Dividing fractions by each other. Dividing a fraction by a natural number

Ordinary fractional numbers They first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to examine 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 multiplication occurs mixed fractions 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 easy to formulate independently: the result of multiplication simple fractions with the same 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, it can also be represented as general formula:

a bc = a*b+ c/c, where is the denominator 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 reverse side. 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 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.

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 learned basic knowledge give 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.

Sooner or later, all children at school begin to learn fractions: their addition, division, multiplication and everything possible actions, which are only possible to perform 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 need to remember this action, but you just can’t put 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 you separate the natural number being divided with a comma, getting a decimal fraction. 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 of 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, know these rules and simple principles the child needs ideally so that in higher grades they do not become a stumbling block, due to which the child cannot solve more complex problems.

) 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 multiplication rule 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 fractions - go to the form of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.

5. Divide a unit by a fraction in your head, simply turning the fraction over.

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 it is stored given meaning 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, representing a natural number as 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. Need to make a transfer mixed number into 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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Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). Most difficult moment in those actions there was reduction of 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 consider the 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 present in fractions whole part, they must be converted into incorrect 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:

1. Plus by minus gives minus;
2. 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:

1. 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;
2. 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 outside 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 by normal 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 note 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, it was not possible to achieve a complete reduction, 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. Therefore, it is impossible to apply the main property of a fraction, since in this property we're talking about specifically about multiplying numbers.

There are simply no other reasons for reducing fractions, so correct solution 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.