) 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.
Pay attention! 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 rule for multiplying 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.
Pay attention! To multiply a fraction by natural number It is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example 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:
Pay attention! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Please 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.
GET OVER THESE RAKES ALREADY! 🙂
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 so. ")
This operation is much more pleasant than addition and 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:
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.:
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. Take practical advice into account, and there will be fewer of them (mistakes)!
1. The most important thing when working with fractional expressions is accuracy and attentiveness! This is not common 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 of fractions, we move on 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!).
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. Right 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. And only Then look at the answers.
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, separated by 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.
All these (and more!) examples are discussed in Special Section 555 “Fractions”. With detailed explanations of what, why and how. This analysis helps a lot with a lack of knowledge and skills!
Yes, and on the second problem there is something there.) Quite practical advice, how to become more attentive. Yes, yes! Advice that can be applied every.
In addition to knowledge and attentiveness, success requires a certain automaticity. Where can I get it? I hear a heavy sigh... Yes, only in practice, nowhere else.
You can go to the website 321start.ru for training. There in the “Try” option there are 10 examples for everyone. With instant verification. For registered users - 34 examples from simple to severe. This is only in fractions.
If you like this site.
By the way, I have a couple more interesting sites for you.)
Here you can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
And here you can get acquainted with functions and derivatives.
Rule 1.
To multiply a fraction by a natural number, you need to multiply its numerator by this number and leave the denominator unchanged.
Rule 2.
To multiply a fraction by a fraction:
1. find the product of the numerators and the product of the denominators of these fractions
2. Write the first product as the numerator, and the second as the denominator.
Rule 3.
In order to multiply mixed numbers, you need to write them as improper fractions, and then use the rule for multiplying fractions.
Rule 4.
To divide one fraction by another, you must multiply the dividend by the reciprocal of the divisor.
Example 1.
Calculate
Example 2.
Calculate
Example 3.
Calculate
Example 4.
Calculate
Mathematics. Other materials
Raising a number to a rational power. (
Raising a number to a natural power. (
Generalized interval method for solving algebraic inequalities (Author A.V. Kolchanov)
Method for replacing factors when solving algebraic inequalities (Author Kolchanov A.V.)
Signs of divisibility (Lungu Alena)
Test yourself on the topic ‘Multiplication and division of ordinary fractions’
Multiplying fractions
We will consider the multiplication of ordinary fractions in several possible options.
Multiplying a common fraction by a fraction
This is the simplest case in which you need to use the following rules for multiplying fractions.
To multiply fraction by fraction, necessary:
Before multiplying numerators and denominators, check to see if the fractions can be reduced. Reducing fractions in calculations will make your calculations much easier.
Multiplying a fraction by a natural number
To make a fraction multiply by a natural number You need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.
If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, highlight the whole part.
Multiplying mixed numbers
To multiply mixed numbers, you must first turn them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.
Another way to multiply a fraction by a natural number
Sometimes when making calculations it is more convenient to use another method of multiplying a common fraction by a number.
To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.
As can be seen from the example, this version of the rule is more convenient to use if the denominator of the fraction is divisible by a natural number without a remainder.
Dividing a fraction by a number
What is the fastest way to divide a fraction by a number? Let's analyze the theory, draw a conclusion, and use examples to see how dividing a fraction by a number can be done using a new short rule.
Typically, dividing a fraction by a number follows the rule for dividing fractions. We multiply the first number (fraction) by the inverse of the second. Since the second number is an integer, its inverse is a fraction whose numerator is one and the denominator is given number. Schematically, dividing a fraction by a natural number looks like this:
From this we conclude:
To divide a fraction by a number, you need to multiply the denominator by that number and leave the numerator the same. The rule can be formulated even more briefly:
When dividing a fraction by a number, the number goes into the denominator.
Divide a fraction by a number:
To divide a fraction by a number, we rewrite the numerator unchanged, and multiply the denominator by this number. We reduce 6 and 3 by 3.
When dividing a fraction by a number, we rewrite the numerator and multiply the denominator by that number. We reduce 16 and 24 by 8.
When dividing a fraction by a number, the number goes into the denominator, so we leave the numerator the same and multiply the denominator by the divisor. We reduce 21 and 35 by 7.
Multiplying and dividing 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 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.
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.
Task. Find the meaning of the expression:
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:
- Plus by minus gives minus;
- Two negatives make an affirmative.
- 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;
- 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.
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:
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 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:
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 due to the fact that when adding the numerator of a fraction, the sum appears, and not the product of numbers. Therefore, it is impossible to apply the basic 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 the correct solution to the previous problem looks like this:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
Dividing fractions.
Dividing a fraction by a natural number.
Examples of dividing a fraction by a natural number
Dividing a natural number by a fraction.
Examples of dividing a natural number by a fraction
Division of ordinary fractions.
Examples of dividing ordinary fractions
Dividing mixed numbers.
- To divide one mixed number by another, you need to:
- convert mixed fractions to improper fractions;
- multiply the first fraction by the reciprocal of the second;
- reduce the resulting fraction;
- If you get an improper fraction, convert the improper fraction into a mixed fraction.
- 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.
- Under- and under- Reworked song “Spring Tango” (The time comes - birds fly from the south) - music. Valery Milyaev I didn’t hear enough, I didn’t understand, I didn’t get it, in the sense that I didn’t guess, I wrote all the verbs with inseparably, I didn’t know about the prefix nedo. It happens that [...]
- Page not found In the third final reading, a package of Government documents providing for the creation of special administrative regions (SAR) was adopted. As a result of leaving the European Union, the UK will not be included in the European VAT area and […]
- The Joint Investigative Committee will appear in the fall The Joint Investigative Committee will appear in the fall The investigation of all law enforcement agencies will be brought under one roof on the fourth attempt Already in the fall of 2014, according to Izvestia, President Vladimir Putin […]
- Patent for an algorithm What a patent for an algorithm looks like How a patent for an algorithm is prepared Preparation technical descriptions Methods for storing, processing, and transmitting signals and/or data specifically for patenting purposes usually do not present any special difficulties, and […]
- WHAT IS IMPORTANT TO KNOW ABOUT THE NEW BILL ON PENSIONS December 12, 1993 CONSTITUTION OF THE RUSSIAN FEDERATION (taking into account amendments made by the Laws of the Russian Federation on amendments to the Constitution of the Russian Federation dated December 30, 2008 N 6-FKZ, dated December 30, 2008 N 7-FKZ, […]
- Funny ditties about a woman's pension for the hero of the day, men for the hero of the day, men - in chorus for the hero of the day, women - dedication to pensioners, women, humorous. Competitions for pensioners will be interesting. Presenter: Dear friends! Just a moment! Sensation! Only […]
Examples of dividing mixed numbers
1 1 2: 2 2 3 = 1 2 + 1 2: 2 3 + 2 3 = 3 2: 8 3 = 3 2 3 8 = 3 3 2 8 = 9 16
2 1 7: 3 5 = 2 7 + 1 7: 3 5 = 15 7: 3 5 = 15 7 5 3 = 15 5 7 3 = 5 5 7 = 25 7 = 7 3 + 4 7 = 3 4 7
Any obscene comments will be deleted and their authors will be blacklisted!
Welcome to OnlineMSchool.
My name is Dovzhik Mikhail Viktorovich. I am the owner and author of this site, I wrote all the theoretical material, and also developed online exercises and calculators that you can use to study mathematics.
Fractions. Multiplying and dividing fractions.
Multiplying a common fraction by a fraction.
To multiply ordinary fractions, you need to multiply the numerator by the numerator (we get the numerator of the product) and the denominator by the denominator (we get the denominator of the product).
Formula for multiplying fractions:
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.
Pay attention! There is no need to look for a common denominator here!!
Dividing a common fraction by a fraction.
Dividing an ordinary fraction by a fraction occurs like this: you turn the second fraction over (i.e., change the numerator and denominator) and after that the fractions are multiplied.
Formula for dividing ordinary fractions:
Multiplying a fraction by a natural number.
Pay attention! When multiplying a fraction by a natural number, the numerator of the fraction is multiplied by our natural number, and the denominator of the fraction is left the same. If the result of the product is an improper fraction, then be sure to highlight the whole part, turning the improper fraction into a mixed fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As with addition, we convert the whole number into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
Pay attention! 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 rule for multiplying 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.
Pay attention! 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 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:
Pay attention! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Please 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 of fractions, go to the type 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.
In the middle and high school courses, students covered 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 unit segment is divided into 4 equal parts, label each part Latin letter, then the result can be 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 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 integer part in 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. Present 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. IN mandatory you need 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 a common fraction by a number, you need to 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
The following rule follows from the previous paragraph. 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 after the one in the factor.
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 difficulties, except perhaps calculating the result; in this case, you simply cannot do without a calculator.
Another operation that can be performed with ordinary fractions is multiplication. We will try to explain its basic rules when solving problems, show how an ordinary fraction is multiplied by a natural number and how to correctly multiply three ordinary fractions or more.
Let's first write down the basic rule:
Definition 1
If we multiply one common fraction, then the numerator of the resulting fraction will be equal to the product numerators of the original fractions, and the denominator is the product of their denominators. In literal form, for two fractions a / b and c / d, this can be expressed as a b · c d = a · c b · d.
Let's look at an example of how to correctly apply this rule. Let's say we have a square whose side is equal to one numerical unit. Then the area of the figure will be 1 square. unit. If we divide the square into equal rectangles with sides equal to 1 4 and 1 8 numerical units, we get that it now consists of 32 rectangles (because 8 4 = 32). Accordingly, the area of each of them will be equal to 1 32 of the area of the entire figure, i.e. 1 32 sq. units.
We have a shaded fragment with sides equal to 5 8 numerical units and 3 4 numerical units. Accordingly, to calculate its area, you need to multiply the first fraction by the second. It will be equal to 5 8 · 3 4 sq. units. But we can simply count how many rectangles are included in the fragment: there are 15 of them, which means the total area is 15 32 square units.
Since 5 3 = 15 and 8 4 = 32, we can write the following equality:
5 8 3 4 = 5 3 8 4 = 15 32
It confirms the rule we formulated for multiplying ordinary fractions, which is expressed as a b · c d = a · c b · d. It works the same for both proper and improper fractions; It can be used to multiply fractions with both different and identical denominators.
Let's look at solutions to several problems involving multiplication of ordinary fractions.
Example 1
Multiply 7 11 by 9 8.
Solution
First, let's calculate the product of the numerators of the indicated fractions by multiplying 7 by 9. We got 63. Then we calculate the product of the denominators and get: 11 · 8 = 88. Let's combine two numbers and the answer is: 63 88.
The whole solution can be written like this:
7 11 9 8 = 7 9 11 8 = 63 88
Answer: 7 11 · 9 8 = 63 88.
If we get a reducible fraction in our answer, we need to complete the calculation and perform its reduction. If we get an improper fraction, we need to separate out the whole part from it.
Example 2
Calculate product of fractions 4 15 and 55 6 .
Solution
According to the rule studied above, we need to multiply the numerator by the numerator, and the denominator by the denominator. The solution record will look like this:
4 15 55 6 = 4 55 15 6 = 220 90
We got a reducible fraction, i.e. one that is divisible by 10.
Let's reduce the fraction: 220 90 gcd (220, 90) = 10, 220 90 = 220: 10 90: 10 = 22 9. As a result, we got an improper fraction, from which we select the whole part and get a mixed number: 22 9 = 2 4 9.
Answer: 4 15 55 6 = 2 4 9.
For ease of calculation, we can also reduce the original fractions before performing the multiplication operation, for which we need to reduce the fraction to the form a · c b · d. Let us decompose the values of the variables into simple factors and reduce the same ones.
Let's explain what this looks like using data from a specific task.
Example 3
Calculate the product 4 15 55 6.
Solution
Let's write down the calculations based on the multiplication rule. We will get:
4 15 55 6 = 4 55 15 6
Since 4 = 2 2, 55 = 5 11, 15 = 3 5 and 6 = 2 3, then 4 55 15 6 = 2 2 5 11 3 5 2 3.
2 11 3 3 = 22 9 = 2 4 9
Answer: 4 15 · 55 6 = 2 4 9 .
Numeric expression, in which the multiplication of ordinary fractions takes place, has a commutative property, that is, if necessary, we can change the order of the factors:
a b · c d = c d · a b = a · c b · d
How to multiply a fraction with a natural number
Let's write down the basic rule right away, and then try to explain it in practice.
Definition 2
To multiply a common fraction by a natural number, you need to multiply the numerator of that fraction by that number. In this case, the denominator of the final fraction will be equal to the denominator of the original ordinary fraction. Multiplication of a certain fraction a b by a natural number n can be written as the formula a b · n = a · n b.
It’s easy to understand this formula if you remember that any natural number can be represented as an ordinary fraction with a denominator equal to one, that is:
a b · n = a b · n 1 = a · n b · 1 = a · n b
Let us explain our idea with specific examples.
Example 4
Calculate the product 2 27 times 5.
Solution
As a result of multiplying the numerator of the original fraction by the second factor, we get 10. By virtue of the rule stated above, we will get 10 27 as a result. The entire solution is given in this post:
2 27 5 = 2 5 27 = 10 27
Answer: 2 27 5 = 10 27
When we multiply a natural number with a fraction, we often have to abbreviate the result or represent it as a mixed number.
Example 5
Condition: calculate the product 8 by 5 12.
Solution
According to the rule above, we multiply the natural number by the numerator. As a result, we get that 5 12 8 = 5 8 12 = 40 12. The final fraction has signs of divisibility by 2, so we need to reduce it:
LCM (40, 12) = 4, so 40 12 = 40: 4 12: 4 = 10 3
Now all we have to do is select the whole part and write down the ready answer: 10 3 = 3 1 3.
In this entry you can see the entire solution: 5 12 8 = 5 8 12 = 40 12 = 10 3 = 3 1 3.
We could also reduce the fraction by factoring the numerator and denominator, and the result would be exactly the same.
Answer: 5 12 8 = 3 1 3.
A numerical expression in which a natural number is multiplied by a fraction also has the property of displacement, that is, the order of the factors does not affect the result:
a b · n = n · a b = a · n b
How to multiply three or more common fractions
We can extend to the action of multiplying ordinary fractions the same properties that are characteristic of multiplying natural numbers. This follows from the very definition of these concepts.
Thanks to knowledge of the combining and commutative properties, you can multiply three or more ordinary fractions. It is acceptable to rearrange the factors for greater convenience or arrange the brackets in a way that makes it easier to count.
Let's show with an example how this is done.
Example 6
Multiply the four common fractions 1 20, 12 5, 3 7 and 5 8.
Solution: first, let's record the work. We get 1 20 · 12 5 · 3 7 · 5 8 . We need to multiply all the numerators and all the denominators together: 1 20 · 12 5 · 3 7 · 5 8 = 1 · 12 · 3 · 5 20 · 5 · 7 · 8 .
Before we start multiplying, we can make things a little easier on ourselves and factor some numbers into prime factors for further reduction. This will be easier than reducing the resulting fraction that is already ready.
1 12 3 5 20 5 7 8 = 1 (2 2 3) 3 5 2 2 5 5 7 (2 2 2) = 3 3 5 7 2 2 2 = 9,280
Answer: 1 · 12 · 3 · 5 20 · 5 · 7 · 8 = 9,280.
Example 7
Multiply 5 numbers 7 8 · 12 · 8 · 5 36 · 10 .
Solution
For convenience, we can group the fraction 7 8 with the number 8, and the number 12 with the fraction 5 36, since future abbreviations will be obvious to us. As a result, we will get:
7 8 12 8 5 36 10 = 7 8 8 12 5 36 10 = 7 8 8 12 5 36 10 = 7 1 2 2 3 5 2 2 3 3 10 = 7 5 3 10 = 7 5 10 3 = 350 3 = 116 2 3
Answer: 7 8 12 8 5 36 10 = 116 2 3.
If you notice an error in the text, please highlight it and press Ctrl+Enter
