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

What is a fraction?

It so happened historically that fractional numbers appeared due to the need to measure. As practice shows, there are often examples for determining the length of a segment, the volume of a rectangular rectangle.

Initially, students are introduced to such a concept as a share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of a watermelon. This one part of eight is called a share.

A share equal to ½ of any value is called a half; ⅓ - third; ¼ - a quarter. Entries like 5 / 8 , 4 / 5 , 2 / 4 are called ordinary fractions. An ordinary fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A fractional bar can be drawn as either a horizontal or a slanted line. In this case, it stands for the division sign.

The denominator represents how many equal shares the value, object is divided into; and the numerator is how many equal shares are taken. The numerator is written above the fractional bar, the denominator below it.

It is most convenient to show ordinary fractions on a coordinate ray. If a single segment is divided into 4 equal parts, each part is designated with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.

Varieties of fractions

Fractions are common, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.

Under proper fraction understand a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer part and a fractional part. For example, 1½. 1 - integer part, ½ - fractional. However, if you need to perform some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.

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

As for this expression, they understand a record in which any number is represented, the denominator of the fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the whole part in decimal notation will be equal to zero.

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

Example. Represent the fraction 7 21 / 1000 in decimal notation.

Algorithm for converting an improper fraction to a mixed number and vice versa

It is incorrect to write down an improper fraction in the answer of the problem, so it must be converted to a mixed number:

• 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, with the denominator remaining unchanged.

Example. Convert improper fraction to mixed number: 47 / 5 .

Solution. 47: 5. The incomplete quotient is 9, the remainder = 2. Hence, 47 / 5 = 9 2 / 5.

Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:

• the integer 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.

Example. Express the number in mixed form as an improper fraction: 9 8 / 10 .

Solution. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.

Answer: 98 / 10.

Multiplication of ordinary fractions

You can perform various algebraic operations on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the 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. AT without fail the resulting expression should be simplified as much as possible. Of course, it cannot be said that an improper fraction in the answer is a mistake, but it is also difficult to call it the correct answer.

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 notation is obtained. Both the numerator and the denominator in this case are divisible by 4, and the result is the answer 5 / 9.

Multiplying decimal fractions

The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:

• two decimal fractions must be written under each other so that the rightmost digits are one under the other;
• you need to multiply the written numbers, despite the commas, that is, as natural numbers;
• count the number of digits after the comma in each of the numbers;
• in the result obtained after multiplication, you need to count as many digital characters on the right as are contained in the sum in both factors after the decimal point, and put a separating sign;
• if there are fewer digits in the product, then so many zeros must be written in front of them to cover this number, put a comma and assign an integer part equal to zero.

Example. Calculate the product of two decimals: 2.25 and 3.6.

Solution.

Multiplication of mixed fractions

To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:

• convert mixed numbers to improper fractions;
• find the product of numerators;
• find the product of the denominators;
• write down the result;
• simplify the expression as much as possible.

Example. Find the product of 4½ and 6 2 / 5.

Multiplying a number by a fraction (fractions by a number)

In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply by a fraction.

So, to find the work 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 the work, despite the comma;
• in the result obtained, separate the integer part from the fractional part using a comma, counting to the right the number of characters that is after the decimal point in the fraction.

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

Example. Calculate the product of 5 / 8 and 12.

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 reduce the resulting result and convert the incorrect fractional expression into a mixed number.

Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the result as much as possible.

Example. Find the product of 9 5 / 6 and 9.

Solution. 9 5 / 6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45 / 6 \u003d 81 + 7 3 / 6 \u003d 88 1 / 2.

Answer: 88 1 / 2.

Multiplication by factors 10, 100, 1000 or 0.1; 0.01; 0.001

The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digit characters as there are zeros in the multiplier after one.

Example 1. Find the product of 0.065 and 1000.

Solution. 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2. Find the product of 3.9 and 1000.

Solution. 3.9 x 1000 = 3.900 x 1000 = 3900.

Answer: 3900.

If you need to multiply natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma to the left in the resulting product by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written in front of a natural number.

Example 1. Find the product of 56 and 0.01.

Solution. 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2. Find the product of 4 and 0.001.

Solution. 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

So, finding the product of various fractions should not cause difficulties, except perhaps the calculation of the result; In this case, you simply cannot do without a calculator.

) and the denominator by the denominator (we get the denominator of the product).

Fraction multiplication formula:

For example:

Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:

Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper fraction to a mixed one.

Note! To multiply a mixed fraction by another mixed fraction, you first need to bring 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 is more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the above example, 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.

Multilevel fractions.

In high school, three-story (or more) fractions are often found. Example:

To bring such a fraction to its usual form, division through 2 points is used:

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 in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

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. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.

5. We divide the unit into a fraction in our mind, simply by 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 some object not entirely, but in separate pieces. The beginning of the study of this topic - share. Shares are equal parts into which an object is divided. After all, it is not always possible to express, for example, the length or price of a product as an integer; one should take into account parts or shares of any measure. Formed from the verb "to crush" - to divide into parts, and having Arabic roots, in the VIII century the word "fraction" itself appeared in Russian.

Fractional expressions have long been considered the most difficult section of mathematics. In the 17th century, when first textbooks in mathematics appeared, they were called "broken numbers", which was very difficult to display in people's understanding.

The modern form of simple fractional residues, parts of which are separated precisely by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His writings are dated 1202. But the purpose of this article is to simply and clearly explain to the reader how the multiplication of mixed fractions with different denominators occurs.

Multiplying fractions with different denominators

Initially, it is necessary to determine varieties of fractions:

• correct;
• wrong;
• 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 multiplying 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 existing ones initially.

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 bar will be the 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 ways to reduce fractional expressions. You can reduce only the numbers of the numerator with the numbers of the denominator; adjacent factors above or below the fractional bar cannot be reduced.

Along with simple fractional numbers, 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 the multiplication of a number by ordinary fractional part, you can write down the rule for this action by the formula:

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 option for solving the multiplication of a number by a fractional remainder. You just need to divide the denominator by this number:

d* e/f = e/f: d.

It is useful to use this technique when the denominator is divided by a natural number without a remainder or, as they say, completely.

Convert mixed numbers to improper fractions and get 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 to represent a mixed fraction as an improper fraction, it can also be represented as general formula:

a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and adding it to the numerator of the original fractional remainder, and the denominator remains the same.

This process also works in reverse. To select the integer part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator with a “corner”.

Multiplication of improper fractions produced in the usual way. When the entry goes under a single fractional line, as necessary, you need to reduce the fractions in order to reduce the numbers using this method and it is easier to calculate the result.

There are many helpers on the Internet to solve even complex problems. math problems in various programs. A sufficient number of such services offer their help 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 is not difficult to work with it, the corresponding fields are filled in on the site page, the sign of the mathematical action is selected and “calculate” is pressed. The program counts automatically.

The topic of arithmetic operations with fractional numbers is relevant throughout the education of middle and senior schoolchildren. In high school, they are no longer considering 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 gives full confidence in the successful solution of the most complex tasks.

In conclusion, it makes sense to cite the words of Leo Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his own merits, but anyone can decrease his denominator - his opinion of himself, and by this decrease come closer to his perfection.