How to multiply fractions with different denominators. Drawing up a system of equations
Multiplication of ordinary fractions
Consider an example.
Let there be $\frac(1)(3)$ part of an apple on the plate. We need to find the $\frac(1)(2)$ part of it. The required part is the result of multiplying the fractions $\frac(1)(3)$ and $\frac(1)(2)$. The result of multiplying two common fractions is a common fraction.
Multiplying two common fractions
Rule for multiplying ordinary fractions:
The result of multiplying a fraction by a fraction is a fraction whose numerator is equal to the product of the numerators of the multiplied fractions, and the denominator is equal to the product of the denominators:
Example 1
Multiply ordinary fractions $\frac(3)(7)$ and $\frac(5)(11)$.
Solution.
Let's use the rule of multiplication of ordinary fractions:
\[\frac(3)(7)\cdot \frac(5)(11)=\frac(3\cdot 5)(7\cdot 11)=\frac(15)(77)\]
Answer:$\frac(15)(77)$
If as a result of multiplying fractions a cancellable or improper fraction is obtained, then it is necessary to simplify it.
Example 2
Multiply fractions $\frac(3)(8)$ and $\frac(1)(9)$.
Solution.
We use the rule for multiplying ordinary fractions:
\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)\]
As a result, we got a reducible fraction (on the basis of division by $3$. Divide the numerator and denominator of the fraction by $3$, we get:
\[\frac(3)(72)=\frac(3:3)(72:3)=\frac(1)(24)\]
Short solution:
\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)=\frac(1) (24)\]
Answer:$\frac(1)(24).$
When multiplying fractions, you can reduce the numerators and denominators to find their product. In this case, the numerator and denominator of the fraction is decomposed into simple factors, after which the repeating factors are reduced and the result is found.
Example 3
Calculate the product of fractions $\frac(6)(75)$ and $\frac(15)(24)$.
Solution.
Let's use the formula for multiplying ordinary fractions:
\[\frac(6)(75)\cdot \frac(15)(24)=\frac(6\cdot 15)(75\cdot 24)\]
Obviously, the numerator and denominator contain numbers that can be reduced in pairs by the numbers $2$, $3$, and $5$. We decompose the numerator and denominator into simple factors and make the reduction:
\[\frac(6\cdot 15)(75\cdot 24)=\frac(2\cdot 3\cdot 3\cdot 5)(3\cdot 5\cdot 5\cdot 2\cdot 2\cdot 2\cdot 3)=\frac(1)(5\cdot 2\cdot 2)=\frac(1)(20)\]
Answer:$\frac(1)(20).$
When multiplying fractions, the commutative law can be applied:
Multiplying a fraction by a natural number
The rule for multiplying an ordinary fraction by a natural number:
The result of multiplying a fraction by a natural number is a fraction in which the numerator is equal to the product of the numerator of the multiplied fraction by the natural number, and the denominator is equal to the denominator of the multiplied fraction:
where $\frac(a)(b)$ is a common fraction, $n$ is a natural number.
Example 4
Multiply the fraction $\frac(3)(17)$ by $4$.
Solution.
Let's use the rule of multiplying an ordinary fraction by a natural number:
\[\frac(3)(17)\cdot 4=\frac(3\cdot 4)(17)=\frac(12)(17)\]
Answer:$\frac(12)(17).$
Do not forget about checking the result of multiplication for the contractibility of a fraction or for an improper fraction.
Example 5
Multiply the fraction $\frac(7)(15)$ by $3$.
Solution.
Let's use the formula for multiplying a fraction by a natural number:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)\]
By the criterion of division by the number $3$), it can be determined that the resulting fraction can be reduced:
\[\frac(21)(15)=\frac(21:3)(15:3)=\frac(7)(5)\]
The result is an improper fraction. Let's take the whole part:
\[\frac(7)(5)=1\frac(2)(5)\]
Short solution:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)=\frac(7)(5)=1\frac(2) (5)\]
It was also possible to reduce fractions by replacing the numbers in the numerator and denominator with their expansions into prime factors. In this case, the solution could be written as follows:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(7\cdot 3)(3\cdot 5)=\frac(7)(5)= 1\frac(2)(5)\]
Answer:$1\frac(2)(5).$
When multiplying a fraction by a natural number, you can use the commutative law:
Division of ordinary fractions
The division operation is the inverse of multiplication and its result is a fraction by which you need to multiply a known fraction to get a known product of two fractions.
Division of two common fractions
The rule for dividing ordinary fractions: Obviously, the numerator and denominator of the resulting fraction can be decomposed into simple factors and reduce:
\[\frac(8\cdot 35)(15\cdot 12)=\frac(2\cdot 2\cdot 2\cdot 5\cdot 7)(3\cdot 5\cdot 2\cdot 2\cdot 3)= \frac(2\cdot 7)(3\cdot 3)=\frac(14)(9)\]
As a result, we got an improper fraction, from which we select the integer part:
\[\frac(14)(9)=1\frac(5)(9)\]
Answer:$1\frac(5)(9).$
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 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.
\(\bf \frac(a)(b) \times \frac(c)(d) = \frac(a \times c)(b \times d)\\\)
Consider 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 \(\frac(12)(21) = \frac(4 \times 3)(7 \times 3) = \frac(4)(7)\\\) has been reduced by 3.
Multiplying a fraction by a number.
Let's start with the rule any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .
Let's use this rule for multiplication.
\(5 \times \frac(4)(7) = \frac(5)(1) \times \frac(4)(7) = \frac(5 \times 4)(1 \times 7) = \frac (20)(7) = 2\frac(6)(7)\\\)
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, multiply the number by the numerator and leave the denominator unchanged. Example:
\(\frac(2)(5) \times 3 = \frac(2 \times 3)(5) = \frac(6)(5) = 1\frac(1)(5)\\\\\) \(\bf \frac(a)(b) \times c = \frac(a \times c)(b)\\\)
Multiplication of mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied 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 reciprocals. The product of reciprocal fractions is 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 the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter if the denominators of fractions are the same or different, multiplication occurs according to the rule for 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 to an improper fraction and then find the product according to the rules of multiplication.
How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and 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 product 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 \(\frac(1)(3)\)?
Answer: \(\frac(3)(1) = 3\)
Example #4:
Calculate the product of two reciprocal fractions: a) \(\frac(104)(215) \times \frac(215)(104)\)
Solution:
a) \(\frac(104)(215) \times \frac(215)(104) = 1\)
Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?
Solution:
a) Let's use an example to answer the first question. The fraction \(\frac(2)(3)\) is proper, its reciprocal 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 an improper fraction at the same time. For example, the improper fraction is \(\frac(3)(3)\) , its reciprocal is \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.
c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac(3)(1)\), then its reciprocal will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac(1)(1) = \frac(1)(1) = 1\). The number 1 is a natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.
Example #6:
Perform 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 be simultaneously mixed numbers?
Let's look at an example. Let's take a mixed fraction \(1\frac(1)(2)\), find its reciprocal, for this we translate it into an improper fraction \(1\frac(1)(2) = \frac(3)(2) \) . Its reciprocal will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.
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 or more ordinary fractions.
Let's write down the basic rule first:
Definition 1
If we multiply one ordinary fraction, then the numerator of the resulting fraction will be equal to the product of the numerators of the original fractions, and the denominator to 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 apply this rule correctly. 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 of the numerical unit, 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, it is necessary to multiply the first fraction by the second. It will be equal to 5 8 3 4 square meters. units. But we can simply count how many rectangles are included in the fragment: there are 15 of them, which means that the total area is 1532 square units.
Since 5 3 = 15 and 8 4 = 32 we can write the following equation:
5 8 3 4 = 5 3 8 4 = 15 32
It is a confirmation of the rule we have 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 different and the same denominators.
Let's analyze the solutions of several problems for the multiplication of ordinary fractions.
Example 1
Multiply 7 11 by 9 8 .
Solution
To begin with, we 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 the answer from two numbers: 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 in the answer we got a reducible fraction, we need to complete the calculation and perform its reduction. If we get an improper fraction, we need to select 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 entry will look like this:
4 15 55 6 = 4 55 15 6 = 220 90
We have obtained a reduced fraction, i.e. one that has a sign of divisibility by 10.
Let's reduce the fraction: 220 90 GCD (220, 90) \u003d 10, 220 90 \u003d 220: 10 90: 10 \u003d 22 9. As a result, we got an improper fraction, from which we select the whole part and get a mixed number: 22 9 \u003d 2 4 9.
Answer: 4 15 55 6 = 2 4 9 .
For the convenience of calculation, we can also reduce the original fractions before performing the multiplication operation, for which we need to bring the fraction to the form a · c b · d. We decompose the values of the variables into simple factors and cancel the same ones.
Let us explain how this looks like using the data of a specific problem.
Example 3
Calculate the product 4 15 55 6 .
Solution
Let's write the calculations based on the multiplication rule. We will be able to:
4 15 55 6 = 4 55 15 6
Since as 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 .
A numerical 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 an ordinary fraction by a natural number, you need to multiply the numerator of this fraction by this number. In this case, the denominator of the final fraction will be equal to the denominator of the original ordinary fraction. The multiplication of some fraction a b by a natural number n can be written as a formula a b · n = a · n b .
It is 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
Compute the product of 2 27 by 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 above, we will get 10 27 as a result. The whole 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 common fraction, we often have to reduce the result or represent it as a mixed number.
Example 5
Condition: Calculate the product of 8 times 5 12 .
Solution
According to the rule above, we multiply a 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) \u003d 4, so 40 12 \u003d 40: 4 12: 4 \u003d 10 3
Now we only have to select the integer part and write down the finished 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 displacement property, 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 multiplication of ordinary fractions the same properties that are characteristic of the multiplication of natural numbers. This follows from the very definition of these concepts.
Thanks to the knowledge of the associative and commutative properties, it is possible to multiply three or more ordinary fractions. It is permissible to rearrange the factors in places for greater convenience or arrange the brackets in a way that will make it easier to count.
Let's show an example how this is done.
Example 6
Multiply 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 multiplication, we can make it a little easier for ourselves and decompose some numbers into prime factors for further reduction. This will be easier than reducing the finished fraction resulting from it.
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 = 9280.
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 this will make future reductions clear 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 a mistake in the text, please highlight it and press Ctrl+Enter
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:
A flying arrow is motionless, since at each 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 the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can 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 elementarily.
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's say we 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 to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, 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 of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 would get completely different results when determining the area of a rectangle in meters and centimeters.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of 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 action does not depend on the value of the number, the unit of measurement used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
In this article, we will analyze multiplication of mixed numbers. First, let's voice the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next, we will talk about the multiplication of a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and an ordinary fraction.
Page navigation.
Multiplication of mixed numbers.
Multiplication of mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers into improper fractions.
Let's write down multiplication rule for mixed numbers:
- First, the mixed numbers to be multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule of multiplying a fraction by a fraction.
Consider examples of applying this rule when multiplying a mixed number by a mixed number.
Perform mixed number multiplication and .
First, we represent the multiplied mixed numbers as improper fractions: 
and 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule of multiplication of fractions, we get 
. The resulting fraction is irreducible (see reducible and irreducible fractions), but it is incorrect (see regular and improper fractions), therefore, to get the final answer, it remains to extract the integer part from the improper fraction: .
Let's write the whole solution in one line: .
.
To consolidate the skills of multiplying mixed numbers, consider the solution of another example.
Do the multiplication.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it's time to remember about fraction reduction: we will replace all the numbers in the fraction with their expansions into prime factors, and we will perform the reduction of the same factors.
Multiplication of a mixed number and a natural number
After replacing the mixed number with an improper fraction, multiplying a mixed number and a natural number is reduced to the multiplication of an ordinary fraction and a natural number.
Multiply the mixed number and the natural number 45 .
A mixed number is a fraction, then 
. Let's replace the numbers in the resulting fraction with their expansions into prime factors, make a reduction, after which we select the integer part: .
.
Multiplication of a mixed number and a natural number is sometimes conveniently done using the distributive property of multiplication with respect to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, 
.
Compute the product.
We replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .
Multiplying a mixed number and a common fraction it is most convenient to reduce to the multiplication of ordinary fractions, representing the multiplied mixed number as an improper fraction.
Multiply the mixed number by the common fraction 4/15.
Replacing the mixed number with a fraction, we get 
.
www.cleverstudents.ru
Multiplication of fractional numbers
§ 140. Definitions. 1) The multiplication of a fractional number by an integer is defined in the same way as the multiplication of integers, namely: to multiply some number (multiplier) by an integer (multiplier) means to make a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
So multiplying by 5 means finding the sum:
2) To multiply some number (multiplier) by a fraction (multiplier) means to find this fraction of the multiplicand.
Thus, finding a fraction of a given number, which we considered before, we will now call multiplication by a fraction.
3) To multiply some number (multiplier) by a mixed number (factor) means to multiply the multiplier first by the integer of the factor, then by the fraction of the factor, and add the results of these two multiplications together.
For example:
The number obtained after multiplication is in all these cases called work, i.e., in the same way as when multiplying integers.
From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.
§ 141. Expediency of these definitions. To understand the expediency of introducing the last two definitions of multiplication into arithmetic, let us take the following problem:
A task. The train, moving evenly, travels 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?
If we remained with the same definition of multiplication, which is indicated in the arithmetic of integers (addition of equal terms), then our problem would have three different solutions, namely:
If the given number of hours is an integer (for example, 5 hours), then to solve the problem, 40 km must be multiplied by this number of hours.
If a given number of hours is expressed as a fraction (for example, hours), then you will have to find the value of this fraction from 40 km.
Finally, if the given number of hours is mixed (for example, hours), then it will be necessary to multiply 40 km by an integer contained in the mixed number, and add to the result such a fraction from 40 km as is in the mixed number.
The definitions we have given allow us to give one general answer to all these possible cases:
40 km must be multiplied by the given number of hours, whatever it may be.
Thus, if the problem is presented in general form as follows:
A train moving uniformly travels v km per hour. How many kilometers will the train cover in t hours?
then, whatever the numbers v and t, we can express one answer: the desired number is expressed by the formula v · t.
Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, to find 5% (i.e. five hundredths) of a given number means the same as multiplying the given number by or by; finding 125% of a given number is the same as multiplying that number by or by , etc.
§ 142. A note about when a number increases and when it decreases from multiplication.
From multiplication by a proper fraction, the number decreases, and from multiplication by an improper fraction, the number increases if this improper fraction is greater than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so,.
§ 143. Derivation of multiplication rules.
1) Multiplying a fraction by an integer. Let the fraction be multiplied by 5. This means to increase by 5 times. To increase a fraction by 5, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).
That's why:
Rule 1. To multiply a fraction by an integer, you must multiply the numerator by this integer, and leave the denominator the same; instead, you can also divide the denominator of the fraction by the given integer (if possible), and leave the numerator the same.
Comment. The product of a fraction and its denominator is equal to its numerator.
So:
Rule 2. To multiply an integer by a fraction, you need to multiply the integer by the numerator of the fraction and make this product the numerator, and sign the denominator of the given fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to the multiplication of a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:
Thus, the three rules now stated are contained in one, which can be expressed in general terms as follows:
4) Multiplication of mixed numbers.
Rule 4. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For example:
§ 144. Reduction in multiplication. When multiplying fractions, if possible, a preliminary reduction should be done, as can be seen from the following examples:
Such a reduction is possible because the value of a fraction will not change if the numerator and denominator are reduced by the same number of times.
§ 145. Change of product with change of factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .
So, if in the example:
in order to multiply several fractions, it is necessary to multiply their numerators among themselves and the denominators among themselves and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to such products in which some factors of the number are integer or mixed, if only we consider the whole number as a fraction whose denominator is one, and we turn mixed numbers into improper fractions. For example:
§ 147. Basic properties of multiplication. Those properties of multiplication that we have indicated for integers (§ 56, 57, 59) also belong to the multiplication of fractional numbers. Let's specify these properties.
1) The product does not change from changing the places of the factors.
For example:
Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their terms differ only in the order of the integer factors, and the product of integers does not change when the places of the factors change.
2) The product will not change if any group of factors is replaced by their product.
For example:
The results are the same.
From this property of multiplication, we can deduce the following conclusion:
to multiply a number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, and so on.
For example:
3) The distributive law of multiplication (with respect to addition). To multiply the sum by some number, you can multiply each term by this number separately and add the results.
This law has been explained by us (§ 59) as applied to whole numbers. It remains true without any changes for fractional numbers.
Let us show, in fact, that the equality
(a + b + c + .)m = am + bm + cm + .
(the distributive law of multiplication with respect to addition) remains true even when the letters mean fractional numbers. Let's consider three cases.
1) Suppose first that the factor m is an integer, for example m = 3 (a, b, c are any numbers). According to the definition of multiplication by an integer, one can write (limited for simplicity to three terms):
(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).
On the basis of the associative law of addition, we can omit all brackets on the right side; applying the commutative law of addition, and then again the combination law, we can obviously rewrite the right-hand side as follows:
(a + a + a) + (b + b + b) + (c + c + c).
(a + b + c) * 3 = a * 3 + b * 3 + c * 3.
Hence, the distributive law in this case is confirmed.
Multiplication and division of fractions
Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.
To multiply two fractions, you need to 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.
From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what definitely won’t happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.
By definition we have:
Multiplication of fractions with an integer part and negative fractions
If there is an integer part in the fractions, they must be converted to improper 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 limits of multiplication or removed altogether according to the following rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:
- We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.
A task. Find the value of the expression:
We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:
Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more 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:
A task. Find the value of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what is left of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, in no case do not 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 a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.
There is simply no other reason to reduce 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.
Multiplication of fractions.
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 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.
Consider 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.
Multiplying a fraction by a number.
Let's start with the rule any number can be represented as a fraction \(\bf n = \frac \) .
Let's use this rule for multiplication.
The improper fraction \(\frac = \frac = \frac + \frac = 2 + \frac = 2\frac \\\) was converted to a mixed fraction.
In other words, When multiplying a number by a fraction, multiply the number by the numerator and leave the denominator unchanged. Example:
Multiplication of mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied with the denominator.
Multiplication of reciprocal fractions and numbers.
Related questions:
How to multiply a fraction by a fraction?
Answer: the product of ordinary fractions is the multiplication of the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter if the denominators of fractions are the same or different, multiplication occurs according to the rule for 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 to an improper fraction and then find the product according to the rules of multiplication.
How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac \times \frac \) b) \(\frac \times \frac \)
Example #2:
Calculate the product of a number and a fraction: a) \(3 \times \frac \) b) \(\frac \times 11\)
Example #3:
Write the reciprocal of the fraction \(\frac \)?
Answer: \(\frac = 3\)
Example #4:
Calculate the product of two reciprocals: a) \(\frac \times \frac \)
Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?
Solution:
a) Let's use an example to answer the first question. The fraction \(\frac \) is correct, its reciprocal will be equal to \(\frac \) - 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 an improper fraction at the same time. For example, the improper fraction is \(\frac \) , its reciprocal is \(\frac \). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.
c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac \), then its reciprocal will be \(\frac \). The fraction \(\frac \) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac = \frac = 1\). The number 1 is a natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.
Example #6:
Perform the product of mixed fractions: a) \(4 \times 2\frac \) b) \(1\frac \times 3\frac \)
Solution:
a) \(4 \times 2\frac = \frac \times \frac = \frac = 11\frac \\\\ \)
b) \(1\frac \times 3\frac = \frac \times \frac = \frac = 4\frac \)
Example #7:
Can two reciprocal numbers be simultaneously mixed numbers?
Let's look at an example. Let's take a mixed fraction \(1\frac \), find its reciprocal, to do this we translate it into an improper fraction \(1\frac = \frac \) . Its reciprocal will be equal to \(\frac \) . The fraction \(\frac \) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.
Multiplying a decimal by a natural number
Presentation for the lesson
Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.
- In a fun way, introduce students to the rule of multiplying a decimal fraction by a natural number, by a bit unit and the rule of expressing a decimal fraction as a percentage. Develop the ability to apply the acquired knowledge in solving examples and problems.
- To develop and activate the logical thinking of students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their work and the work of each other.
- To cultivate interest in mathematics, activity, mobility, ability to communicate.
Equipment: interactive board, a poster with a cyphergram, posters with mathematicians' statements.
- Organizing time.
- Oral counting is a generalization of previously studied material, preparation for the study of new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of the acquired knowledge in a playful way with the help of a computer.
- Grading.
2. Guys, today we will have a somewhat unusual lesson, because I will not spend it alone, but with my friend. And my friend is also unusual, now you will see him. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, friend? Komposha replies: "My name is Komposha." Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is posted on the board with an oral account for adding and subtracting decimal fractions, as a result of which the guys get the following code 523914687. )
Komposha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is the keyword of the topic of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how the multiplication of natural numbers is performed. Today we will consider the multiplication of decimal numbers by a natural number. The multiplication of a decimal fraction by a natural number can be considered as the sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 3 = 5.21 + 5, 21 + 5.21 = 15.63 So 5.21 3 = 15.63. Representing 5.21 as an ordinary fraction of a natural number, we get
And in this case, we got the same result of 15.63. Now, ignoring the comma, let's take the number 521 instead of the number 5.21 and multiply by the given natural number. Here we must remember that in one of the factors the comma is moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now, in this example, we will move the comma to the left by two digits. Thus, by how many times one of the factors was increased, the product was reduced by so many times. Based on the similar points of these methods, we draw a conclusion.
To multiply a decimal by a natural number, you need:
1) ignoring the comma, perform the multiplication of natural numbers;
2) in the resulting product, separate with a comma on the right as many characters as there are in a decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 3 = 15.63 and 7.624 15 = 114.34. After I show multiplication by a round number 12.6 50 \u003d 630. Next, I turn to the multiplication of a decimal fraction by a bit unit. I show the following examples: 7.423 100 \u003d 742.3 and 5.2 1000 \u003d 5200. So, I introduce the rule for multiplying a decimal fraction by a bit unit:
To multiply a decimal fraction by bit units 10, 100, 1000, etc., it is necessary to move the comma to the right in this fraction by as many digits as there are zeros in the bit unit record.
I end the explanation with the expression of a decimal fraction as a percentage. I enter the rule:
To express a decimal as a percentage, multiply it by 100 and add the % sign.
I give an example on a computer 0.5 100 = 50 or 0.5 = 50%.
4. At the end of the explanation, I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, to consolidate the topic, we do a mathematical physical education session together with Komposha. Everyone stands up, shows the class the solved examples and they must answer whether the example is correct or incorrect. If the example is solved correctly, then they raise their hands above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and knead their fingers.
6. And now you have a little rest, you can solve the tasks. Open your textbook to page 205, № 1029. in this task it is necessary to calculate the value of expressions:
Tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when fully assembled, sails away.
Solving this task on a computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year, spaceships take off from the Baikonur cosmodrome from Kazakhstan to the stars. Near Baikonur, Kazakhstan is building its new Baiterek cosmodrome.
How far will a car travel in 4 hours if the speed of the car is 74.8 km/h.
Gift certificate Don't know what to give to your significant other, friends, employees, relatives? Take advantage of our special offer: "Gift certificate of the Blue Osoka Country Hotel". The certificate […]