Ratio 1 0. How to calculate ratios. How to calculate proportion

Proportions are such a familiar combination that is probably known from primary classes secondary school. In the most general sense, proportion is the equality of two or more ratios.

That is, if there are certain numbers A, B and C

then the proportion

if there are four numbers A, B, C and D

then or are also proportions

The simplest example where proportion is used is to calculate percentages.

In general, the use of proportions is so wide that it is easier to say where they are not used.

Proportions can be used to determine distances, masses, volumes, as well as quantities of anything, with one important condition: in proportion, there should be linear relationships between different objects. Below, using the example of constructing a model of the Bronze Horseman, you will see how to calculate proportions where there are nonlinear dependencies.

Determine how many kilograms of rice there will be if you take 17 percent of the total volume of rice of 150 kilograms?

Let's make a proportion in words: 150 kilograms is the total volume of rice. So let's take it as 100%. Then 17% of 100% will be calculated as a proportion of two ratios: 100 percent is to 150 kilograms the same as 17 percent is to an unknown number.

Now the unknown number can be easily calculated

That is, our answer is 25.5 kilograms of rice.

Also related to proportions interesting riddles, which show that there is no need to rashly apply proportions for all occasions.

Here is one of them, slightly modified:

For display in the company's office, the director ordered the creation of a model of the Bronze Horseman sculpture without a granite pedestal. One of the conditions is that the layout must be made from the same materials as the original, the proportions must be respected and the height of the layout must be exactly 1 meter. Question: What will be the mass of the model?

First, let's look at the reference books.

The rider's height is 5.35 meters and his weight is 8,000 kg.

If we use the very first thought - to make a proportion: 5.35 meters is related to 8,000 kilograms as 1 meter is to an unknown quantity, then we may not even start the calculation, since the answer will be incorrect.

It's all about a small nuance that must be taken into account. It's all about the connection between mass and height sculptors nonlinear, that is, it cannot be said that by increasing, for example, a cube by 1 meter (observing the proportions so that it remains a cube), we will increase its weight by the same amount.

This is easy to check with examples:

1. glue a cube with an edge length of 10 centimeters. How much water will go in there? It is logical that 10*10*10 = 1000 cubic centimeters, that is, 1 liter. Well, since water was poured there (density is equal to unity), and not another liquid, then the mass will be equal to 1 kg.

2. glue a similar cube but with an edge length of 20 cm. The volume of water poured there will be equal to 20*20*20=8000 cubic centimeters, that is, 8 liters. Well, the weight is naturally 8 kg.

It is easy to notice that the relationship between mass and change in the length of a cube edge is nonlinear, or rather cubic.

Recall that volume is the product of height, width and depth.

That is, when changing the figure (subject to proportions/shape) linear size (height, width, depth) mass/volume volumetric figure changes cubically.

We reason:

Our linear size has changed from 5.35 meters to 1 meter, then the mass (volume) will change as the cube root of 8000/x

And we get that the layout Bronze Horseman in the company office with a height of 1 meter it will weigh 52 kilograms 243 grams.

But on the other hand, if the task was posed like this" the layout must be made from the same materials as the original, the proportions must be respected and volume 1 cubic meter "knowing that between volume and mass linear dependence- we would just use the standard ratio of the old volume to the new, and the old mass to the unknown number.

But our bot helps calculate proportions in other, more common and practical cases.

Surely, it will be useful to all housewives who prepare food.

Situations arise when a recipe for an amazing 10 kg cake has been found, but its volume is too large to prepare. I would like it to be smaller, for example, only two kilograms, but how to calculate all the new weights and volumes of ingredients?

This is where a bot will help you, which can calculate the new parameters of a 2-kilogram cake.

The bot will also help in calculations for hard-working men who are building a house and need to calculate how much ingredients they need to take for concrete if they only have 50 kilograms of sand.

Syntax

For XMPP client users: pro<строка>

where the string has required elements

number1/number2 - finding the proportion.

So that you don’t get scared by such a short description, let’s give an example here

200 300 100 3 400/100

What does it say, for example:

200 grams of flour, 300 milliliters of milk, 100 grams of butter, 3 eggs - pancake yield 400 grams.

How many ingredients do you need to take to bake just 100 grams of pancakes?

How easy it is to notice

400/100 is the ratio of a typical recipe and the yield we want to get.

We will look at examples in more detail in the corresponding section.

Examples

A friend shared a wonderful recipe

Dough: 200 grams of poppy seeds, 8 eggs, 200 icing sugar, 50 grams of grated bread, 200 grams of ground nuts, 3 cups of honey.
Boil the poppy seeds for 30 minutes over low heat, grind with a pestle, add melted honey, ground crackers, and nuts.
Beat eggs with powdered sugar and add to mixture.
Mix the dough carefully, pour into the mold, and bake.
Cut the cooled cake into 2 layers, coat with sour jam, then cream.
Decorate with jam berries.
Cream: 1 cup sour cream, 1/2 cup sugar, beat.

basis mathematical research is the ability to gain knowledge about certain quantities by comparing them with other quantities that either equal, or more or less than those that are the subject of research. This is usually done using a series equations And proportions. When we use equations, we determine the quantity we are looking for 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 than her. This requires a different approach to data processing. We may need to know, for example, for how long one quantity is greater than the other, or how many times one contains the other. To find the answer to these questions, we will find out what it is ratio two sizes. One ratio is called arithmetic, and the other geometric. Although it is worth noting that both of these terms were not adopted by chance or merely for the purpose of distinction. Both arithmetic and geometric relations apply to both arithmetic and geometry.

As a component of a broad and important subject, proportion depends on ratios, so a clear and complete understanding of these concepts is necessary.

338. Arithmetic relation This differencebetween two quantities or a series of quantities. The quantities themselves are called members relationships, that is, terms between which there is a relationship. Thus, 2 is the 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 description point by point is practically useless, since only a word is replaced difference by the minus sign in the expression.

339. If both terms of an arithmetic relation multiply or divide by the same amount, then ratio, will ultimately be multiplied or divided by this amount.
Thus, if we have a - b = r
Then 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 terms of an arithmetic relation add or subtract 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.
Thus the arithmetic ratio 11 - 4 is equal to 7
And the arithmetic relation 5 - 2 is 3
The ratio of the sum of terms 16 - 6 is 10, - the sum of the ratios.
The ratio of the difference of terms 6 - 2 is 4, - the difference of 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 divided 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, in principle, is the same thing, 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. A geometric relationship is also written by placing two points one above the other between the quantities being compared.
Thus a:b is the ratio of a to b, and 12:4 is the ratio of 12 to 4. The two quantities together form a couple, in which the first term is called antecedent, and the last one - consequential.

343. This notation in dotted form and the other in fractional form are interchangeable as necessary, the antecedent becoming the numerator of the fraction and the consequent the denominator.
So 10:5 is the same as $\frac(10)(5)$ and b:d is the same as $\frac(b)(d)$.

344. If any of these three meanings: antecedent, consequent and ratio are given 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 times the ratio.
Let's divide by r, $c=\frac(a)(r)$, that is, the consequent is equal to the antecedent divided by the ratio.

Resp. 1. If two pairs have equal antecedents and consequents, then their ratios are also equal.

Resp. 2. If two pairs have equal ratios and antecedents, then the consequents are equal, and if the ratios and consequents are equal, then the antecedents are equal.

345. If two quantities being compared equal, then their ratio is equal to one or the equality ratio. 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, then the ratio is greater than one. Since the dividend is greater than the divisor, the quotient is greater than one. So the ratio 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 one and this is called the ratio less inequality. So the ratio 2:3 is less than one because the dividend is less than the divisor.

346. Reverse a ratio is the ratio of two reciprocals.
So the inverse ratio is 6 to 3 is to, that is:.
The direct relation of a to b is $\frac(a)(b)$, that is, the antecedent divided by the consequent.
The inverse relation 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 relationship is expressed by inverting the fraction, which displays a direct relationship, or, when recording is done using points, inverting the order of writing the members.
Thus a is to b in the opposite way as b is to a.

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

Here a complex relation is obtained by multiplying two antecedents and also two consequents of simple relations.
So the ratio is drawn up
From the ratio a:b
And c:d ratios
and h:y ratios
This is the relation $ach:bdy=\frac(ach)(bdy)$.
The complex relationship is no different in its nature from any other ratio. This term is used to show the origin of a relationship in certain cases.

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

348. If in the sequence of relations in each previous pair the consequent is the antecedent in the subsequent one, then the ratio of the first antecedent and the last consequent is equal to that obtained from the intermediate ratios.
So in a number of ratios
a:b
b:c
c:d
d:h
the ratio a:h is equal to the ratio added up 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 will disappear, when the product of fractions will be simplified to its lower terms and the remainder of the complex relationship 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 yourself or to another equal ratio. These relations are called double, triple, quadruple, and so on, in accordance with the number of multiplication operations.

A ratio made up of two equal proportions, that is, square double ratio.

Composed of three, that is, cube simple relation is called triple, and so on.

Similar ratio square roots two quantities is called the ratio square root , and the ratio cubic roots- ratio cube root, and so on.
So the simple ratio of a to b is a:b
The double ratio of a to b is a 2:b 2
The triple ratio of a to b is 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 do not need to be mixed with doubled, tripled, and so on.
The ratio of 6 to 2 is 6:2 = 3
We double this ratio, that is, the ratio twice, then we get 12:2 = 6
Triple this ratio, that is, this ratio three times, we get 18:2 = 9
A double ratio, that is square ratio is equal to 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 quantities to be correlated with each other, they must be of the same kind, so that one can confidently say whether they are equal to each other, or whether one of them is greater or less. A foot is to an inch as 12 is to 1: 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 more 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 in an hour and the number of steps in a 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 that are compared with each other will affect the ratio itself. Recall that the direct relationship is expressed as a fraction, where antecedet couples are always this 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 quantities. The ratio of the two quantities is the same as meaning fractions, each of which represents private: numerator divided by denominator. (Art. 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. That's why,

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

Resp. Given a known consequent, the more antecedent, the more ratio, and, conversely, the larger the ratio, the larger the antecedent.

353. By multiplying the consequent of a pair by any value, the result is dividing 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 12:2 is 6
And the 12:4 ratio is 3.
Here is the consequent of the second pair in twice more, and the ratio twice less than the first.
The ratio a:b is equal to $\frac(a)(b)$
And the ratio a:nb is equal to $\frac(a)(nb)$.

Resp. Given an antecedent, the larger the consequent, the smaller the ratio. Conversely, the larger the ratio, the smaller the consequent.

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

Resp. Any factor or divider can be transferred from the antecedent of a pair to the consequent or from the consequent to the antecedent without changing the relationship.

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

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

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

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

Resp. 2. Direct the ratio of two fractions that have a common numerator is equal to the inverse of their ratio 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 total values from the numerators and denominators.

356. b. Ratio greater inequality increases his
Let the ratio of greater inequality be given as 1+n:1
And any ratio like a:b
The complex ratio will be (Article 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 be 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 remainders will have the same ratio.
Let the ratio a:b
It will be the same as c:d
Then the ratio amounts antecedents to the sum of 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 antecedents to the difference in consequents are also the same.

358. If in several pairs the ratios are equal, then the sum of all antecedents is related to the sum of all consequents, just as any antecedent is to its consequent.
So 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 amount to both members.
Let the given ratio a+b:a or $\frac(a+b)(a)$
By adding x to both terms 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. (Article 351 resp.)

But the ratio less inequality increases, adding the same amount to both terms.
Let the given ratio be (a-b):a, or $\frac(a-b)(a)$.
By adding x to both terms, it becomes (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, then it is obvious that the effect on the ratio will be the opposite.

Examples.

1. Which is larger: 11:9 ratio or 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 antecedent of a pair 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 look like made up of 8:7, and 2a:5b, as well as (7x+1):(3y-2)?

6. What does a complex relationship look like composed of (x+y):b, and (x-y):(a + b), as well as (a+b):h? Rep. (x 2 - y 2):bh.

7. If the relations (5x+7):(2x-3), and $(x+2):\left(\frac(x)(2)+3\right)$ form a complex relation, then what relation will be obtained: More or less inequality? Rep. The ratio of greater inequality.

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

9. What is the ratio of 7:5, double the ratio 4:9, and triple the ratio 3:2?
Rep. 14:15.

10. What is the ratio made from 3:7, and triple the x:y ratio, and taking the root of the ratio 49:9?
Rep. x 3:y 3 .

For solving most problems in mathematics high school Knowledge of drawing up proportions is required. This simple skill will help you not only perform complex exercises from the textbook, but also delve into the very essence of mathematical science. How to make a proportion? Let's figure it out now.

The most simple example is a problem where three parameters are known, and the fourth needs to be found. The proportions are, of course, different, but often you need to find some number using percentages. For example, the boy had ten apples in total. He gave the fourth part to his mother. How many apples does the boy have left? This is the simplest example that will allow you to create a proportion. The main thing is to do this. Initially there were ten apples. Let it be 100%. We marked all his apples. He gave one-fourth. 1/4=25/100. This means he has left: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the amount of fruit remaining compared to the amount initially available. Now we have three numbers by which we can already solve the proportion. 10 apples - 100%, X apples - 75%, where x is the required amount of fruit. How to make a proportion? You need to understand what it is. Mathematically it looks like this. The equal sign is placed for your understanding.

10 apples = 100%;

x apples = 75%.

It turns out that 10/x = 100%/75. This is the main property of proportions. After all, the larger x, the greater the percentage of this number from the original. We solve this proportion and find that x = 7.5 apples. We do not know why the boy decided to give away an integer amount. Now you know how to make a proportion. The main thing is to find two relationships, one of which contains the unknown unknown.

Solving a proportion often comes down to simple multiplication and then division. Schools do not explain to children why this is so. Although it is important to understand that proportional relationships are mathematical classics, the very essence of science. To solve proportions, you need to be able to handle fractions. For example, you often need to convert percentages to fractions. That is, recording 95% will not work. And if you immediately write 95/100, then you can make significant reductions without starting the main calculation. It’s worth saying right away that if your proportion turns out to be with two unknowns, then it cannot be solved. No professor will help you here. And your task most likely has a more complex algorithm for correct actions.

Let's look at another example where there are no percentages. A motorist bought 5 liters of gasoline for 150 rubles. He thought about how much he would pay for 30 liters of fuel. To solve this problem, let's denote by x the required amount of money. You can solve this problem yourself and then check the answer. If you have not yet understood how to make a proportion, then take a look. 5 liters of gasoline is 150 rubles. As in the first example, we write down 5l - 150r. Now let's find the third number. Of course, this is 30 liters. Agree that a pair of 30 l - x rubles is appropriate in this situation. Let's move on to mathematical language.

5 liters - 150 rubles;

30 liters - x rubles;

Let's solve this proportion:

x = 900 rubles.

So we decided. In your task, do not forget to check the adequacy of the answer. It happens that with the wrong decision, cars reach unrealistic speeds of 5000 kilometers per hour and so on. Now you know how to make a proportion. You can also solve it. As you can see, there is nothing complicated about this.

Proportion formula

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

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

Basic properties of proportion

Work extreme members 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

Rearrangement of middle terms: if a:b=c:d then a:c=b:d

1 10 7 70 1 7 = 10 70

Rearrangement of extreme terms: if a:b=c:d then d:b=c:a

1 10 7 70 70 10 = 7 1

Solving a proportion 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 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 tablets✕ 10 kg 70 kg x = 1 ⋅ 70 : 10 = 7 Answer: 7 tablets

Task: in five hours Vasya writes two articles. 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 tell future school graduates that the ability to draw up proportions was useful to me both in order to proportionally reduce pictures, and in the HTML layout of an Internet page, and in everyday situations.

A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute values or parts of the whole. Ratios are calculated and written in different ways, but the basic principles are the same for all ratios.

Steps

Part 1

Definition of ratios

Using ratios. Ratios are used both in science and in Everyday life to compare values. The simplest relationships connect only two numbers, but there are relationships that compare three or more values. In any situation in which more than one quantity is present, a relationship can be written down. By connecting certain values, ratios can, for example, suggest how to increase the amount of ingredients in a recipe or substances in a chemical reaction.

Definition of ratios. A ratio is a relationship between two (or more) values of the same kind. For example, if you need 2 cups of flour and 1 cup of sugar to make a cake, then the ratio of flour to sugar is 2 to 1.
- Ratios can also be used in cases where two quantities are not related to each other (as in the cake example). For example, if there are 5 girls and 10 boys in a class, then the ratio of girls to boys is 5 to 10. These values (the number of boys and the number of girls) are independent of each other, that is, their values will change if someone leaves the class or a new student will come to the class. Ratios simply compare the values of quantities.
pay attention to different ways presentation of ratios. Relationships can be represented in words or using mathematical symbols.
- Very often relationships are expressed in words (as shown above). This form of representing relationships is especially used in everyday life, far from science.
- Relationships can also be expressed using a colon. When comparing two numbers in a ratio, you will use a single colon (for example, 7:13); When comparing three or more values, place a colon between each pair of numbers (for example, 10:2:23). In our class example, you could express the ratio of girls to boys as 5 girls: 10 boys. Or like this: 5:10.
- Less commonly, relationships are expressed using a slash. In the class example, it could be written like this: 5/10. Nevertheless, this is not a fraction and such a ratio is not read as a fraction; Moreover, remember that in a ratio, the numbers do not represent part of a whole.
Part 2
Using ratios
1. Simplify the ratio. The ratio can be simplified (similar to fractions) by dividing each term (number) of the ratio by . However, do not lose sight of the original ratio values.
  - In our example, there are 5 girls and 10 boys in the class; the ratio is 5:10. The greatest common divisor of the terms in the ratio is 5 (since both 5 and 10 are divisible by 5). Divide each ratio number by 5 to get a ratio of 1 girl to 2 boys (or 1:2). However, when simplifying the ratio, keep the original values in mind. In our example, there are not 3 students in the class, but 15. A simplified ratio compares the number of boys and the number of girls. That is, for every girl there are 2 boys, but there are not 2 boys and 1 girl in the class.
  - Some relationships cannot be simplified. For example, the ratio 3:56 is not simplified because these numbers do not have common divisors(3 is a prime number, and 56 is not divisible by 3).
2. Use multiplication or division to increase or decrease a ratio. Common problems involve increasing or decreasing two values that are proportional to each other. If you are given a ratio and need to find a corresponding greater or lesser ratio, multiply or divide the original ratio by some given number.
  - For example, a baker needs to triple the amount of ingredients given in a recipe. If a recipe calls for a flour to sugar ratio of 2 to 1 (2:1), then the baker will multiply each term in the ratio by 3 to get a ratio of 6:3 (6 cups flour to 3 cups sugar).
  - On the other hand, if the baker needs to halve the amount of ingredients given in a recipe, then the baker will divide each term of the ratio by 2 and get a ratio of 1:½ (1 cup flour to 1/2 cup sugar).
3. Search unknown value, when two equivalent relations are given. This is a problem in which you need to find an unknown variable in one relation using a second relation that is equivalent to the first. To solve such problems, use . Write each relationship as common fraction, put an equal sign between them and multiply their terms crosswise.
  - For example, given a group of students in which there are 2 boys and 5 girls. What will be the number of boys if the number of girls is increased to 20 (the proportion remains the same)? First, write down two ratios - 2 boys:5 girls and X boys:20 girls. Now write these ratios as fractions: 2/5 and x/20. Multiply the terms of the fractions crosswise and get 5x = 40; therefore x = 40/5 = 8.
Part 3
Common Mistakes
1. Avoid addition and subtraction in ratio word problems. Many word problems look something like this: “The recipe calls for 4 potato tubers and 5 carrot roots. If you want to add 8 potatoes, how many carrots will you need to keep the ratio the same? When solving problems like this, students often make the mistake of adding the same number of ingredients to the original number. However, to maintain the ratio, you need to use multiplication. Here are examples of correct and incorrect solutions:
  - Incorrect: “8 - 4 = 4 - so we added 4 potato tubers. This means you need to take 5 carrot roots and add 4 more to them... Stop! Ratios are not calculated that way. It's worth trying again."
  - Correct: “8 ÷ 4 = 2 - which means we multiplied the amount of potatoes by 2. Accordingly, 5 carrot roots also need to be multiplied by 2. 5 x 2 = 10 - you need to add 10 carrot roots to the recipe.”