represent the multiplied mixed numbers as improper fractions: 
And 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule of multiplication of fractions, we get 
. The resulting fraction is irreducible (see reduced and irreducible fractions), but it is incorrect (see regular and improper fractions), therefore, to get the final answer, it remains to perform the extraction of the integer part from the improper fraction: .
Let's write the whole solution in one line: .
Answer:
.
To consolidate the skills of multiplying mixed numbers, consider the solution of another example.
Example.
Do the multiplication.
Solution.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it's time to remember about fraction reduction: replace all the numbers in the fraction with their expansions into prime factors, and perform the reduction of the same factors .
Answer:
Multiplication of a mixed number and a natural number
After replacing the mixed number with an improper fraction, multiplying a mixed number and a natural number is reduced to the multiplication of an ordinary fraction and a natural number.
Example.
Multiply the mixed number and the natural number 45 .
Solution.
A mixed number is a fraction, then 
. Let's replace the numbers in the resulting fraction with their expansions into prime factors, make a reduction, after which we select the integer part: .
Answer:
Multiplication of a mixed number and a natural number is sometimes conveniently done using the distributive property of multiplication with respect to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part and the given natural number and the fractional part of the given natural number, that is, 
.
Example.
Compute the product.
Lesson topic: "Multiplication and division of mixed fractions"Purpose: to develop in students the ability and skills to apply the rule of multiplication and division of mixed fractions;
development of analytical thinking of students, the formation of students' ability to highlight the main thing and generalize.
Tasks: repeat the rule of multiplication and division of ordinary fractions.
To test the ability to apply the rules of multiplication and division of ordinary fractions,
rule for multiplying a fraction by a natural number and vice versa. Test the ability to convert an improper fraction to a mixed number and vice versa.
Derive a new rule and algorithm for multiplying and dividing mixed numbers.
Work out a new rule for completing tasks.
Subject results: an algorithm for multiplying and dividing mixed fractions (reminder)
Meta-subject and personal results :
Regulatory UUD: goal setting; plan, result
Cognitive UUD: general educational, logical, problem setting and solving
Communicative UUD: work in pairs
Equipment: math textbook grade 6
Handout.
Projector.
During the classes:
I. Problem situation and updating of knowledge
1. A survey of children to repeat the studied material on the topic of multiplication and division of fractions (execution algorithm, rule for multiplying a fraction by a natural number).
2. Illustration of examples on the projector. Types of ordinary fractions. How to get a mixed fraction from an improper fraction and vice versa.
3. At the end of the survey, independent work, including examples on the multiplication and division of ordinary fractions and containing two examples on the multiplication and division of mixed fractions, where children are faced with a problem. The correct answers for checking with students are reflected on the projector.
4. Discussion of the problem. Lead to the topic of the lesson.
II. Joint discovery of knowledge.
1/ It is proposed to discuss in pairs, to voice the version of the solution to the problem. Versions write on the blackboard. How do you know which version is correct?
2/ Invite students to refer to the textbook on the relevant topic.
3 / Perform an introductory reading, find the desired paragraph and study it to compile an algorithm for multiplying and dividing mixed fractions. Control over the execution of the task.
4/ Listen to versions compose from main general algorithm. Reflect it on the projector and distribute to the students in the form of a memo.
III.Independent application of knowledge
1/Return to the problem with solving examples from independent work and using the resulting algorithm to solve them. Check in pairs. Reflect the results on the projector for verification.
2/ Give a task from the textbook. Execution control.
IV. Lesson summary
Start with the problem that arose at the beginning of the lesson, talk about the ways to solve it and the result.
Evaluation of student work.
Assignment for homework.
) and the denominator by the denominator (we get the denominator of the product).
Fraction multiplication formula:
For example:
Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.
Division of an ordinary fraction by a fraction.
Division of fractions involving a natural number.
It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:
Multiplication of mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper;
- multiply the numerators and denominators of fractions;
- we reduce the fraction;
- if we get an improper fraction, then we convert the improper fraction to a mixed one.
Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It is more convenient to use the second method of multiplying an ordinary fraction by a number.
Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the above example, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multilevel fractions.
In high school, three-story (or more) fractions are often found. Example:
To bring such a fraction to its usual form, division through 2 points is used:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.
2. In tasks with different types fractions - go to the form of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.
