To solve various tasks from a course in mathematics and physics you have to divide fractions. This is very easy to do if you know certain rules for performing this mathematical operation.
Before we move on to formulating the rule for dividing fractions, let's remember some mathematical terms:
- The top part of the fraction is called the numerator, and the bottom part is called the denominator.
- When dividing, numbers are called as follows: dividend: divisor = quotient
How to divide fractions: simple fractions
To divide two simple fractions, multiply the dividend by the reciprocal of the divisor. This fraction is also called inverted because it is obtained by swapping the numerator and denominator. For example:
3/77: 1/11 = 3 /77 * 11 /1 = 3/7
How to divide fractions: mixed fractions
If we have to divide mixed fractions, then everything here is also quite simple and clear. First, we convert the mixed fraction to a regular improper fraction. To do this, multiply the denominator of such a fraction by an integer and add the numerator to the resulting product. As a result, we received a new numerator of the mixed fraction, but its denominator will remain unchanged. Further, the division of fractions will be carried out in exactly the same way as the division of simple fractions. For example:
10 2/3: 4/15 = 32/3: 4/15 = 32/3 * 15 /4 = 40/1 = 40
How to divide a fraction by a number
In order to divide a simple fraction by a number, the latter should be written as a fraction (irregular). This is very easy to do: this number is written in place of the numerator, and the denominator of such a fraction is equal to one. Further division is performed in the usual way. Let's look at this with an example:
5/11: 7 = 5/11: 7/1 = 5/11 * 1/7 = 5/77
How to divide decimals
Often an adult has difficulty dividing a whole number or a decimal fraction by a decimal fraction without the help of a calculator.
So to do the division decimals, you just need to cross out the comma in the divisor and stop paying attention to it. In the dividend, the comma must be moved to the right exactly as many places as it was in the fractional part of the divisor, adding zeros if necessary. And then they perform the usual division by an integer. To make this more clear, consider the following example.
Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult moment in those actions was reducing fractions to common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's look at simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.
Designation:
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, because there was no pair for it, we take it out of the limits of multiplication. The result is a negative fraction.
Task. Find the meaning of the expression:
We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the entire notation more accurate.
Reducing fractions on the fly
Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the meaning of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what remains of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example complete reduction It was not possible to achieve this, but the total amount of calculations still decreased.
However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.
There are simply no other reasons for reducing fractions, so the right decision the previous task looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
) and denominator by denominator (we get the denominator of the product).
Formula for multiplying fractions:
For example:
Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.
Dividing a common fraction by a fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper fractions;
- multiplying the numerators and denominators of fractions;
- reduce the fraction;
- If you get an improper fraction, then we convert the improper fraction into a mixed fraction.
Pay attention! To multiply a mixed fraction by another mixed fraction, you first need to convert 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 may be more convenient to use the second method of multiplying a common fraction by a number.
Pay attention! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example given above, 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.
Multistory fractions.
In high school, three-story (or more) fractions are often encountered. Example:
To bring such a fraction to its usual form, use division through 2 points:
Pay attention! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Please 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 when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.
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. Multi-storey fractional expressions we bring them into ordinary form, using division through 2 points.
5. Divide a unit by a fraction in your head, simply turning the fraction over.
T lesson type: ONZ (discovery of new knowledge - using the technology of the activity-based teaching method).
Main goals:
- Deduce methods for dividing a fraction by a natural number;
- Develop the ability to divide a fraction by a natural number;
- Repeat and reinforce division of fractions;
- Train the ability to reduce fractions, analyze and solve problems.
Equipment demonstration material:
1. Tasks for updating knowledge:
Compare expressions:
Reference:
2. Trial (individual) task.
1. Perform division:
2. Perform division without performing the entire chain of calculations: .
Standards:
- When dividing a fraction by a natural number, you can multiply the denominator by that number, but leave the numerator the same.
- If the numerator is divisible by a natural number, then when dividing a fraction by this number, you can divide the numerator by the number and leave the denominator the same.
Lesson progress
I. Motivation (self-determination) to educational activities.
Purpose of the stage:
- Organize the updating of requirements for the student in terms of educational activities (“must”);
- Organize student activities to establish thematic frameworks (“I can”);
- Create conditions for the student to develop an internal need for inclusion in educational activities (“I want”).
Organization of the educational process at stage I.
Hello! I'm glad to see you all at the math lesson. I hope it's mutual.
Guys, what new knowledge did you acquire in the last lesson? (Divide fractions).
Right. What helps you do division of fractions? (Rule, properties).
Where do we need this knowledge? (In examples, equations, problems).
Well done! You did well on the assignments in the last lesson. Do you want to discover new knowledge yourself today? (Yes).
Then - let's go! And the motto of the lesson will be the statement “You can’t learn mathematics by watching your neighbor do it!”
II. Updating knowledge and fixing individual difficulties in a trial action.
Purpose of the stage:
- Organize the updating of learned methods of action sufficient to build new knowledge. Record these methods verbally (in speech) and symbolically (standard) and generalize them;
- Organize the actualization of mental operations and cognitive processes, sufficient for the construction of new knowledge;
- Motivate for a trial action and its independent implementation and justification;
- Present individual assignment for a trial action and analyze it in order to identify new educational content;
- Organize fixation of the educational goal and topic of the lesson;
- Organize the implementation of a trial action and fix the difficulty;
- Organize an analysis of the responses received and record individual difficulties in performing a trial action or justifying it.
Organization of the educational process at stage II.
Frontally, using tablets (individual boards).
1. Compare expressions:
(These expressions are equal)
What interesting things did you notice? (The numerator and denominator of the dividend, the numerator and denominator of the divisor in each expression increased by the same number of times. Thus, the dividends and divisors in the expressions are represented by fractions that are equal to each other).
Find the meaning of the expression and write it down on your tablet. (2)
How can I write this number as a fraction?
How did you perform the division action? (Children recite the rule, the teacher hangs it on the board letter designations)
2. Calculate and record the results only:
3. Add up the results and write down the answer. (2)
What is the name of the number obtained in task 3? (Natural)
Do you think you can divide a fraction by a natural number? (Yes, we'll try)
Try this.
4. Individual (trial) task.
Perform division: (example a only)
What rule did you use to divide? (According to the rule of dividing fractions by fractions)
Now divide the fraction by a natural number greater than in a simple way, without performing the entire chain of calculations: (example b). I'll give you 3 seconds for this.
Who couldn't complete the task in 3 seconds?
Who did it? (No such thing)
Why? (We don't know the way)
What did you get? (Difficulty)
What do you think we will do in class? (Divide fractions by natural numbers)
That's right, open your notebooks and write down the topic of the lesson: “Dividing a fraction by a natural number.”
Why does this topic sound new when you already know how to divide fractions? (Need a new way)
Right. Today we will establish a technique that simplifies the division of a fraction by a natural number.
III. Identifying the location and cause of the problem.
Purpose of the stage:
- Organize the restoration of completed operations and record (verbal and symbolic) the place - step, operation - where the difficulty arose;
- Organize the correlation of students’ actions with the method (algorithm) used and fixation in external speech of the cause of the difficulty - that specific knowledge, skills or abilities that are lacking to solve the initial problem of this type.
Organization of the educational process at stage III.
What task did you have to complete? (Divide a fraction by a natural number without going through the entire chain of calculations)
What caused you difficulty? (Couldn't decide for short time fast way)
What goal do we set for ourselves in the lesson? (Find quick way dividing a fraction by a natural number)
What will help you? (Already known rule for dividing fractions)
IV. Building a project for getting out of a problem.
Purpose of the stage:
- Clarification of the project goal;
- Choice of method (clarification);
- Determination of means (algorithm);
- Building a plan to achieve the goal.
Organization of the educational process at stage IV.
Let's return to the test task. You said you divided according to the rule for dividing fractions? (Yes)
To do this, replace the natural number with a fraction? (Yes)
What step (or steps) do you think can be skipped?
(The solution chain is open on the board: 
Analyze and draw a conclusion. (Step 1)
If there is no answer, then we lead you through questions:
Where did the natural divisor go? (Into the denominator)
Has the numerator changed? (No)
So which step can you “omit”? (Step 1)
Action plan:
- Multiply the denominator of a fraction by a natural number.
- We do not change the numerator.
- We get a new fraction.
V. Implementation of the constructed project.
Purpose of the stage:
- Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
- Organize the recording of the constructed method of action in speech and signs (using a standard);
- Organize the solution to the initial problem and record how to overcome the difficulty;
- Organize clarification general new knowledge.
Organization of the educational process at stage V.
Now run the test case in a new way quickly.
Now you were able to complete the task quickly? (Yes)
Explain how you did this? (Children talk)
This means that we have gained new knowledge: the rule for dividing a fraction by a natural number.
Well done! Say it in pairs.
Then one student speaks to the class. We fix the rule-algorithm verbally and in the form of a standard on the board.
Now enter the letter designations and write down the formula for our rule.
The student writes on the board, saying the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, but leave the numerator the same.
(Everyone writes the formula in their notebooks).
Now analyze the chain of solving the test task again, paying special attention to the answer. What did you do? (The numerator of the fraction 15 was divided (reduced) by the number 3)
What is this number? (Natural, divisor)
So how else can you divide a fraction by a natural number? (Check: if the numerator of a fraction is divisible by this natural number, then you can divide the numerator by this number, write the result in the numerator of the new fraction, and leave the denominator the same)
Write this method down as a formula. (The student writes the rule on the board while pronouncing it. Everyone writes the formula in their notebooks.)
Let's go back to the first method. You can use it if a:n? (Yes it is general method)
And when is it convenient to use the second method? (When the numerator of a fraction is divided by a natural number without a remainder)
VI. Primary consolidation with pronunciation in external speech.
Purpose of the stage:
- Organize children’s assimilation of a new method of action when solving standard problems with their pronunciation in external speech (frontally, in pairs or groups).
Organization of the educational process at stage VI.
Calculate in a new way:
- No. 363 (a; d) - performed at the board, pronouncing the rule.
- No. 363 (e; f) - in pairs with checking according to the sample.
VII. Independent work with self-test according to the standard.
Purpose of the stage:
- Organize students’ independent completion of tasks for a new way of action;
- Organize self-test based on comparison with the standard;
- Based on the results of execution independent work organize reflection on the assimilation of a new way of action.
Organization of the educational process at stage VII.
Calculate in a new way:
- No. 363 (b; c)
Students check against the standard and mark the correctness of execution. The causes of errors are analyzed and errors are corrected.
The teacher asks those students who made mistakes, what is the reason?
At this stage, it is important that each student independently checks their work.
VIII. Inclusion in the knowledge system and repetition.
Purpose of the stage:
- Organize the identification of the boundaries of application of new knowledge;
- Organize repetition of educational content necessary to ensure meaningful continuity.
Organization of the educational process at stage VIII.
Organization of the educational process at stage IX.
1. Dialogue:
Guys, what new knowledge have you discovered today? (Learned how to divide a fraction by a natural number in a simple way)
Formulate a general method. (They say)
In what way and in what cases can you use it? (They say)
What is the advantage of the new method?
Have we achieved our lesson goal? (Yes)
What knowledge did you use to achieve your goal? (They say)
Did everything work out for you?
What were the difficulties?
2. Homework: clause 3.2.4.; No. 365(l, n, o, p); No. 370.
3. Teacher: I’m glad that everyone was active today and managed to find a way out of the difficulty. And most importantly, they were not neighbors when opening a new one and establishing it. Thanks for the lesson, kids!
You can do everything with fractions, including division. This article shows the division of ordinary fractions. Definitions will be given and examples will be discussed. Let us dwell in detail on dividing fractions by natural numbers and vice versa. Dividing a common fraction by a mixed number will be discussed.
Dividing fractions
Division is the inverse of multiplication. When dividing, the unknown factor is found at famous work and another factor, where its given meaning is preserved with ordinary fractions.
If it is necessary to divide a common fraction a b by c d, then to determine such a number you need to multiply by the divisor c d, this will ultimately give the dividend a b. Let's get a number and write it a b · d c , where d c is the inverse of the c d number. Equalities can be written using the properties of multiplication, namely: a b · d c · c d = a b · d c · c d = a b · 1 = a b, where the expression a b · d c is the quotient of dividing a b by c d.
From here we obtain and formulate the rule for dividing ordinary fractions:
Definition 1
To divide a common fraction a b by c d, you need to multiply the dividend by the reciprocal of the divisor.
Let's write the rule in the form of an expression: a b: c d = a b · d c
The rules of division come down to multiplication. To stick with it, you need to have a good understanding of multiplying fractions.
Let's move on to considering the division of ordinary fractions.
Example 1
Divide 9 7 by 5 3. Write the result as a fraction.
Solution
The number 5 3 is the reciprocal fraction 3 5. It is necessary to use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 = 9 7 · 3 5 = 9 · 3 7 · 5 = 27 35.
Answer: 9 7: 5 3 = 27 35 .
When reducing fractions, separate out the whole part if the numerator is greater than the denominator.
Example 2
Divide 8 15: 24 65. Write the answer as a fraction.
Solution
To solve, you need to move from division to multiplication. Let's write it in this form: 8 15: 24 65 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
It is necessary to make a reduction, and this is done as follows: 8 65 15 24 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
Select the whole part and get 13 9 = 1 4 9.
Answer: 8 15: 24 65 = 1 4 9 .
Dividing an extraordinary fraction by a natural number
We use the rule for dividing a fraction by a natural number: to divide a b by a natural number n, you only need to multiply the denominator by n. From here we get the expression: a b: n = a b · n.
The division rule is a consequence of the multiplication rule. Therefore the presentation natural number in the form of a fraction will give an equality of this type: a b: n = a b: n 1 = a b · 1 n = a b · n.
Consider this division of a fraction by a number.
Example 3
Divide the fraction 16 45 by the number 12.
Solution
Let's apply the rule for dividing a fraction by a number. We obtain an expression of the form 16 45: 12 = 16 45 · 12.
Let's reduce the fraction. We get 16 45 12 = 2 2 2 2 (3 3 5) (2 2 3) = 2 2 3 3 3 5 = 4 135.
Answer: 16 45: 12 = 4 135 .
Dividing a natural number by a fraction
The division rule is similar O the rule for dividing a natural number by an ordinary fraction: in order to divide a natural number n by an ordinary fraction a b, it is necessary to multiply the number n by the reciprocal of the fraction a b.
Based on the rule, we have n: a b = n · b a, and thanks to the rule of multiplying a natural number by an ordinary fraction, we get our expression in the form n: a b = n · b a. It is necessary to consider this division with an example.
Example 4
Divide 25 by 15 28.
Solution
We need to move from division to multiplication. Let's write it in the form of the expression 25: 15 28 = 25 28 15 = 25 28 15. Let's reduce the fraction and get the result in the form of the fraction 46 2 3.
Answer: 25: 15 28 = 46 2 3 .
Dividing a fraction by a mixed number
When dividing a common fraction by a mixed number, you can easily begin to divide common fractions. You need to convert a mixed number to an improper fraction.
Example 5
Divide the fraction 35 16 by 3 1 8.
Solution
Since 3 1 8 is a mixed number, let's represent it as an improper fraction. Then we get 3 1 8 = 3 8 + 1 8 = 25 8. Now let's divide fractions. We get 35 16: 3 1 8 = 35 16: 25 8 = 35 16 8 25 = 35 8 16 25 = 5 7 2 2 2 2 2 2 2 (5 5) = 7 10
Answer: 35 16: 3 1 8 = 7 10 .
Dividing a mixed number is done in the same way as ordinary numbers.
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