Dividing a common fraction by a number. Dividing a fraction by a fraction

Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of those actions was bringing fractions to a 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 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:

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 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, 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. 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 correct solution to the previous problem looks like this:

Correct solution:

As you can see, the correct answer turned out to be not so beautiful. In general, be careful.

A fraction is one or more parts of a whole, which is 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 of fractions.

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 operations with decimals are 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 divide a decimal fraction by a natural number, long division is used;
  
• A comma is placed in a quotient when the division of the whole part of the dividend is completed.
  

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.

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 with different denominators are multiplied.

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 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 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 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 of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the 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-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.

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.

To solve various problems from mathematics and physics courses, 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:

1. The top part of the fraction is called the numerator, and the bottom part is called the denominator.
2. 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 divide 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.

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 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 how 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 this action you will end up with a very large 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:

• First, convert both numbers to 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 is the fastest way to divide and will take you seconds after a few practice sessions. 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, 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.