How to make the correct proportion from numbers. Drawing up a system of equations

Proportion translated from Latin language(proportio) denotes the relationship, the evenness of the parts, that is, the equality of 2 relations. Knowing how to calculate proportions is often necessary in everyday situations.

Instructions

1. Easy example when you need to apply your knowledge of solving proportions: how to calculate 13% of your wages- the same interest that goes to the Pension Fund.

2. Write two lines of proportion. In the first, indicate the total salary amount, which represents 100%, that is, say, 15,000 (rubles) = 100%.

3. In the line below, indicate the amount that needs to be calculated with the sign “X”, the one that is equal to 13%, that is, X = 13%.

4. The main quality of proportion sounds like this: the product of the extreme members of the proportion is equal to the product of its middle members. This means that if you multiply 15,000 by 13, the resulting number will be equal to the value of X multiplied by 100. That is, multiplying the terms of the proportion crosswise, you will get an identical value.

5. In order to calculate what X is equal to in the final result, multiply 15,000 by 13 and divide by 100. You will get that 13 percent of your salary is 1,950 rubles, so you get 15,000 - 1,950 = 13,050 net rubles in your hands salaries.

6. If you need to take 100 grams of powdered sugar for a pie, and you know that 140 grams fit in one faceted glass, make the following proportion: 100 = X140 = 1

7. Calculate what X.X = 100 x 1 / 140 = 0.7 That is, you will need 0.7 cups of powdered sugar.

8. It happens that you need to calculate an integer, knowing only percentage part. Let's say you know that 21 people at the enterprise, which is 5% of the total number of employees, have secondary special education. Make up a proportion to calculate the total number of workers: X (person) = 100%, 21 = 5%. 21 x 100 / 5 = 420 people.

9. Thus, having written down the available data in two lines, the value of the unknown term must be found as follows: multiply among themselves those terms of the proportion that are next to and above the unknown one and divide the resulting number by the value that is diagonally from the unknown. A = BS = YES = B x S/D; B = A x D / C; C = A x D / B; D = C x B / A

There are several types of diagonals in geometry. Diagonal called a segment that connects two non-adjacent (not belonging to the same side or edge) vertices of a polygon or polyhedron. There are also diagonals of faces considered as polygons and spatial diagonals connecting the vertices of different faces of a polyhedron. There are figures in which all diagonals are equal to each other. On the plane it is a regular pentagon and a square, in space it is a positive octahedron. Knowing the lengths of the sides of a positive polygon or the lengths of the edges of a positive polyhedron, you can calculate the length of each diagonal.

Instructions

1. In any regular polygon, the angles are equal to each other and are calculated by the formula ?? = (N - 2) * 180?/N, where?? – each of the angles of a positive polygon, N – the number of vertices. Knowing the angles at the vertices of a polygon, its diagonals can be calculated using the cosine theorem BE = v(AB? + AE? – 2 * AB * AE * cos??)

2. If the number of vertices is greater than five, then to calculate the diagonals that connect the vertices lying on different sides You can use the same cosine theorem to calculate the angles of the resulting triangles. Let's say, in the hexagon ABCDEF, to find the diagonal BE, you need to calculate the diagonal CE, then use the same cosine theorem to calculate the angle??, then?? = ?? - ??. Thus, BE = v(BC? + CE? – 2 * BC * CE * cos??).

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Note!
To calculate the spatial diagonal of a polyhedron, you need to construct a section containing this diagonal, calculate the angles at the vertices of this section, considering the section as a flat polygon. Then the diagonal can be calculated using the above diagram.

What is proportion? From a mathematical point of view, proportion is the equality of 2 ratios. All parts of the proportion are interdependent, and their result is unshakable.

You will need

• — Algebra textbook for 7th grade.

Instructions

1. The numbers that are on the edges of the equality are called extreme. Accordingly, those in the middle are average. The main property of proportion is that the extreme and middle parts of an equality can be multiplied among themselves. Take the proportion 6:3=8:4. Multiply the extreme parts together, you get 6 * 4 = 24, the product of the middle parts will also be equal to 24. Hence the result: the product of some parts of the proportion must be equal to the product of other parts (extreme = middle).

2. Take this quality of proportion into service, calculate the unfamiliar term of the equation x: 4 = 15: 3. In order to discover the unknown part of the proportion, use the rule of equivalence of the extreme and middle parts. Write this equation like this: x*3=4*15. Solving this equation will give you the correct proportion.

3. If the proportion consists of huge or fractional numbers, it can be simplified. Reduce both terms of the ratio by an identical number of times. To avoid any violation of the proportion, do this: 40:10=60:15. Increase both terms of the ratio by three times (120:30=60:15) or decrease parts of the second ratio (40:10=12:3). Both proportions will be positive.

4. Increase or decrease the proportions only by an identical number of times. Having obtained simplified reformations, you free the proportion from fractional terms and simplify the equation. Take an example: 200:25=56:x. To avoid performing calculations with huge numbers, divide them by the same number. If we take 25 as this number, the equation will take the following form: 8:1=56:x. The unknown part of this proportion can be determined in the mind without resorting to difficult calculations.

5. Parts of the proportions can be rearranged. Take the proportion 3:5=12:20. Rearrange the outer parts (20:5=12:3), simultaneous rearrangement of all parts is also possible (20:12=5:3). All proportions will be correct. So from one proportion you will get several, and they will all be positive.

Note!
Regrouping parts of proportions in places is convenient when solving problems.

Helpful advice
The basic quality of all proportions: ab = bc.

In mathematics, a proportion is the equality of two ratios. All its parts are characterized by interdependence and a constant outcome. It is enough to look at one example in order to understand the thesis of solving proportions.

Instructions

1. Study the properties of proportions. The numbers on the edges of the equality are called extreme, and those in the middle are called average. The main quality of proportion is that the middle and extreme parts of an equality can be multiplied among themselves. It is enough to take the proportion 8:4 = 6:3. If you multiply the extreme parts together, you get 8*3=24, as when multiplying the middle numbers. This means that the product of the extreme parts of a proportion is invariably equal to the product of its middle parts.

2. Take into account the basic quality of proportion in order to calculate the unknown term in the equation x: 4 = 8: 2. To find an unfamiliar part of a proportion, you should use the rule of equivalence of the middle and extreme parts. Write the equation in the form x*2=4*8, that is, x*2=32. Solve this equation (32/2), you will get the missing term of the proportion (16).

3. Simplify the proportion if it consists of fractional or large numbers. To do this, divide or multiply both of its terms by an identical number. For example, the combined parts of the proportion 80:20=120:30 can be simplified by dividing its terms by 10 (8:2=12:3). You will get equivalent equality. The same will happen if you increase all terms of the proportion, say, by 2, so 160:40 = 240:60.

4. Try rearranging parts of the proportions. For example, 6:10=24:40. Swap the outer parts (40:10=24:6) or rearrange all parts at once (40:24=10:6). All resulting proportions will be equivalent. This way you can get several equalities from one.

5. Solve the proportion with percentages. Write it down, say, in the form: 25=100%, 5=x. Now you need to multiply the middle terms (5 * 100) and divide by the famous extreme (25). The result is that x=20%. In the same way, you can multiply the famous extreme terms and divide them by the existing average, obtaining the desired result.

One percent is a hundredth of a number. This concept used when you need to indicate the ratio of a part to the whole. In addition, several values ​​can be compared as percentages, but be sure to indicate relative to which integer the percentages are calculated. For example, expenses are 10% higher than income or the price of train tickets has increased by 15% compared to last year's tariffs. A percent number above 100 means that the proportion is greater than the whole, as is often the case in statistical calculations.

Interest as a financial concept is a payment from a borrower to a lender for providing money for temporary use. In business, the expression “work for interest” is common. In this case, it is understood that the amount of remuneration depends on profit or turnover (commissions). It is impossible to do without calculating percentages in accounting, business, and banking. To simplify calculations, an online interest calculator has been developed.

The calculator allows you to calculate:

• Percentage of the set value.
• Percentage of the amount (tax on actual salary).
• Percentage of the difference (VAT from ).
• And much more...

When solving problems using a percentage calculator, you need to operate with three values, one of which is unknown (a variable is calculated using the given parameters). The calculation scenario should be selected based on the specified conditions.

Examples of calculations

1. Calculating the percentage of a number

To find a number that is 25% of 1,000 rubles, you need:

• 1,000 × 25 / 100 = 250 rub.
• Or 1,000 × 0.25 = 250 rubles.

To calculate using a regular calculator, you need to multiply 1,000 by 25 and press the % button.

2. Definition of an integer (100%)

We know that 250 rub. is 25% of a certain number. How to calculate it?

Let's make a simple proportion:

• 250 rub. - 25%
• Y rub. - 100 %
• Y = 250 × 100 / 25 = 1,000 rub.

3. Percentage between two numbers

Let's say a profit of 800 rubles was expected, but we received 1,040 rubles. What is the percentage of excess?

The proportion will be like this:

• 800 rub. - 100 %
• RUB 1,040 – Y%
• Y = 1,040 × 100 / 800 = 130%

Exceeding the profit plan is 30%, that is, fulfillment is 130%.

4. Calculation is not based on 100%

For example, a store consisting of three departments receives 100% of customers. In the grocery department - 800 people (67%), in the department household chemicals- 55. What percentage of buyers come to the household chemicals department?

Proportion:

• 800 visitors – 67%
• 55 visitors - Y%
• Y = 55 × 67 / 800 = 4.6%

5. By what percentage is one number less than another?

The price of the product dropped from 2,000 to 1,200 rubles. By what percentage has the price of the product fallen or by what percentage is 1,200 less than 2,000?

• 2 000 - 100 %
• 1,200 – Y%
• Y = 1,200 × 100 / 2,000 = 60% (60% to the figure 1,200 from 2,000)
• 100% − 60% = 40% (the number 1,200 is 40% less than 2,000)

6. By what percentage is one number greater than another?

The salary increased from 5,000 to 7,500 rubles. By what percentage did the salary increase? What percentage is 7,500 greater than 5,000?

• 5,000 rub. - 100 %
• 7,500 rub. - Y%
• Y = 7,500 × 100 / 5,000 = 150% (in numbers 7,500 is 150% of 5,000)
• 150% − 100% = 50% (the number 7,500 is 50% greater than 5,000)

7. Increase the number by a certain percentage

The price of product S is above 1,000 rubles. by 27%. What is the price of the product?

• 1,000 rub. - 100 %
• S - 100% + 27%
• S = 1,000 × (100 + 27) / 100 = 1,270 rub.

The online calculator makes calculations much simpler: you need to select the type of calculation, enter the number and percentage (in the case of calculating a percentage, the second number), indicate the accuracy of the calculation and give the command to begin the action.

Problem 1. The thickness of 300 sheets of printer paper is 3.3 cm. What thickness will a pack of 500 sheets of the same paper be?

Solution. Let x cm be the thickness of a stack of paper of 500 sheets. There are two ways to 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. 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 have a thickness 5.5 cm.

This is a classic reasoning and design of a solution to a problem. Such tasks are often included in test tasks for graduates who usually write the solution in this form:

or they decide orally, reasoning like this: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times less. Divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100-sheet pack of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and 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 down 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 the watermelon (5 kg) is 100%. Water will be x kg or 98%. There are two ways to find how many kg are in 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: 5kg watermelon contains 4.9 kg 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 you can find the mass of 1 liter of oil in two ways:

16,8: 21 or x : 35. We get the proportion:

16,8: 21=x : 35.

We find average member proportions. To do this, we multiply the extreme terms of the proportion ( 16,8 And 35 ) and divide by the known average term ( 21 ). Let's 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 mass 28 kg.

After 82% of the entire field had been plowed, there was still 9 hectares left to plow. What is the area of ​​the entire field?

Solution.

Let the area of ​​the entire field be x hectares, which is 100%. There are 9 hectares left to plow, which is 100% - 82% = 18% of the entire field. We can express 1% of the field area in two ways. This:

X : 100 or 9 : 18. We make up the proportion:

X : 100 = 9: 18.

We find the unknown extreme term of the proportion. To do this, 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 entire field 50 hectares.

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But not everything is as complicated and incomprehensible as it seems at first glance. Why is all this needed? Here is the most common example.

Let's say we have images uploaded on our website, and we want that after loading we create a miniature copy, a preview of the image. This is often necessary to announce news, for example. And the script requires that you specify at least the approximate dimensions of the miniature image - its width and height.

Let's also say that you have already outlined its width, but what about the height? How to calculate it so that the picture seems more or less proportional to the original one.

Calculation formula

Everything is done in two stages:

• 1 - Divide the original width by the required width;
• 2 - We obtain the required height by dividing the original height by the result of dividing the two widths (step 1).

Example. Let’s take the image sizes already known to everyone: 1024x768 and 800x600. Let's imagine that we don't know the height of the second picture. The formula gives the following: 768/(1024/800) = 600 . This is the height we need.

If we know the height, but we need to get the width, then we need to do everything as in the first formula, only in reverse.

To get the required width you need:

• 1 - Divide the original height by the required height;
• 2 - We obtain the required width by dividing the original width by the result of dividing the two heights (step 1).

That is, 1024/(768/600) = 800 .

§ 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 to 1 (unit) as 10 is to 5 (the first proportion). You can read it differently, for example: 2 is as many times more than 1, how many times is 10 more than 5. The third proportion can be read like this: - 0.5 is as many times less than 2, how many times 0.75 is less than 3.

The numbers included in the proportion are called members of the proportion. This means that 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 located in the middle are called average members. This means that in the first proportion the numbers 2 and 5 will be the extreme terms, and the numbers 1 and 10 will be the middle terms of the proportion.

§ 126. The main property of proportion.

Consider the proportion:

Let us multiply its extreme and middle terms separately. The product of the extremes is 6 4 = 24, the product of the middle ones is 3 8 = 24.

Let's consider another proportion: 10: 5 = 12: 6. Let's multiply the extreme and middle terms separately here too.

The product of the extremes is 10 6 = 60, the product of the middle ones is 5 12 = 60.

The main property of proportion: the product of the extreme terms of a proportion is equal to the product of its middle terms.

IN general view the basic property of proportion is written as follows: ad = bc .

Let's check it on several proportions:

1) 12: 4 = 30: 10.

This proportion is correct, since the ratios from 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 ratios. The main property of proportion will take the form:

1 / 2 5 / 6 = 1 / 48 20

It is not difficult to verify that if we write an equality in which on the left side there is the product of two numbers, and on the right side the product of two other numbers, then from these four numbers you can make a proportion.

Let us have an equality that includes four numbers multiplied in pairs:

these four numbers can be terms 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 terms. The published equality can be compiled, for example, into the following proportion:

In general, from equality ad = bc the following proportions can be obtained:

Do the following exercise yourself. 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 terms of proportion.

The basic property of 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 member is unknown. We know that in any 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 consider 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 result found 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 terms of the proportion, only the extreme one was unknown to us; the middle two and the 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 found product 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 proportion: 70 X = 10 21.

Multiplying the numbers 10 and 21, we rewrite the equality as follows:

70 X = 210.

Here one factor is unknown; to calculate it, it is enough to divide the product (210) by another factor (70),

X = 210: 70; X = 3.

So we can say that each extreme term of the proportion is equal to the product of the averages divided by the other extreme.

Let us now move on to calculating the unknown average term. Let's take the proportion:

30: X = 27: 9.

Let's write the main property of proportion:

30 9 = X 27.

Let's 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 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 proportion:

12 . 8 = 6 X . Multiplying 12 and 8 and rearranging the parts of the equality, we get:

6 X = 96. Find the unknown factor:

X = 96:6, or X = 16.

Thus, each middle term of the proportion is equal to the product of the extremes divided by the other middle.

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 .

Two latest rules in general form it can be written as follows:

1) If the proportion looks like:

x: a = b: c , That

2) If the proportion looks like:

a: x = b: c , That

§ 128. Simplification of proportion and rearrangement of its terms.

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. The transformations that do not violate the proportion include the following:

1. Simultaneous increase or decrease of both terms of any ratio in same number once.

EXAMPLE 40:10 = 60:15.

Multiplying both terms of the first relation by 3 times, we get:

120:30 = 60: 15.

The proportion was not violated.

Reducing both terms of the second ratio by 5 times, we get:

We got the correct proportion again.

2. Simultaneous increase or decrease of both previous or both subsequent terms by the same number of times.

Example. 16:8 = 40:20.

Let us double the previous terms of both relations:

We got the correct proportion.

Let us decrease the subsequent terms of both relations by 4 times:

The proportion was not violated.

The two conclusions obtained can be briefly stated as follows: The proportion will not be violated if we simultaneously increase or decrease by the same number of times any extreme term of the proportion and any middle one.

For example, reducing by 4 times the 1st extreme and 2nd middle terms of the proportion 16:8 = 40:20, we get:

3. Simultaneous increase or decrease of all terms 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 was not violated. Let's decrease all four numbers by 4 times:

The proportion is correct.

The listed transformations make it possible, firstly, to simplify proportions, and secondly, to free them from fractional terms. Let's give examples.

1) Let there be a proportion:

200: 25 = 56: x .

In it, the members of the first ratio are relatively large numbers, and if we wanted to find the value X , then we would have to perform calculations on these numbers; but we know that the proportion will not be violated if both terms of the ratio are divided by the same number. Let's 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) Let's 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, you will have to multiply this term, for example, by 2. But we do not have the right to increase one middle term of the proportion; it is necessary to increase one of the extreme members along with it; 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 member:

2: (2 1/2) = 20: (2 5), or 2: 1 = 20: 10.

Let's look at three more examples of freeing proportions from fractional terms.

Example 1. 1 / 4: 3 / 8 = 20:30.

Let's reduce the fractions to common denominator:

2 / 8: 3 / 8 = 20: 30.

Multiplying both terms of the first ratio by 8, we get:

Example 2. 12: 15 / 14 = 16: 10 / 7. Let's bring the fractions to a common denominator:

12: 15 / 14 = 16: 20 / 14

Let's multiply both subsequent terms by 14, we get: 12:15 = 16:20.

Example 3. 1 / 2: 1 / 48 = 20: 5 / 6.

Let's multiply all 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. Let's consider which permutations are legal, i.e., do not violate the proportions. Let's take the proportion:

3: 5 = 12: 20. (1)

Rearranging the extreme terms in it, we get:

20: 5 = 12:3. (2)

Let us now rearrange the middle terms:

3:12 = 5: 20. (3)

Let us rearrange both the extreme and middle terms at the same time:

20: 12 = 5: 3. (4)

All these proportions are correct. Now let's put the first relation in the place of the second, and the second in the place of the first. You get the proportion:

12: 20 = 3: 5. (5)

In this proportion we will make the same rearrangements as we did before, that is, we will first rearrange the extreme terms, then the middle ones, and finally, both the extremes and the middle ones at the same time. You will get three more proportions, 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.

The validity of all these proportions is especially easy to discover when writing 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 = bc.

Thus, these permutations do not violate the fairness of the proportion and can be used if necessary.