Multiplying and dividing fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:
For example:
Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...
To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:
For example:
If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:
In high school, you often have to deal with three-story (or even four-story!) fractions. For example:
How can I make this fraction look decent? Yes, very simple! Use two-point division:
But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:
then divide and multiply in order, from left to right!
And also very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:
The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.
That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer of them (errors)!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.
2. In examples with different types fractions - go to ordinary fractions.
3. We reduce all fractions until they stop.
4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.
So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.
Calculate:
Have you decided?
We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.
0; 17/22; 3/4; 2/5; 1; 25.
Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Sooner or later, all children at school begin to learn fractions: their addition, division, multiplication and all the possible operations that can be performed with fractions. In order to provide proper assistance to the child, parents themselves should not forget how to divide integers into fractions, otherwise you will not be able to help him in any way, but will only confuse him. If you needed to remember this action, but you just can’t bring all the information in your head into a single rule, then this article will help you: you will learn to divide a number by a fraction and see clear examples.
How to divide a number into a fraction
Write your example down as a rough draft so you can make notes and erasures. Remember that the integer number is written between the cells, right at their intersection, and fractional numbers are written each in its own cell.
- IN this method you need to turn the fraction upside down, that is, write the denominator into the numerator, and the numerator into the denominator.
- The division sign must be changed to multiplication.
- Now all you have to do is perform the multiplication according to the rules you have already learned: the numerator is multiplied by an integer, but you do not touch the denominator.
Of course, as a result of such an action you will get very big number in the numerator. You cannot leave a fraction in this state - the teacher simply will not accept this answer. Reduce the fraction by dividing the numerator by the denominator. Write the resulting integer to the left of the fraction in the middle of the cells, and the remainder will be the new numerator. The denominator remains unchanged.
This algorithm is quite simple, even for a child. After completing it five or six times, the child will remember the procedure and will be able to apply it to any fractions.
How to divide a number by a decimal
There are other types of fractions - decimals. The division into them occurs according to a completely different algorithm. If you encounter such an example, then follow the instructions:
- To begin, turn both numbers into decimals. This is easy to do: your divisor is already represented as a fraction, and the dividend natural number you separate with a comma to get a decimal. That is, if the dividend was 5, you get the fraction 5.0. You need to separate a number by as many digits as there are after the decimal point and divisor.
- After this, you must make both decimal fractions natural numbers. It may seem a little confusing at first, but it's the most quick way division, which will take you seconds after a few practices. The fraction 5.0 will become the number 50, the fraction 6.23 will become 623.
- Do the division. If the numbers are large, or the division will occur with a remainder, do it in a column. This way you can clearly see all the actions this example. You don't need to put a comma on purpose, as it will appear on its own during the long division process.
This type of division initially seems too confusing, since you need to turn the dividend and divisor into a fraction, and then back into natural numbers. But after a short practice, you will immediately begin to see those numbers that you simply need to divide by each other.
Remember that the ability to correctly divide fractions and whole numbers by them can come in handy many times in life, therefore, a child needs to know these rules and simple principles perfectly so that in higher grades they do not become a stumbling block because of which the child cannot solve more complex tasks.
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
Modern look simple fractional remainders, the parts of which are separated by a horizontal line, were first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions are multiplied with different denominators.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers are multiplied with same denominators. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.
When multiplying simple fractions with different denominators for two or more factors the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional line will be a product of different numbers and, naturally, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves the presentation method mixed fraction incorrectly, it can also be represented in the form general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations programs. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
Subject arithmetic operations with fractional numbers is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in successful decision most complex tasks.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
SUBJECT: Dividing fractions.
- Learning the rules for dividing fractions; Formation of basic skills in dividing fractions;
- development of basic skills to divide fractions using the basic algorithm; Development of attention logical thinking;
- nurturing interest in studying the subject and the ability to work in groups.
LESSON PLAN:
1. Organizational moment.
2. Oral work leading to a new rule.
3. Introduction of the definition.
4. Work with cards for assimilation.
5. Physical exercises.
6. Oral work “find the mistake.”
7. Pinning: chain calculations.
8. Summing up the lesson.
DURING THE CLASSES
1) Today in class, guys, we have to do some serious work. You will need perseverance, desire, attention, consistency and correctness in completing tasks.
Oral work: write the inverse of this number:
2) How can you check whether the multiplication operation is performed correctly? (By the action of division).
We don't know how fractions are divided. It's time to get acquainted with this new action.
Dividing and dividing can sometimes be difficult, so the operation of dividing fractions itself requires special attention.
Let's remember what division is as a mathematical operation? (action inverse to multiplication; action when one of the factors and the product is used to find another factor).
Now together we will try to see a rule for dividing fractions that is new to us while considering the next problem.
Now our solutions will diverge.
What suggestions do you have to solve this equation?
Firstly, we know how to solve such equations using the concept of reciprocal numbers (it is enough to multiply both sides of the equation by the inverse of the coefficient of the variable X).
Secondly, we know the standard rule for finding an unknown factor (the product must be divided by a known factor).
Let's consider both of these cases:
Look carefully at the two resulting expressions for finding the value of X. These are answers to the same problem, which means the answers must be the same. In one case we multiply by 7/6, and in the other we divide by 6/7.
We find that when divided by 6/7, the same answer should be obtained if multiplied by 7/6. This means that the meaning of dividing fractions comes down to multiplying by the reciprocal of the divisor. This is not a random feature we noticed.
Introduce the new rule on page 100 of the textbook, repeat several times, ask several students from memory.
3) Using the learned rule, consider its application in various examples .
Children receive special cards, which they fill out together with the teacher, with comments from the place. You should consider dividing a fraction by a fraction, dividing a natural number by a fraction and fractions by a natural number, dividing mixed numbers. When filling out, the children say the rule again. Pay special attention to three stages when performing division: the dividend remains unchanged; division is replaced by multiplication; multiply by the inverse of the divisor.
Division |
Application |
Rule |
Conversion |
||
| 5/7: 3/4 = | 5/7 * 4/3= | (5*4) / (7*3) = | 20/21 | 20/21 | |
| 5: 2/5 = | 5 * | ||||
| 7/8: 2 = | 7/8: 2/1= | 7/8 * | |||
| 4 1/2: 1 1/2= | 9/2: 3/2 = | 9/2 * |
On back side The cards have three tasks that children solve after filling out the cards on the spot, then check the solutions and results obtained.
DECIDE FOR YOURSELF |
| 1. 4/6: 3 = |
| 2. 8: 4/5 = |
| 3 . 1 2/3: 1 1/10 = |
4) Conducting physical exercises.
5) Stage of mastering the definition.
Let's check how you have learned today's rule and find out how attentive you are: “FIND THE ERROR”
6) Solving problems from the textbook: No. 619 (a, b, d).
7) Work in groups. Children take turns going to the board and writing down the solution to the example.
8) Well done. Well done. Let's summarize:
What new did you learn in class today?
How are fractions divided?
What are reciprocal numbers?
At home: Rule No. 617.