### Education, Secondary education and schools

# Fraction. Multiplication of fractions of ordinary, decimal, mixed

In the course of middle and high school students were taught the topic "Fractions". However, this concept is much broader than is given in the learning process. Today the concept of a fraction occurs quite often, and not everyone can calculate any expression, for example, multiplying fractions.

## What is a fraction?

So historically, fractional numbers appeared because of the need to measure. As practice shows, there are often examples of determining the length of a segment, the volume of a rectangular parallelepiped, the area of a rectangle.

Initially, students get acquainted with such a notion as share. For example, if you divide a watermelon into 8 parts, then each will get one eighth watermelon. This one part of the eight is called the share.

A share equal to ½ of any value is called a half; ⅓ - the third; ¼ - a quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. Ordinary fraction is divided into numerator and denominator. Between them is a feature of a fraction, or a fractional line. A fractional line can be drawn as a horizontal or inclined line. In this case, it denotes the division sign.

The denominator represents how many equal parts a quantity is divided by an object; And the numerator - how many identical shares are taken. The numerator is written above the fractional line, the denominator is written below it.

It is most convenient to show ordinary fractions on a coordinate ray. If a single segment is divided into 4 equal parts, designate each share with a Latin letter, then as a result, you can get an excellent visual aid. Thus, point A shows a fraction equal to _{1/4} of the whole unit interval, and point B marks _{2/8} of the given segment.

## Variation of fractions

Fractions are ordinary, decimal, and also mixed numbers. In addition, fractions can be divided into correct and incorrect. This classification is more suitable for ordinary fractions.

A correct fraction is a number whose numerator is less than the denominator. Accordingly, the irregular fraction is a number whose numerator is greater than the denominator. The second kind is usually written in the form of a mixed number. This expression consists of an integer and a fractional part. For example, 1½. 1 - integer part, ½ - fractional. However, if you need to do some manipulation with the expression (dividing or multiplying fractions, reducing them or converting them), the mixed number is converted to the wrong fraction.

The correct fractional expression is always less than one, and the incorrect fractional expression is greater than or equal to 1.

As for decimals, this expression means a record in which any number is represented, the denominator of a fractional expression of which can be expressed in terms of a unit with several zeros. If the fraction is correct, then the integer part in the decimal notation will be zero.

To write a decimal, you must first write the whole part, separate it from fractional with a comma and then write a fractional expression. It must be remembered that, after a comma, the numerator must contain as many numeric characters as the zeros in the denominator.

*An example* . Provide a fraction of 7 _{21/1000} in the decimal notation.

## The algorithm for converting an irregular fraction to a mixed number and vice versa

To record the wrong fraction in the answer of the problem is incorrect, therefore it needs to be translated into a mixed number:

- To divide the numerator by the existing denominator;
- In a specific example, an incomplete quotient is an integer;
- And the remainder is the numerator of the fractional part, and the denominator remains unchanged.

*An example* . Transfer the wrong fraction to a mixed number: ^{47/5} . ** **

*The solution* . 47: 5. The partial partial equals 9, the remainder = 2. Hence, ^{47/5} = 9 _{2/5} .

Sometimes it is necessary to present a mixed number as an irregular fraction. Then you need to use the following algorithm:

- The whole part is multiplied by the denominator of the fractional expression;
- The resulting product is added to the numerator;
- The result is written in the numerator, the denominator remains unchanged.

*An example* . Represent the number in a mixed form as an irregular fraction: 9 ^{8/10} . ** **

*The solution* . 9 x 10 + 8 = 90 + 8 = 98 is the numerator.

*The answer is* : ^{98/10} _{.}

## Multiplication of fractions of ordinary

Over ordinary fractions, you can perform various algebraic operations. To multiply two numbers, the numerator must be multiplied with the numerator, and the denominator with the denominator. And multiplication of fractions with different denominators ** ** Does not differ from the product of fractional numbers with the same denominators.

It happens that after finding the result, you need to reduce the fraction. It is necessary to simplify the resulting expression as much as possible. Of course, we can not say that the wrong fraction in the answer is a mistake, but it is also difficult to call a correct answer to it.

*An example* . Find the product of two ordinary fractions: ½ and _{20/18} .

As can be seen from the example, after finding the product, a reducible fractional record was obtained. Both the numerator and denominator in this case is divided by 4, and the result is the answer _{5/9} .

## Multiplication of decimals

The product of decimal fractions differs quite a bit from the product of ordinary ones in its own way. Thus, the multiplication of fractions is as follows:

- Two decimal fractions must be written one below the other so that the rightmost digits are one under the other;
- It is necessary to multiply the recorded numbers, in spite of commas, that is, as natural numbers;
- Count the number of digits after the decimal point in each of the numbers;
- In the resulting result after multiplying the result, it is necessary to count out as many digital symbols as there are in the sum in both factors after the comma, and put the separating sign;
- If the numbers in the work were smaller, then before them you need to write as many zeros to cover this number, put a comma and assign the whole part equal to zero.

*An example* . Calculate the product of two decimal fractions: 2.25 and 3.6.

*The solution* .

## Multiplication of mixed fractions

To calculate the product of two mixed fractions, you need to use the multiplier of fractions:

- Translate numbers in mixed form into irregular fractions;
- Find the product of numerators;
- Find the product of denominators;
- Record the result;
- To simplify the expression as much as possible.

*An example* . Find the product 4½ and 6 2/5 _{.}

## Multiplication of a number by a fraction (fractions by a number)

In addition to finding the product of two fractions, mixed numbers, there are jobs where it is necessary to multiply the natural number by a fraction.

So, to find the product of a decimal fraction and a natural number, you need:

- Write the number under the fraction so that the rightmost digits are one above the other;
- Find a work, despite the comma;
- In the resulting result, to separate the whole part from the fractional part with a comma, counting the number of digits after the decimal point in the fraction.

To multiply an ordinary fraction by a number, it is necessary to find the product of the numerator and the natural factor. If the answer is a reducible fraction, it should be converted.

*An example* . Calculate the product 5/8 and 12.

*The solution* . _{5/8} * 12 = ^{(5 * 12)} / _{8} = ^{60/8} = ^{30/4} = ^{15/2} = 7 ^{1/2} _{.}

*Answer* : 7 ^{1/2} _{.}

As you can see from the previous example, it was necessary to shorten the result and convert the incorrect fractional expression to a mixed number.

Also, the multiplication of fractions concerns the finding of the product of a number in a mixed form and the natural factor. To multiply these two numbers, multiply the whole part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If required, the result should be simplified as much as possible.

*An example* . Find the product 9 _{5/6} and 9.

*The solution* . 9 _{5/6} x 9 = 9 x 9 + ^{(5 x 9)} / _{6} = 81 + ^{45/6} = 81 + 7 _{3/6} = 88 ^{1/2} _{.}

*Answer* : 88 ^{1/2} _{.}

## Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0,001

The preceding rule implies the following. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., move the comma to the right by as many digits of the digits as the zeros in the multiplier after one.

*Example 1* . Find the product of 0.065 and 1000.

*The solution* . 0.065 x 1000 = 0065 = 65.

*Answer* : 65.

*Example 2* . Find the product 3.9 and 1000.

*The solution* . 3.9 x 1000 = 3.900 x 1000 = 3900.

*Answer* : 3900.

If it is necessary to multiply the natural number and 0,1; 0.01; 0.001; 0.0001, etc., it is necessary to move left a comma in the resulting product by as many digits of digits as the number of zeros is up to one. If necessary, zeros are written before the natural number in sufficient quantity.

*Example 1* . Find the product of 56 and 0.01.

*The solution* . 56 x 0.01 = 0056 = 0.56.

*Answer* : 0.56.

*Example 2* . Find the product 4 and 0,001.

*The solution* . 4 x 0.001 = 0004 = 0.004.

*Answer* : 0.004.

So, finding the product of different fractions should not cause difficulties, except that the calculation of the result; In this case, you can not do without a calculator.

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