How are fractional numbers multiplied? Fraction. Multiplication of common, decimal, mixed fractions
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not jump to reciprocals. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
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?
It so happened historically that 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 obtained visual material. So, point A shows a share equal to 1/4 of the total unit segment, and point B marks 2/8 of this 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 to improper fraction.
Correct fractional expression always less than one, and incorrect - 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;
- in a specific example, an incomplete quotient is a 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 multiplying fractions 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 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.
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 compose 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's 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 into prime factors, 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 the 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 prepared.
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
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 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) at the same time 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 a 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 reciprocal numbers exist at the same time? mixed numbers?
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.