INSTRUCTIONS AND PROPHECIES OF THE Blessed MOTHER ALIPIA GOLOSEEVSKY, Kyiv...
Proportion - equality of two relations, i.e. equality of the form a:b = c:d , or, in other notation, the equality
If a a : b = c : d, then a and d called extreme, a b and c - averagemembers proportions.
There is no getting away from the “proportion”, it is indispensable in many tasks. There is only one way out - to deal with this ratio and use the proportion as a lifesaver.
Before proceeding with the consideration of proportion problems, it is important to remember the basic rule of proportion:
in proportion
the product of the extreme terms is equal to the product of the average
If some value in the proportion is unknown, it will be easy to find it based on this rule.
For example,
That is, the unknown value of the proportion - the value of the fraction, in the denominator
which is the number opposite the unknown value
, in the numerator - the product of the remaining members of the proportion
(regardless of where this unknown value stands
). 
Task 1.
From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
Solution:
We understand that a decrease in the weight of a seed by a factor of several times entails a decrease in the weight of the resulting oil by the same amount. That is, the quantities are directly related.
Let's fill in the table: 
Unknown value - the value of the fraction, in the denominator of which - 21 - the value opposite the unknown in the table, in the numerator - the product of the remaining members of the table-proportion.
Therefore, we get that 1.7 kg of oil will come out of 7 kg of seed.
To right fill in the table, it is important to remember the rule:
Identical names must be written under each other. We write percentages under percentages, kilograms under kilograms, etc.
Task 2.
Convert to radians.
Solution:
We know that . Let's fill in the table:
Task 3.
A circle is depicted on checkered paper. What is the area of the circle if the area of the shaded sector is 27?
Solution:
It is clearly seen that the unshaded sector corresponds to the angle at (for example, because the sides of the sector are formed by the bisectors of two adjacent right angles). And since the whole circle is , then the shaded sector accounts for .
Let's make a table:
Where does the area of the circle come from?
Task 4. After 82% of the entire field had been plowed, 9 hectares remained to be plowed. What is the area of the entire field?
Solution:
The whole field is 100%, and since 82% is plowed, then 100%-82%=18% of the field remains to be plowed.
Fill in the table: 
Where do we get that the whole field is (ha).
And the next task is with an ambush.
Task 5.
The distance between two cities is traveled by a passenger train at a speed of 80 km/h in 3 hours. How many hours will it take for a freight train to travel the same distance at a speed of 60 km/h?
If you solve this problem in the same way as the previous one, you will get the following:
the time it takes for a freight train to travel the same distance as a passenger train is hours. That is, it turns out that, going at a lower speed, it overcomes (in the same time) the distance faster than a train with a higher speed.
What is the reasoning error?
So far, we have considered problems where the quantities were directly proportional to each other , that is growth of the same magnitude by a certain amount, gives growth the second quantity associated with it by the same number of times (similarly with a decrease, of course). And here we have a different situation: the speed of a passenger train more the speed of a freight train by a factor of several times, but the time required to overcome the same distance is required by a passenger train lesser as much as a freight train. That is, values to each other inversely proportional .
The scheme that we have used so far needs to be slightly modified in this case.
Solution:
We reason like this:
A passenger train traveled 3 hours at a speed of 80 km/h, so it traveled km. This means that a freight train will cover the same distance in one hour.
That is, if we were to make up a proportion, we should have swapped the cells of the right column first. Would have received:
That's why, please be careful when drawing up the proportion. First, figure out what kind of addiction you are dealing with - direct or reverse.
Task 1. The thickness of 300 sheets of printer paper is 3.3 cm. How thick would a stack of 500 sheets of the same paper be?
Solution. Let x cm be the thickness of a 500-sheet paper ream. In two ways we find the thickness of one sheet of paper:
3,3: 300 or x : 500.
Since the sheets of paper are the same, these two ratios are equal to each other. We get the proportion reminder: proportion is the equality of two ratios):
x=(3.3 · 500): 300;
x=5.5. Answer: pack 500 sheets of paper has a thickness 5.5 cm.
This is a classic reasoning and formulation of a solution to a problem. Such problems are often included in graduate tests, which usually write the solution in this form:
or they decide orally, arguing as follows: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times smaller. We divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100 sheet of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and we get the answer: 5.5 cm.
Of course, this is justified, since the time for testing graduates and applicants is limited. However, in this lesson we will reason and write the solution as it should be done in 6 class.
Task 2. How much water is contained in 5 kg of watermelon if it is known that watermelon consists of 98% water?
Solution.
The entire mass of watermelon (5 kg) is 100%. Water will be x kg or 98%. In two ways, you can find how many kg fall on 1% of the mass.
5: 100 or x : 98. We get the proportion:
5: 100 = x : 98.
x=(5 · 98): 100;
x=4.9 Answer: in 5kg watermelon contains 4.9 kg of water.
The mass of 21 liters of oil is 16.8 kg. What is the mass of 35 liters of oil?
Solution.
Let the mass of 35 liters of oil be x kg. Then in two ways you can find the mass of 1 liter of oil:
16,8: 21 or x : 35. We get the proportion:
16,8: 21=x : 35.
Find the middle term of the proportion. To do this, we multiply the extreme terms of the proportion ( 16,8 and 35 ) and divide by the known middle term ( 21 ). Reduce the fraction by 7 .
Multiply the numerator and denominator of the fraction by 10 so that the numerator and denominator contain only integers. We reduce the fraction by 5 (5 and 10) and on 3 (168 and 3).
Answer: 35 liters of oil have a mass 28 kg.
After 82% of the entire field had been plowed, 9 hectares remained to be plowed. What is the area of the entire field?
Solution.
Let the area of the entire field be x ha, which is 100%. It remains to plow 9 hectares, which is 100% - 82% = 18% of the entire field. Let's express 1% of the field area in two ways. It:
X : 100 or 9 : 18. We make a proportion:
X : 100 = 9: 18.
We find the unknown extreme term of the proportion. To do this, we multiply the average terms of the proportion ( 100
and 9
) and divide by the known extreme term ( 18
). We reduce the fraction.
Answer: area of the whole field 50 ha.
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§ 125. The concept of proportion.
Proportion is the equality of two ratios. Here are examples of equalities called proportions:
Note. The names of the quantities in the proportions are not indicated.
Proportions are usually read as follows: 2 is related to 1 (one), as 10 is related to 5 (the first proportion). You can read it differently, for example: 2 is so many times greater than 1, how many times 10 is greater than 5. The third proportion can be read as follows: - 0.5 is so many times less than 2, how many times 0.75 is less than 3.
The numbers in a proportion are called members of the proportion. Hence, the proportion consists of four terms. The first and last members, i.e., the members standing at the edges, are called extreme, and the terms of the proportion that are in the middle are called average members. This means that in the first proportion, the numbers 2 and 5 will be the extreme members, and the numbers 1 and 10 will be the middle members of the proportion.
§ 126. The main property of proportion.
Consider the proportion:
We multiply its extreme and middle terms separately. The product of the extreme 6 4 \u003d 24, the product of the average 3 8 \u003d 24.
Consider another proportion: 10: 5 \u003d 12: 6. We also multiply here separately the extreme and middle terms.
The product of the extreme 10 6 \u003d 60, the product of the average 5 12 \u003d 60.
The main property of proportion: the product of the extreme terms of the proportion is equal to the product of its middle terms.
AT general view the main property of the proportion is written as follows: ad = bc .
Let's check it on several proportions:
1) 12: 4 = 30: 10.
This proportion is true, since the ratios of which it is composed are equal. At the same time, taking the product of the extreme terms of the proportion (12 10) and the product of its middle terms (4 30), we will see that they are equal to each other, i.e.
12 10 = 4 30.
2) 1 / 2: 1 / 48 = 20: 5 / 6
The proportion is correct, which is easy to verify by simplifying the first and second relations. The main property of the proportion will take the form:
1 / 2 5 / 6 = 1 / 48 20
It is easy to make sure that if we write such an equality, which has the product of any two numbers on the left side, and the product of two other numbers on the right side, then a proportion can be made from these four numbers.
Let us have an equality, which includes four numbers, multiplied in pairs:
these four numbers may be members of a proportion, which is not difficult to write, if we take the first product as the product of the extreme terms, and the second as the product of the middle ones. The published equality can be made, for example, the following proportion:
In general, from equality ad = bc you can get the following proportions:
Do the following exercise on your own. Given the product of two pairs of numbers, write the proportion corresponding to each equality:
a) 1 6 = 2 3;
b) 2 15 = b 5.
§ 127. Calculation of unknown members of the proportion.
The main property of the proportion allows you to calculate any of the terms of the proportion if it is unknown. Let's take the proportion:
X : 4 = 15: 3.
In this proportion, one extreme term is unknown. We know that in every proportion the product of the extreme terms is equal to the product of the middle terms. On this basis, we can write:
x 3 = 4 15.
After multiplying 4 by 15, we can rewrite this equation as follows:
X 3 = 60.
Let's look at this equality. In it, the first factor is unknown, the second factor is known, and the product is known. We know that to find an unknown factor, it is enough to divide the product by another (known) factor. Then it will turn out:
X = 60:3, or X = 20.
Let's check the found result by substituting the number 20 instead of X in this proportion:
The proportion is correct.
Let's think about what actions we had to perform to calculate the unknown extreme term of the proportion. Of the four members of the proportion, only one extreme was unknown to us; two middle and second extreme were known. To find the extreme term of the proportion, we first multiplied the middle terms (4 and 15), and then divided the product found by the known extreme term. Now we will show that the actions would not change if the desired extreme term of the proportion were not in the first place, but in the last. Let's take the proportion:
70: 10 = 21: X .
Let's write down the main property of the proportion: 70 X = 10 21.
Multiplying the numbers 10 and 21, we rewrite the equality in this form:
70 X = 210.
One factor is unknown here, to calculate it, it is enough to divide the product (210) by another factor (70),
X = 210: 70; X = 3.
Thus, we can say that each extreme member of the proportion is equal to the product of the averages divided by the other extreme.
Let us now proceed to the calculation of the unknown mean term. Let's take the proportion:
30: X = 27: 9.
Let's write the main property of the proportion:
30 9 = X 27.
We calculate the product of 30 by 9 and rearrange the parts of the last equality:
X 27 = 270.
Let's find the unknown factor:
X = 270: 27, or X = 10.
Let's check with a substitution:
30:10 = 27:9. The proportion is correct.
Let's take another proportion:
12:b= X : 8. Let's write the main property of the proportion:
12 . 8 = 6 X . Multiplying 12 and 8 and rearranging the parts of the equation, we get:
6 X = 96. Find the unknown factor:
X = 96:6, or X = 16.
In this way, each middle member of the proportion is equal to the product of the extremes divided by the other middle one.
Find the unknown terms of the following proportions:
1) a : 3= 10:5; 3) 2: 1 / 2 = x : 5;
2) 8: b = 16: 4; 4) 4: 1 / 3 = 24: X .
The last two rules can be written in general form as follows:
1) If the proportion looks like:
x: a = b: c , then
2) If the proportion looks like:
a: x = b: c , then
§ 128. Simplification of proportion and rearrangement of its members.
In this section, we will derive rules that allow us to simplify the proportion in the case when it includes large numbers or fractional terms. Transformations that do not violate the proportion include the following:
1. Simultaneous increase or decrease of both members of any relation in the same number once.
EXAMPLE 40:10 = 60:15.
By multiplying both terms of the first relation by 3 times, we get:
120:30 = 60: 15.
The proportion has not changed.
Decreasing both terms of the second relation by 5 times, we get:
We got the correct proportion again.
2. Simultaneous increase or decrease of both previous or both subsequent terms in the same number of times.
Example. 16:8 = 40:20.
Let's double the previous members of both relations:
Got the right proportion.
Let us reduce the next terms of both relations by 4 times:
The proportion has not changed.
The two conclusions obtained can be summarized as follows: The proportion will not be violated if we simultaneously increase or decrease any extreme member of the proportion and any middle one by the same number of times.
For example, by reducing the 1st extreme and 2nd middle members of the proportion 16:8 = 40:20 by 4 times, we get:
3. Simultaneous increase or decrease of all members of the proportion by the same number of times. Example. 36:12 = 60:20. Let's increase all four numbers by 2 times:
The proportion has not changed. Let's reduce all four numbers by 4 times:
The proportion is correct.
The listed transformations make it possible, firstly, to simplify the proportions, and secondly, to free them from fractional members. Let's give examples.
1) Let there be a proportion:
200: 25 = 56: x .
In it, the terms of the first relation are relatively large numbers, and if we wished to find the value X , then we would have to perform calculations on these numbers; but we know that the proportion is not violated if both terms of the ratio are divided by the same number. Divide each of them by 25. The proportion will take the form:
8:1 = 56: x .
We have thus obtained a more convenient proportion, from which X can be found in the mind:
2) Take the proportion:
2: 1 / 2 = 20: 5.
In this proportion there is a fractional term (1 / 2), from which you can get rid of. To do this, we will have to multiply this term, for example, by 2. But we do not have the right to increase the middle term of the proportion; it is necessary, together with it, to increase one of the extreme terms; then the proportion will not be violated (based on the first two points). Let's increase the first of the extreme terms
(2 2) : (2 1 / 2) = 20: 5, or 4: 1 = 20:5.
Let's increase the second extreme term:
2: (2 1 / 2) = 20: (2 5), or 2: 1 = 20: 10.
Let's consider three more examples of freeing the proportion from fractional terms.
Example 1. 1/4: 3/8 = 20:30.
Let's bring the fractions to a common denominator:
2 / 8: 3 / 8 = 20: 30.
Multiplying both terms of the first relation by 8, we get:
Example 2. 12: 15 / 14 \u003d 16: 10 / 7. Let's bring the fractions to a common denominator:
12: 15 / 14 = 16: 20 / 14
We multiply both subsequent terms by 14, we get: 12:15 \u003d 16:20.
Example 3. 1/2: 1/48 = 20: 5/6.
Let's multiply all the terms of the proportion by 48:
24: 1 = 960: 40.
When solving problems in which some proportions occur, it is often necessary to rearrange the terms of the proportion for different purposes. Consider which permutations are legal, i.e., do not violate proportions. Let's take the proportion:
3: 5 = 12: 20. (1)
Rearranging the extreme terms in it, we get:
20: 5 = 12:3. (2)
We now rearrange the middle terms:
3:12 = 5: 20. (3)
We rearrange both the extreme and middle terms at the same time:
20: 12 = 5: 3. (4)
All of these proportions are correct. Now let's put the first relation in place of the second, and the second in place of the first. Get the proportion:
12: 20 = 3: 5. (5)
In this proportion, we will make the same permutations as we did before, i.e., we will first rearrange the extreme terms, then the middle ones, and, finally, both the extreme and the middle ones at the same time. Three more proportions will turn out, which will also be fair:
5: 20 = 3: 12. (6)
12: 3 = 20: 5. (7)
5: 3 = 20: 12. (8)
So, from one given proportion, by rearranging, you can get 7 more proportions, which together with this one makes 8 proportions.
It is especially easy to find out the validity of all these proportions when written in letters. The 8 proportions obtained above take the form:
a: b = c: d; c:d = a:b;
d:b = c:a; b:d = a:c;
a:c = b:d; c:a = d:b;
d:c=b:a; b:a = d:c.
It is easy to see that in each of these proportions the main property takes the form:
ad = b.c.
Thus, these permutations do not violate the fairness of the proportion and they can be used if necessary.
Set up a proportion. In this article I want to talk to you about proportions. To understand what proportion is, to be able to compose it - this is very important, it really saves. It seems to be a small and insignificant “letter” in the big alphabet of mathematics, but without it, mathematics is doomed to be lame and inferior.First, let me remind you what proportion is. This is an equality of the form:
which is the same (this different shape records).
Example:
They say one is to two as four is to eight. That is, this is the equality of two relations (in this example, the relations are numeric).
Basic rule of proportion:
a:b=c:d
the product of the extreme terms is equal to the product of the average
that is
a∙d=b∙c
*If any value in the proportion is unknown, it can always be found.
If we consider the form of the record of the form:
then you can use the following rule, it is called the “rule of the cross”: the equality of the products of elements (numbers or expressions) standing diagonally is written
a∙d=b∙c
As you can see the result is the same.
If the three elements of the proportion are known, thenwe can always find a fourth.
This is the essence of the benefit and necessityproportions in problem solving.
Let's look at all the options where the unknown value x is in "any place" of the proportion, where a, b, c are numbers:
The value standing on the diagonal from x is written in the denominator of the fraction, and the known values standing on the diagonal are written in the numerator as a product. It is not necessary to memorize it, you will calculate everything correctly if you have mastered the basic rule of proportion.
Now main question The associated with the title of the article. When does proportion save and where is it used? For example:
1. First of all, these are tasks for interest. We considered them in the articles "" and "".
2. Many formulas are given as proportions:
> sine theorem
> ratio of elements in a triangle
> tangent theorem
> Thales' theorem and others.
3. In tasks on geometry, the ratio of sides (of other elements) or areas is often set in the condition, for example, 1:2, 2:3, and others.
4. Conversion of units of measurement, and the proportion is used to convert units both in one measure, and to convert from one measure to another:
hours to minutes (and vice versa).
units of volume, area.
— lengths, such as miles to kilometers (and vice versa).
degrees to radians (and vice versa).
here without compiling a proportion is indispensable.
The key point is that you need to correctly establish the correspondence, consider simple examples:
It is necessary to determine the number that is 35% of 700.
In problems with percentages, the value with which we compare is taken as 100%. Let's denote the unknown number as x. Let's match:
We can say that seven hundred thirty-five corresponds to 100 percent.
X corresponds to 35 percent. Means,
700 – 100%
x - 35%
We decide
Answer: 245
Convert 50 minutes to hours.
We know that one hour corresponds to 60 minutes. Let's denote the correspondence -x hours is 50 minutes. Means
1 – 60
x - 50
We decide:
That is, 50 minutes is five-sixths of an hour.
Answer: 5/6
Nikolai Petrovich drove 3 kilometers. How much will it be in miles (note that 1 mile is 1.6 km)?
We know that 1 mile is 1.6 kilometers. Let us take the number of miles that Nikolai Petrovich traveled as x. We can match:
One mile corresponds to 1.6 kilometers.
X miles is three kilometers.
1 – 1,6
x - 3
Answer: 1,875 miles
You know that there are formulas to convert degrees to radians (and vice versa). I do not write them down, because I think it is superfluous to memorize them, and so you have to keep a lot of information in memory. You can always convert degrees to radians (and vice versa) if you use proportion.
Convert 65 degrees to radians.
The main thing to remember is that 180 degrees is Pi radians.
Let's denote the desired value as x. Set up a match.
One hundred and eighty degrees corresponds to Pi radians.
Sixty-five degrees corresponds to x radians. study the article on this blog topic. The material is presented in a slightly different way, but the principle is the same. I'll finish with this. There will definitely be something more interesting, do not miss it!
If we recall the very definition of mathematics, then it contains the following words: mathematics studies quantitative RELATIONS (RELATIONSHIPS- key word here). As you can see, the very definition of mathematics contains a proportion. In general, mathematics without proportion is not mathematics!!!
All the best!
Sincerely, Alexander
P.S: I would be grateful if you tell about the site in social networks.
From the point of view of mathematics, a proportion is the equality of two ratios. Interdependence is characteristic of all parts of the proportion, as well as their unchanging result. You can understand how to make a proportion by familiarizing yourself with the properties and formula of proportion. To understand the principle of solving proportions, it will be sufficient to consider one example. Only directly solving proportions, you can easily and quickly learn these skills. And this article will help the reader in this.
Proportion properties and formula
- Reversal of proportion. In the case when the given equality looks like 1a: 2b = 3c: 4d, write 2b: 1a = 4d: 3c. (Moreover, 1a, 2b, 3c and 4d are prime numbers, other than 0).
- Multiplying the given members of the proportion crosswise. In literal terms, this looks like this: 1a: 2b \u003d 3c: 4d, and writing 1a4d \u003d 2b3c will be equivalent to it. Thus, the product of the extreme parts of any proportion (the numbers at the edges of the equality) is always equal to the product of the middle parts (the numbers located in the middle of the equality).
- When compiling a proportion, such a property of it as a permutation of the extreme and middle terms can also be useful. The equality formula 1a: 2b = 3c: 4d can be displayed in the following ways:
- 1a: 3c = 2b: 4d (when the middle members of the proportion are rearranged).
- 4d: 2b = 3c: 1a (when the extreme members of the proportion are rearranged).
- Perfectly helps in solving the proportion of its property of increase and decrease. With 1a: 2b = 3c: 4d, write:
- (1a + 2b) : 2b = (3c + 4d) : 4d (equality by increasing proportion).
- (1a - 2b) : 2b = (3c - 4d) : 4d (equality by decreasing proportion).
- You can create proportions by adding and subtracting. When the proportion is written as 1a:2b = 3c:4d then:
- (1a + 3c) : (2b + 4d) = 1a: 2b = 3c: 4d (the proportion is added).
- (1a - 3c) : (2b - 4d) = 1a: 2b = 3c: 4d (the proportion is subtracted).
- Also, when solving a proportion containing fractional or large numbers, you can divide or multiply both of its members by the same number. For example, the components of the proportion 70:40=320:60 can be written like this: 10*(7:4=32:6).
- The variant of solving the proportion with percentages looks like this. For example, write down, 30=100%, 12=x. Now you should multiply the middle terms (12 * 100) and divide by the known extreme (30). Thus, the answer is: x=40%. In a similar way, if necessary, you can multiply the known extreme terms and divide them by a given average number, obtaining the desired result.
If you are interested in a specific proportion formula, then in the simplest and most common version, the proportion is such an equality (formula): a / b \u003d c / d, in which a, b, c and d are four non-zero numbers.
