Multiplying a mixed fraction by a common fraction. Multiplying fractions

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. Good news is that these operations are even simpler than addition and subtraction. First, let's look at 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 complete reduction It was not possible to achieve this, 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 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.

Multiplying a whole number by a fraction is not a difficult task. But there are subtleties that you probably understood at school, but have since forgotten.

How to multiply a whole number by a fraction - a few terms

If you remember what a numerator and denominator are and how a proper fraction differs from an improper fraction, skip this paragraph. It is for those who have completely forgotten the theory.

The numerator is top part fractions are what we divide. The denominator is lower. This is what we divide by.
A proper fraction is one whose numerator is less than its denominator. An improper fraction is one whose numerator is greater than or equal to its denominator.

How to multiply a whole number by a fraction

The rule for multiplying an integer by a fraction is very simple - we multiply the numerator by the integer, but do not touch the denominator. For example: two multiplied by one fifth - we get two fifths. Four multiplied by three sixteenths equals twelve sixteenths.


Reduction

In the second example, the resulting fraction can be reduced.
What does it mean? Please note that both the numerator and denominator of this fraction are divisible by four. Dividing both numbers by a common divisor is called reducing the fraction. We get three quarters.


Improper fractions

But suppose we multiply four by two fifths. It turned out to be eight-fifths. This is an improper fraction.
She definitely needs to be brought to the right kind. To do this, you need to select an entire part from it.
Here you need to use division with a remainder. We get one and three as a remainder.
One whole and three fifths is our proper fraction.

Bringing thirty-five eighths to the correct form is a little more difficult. The closest number to thirty-seven that is divisible by eight is thirty-two. When divided we get four. Subtract thirty-two from thirty-five and we get three. Result: four whole and three eighths.


Equality of numerator and denominator. And here everything is very simple and beautiful. If the numerator and denominator are equal, the result is simply one.

To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.

Multiplying a common fraction by a fraction.

To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.

\(\bf \frac(a)(b) \times \frac(c)(d) = \frac(a \times c)(b \times d)\\\)

Let's look at an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.

\(\frac(6)(7) \times \frac(2)(3) = \frac(6 \times 2)(7 \times 3) = \frac(12)(21) = \frac(4 \ times 3)(7 \times 3) = \frac(4)(7)\\\)

The fraction \(\frac(12)(21) = \frac(4 \times 3)(7 \times 3) = \frac(4)(7)\\\) was reduced by 3.

Multiplying a fraction by a number.

First, let's remember the rule, any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .

Let's use this rule when multiplying.

\(5 \times \frac(4)(7) = \frac(5)(1) \times \frac(4)(7) = \frac(5 \times 4)(1 \times 7) = \frac (20)(7) = 2\frac(6)(7)\\\)

Improper fraction \(\frac(20)(7) = \frac(14 + 6)(7) = \frac(14)(7) + \frac(6)(7) = 2 + \frac(6)( 7)= 2\frac(6)(7)\\\) converted to a mixed fraction.

In other words, When multiplying a number by a fraction, we multiply the number by the numerator and leave the denominator unchanged. Example:

\(\frac(2)(5) \times 3 = \frac(2 \times 3)(5) = \frac(6)(5) = 1\frac(1)(5)\\\\\) \(\bf \frac(a)(b) \times c = \frac(a \times c)(b)\\\)

Multiplying mixed fractions.

To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. We multiply the numerator with the numerator, and multiply the denominator with the denominator.

Example:
\(2\frac(1)(4) \times 3\frac(5)(6) = \frac(9)(4) \times \frac(23)(6) = \frac(9 \times 23) (4 \times 6) = \frac(3 \times \color(red) (3) \times 23)(4 \times 2 \times \color(red) (3)) = \frac(69)(8) = 8\frac(5)(8)\\\)

Multiplication of reciprocal fractions and numbers.

The fraction \(\bf \frac(a)(b)\) is the inverse of the fraction \(\bf \frac(b)(a)\), provided a≠0,b≠0.
The fractions \(\bf \frac(a)(b)\) and \(\bf \frac(b)(a)\) are called reciprocal fractions. The product of reciprocal fractions is equal to 1.
\(\bf \frac(a)(b) \times \frac(b)(a) = 1 \\\)

Example:
\(\frac(5)(9) \times \frac(9)(5) = \frac(45)(45) = 1\\\)

Related questions:
How to multiply a fraction by a fraction?
Answer: The product of ordinary fractions is the multiplication of a numerator with a numerator, a denominator with a denominator. To get the product of mixed fractions, you need to convert them into an improper fraction and multiply according to the rules.

How to multiply fractions with different denominators?
Answer: it doesn’t matter whether they are the same or different denominators For fractions, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.

How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction into an improper fraction and then find the product using the rules of multiplication.

How to multiply a number by a fraction?
Answer: we multiply the number with the numerator, but leave the denominator the same.

Example #1:
Calculate the product: a) \(\frac(8)(9) \times \frac(7)(11)\) b) \(\frac(2)(15) \times \frac(10)(13)\ )

Solution:
a) \(\frac(8)(9) \times \frac(7)(11) = \frac(8 \times 7)(9 \times 11) = \frac(56)(99)\\\\ \)
b) \(\frac(2)(15) \times \frac(10)(13) = \frac(2 \times 10)(15 \times 13) = \frac(2 \times 2 \times \color( red) (5))(3 \times \color(red) (5) \times 13) = \frac(4)(39)\)

Example #2:
Calculate the products of a number and a fraction: a) \(3 \times \frac(17)(23)\) b) \(\frac(2)(3) \times 11\)

Solution:
a) \(3 \times \frac(17)(23) = \frac(3)(1) \times \frac(17)(23) = \frac(3 \times 17)(1 \times 23) = \frac(51)(23) = 2\frac(5)(23)\\\\\)
b) \(\frac(2)(3) \times 11 = \frac(2)(3) \times \frac(11)(1) = \frac(2 \times 11)(3 \times 1) = \frac(22)(3) = 7\frac(1)(3)\)

Example #3:
Write the reciprocal of the fraction \(\frac(1)(3)\)?
Answer: \(\frac(3)(1) = 3\)

Example #4:
Calculate the product of two mutually inverse fractions: a) \(\frac(104)(215) \times \frac(215)(104)\)

Solution:
a) \(\frac(104)(215) \times \frac(215)(104) = 1\)

Example #5:
Can reciprocal fractions be:
a) simultaneously with proper fractions;
b) simultaneously improper fractions;
c) simultaneously natural numbers?

Solution:
a) to answer the first question, let's give an example. The fraction \(\frac(2)(3)\) is proper, its inverse fraction will be equal to \(\frac(3)(2)\) - an improper fraction. Answer: no.

b) in almost all enumerations of fractions this condition is not met, but there are some numbers that fulfill the condition of being simultaneously an improper fraction. For example, the improper fraction is \(\frac(3)(3)\), its inverse fraction is equal to \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions when the numerator and denominator are equal.

c) natural numbers are numbers that we use when counting, for example, 1, 2, 3, …. If we take the number \(3 = \frac(3)(1)\), then its inverse fraction will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not natural number. If we go through all the numbers, the reciprocal of the number is always a fraction, except for 1. If we take the number 1, then its reciprocal fraction will be \(\frac(1)(1) = \frac(1)(1) = 1\). Number 1 is a natural number. Answer: they can simultaneously be natural numbers only in one case, if this is the number 1.

Example #6:
Do the product of mixed fractions: a) \(4 \times 2\frac(4)(5)\) b) \(1\frac(1)(4) \times 3\frac(2)(7)\)

Solution:
a) \(4 \times 2\frac(4)(5) = \frac(4)(1) \times \frac(14)(5) = \frac(56)(5) = 11\frac(1 )(5)\\\\ \)
b) \(1\frac(1)(4) \times 3\frac(2)(7) = \frac(5)(4) \times \frac(23)(7) = \frac(115)( 28) = 4\frac(3)(7)\)

Example #7:
Can two mutually reciprocal numbers be mixed numbers at the same time?

Let's look at an example. Let's take a mixed fraction \(1\frac(1)(2)\), find its inverse fraction, to do this we convert it into an improper fraction \(1\frac(1)(2) = \frac(3)(2) \) . Its inverse fraction will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two fractions that are mutually inverse cannot be mixed numbers at the same time.

In the course of secondary and high school Students studied the topic “Fractions”. However, this concept is much broader than what is 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?

Historically, fractional numbers arose out of the need to measure. As practice shows, there are often examples of determining the length of a segment and the volume of a rectangular rectangle.

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

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

The denominator represents how many equal parts the quantity or object is divided into; and the numerator is how many identical shares are taken. The numerator is written above the fraction 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, each part is designated by a Latin letter, then the result can be excellent visual material. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of a given segment.

Types of fractions

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

A proper fraction is a number whose numerator is less than its denominator. Accordingly, an improper fraction is a number whose numerator is greater than its denominator. The second type is usually written as a mixed number. This expression consists of an integer and a fractional part. For example, 1½. 1 is an integer part, ½ is a fractional part. However, if you need to carry out 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, we mean a record in which any number is represented, the denominator of the fractional expression of which can be expressed in terms of one with several zeros. If the fraction is proper, then the whole part is decimal notation will be equal to zero.

To write a decimal fraction, you must first write the whole part, separate it from the fraction using a comma, and then write the fraction expression. It must be remembered that after the decimal point the numerator must contain the same number of digital characters as there are zeros in the denominator.

Example. Express 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 an improper fraction in the answer to a problem, so it needs to be converted to a mixed number:

  • divide the numerator by the existing denominator;
  • V specific example incomplete quotient - whole;
  • 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 partial quotient is 9, the remainder = 2. So, 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. Represent 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.

Multiplying fractions

Various algebraic operations can be performed on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, multiplying fractions with different denominators is no different from the product fractional numbers with the same denominators.

It happens that after finding the result you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, one cannot say that an improper fraction in an answer is an error, but it is also difficult to call it a 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 divided 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 one under the 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 decimal point in each number;
  • in the result obtained after multiplication, you need to count from the right as many digital symbols as are contained in the sum in both factors after the decimal point, and put a separating sign;
  • if there are fewer numbers in the product, then you need to write as many zeros in front of them to cover this number, put a comma and add the whole part equal to zero.

Example. Calculate the product of two decimal fractions: 2.25 and 3.6.

Solution.

Multiplying mixed fractions

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

  • convert mixed numbers into improper fractions;
  • find the product of the numerators;
  • find the product of 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 and mixed numbers, there are tasks where you need to multiply by a fraction.

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

  • write the number under the fraction so that the rightmost digits are one above the other;
  • find the product despite the comma;
  • in the resulting result, separate the integer part from the fractional part using a comma, counting from the right the number of digits that are located after the decimal point in the fraction.

To multiply common fraction to a number, you should find the product of the numerator and the natural factor. If the answer produces a fraction that can be reduced, 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.

Multiplication of fractions also concerns finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the whole 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 resulting result as much as possible.

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

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

It follows from the previous paragraph next rule. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the decimal point to the right by as many digits as there are zeros in the factor after the 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 a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma in the resulting product to the left by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written before the 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 different fractions should not cause any difficulties, except perhaps calculating the result; in this case, you simply cannot do without a calculator.

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 fractions - go 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 must definitely complete. 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.)

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