) 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.
Note! 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.
Note! 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:
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 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 of fractions, go to the type 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.
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.
In contact with
Fractional Expressions long time 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.
The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was 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 with the same denominators are multiplied. 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 resulting number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.
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 a way of representing a mixed fraction as an improper fraction, and can also be represented as a 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 improper fraction to its denominator with 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 of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the denominators - the 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 easy 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.
The topic of arithmetic operations with fractions 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 the most difficult 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.
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 another 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 is a 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 of fractions, we move on 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 definitely need to solve. 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.
A fraction is one or more parts of a whole, usually taken to be one (1). As with natural numbers, you can perform all basic arithmetic operations (addition, subtraction, division, multiplication) with fractions; to do this, you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fraction has its own specifics, but once you thoroughly understand how to handle them, you will be able to solve any examples with fractions, since you will know the basic principles of performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by a whole number using different types fractions
How to divide a simple fraction by a natural number?Ordinary or simple fractions are fractions that are written in the form of a ratio of numbers in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated at the bottom. How to divide such a fraction by a whole number? Let's look at an example! Let's say we need to divide 8/12 by 2.
To do this we must perform a number of actions:
Thus, if we are faced with the task of dividing a fraction by a whole number, the solution diagram will look something like this: 
In a similar way, you can divide any ordinary (simple) fraction by an integer.
How to divide a decimal by a whole number?
A decimal is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic with decimals is quite simple.
Let's look at an example of how to divide a fraction by a whole number. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.
To summarize, let us dwell on two main points that are important when performing the operation of dividing decimal fractions by an integer: - for separation decimal Column division is used for a natural number;
- A comma is placed in a quotient when the division of the whole part of the dividend is completed.
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 with the known product of 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
We 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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