How to calculate ratios. How to make the proportion? Any student and adult will understand Ratio 1 0

Formula of proportions

Proportion is the equality of two ratios when a: b = c: d

ratio 1 : 10 is equal to the ratio 7 : 70, which can also be written as a fraction: 1 10 = 7 70 reads as: "one refers to ten as well as seven refers to seventy"

Basic proportion properties

The product of the extreme terms is equal to the product of the middle terms (crosswise): if a: b = c: d, then a⋅d = b⋅c

1 10 ✕ 7 70 1 ⋅ 70 = 10 ⋅ 7

Inversion of proportion: if a: b = c: d then b: a = d: c

1 10 7 70 10 1 = 70 7

Permutation of middle terms: if a: b = c: d, then a: c = b: d

1 10 7 70 1 7 = 10 70

Permutation of extreme terms: if a: b = c: d, then d: b = c: a

1 10 7 70 70 10 = 7 1

Solving proportions with one unknown | The equation

1 : 10 = x : 70 or 1 10 = x 70

To find x, you need to multiply two known numbers crosswise and divide by the opposite value

x = 1 ⋅ 70 10 = 7

How to calculate the proportion

Task: you need to drink 1 tablet of activated carbon per 10 kilograms of weight. How many tablets should you take if a person weighs 70 kg?

Let's make a proportion: 1 tablet - 10 kg x tablets - 70 kg To find x, you need to multiply two known numbers crosswise and divide by the opposite value: 1 tablet x pills✕ 10 kg 70 Kg x = 1 ⋅ 70 : 10 = 7 Answer: 7 tablets

Task: Vasya writes two articles in five hours. How many articles will he write in 20 hours?

Let's make a proportion: 2 articles - 5 hours x articles - 20 hours x = 2 ⋅ 20 : 5 = 8 Answer: 8 articles

I can say to future school graduates that the ability to make proportions came in handy for me, and in order to proportionally reduce pictures, and in HTML layout of a web page, and in everyday situations.

A relationship is called a certain relationship between the entities of our world. These can be numbers, physical quantities, objects, products, phenomena, actions, and even people.

In everyday life, when it comes to ratios, we say "The ratio of this and that"... For example, if there are 4 apples and 2 pears in a vase, then we say "The ratio of apples and pears" "The ratio of pears and apples".

In mathematics, the ratio is often used as "The attitude of so-and-so to that-and-so"... For example, the ratio of four apples and two pears, which we considered above, in mathematics will read as "The ratio of four apples to two pears" or if you swap apples and pears, then "The ratio of two pears to four apples".

The ratio is expressed as a To b(where instead of a and b any numbers), but more often you can find an entry that is composed using a colon as a: b... You can read this entry in different ways:

• a To b
• a refers to b
• attitude a To b

Let's write the ratio of four apples to two pears using the ratio symbol:

4: 2

If we swap the places of apples and pears, then we will have a ratio of 2: 4. This ratio can be read as "Two to four" or either "Two pears refer to four apples" .

In what follows, we will call the ratio a ratio.

Lesson content

What is an attitude?

The relation, as mentioned earlier, is written in the form a: b... It can also be written as a fraction. And we know that such a notation in mathematics means division. Then the result of the relationship will be the quotient a and b.

A ratio in mathematics is called the quotient of two numbers.

The ratio allows you to find out how much of one entity falls on the unit of another. Let's go back to the ratio of four apples to two pears (4: 2). This ratio will allow us to find out how many apples are there per unit of pear. A unit means one pear. First, let's write the ratio 4: 2 as a fraction:

This ratio is the division of the number 4 by the number 2. If we perform this division, we will get an answer to the question how many apples are there per unit of pear

Received 2. So four apples and two pears (4: 2) correlate (are interconnected with each other) so that there are two apples per pear

The figure shows how four apples and two pears relate to each other. It can be seen that there are two apples for each pear.

The relationship can be reversed by writing as. Then we get the ratio of two pears to four apples, or "the ratio of two pears to four apples." This ratio will show how many pears are there per unit of apple. The unit of apple means one apple.

To find the value of a fraction, you need to remember how to divide a smaller number by a larger one.

Received 0.5. Let's convert this decimal fraction to an ordinary one:

Reduce the resulting fraction by 5

Received an answer (half a pear). This means that two pears and four apples (2: 4) correlate (are interconnected with each other) so that one apple accounts for half of the pear

The figure shows how two pears and four apples relate to each other. It can be seen that for each apple there is a half of a pear.

The numbers that make up the ratio are called members of the relationship... For example, in a 4: 2 ratio, the members are the numbers 4 and 2.

Let's consider other examples of relationships. To prepare something, a recipe is drawn up. The recipe is built from the relationship between the products. For example, making oatmeal usually requires a glass of cereal for two glasses of milk or water. The ratio is 1: 2 ("one to two" or "one glass of cereal for two glasses of milk").

We convert the ratio 1: 2 to a fraction, we get. Calculating this fraction, we get 0.5. This means that one glass of cereal and two glasses of milk are correlated (interconnected with each other) so that one glass of milk accounts for half a glass of cereal.

If you flip the ratio 1: 2, you get a 2: 1 ratio ("two to one" or "two glasses of milk for one glass of cereal"). Convert the ratio 2: 1 to a fraction, we get. Calculating this fraction, we get 2. So two glasses of milk and one glass of cereals are correlated (interconnected with each other) so that there are two glasses of milk for one glass of cereals.

Example 2. There are 15 students in the class. 5 of them are boys, 10 are girls. You can write down the ratio of girls to boys 10: 5 and convert that ratio to a fraction. Calculating this fraction, we get 2. That is, girls and boys are related to each other in such a way that for every boy there are two girls

The figure shows how ten girls and five boys relate to each other. It can be seen that there are two girls for every boy.

The ratio cannot always be converted into a fraction and the quotient can be found. In some cases, this will not be logical.

So, if you turn over the attitude, it turns out, and this is the attitude of boys to girls. If you calculate this fraction, you get 0.5. It turns out that five boys relate to ten girls in such a way that for every girl there is half a boy. Mathematically, this is of course true, but from the point of view of reality it is not entirely reasonable, because a boy is a living person and you cannot just take and divide him, like a pear or an apple.

Building the right attitude is an important problem solving skill. So in physics, the ratio of the distance traveled to time is the speed of movement.

The distance is denoted by the variable S, time - through the variable t, speed - through the variable v... Then the phrase "The ratio of the distance traveled to time is the speed of movement" will be described by the following expression:

Suppose the car has traveled 100 kilometers in 2 hours. Then the ratio of the traversed one hundred kilometers to two hours will be the speed of the car:

It is customary to call speed the distance traveled by the body per unit of time. The unit of time means 1 hour, 1 minute or 1 second. And the relationship, as mentioned earlier, allows you to find out how much of one entity falls on the unit of another. In our example, the ratio of one hundred kilometers to two hours shows how many kilometers are there for one hour of movement. We see that there are 50 kilometers for every hour of movement.

Therefore, the speed is measured in km / h, m / min, m / s... The fraction symbol (/) indicates the ratio of distance to time: kilometers per hour , meters per minute and meters per second respectively.

Example 2... The ratio of the value of a product to its quantity is the price of one unit of a product

If we took 5 chocolate bars from the store and their total cost was 100 rubles, then we can determine the price of one bar. To do this, you need to find the ratio of one hundred rubles to the number of bars. Then we get that there are 20 rubles for one bar.

Comparison of quantities

Earlier we learned that the relationship between quantities of different natures form a new quantity. So, the ratio of the distance traveled to time is the speed of movement. The ratio of the value of a commodity to its quantity is the price of one unit of a commodity.

But the ratio can also be used to compare values. The result of such a relationship is a number showing how many times the first value is greater than the second, or how much of the first value is from the second.

To find out how many times the first value is greater than the second, a larger value must be written in the numerator of the ratio, and a smaller value in the denominator.

To find out what part of the first value is from the second, you need to write a smaller value in the numerator of the ratio, and a larger value in the denominator.

Consider the numbers 20 and 2. Let's find out how many times the number 20 is greater than the number 2. To do this, we find the ratio of the number 20 to the number 2. In the numerator of the ratio we write the number 20, and in the denominator - the number 2

The value of this ratio is ten

The ratio of the number 20 to the number 2 is the number 10. This number shows how many times the number 20 is greater than the number 2. So the number 20 is ten times greater than the number 2.

Example 2. There are 15 students in the class. 5 of them are boys, 10 are girls. Determine how many times there are more girls than boys.

We write down the attitude of girls towards boys. We write down the number of girls in the numerator of the relationship, and the number of boys in the denominator of the relationship:

The value of this ratio is 2. This means that there are twice as many girls in a class of 15 as boys.

There is no longer the question of how many girls there are for one boy. In this case, the ratio is used to compare the number of girls with the number of boys.

Example 3... What part of the number 2 is from the number 20.

We find the ratio of the number 2 to the number 20. In the numerator of the ratio we write the number 2, and in the denominator - the number 20

To find the meaning of this relationship, you need to remember

The value of the ratio of the number 2 to the number 20 is the number 0.1

In this case, the decimal fraction 0.1 can be converted to an ordinary one. This answer will be easier to understand:

So the number 2 of the number 20 is one tenth.

You can check. To do this, we find from the number 20. If we did everything correctly, then we should get the number 2

20: 10 = 2

2 × 1 = 2

We got the number 2. So one tenth of the number 20 is the number 2. Hence we conclude that the problem is solved correctly.

Example 4. There are 15 people in the class. 5 of them are boys, 10 are girls. Determine what proportion of the total number of schoolchildren are boys.

We write down the ratio of boys to the total number of schoolchildren. We write down five boys in the numerator of the relationship, and the total number of students in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write 15 in the denominator of the relationship

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 5 must be divided by the number 15

When you divide 5 by 15, you get a periodic fraction. Let's convert this fraction to an ordinary one

We got the final answer. So boys make up one third of the class.

The figure shows that in a class of 15 students, 5 boys make up a third of the class.

If we find from 15 schoolchildren for verification, then we get 5 boys

15: 3 = 5

5 × 1 = 5

Example 5. How many times is 35 greater than 5?

We write down the ratio of the number 35 to the number 5. In the numerator of the ratio, you need to write the number 35, in the denominator - the number 5, but not vice versa

The value of this ratio is 7. So the number 35 is seven times the number 5.

Example 6. There are 15 people in the class. 5 of them are boys, 10 are girls. Determine what proportion of the total number are girls.

We write down the ratio of girls to the total number of schoolchildren. We write ten girls in the numerator of the relationship, and the total number of schoolchildren in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write 15 in the denominator of the relationship

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 10 must be divided by the number 15

When you divide 10 by 15, you get a periodic fraction. Let's convert this fraction to an ordinary one

Reduce the resulting fraction by 3

We got the final answer. So girls make up two-thirds of the class.

The figure shows that in a class of 15 students, two thirds of the class are 10 girls.

If we find from 15 schoolchildren for verification, then we get 10 girls

15: 3 = 5

5 × 2 = 10

Example 7. What part of 10 cm is 25 cm

We write down the ratio of ten centimeters to twenty-five centimeters. We write 10 cm in the numerator of the ratio, 25 cm in the denominator

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 10 must be divided by the number 25

Let's convert the resulting decimal fraction to an ordinary

Reduce the resulting fraction by 2

We got the final answer. This means that 10 cm are from 25 cm.

Example 8. How many times is 25 cm more than 10 cm

We write down the ratio of twenty-five centimeters to ten centimeters. In the numerator of the ratio we write 25 cm, in the denominator - 10 cm

The answer was 2.5. Means 25 cm more than 10 cm 2.5 times (two and a half times)

Important note. When finding the ratio of physical quantities of the same name, these quantities must necessarily be expressed in one unit of measurement, otherwise the answer will be incorrect.

For example, if we are dealing with two lengths and we want to know how many times the first length is greater than the second, or what part of the first length is from the second, then both lengths must first be expressed in one unit of measurement.

Example 9. How many times is 150 cm more than 1 meter?

First, let's make it so that both lengths are expressed in the same unit of measurement. To do this, let's convert 1 meter to centimeters. One meter is one hundred centimeters

1 m = 100 cm

Now we find the ratio of one hundred and fifty centimeters to one hundred centimeters. In the numerator of the ratio we write 150 centimeters, in the denominator - 100 centimeters

Let's find the value of this ratio

The answer was 1.5. This means that 150 cm is 1.5 times more than 100 cm (one and a half times).

And if they did not convert meters to centimeters and immediately tried to find the ratio of 150 cm to one meter, then we would get the following:

It would turn out that 150 cm is more than one meter one hundred and fifty times, but this is not true. Therefore, it is imperative to pay attention to the units of measurement of physical quantities that are involved in the relationship. If these quantities are expressed in different units of measurement, then to find the ratio of these quantities, you need to go to one unit of measurement.

Example 10. Last month, a person's salary was 25,000 rubles, and this month, the salary has increased to 27,000 rubles. Determine how many times the salary has grown

We write down the ratio of twenty-seven thousand to twenty-five thousand. We write 27000 in the numerator of the ratio, 25000 in the denominator.

Let's find the value of this ratio

The answer was 1.08. This means that the salary has increased by 1.08 times. In the future, when we get to know percentages, we will express such indicators as salaries as percentages.

Example 11... The width of the apartment building is 80 meters and the height is 16 meters. How many times is the house wider than its height?

We write down the ratio of the width of the house to its height:

The value of this ratio is 5. This means that the width of the house is five times its height.

Relationship property

The ratio will not change if its members are multiplied or divided by the same number.

This is one of the most important properties of the relationship follows from the property of the particular. We know that if the dividend and divisor are multiplied or divided by the same number, then the quotient will not change. And since the relation is nothing more than division, the property of the particular works for it too.

Let's go back to girls' attitudes towards boys (10: 5). This attitude showed that there are two girls for every boy. Let's check how the relationship property works, namely, let's try to multiply or divide its members by the same number.

In our example, it is more convenient to divide the members of the relationship by their greatest common divisor (GCD).

The gcd of members 10 and 5 is the number 5. Therefore, the members of the relationship can be divided by the number 5

Got a new attitude. This is a two-to-one ratio (2: 1). This ratio, like the past ratio of 10: 5, shows that there are two girls for one boy.

The figure shows a 2: 1 (two to one) ratio. As in the past, the ratio of 10: 5 per boy has two girls. In other words, the attitude has not changed.

Example 2... There are 10 girls and 5 boys in one class. In another class there are 20 girls and 10 boys. How many times are there more girls in the first grade than boys? How many times are there more girls in the second grade than boys?

In both classes, there are twice as many girls as boys, because the relationships and are equal to the same number.

The relationship property allows you to build various models that have similar parameters to the real object. Suppose an apartment building is 30 meters wide and 10 meters high.

To draw a similar house on paper, you need to draw it in the same ratio of 30: 10.

Divide both terms of this ratio by the number 10. Then we get the ratio 3: 1. This ratio is 3, just like the previous ratio is 3

Let's convert meters to centimeters. 3 meters is 300 centimeters, and 1 meter is 100 centimeters

3 m = 300 cm

1 m = 100 cm

We have a ratio of 300 cm: 100 cm. Divide the terms of this ratio by 100. We obtain a ratio of 3 cm: 1 cm. Now we can draw a house with a width of 3 cm and a height of 1 cm.

Of course, the drawn house is much smaller than the real house, but the ratio of width and height remains unchanged. This allowed us to draw a house as close as possible to the real one.

Attitude can be understood in other ways as well. It was originally said that a real house has a width of 30 meters and a height of 10 meters. The total is 30 + 10, that is, 40 meters.

These 40 meters can be understood as 40 parts. A ratio of 30: 10 means there are 30 pieces for the width and 10 pieces for the height.

Further, the members of the ratio 30: 10 were divided by 10. The result was a ratio of 3: 1. This ratio can be understood as 4 parts, three of which are for the width, one for the height. In this case, you usually need to find out how many meters are in the width and height.

In other words, you need to find out how many meters are in 3 parts and how many meters are in 1 part. First you need to find out how many meters are in one part. To do this, the total 40 meters must be divided by 4, since in a 3: 1 ratio there are only four parts

Let's determine how many meters are in the width:

10 m × 3 = 30 m

Let's determine how many meters are at the height:

10 m × 1 = 10 m

Multiple relationship members

If several members are given in a relation, then they can be understood as parts of something.

Example 1... Purchased 18 apples. These apples were divided between mom, dad and daughter in a ratio of 2: 1: 3. How many apples did each get?

The ratio 2: 1: 3 means that mom got 2 parts, dad - 1 part, daughter - 3 parts. In other words, each member of the 2: 1: 3 ratio is a specific fraction of 18 apples:

If you add up the members of the ratio 2: 1: 3, then you can find out how many parts there are in total:

2 + 1 + 3 = 6 (parts)

Find out how many apples are in one part. To do this, divide 18 apples by 6

18: 6 = 3 (apples per slice)

Now let's determine how many apples each got. By multiplying three apples by each member of the 2: 1: 3 ratio, you can determine how many apples mom got, how much dad got, and how much daughter got.

Let's find out how many apples mom got:

3 × 2 = 6 (apples)

Find out how many apples dad got:

3 × 1 = 3 (apples)

Let's find out how many apples my daughter received:

3 × 3 = 9 (apples)

Example 2... New silver (alpaca) is an alloy of nickel, zinc and copper in a ratio of 3: 4: 13. How many kilograms of each metal do you need to take to get 4 kg of new silver?

4 kilograms of new silver will contain 3 parts nickel, 4 parts zinc and 13 parts copper. First, we find out how many parts there will be in four kilograms of silver:

3 + 4 + 13 = 20 (parts)

Let's determine how many kilograms will be in one part:

4 kg: 20 = 0.2 kg

Let's determine how many kilograms of nickel will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, three parts of the alloy are indicated to contain nickel. Therefore, we multiply 0.2 by 3:

0.2 kg × 3 = 0.6 kg nickel

Now let's determine how many kilograms of zinc will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, the four parts of the alloy are said to contain zinc. Therefore, we multiply 0.2 by 4:

0.2kg × 4 = 0.8kg zinc

Now let's determine how many kilograms of copper will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, thirteen parts of the alloy are said to contain copper. Therefore, we multiply 0.2 by 13:

0.2 kg × 13 = 2.6 kg copper

This means that to get 4 kg of new silver, you need to take 0.6 kg of nickel, 0.8 kg of zinc and 2.6 kg of copper.

Example 3... Brass is an alloy of copper and zinc, the weight of which is 3: 2. To make a piece of brass, 120 g of copper is required. How much zinc does it take to make this piece of brass?

Let's determine how many grams of alloy are in one part. The condition says that 120 g of copper is required to make a piece of brass. It is also said that the three parts of the alloy contain copper. If we divide 120 by 3, we find out how many grams of alloy are in one part:

120: 3 = 40 grams per portion

Now let's determine how much zinc is required to make a piece of brass. To do this, multiply 40 grams by 2, since in the ratio 3: 2 it is indicated that two parts contain zinc:

40 g × 2 = 80 grams of zinc

Example 4... We took two alloys of gold and silver. In one, the amount of these metals is in a ratio of 1: 9, and in the other 2: 3. How much of each alloy should be taken to get 15 kg of a new alloy, in which gold and silver would be in a ratio of 1: 4?

Solution

15 kg of the new alloy should be in a ratio of 1: 4. This ratio suggests that one part of the alloy will be gold, and four parts will be silver. There are five parts in total. This can be schematically represented as follows

Let's determine the mass of one part. To do this, first add all parts (1 and 4), then divide the mass of the alloy by the number of these parts

1 + 4 = 5
15 kg: 5 = 3 kg

One part of the alloy will have a mass of 3 kg. Then 15 kg of the new alloy will contain 3 × 1 = 3 kg of gold and silver 3 × 4 = 12 kg of silver.

Therefore, to obtain an alloy weighing 15 kg, we need 3 kg of gold and 12 kg of silver.

Now let's answer the question of the problem - " How much of each alloy should you take? »

We will take 10 kg of the first alloy, since gold and silver in it are in a ratio of 1: 9. That is, this first alloy will give us 1 kg of gold and 9 kg of silver.

We will take 5 kg of the second alloy, since gold and silver are in it in a ratio of 2: 3. That is, this second alloy will give us 2 kg of gold and 3 kg of silver.

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The basis mathematical research is the ability to gain knowledge about certain quantities by comparing them with other quantities that are either are equal or more or less than those that are the subject of research. This is usually done using the series equations and proportions... When we use equations, we determine the required value by finding it equality with some other already familiar quantity or quantities.

However, it often happens that we compare an unknown quantity with others that not equal her, but more or less of her. This requires a different approach to data processing. We may need to find out, for example, how much one quantity is greater than the other, or how many times one contains the other. To find the answer to these questions, we will learn what is ratio two quantities. One ratio is called arithmetic and the other geometric... It is worth noting, though, that both of these terms were not accidentally adopted or just for purposes of distinction. Both arithmetic and geometric relationships apply to both arithmetic and geometry.

As a component of a vast and important subject, proportion depends on proportions, therefore a clear and complete understanding of these concepts is necessary.

338. Arithmetic ratio it differencebetween two quantities or a series of quantities... The quantities themselves are called members of ratios, that is, terms between which there is a ratio. Thus, 2 is an arithmetic ratio of 5 and 3. This is expressed by placing a minus sign between two values, that is, 5 - 3. Of course, the term arithmetic ratio and its scribbling is practically useless, since only the word is replaced difference by the minus sign in the expression.

339. If both terms of the arithmetic relation multiply or divide by the same amount, then ratio, ultimately, will be multiplied or divided by this value.
Thus, if we have a - b = r
Then we multiply both sides by h, (Ax. 3.) ha - hb = hr
And dividing by h, (Ax. 4.) $ \ frac (a) (h) - \ frac (b) (h) = \ frac (r) (h) $

340. If the terms of an arithmetic ratio are added to or subtracted from the corresponding terms of another, then the ratio of the sum or difference will be equal to the sum or difference of the two ratios.
If a - b
And d - h,
are two relations,
Then (a + d) - (b + h) = (a - b) + (d - h). Which in each case = a + d - b - h.
And (a - d) - (b - h) = (a - b) - (d - h). Which in each case = a - d - b + h.
So the arithmetic ratio 11 - 4 is 7
And the arithmetic ratio of 5 - 2 is 3
The ratio of the sum of members 16 - 6 is 10, is the sum of the ratios.
The ratio of the difference of terms 6 - 2 is 4, is the difference of the ratios.

341. Geometric ratio is the relationship between quantities, which is expressed PRIVATE if one quantity is divided by another.
Thus, the ratio of 8 to 4 can be written as 8/4 or 2. That is, the quotient of 8 by 4. In other words, it shows how many times 4 is contained in 8.

In the same way, the ratio of any quantity to another can be determined by dividing the first by the second or, which is, in principle, the same, by making the first the numerator of the fraction and the second the denominator.
So the ratio of a to b is $ \ frac (a) (b) $
The ratio of d + h to b + c is $ \ frac (d + h) (b + c) $.

342. The geometric relationship is also recorded by placing two points one above the other between the values ​​being compared.
Thus, a: b is a record of the ratio of a to b, and 12: 4 is a ratio of 12 to 4. The two quantities together form couple in which the first term is called antecedent and the last one is consequent.

343. This notation with the help of dots and the other, in the form of a fraction, are interchangeable as necessary, while the antecedent becomes the numerator of the fraction, and the consequent becomes the denominator.
So 10: 5 is the same as $ \ frac (10) (5) $ and b: d, the same as $ \ frac (b) (d) $.

344. If of these three values: antecedent, consequent and ratio, any two then the third can be found.

Let a = antecedent, c = consequent, r = ratio.
By definition, $ r = \ frac (a) (c) $, that is, the ratio is equal to the antecedent divided by the consequent.
Multiplying by c, a = cr, that is, the antecedent is equal to the consequent multiplied by the ratio.
Divide by r, $ c = \ frac (a) (r) $, that is, the consequent is equal to the antecedent divided by the ratio.

Corresponding 1. If two couples have equal antecedents and consequents, then their ratios are also equal.

Corresponding 2. If for two pairs the ratios and antecedents are equal, then the consequents are equal, and if the ratios and the consequents are equal, then the antecedents are also equal.

345. If two compared values are equal, then their ratio is equal to one or the ratio of equality. The ratio 3 * 6: 18 is equal to one, since the quotient of any quantity divided by itself is equal to 1.

If the antecedent of the pair more, than the consequent, the ratio is greater than one. Since the dividend is greater than the divisor, the quotient is greater than one. So the ratio of 18: 6 is 3. This is called the ratio greater inequality.

On the other hand, if the antecedent less than the consequent, then the ratio is less than unity and this is called the ratio less inequality... So the ratio of 2: 3 is less than one, because the dividend is less than the divisor.

346. The reverse ratio is the ratio of two reciprocals.
So the inverse ratio of 6 to 3 is k, that is:.
The direct ratio of a to b is $ \ frac (a) (b) $, that is, the antecedent is divided into a consequent.
The inverse relationship is $ \ frac (1) (a) $: $ \ frac (1) (b) $ or $ \ frac (1) (a). \ Frac (b) (1) = \ frac (b) ( a) $.
that is, the cosequent b divided by the antecedent a.

Hence, the inverse relation is expressed by inverting the fraction, which displays a direct ratio, or, when recording is carried out using dots, inverting the order of the members.
So a refers to b in the same way as b to a.

347. Complex ratio this ratio works corresponding terms with two or more simple relationships.
So the ratio 6: 3 is 2
And the ratio 12: 4 equals 3
The ratio made up of them is 72:12 = 6.

Here, a complex relationship is obtained by multiplying between themselves two antecedents and also two consequents of simple relationships.
So the ratio made
From the ratio a: b
And the ratio c: d
and the ratio h: y
This ratio is $ ach: bdy = \ frac (ach) (bdy) $.
The compound ratio does not differ in its nature from any other ratio. This term is used to show the origin of the relationship in certain cases.

Corresponding A complex ratio is equal to the product of simple ratios.
Ratio a: b is equal to $ \ frac (a) (b) $
The ratio c: d is equal to $ \ frac (c) (d) $
The h: y ratio is $ \ frac (h) (y) $
And the ratio added of these three will be ach / bdy, which is the product of fractions that express simple ratios.

348. If in the sequence of ratios in each previous pair the consequent is the antecedent in the next one, then the ratio of the first antecedent and the last consequent are equal to that obtained from the intermediate ratios.
So in a number of relations
a: b
b: c
c: d
d: h
the ratio a: h is equal to the ratio added from the ratios a: b and b: c and c: d and d: h. So the complex ratio in the last article is $ \ frac (abcd) (bcdh) = \ frac (a) (h) $, or a: h.

In the same way, all quantities that are both antecedents and consequents disappear, when the product of fractions will be simplified to its lowest terms and in the remainder the complex ratio will be expressed by the first antecedent and the last consequent.

349. A special class of complex relations is obtained by multiplying a simple relation by myself or something else equal ratio. These ratios are called double, triple, quads, and so on, according to the number of multiplication operations.

A ratio composed of two equal ratios, that is, square double ratio.

Composed of three, that is, cube simple relation are called triple, etc.

Similarly, the ratio square roots two quantities is called the ratio square root and the ratio cubic roots- the ratio cubic root, etc.
So the simple ratio of a to b is a: b
The double ratio of a to b is equal to a 2: b 2
The triple ratio of a to b is equal to a 3: b 3
The ratio of the square root of a to b is √a: √b
The ratio of the cube root of a to b is 3 √a: 3 √b, and so on.
Terms double, triple, and so on does not need to be mixed with doubled, tripled, etc.
The 6 to 2 ratio is 6: 2 = 3
Doubling this ratio, that is, the ratio twice, we get 12: 2 = 6
We triple this ratio, that is, this ratio three times, then we get 18: 2 = 9
A double ratio, that is square ratio is 6 2: 2 2 = 9
AND triple the ratio, that is, the cube of the ratio, is 6 3: 2 3 = 27

350. In order for the quantities to be correlated with each other, they must be of the same kind, so that one can confidently assert whether they are equal to each other, or one of them is greater or less. A foot is like 12 to 1 in relation to an inch: it is 12 times larger than an inch. But one cannot say, for example, that an hour is longer or shorter than a stick, or an acre is greater or less than a degree. However, if these quantities are expressed in numbers, then there may be a relationship between these numbers. That is, there may be a relationship between the number of minutes per hour and the number of steps per mile.

351. Turning to nature ratios, the next step we need to take into account the way in which the change in one or two terms, which are compared with each other, will affect the ratio itself. Recall that the direct ratio is expressed as a fraction, where antecedet couples always it numerator, a consequent - denominator... Then it will be easy to obtain from the property of fractions that changes in the ratio occur by varying the compared values. The ratio of the two quantities is the same as meaning fractions, each of which represents private: numerator divided by denominator. (Article. 341.) Now it has been shown that multiplying the numerator of a fraction by any value is the same as multiplying meaning by the same amount and dividing the numerator is the same as dividing the values ​​of a fraction. So,

352. Multiplying the antecedent of a pair by any value means multiplying the ratios by this value, and dividing the antecedent is dividing this ratio.
So a 6: 2 ratio is 3
And the ratio of 24: 2 is equal to 12.
Here the antecedent and ratio in the last pair is 4 times greater than in the first.
The a: b ratio is $ \ frac (a) (b) $
And the ratio na: b is equal to $ \ frac (na) (b) $.

Corresponding With a known consequent, the more antecedent, the more ratio, and conversely, the larger the ratio, the larger the antecedent.

353. Multiplying the consequent of the pair by any value, as a result, we get the division of the ratio by this value, and dividing the consequent we multiply the ratio. By multiplying the denominator of a fraction, we divide the value, and by dividing the denominator, the value is multiplied.
So the ratio of 12: 2 is 6
And the ratio of 12: 4 is 3.
Here is the consequent of the second pair in twice more, and the ratio twice less than the first.
The a: b ratio is $ \ frac (a) (b) $
And the a: nb ratio is equal to $ \ frac (a) (nb) $.

Corresponding With a given antecedent, the larger the consequent, the lower the ratio. Conversely, the larger the ratio, the lower the consequent.

354. From the last two articles it follows that multiplication of antecedent pair by any amount will have the same effect on the ratio as division of the consequent by this amount, and division of antecedent, will have the same effect as consequent multiplication.
Therefore, the ratio of 8: 4 is equal to 2
By multiplying the antecedent by 2, the ratio of 16: 4 is 4
By dividing the antecedent by 2, the 8: 2 ratio is 4.

Corresponding Any factor or divider can be transferred from the pair's antecedent to the consequent or from the consequent to the antecedent without changing the ratio.

It is worth noting that when a factor is thus transferred from one term to another, it becomes a divisor, and the transferred divisor becomes a factor.
So the ratio is 3.6: 9 = 2
Carrying over factor 3, $ 6: \ frac (9) (3) = 2 $
the same ratio.

Ratio $ \ frac (ma) (y): b = \ frac (ma) (by) $
Carrying over y $ ma: by = \ frac (ma) (by) $
Moving m, a: $ a: \ frac (m) (by) = \ frac (ma) (by) $.

355. As is obvious from Articles. 352 and 353, if the antecedent and consequent are both multiplied or divided by the same value, then the ratio does not change.

Corresponding 1. The ratio of the two fractions, which have a common denominator, is the same as the ratio of their numerators.
So the a / n: b / n ratio is the same as a: b.

Corresponding 2. Direct the ratio of two fractions that have a common numerator is equal to the inverse ratio of their denominators.

356. From the article it is easy to determine the ratio of any two fractions. If each term is multiplied by two denominators, then the ratio will be given by integral expressions. Thus, multiplying the terms of the pair a / b: c / d by bd, we get $ \ frac (abd) (b) $: $ \ frac (bcd) (d) $, which becomes ad: bc, by reducing the common values ​​from the numerators and denominators.

356. b. Ratio greater inequality increases his
Let the ratio of the greater inequality be given as 1 + n: 1
And any ratio like a: b
The compound ratio would be (Art. 347,) a + na: b
Which is greater than the ratio a: b (Art. 351. resp.)
But the ratio less inequality folded with a different ratio, reduces his.
Let the ratio of the smaller difference 1-n: 1
Any given ratio a: b
Complex ratio a - na: b
Which is less than a: b.

357. If to or from members of any pairadd or subtract two other quantities that are in the same ratio, then the sums or balances will have the same ratio.
Let the ratio a: b
Will be the same as c: d
Then the relation sums antecedents to the sum of the consequents, namely, a + c to b + d, are also the same.
That is, $ \ frac (a + c) (b + d) $ = $ \ frac (c) (d) $ = $ \ frac (a) (b) $.

Proof.

1.According to the assumption, $ \ frac (a) (b) $ = $ \ frac (c) (d) $
2. Multiply by b and d, ad = bc
3. Add cd to both sides, ad + cd = bc + cd
4. Divide by d, $ a + c = \ frac (bc + cd) (d) $
5. Divide by b + d, $ \ frac (a + c) (b + d) $ = $ \ frac (c) (d) $ = $ \ frac (a) (b) $.

Ratio differences the antecedents to the consequent difference are also the same.

358. If in several pairs the ratios are equal, then the sum of all antecedents refers to the sum of all consequent, as any antecedent to its consequent.
Thus, the ratio
|12:6 = 2
|10:5 = 2
|8:4 = 2
|6:3 = 2
Thus, the ratio (12 + 10 + 8 + 6) :( 6 + 5 + 4 + 3) = 2.

358. b. Ratio greater inequalitydecreases adding the same value to both members.
Let the given relation a + b: a or $ \ frac (a + b) (a) $
By adding x to both members, we get a + b + x: a + x or $ \ frac (a + b) (a) $.

The first becomes $ \ frac (a ^ 2 + ab + ax + bx) (a (a + x)) $
And the last one is $ \ frac (a ^ 2 + ab + ax) (a (a + x)) $.
Since the last numerator is obviously less than the other, then ratio should be less. (Art. 351. resp.)

But the ratio less inequality increases by adding the same amount to both terms.
Let the given relation (a-b): a, or $ \ frac (a-b) (a) $.
Adding x to both terms, it takes the form (a-b + x) :( a + x) or $ \ frac (a-b + x) (a + x) $
Bringing them to a common denominator,
The first one becomes $ \ frac (a ^ 2-ab + ax-bx) (a (a + x)) $
And the last one, $ \ frac (a ^ 2-ab + ax) (a (a + x)). \ Frac ((a ^ 2-ab + ax)) (a (a + x)) $.

Since the last numerator is greater than the other, then ratio more.
If instead of adding the same value take away from two terms, it is obvious that the effect on the ratio will be the opposite.

Examples.

1. Which is greater: an 11: 9 ratio, or a 44:35 ratio?

2. Which is greater: the ratio $ (a + 3): \ frac (a) (6) $, or the ratio $ (2a + 7): \ frac (a) (3) $?

3. If the pair's antecedent is 65 and the ratio is 13, what is the consequent?

4. If the consequent of a pair is 7 and the ratio is 18, what is the antecedent?

5. What does a complex ratio composed of 8: 7, and 2a: 5b, and also (7x + 1) :( 3y-2) look like?

6. What does a complex ratio composed of (x + y): b, and (x-y) :( a + b), and also (a + b): h look like? Resp. (x 2 - y 2): bh.

7. If the ratios (5x + 7) :( 2x-3), and $ (x + 2): \ left (\ frac (x) (2) +3 \ right) $ form a complex ratio, then what ratio will be obtained: more or less inequality? Resp. Ratio of greater inequality.

8. What is the ratio composed of (x + y): a and (x - y): b, and $ b: \ frac (x ^ 2-y ^ 2) (a) $? Resp. Equality ratio.

9. What is the ratio of 7: 5 and twice the ratio of 4: 9 and three times the ratio of 3: 2?
Resp. 14:15.

10. What is the ratio made up of 3: 7 and triple the x: y ratio and root extraction from the 49: 9 ratio?
Resp. x 3: y 3.