What does the proportion 1 to 4 mean? Ratios. How to calculate proportion

A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute quantities or parts of a 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.
Notice the different ways of representing 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. Largest common divisor terms of the ratio is equal to 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 factors (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. Finding an unknown value when given two equivalent ratios. 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 ratio as a 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.”

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

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

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 opposite meaning

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.

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 the substitution of a word occurs 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. Of two latest articles 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, which have a common denominator, the same as their ratio 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 not only perform difficult exercises from the textbook, but also to 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.

A relationship is 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 relationship between this and that". For example, if there are 4 apples and 2 pears in a vase, then we say "apple to pear ratio" "ratio of pears and apples".

In mathematics, the ratio is more often used as "the attitude of so-and-so to so-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 "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 post in different ways:

a To b
a refers to b
attitude a To b

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

4: 2

If we swap apples and pears, we will have a ratio of 2: 4. This ratio can be read as "two to four" or either "two pears are equal to four apples" .

In what follows we will call the relationship a ratio.

Lesson content

What is 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 relation will be the quotient of the numbers a And b.

In mathematics, a ratio is the quotient of two numbers.

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

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

We got 2. So four apples and two pears (4: 2) are correlated (interconnected with each other) so that there are two apples for one pear

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

The relationship can be reversed by writing it 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 there are per unit of apple. An apple unit means one apple.

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

We got 0.5. Let's translate this decimal to ordinary:

Let's reduce the resulting common fraction by 5

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

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

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

Let's look at other examples of relationships. To prepare something, a recipe is compiled. A recipe is built from the relationships between products. For example, to prepare oatmeal, you usually need a glass of cereal to two glasses of milk or water. The resulting ratio is 1:2 (“one to two” or “one glass of cereal to two glasses of milk”).

Let's convert the ratio 1:2 into a fraction, we get . Having calculated this fraction, we get 0.5. This means that one glass of cereal and two glasses of milk are correlated (interrelated with each other) so that one glass of milk accounts for half a glass of cereal.

If you flip the 1:2 ratio you get a 2:1 ratio (“two to one” or “two cups of milk to one cup of cereal”). Converting the ratio 2:1 into a fraction, we get . Calculating this fraction, we get 2. This means that two glasses of milk and one glass of cereal are correlated (interrelated with each other) so that for one glass of cereal there are two glasses of milk.

Example 2. There are 15 students in the class. Of these, 5 are boys, 10 are girls. You can write the ratio of girls to boys as 10:5 and convert this ratio to a fraction. Having calculated 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 compare to each other. It can be seen that for every boy there are two girls.

It is not always possible to convert a ratio into a fraction and find the quotient. In some cases this will be counter-intuitive.

So, if you turn the attitude around, it turns out, and this is the attitude of boys to girls. If you calculate this fraction it turns out to be 0.5. It turns out that five boys are related to ten girls so that for each girl there is half a boy. Mathematically, this is certainly true, but from the point of view of reality it is not entirely reasonable, because a boy is a living person and cannot simply be taken and divided, like a pear or an apple.

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

The distance is indicated through 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:

Let's assume that the car traveled 100 kilometers in 2 hours. Then the ratio of one hundred kilometers traveled to two hours will be the speed of the car:

Speed is usually called the distance traveled by a body per unit time. A unit of time means 1 hour, 1 minute or 1 second. And the ratio, as mentioned earlier, allows you to find out how much of one entity is per unit of another. In our example, the ratio of one hundred kilometers to two hours shows how many kilometers there are in one hour of movement. We see that for every hour of movement there are 50 kilometers

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

Example 2. The ratio of the cost of a product to its quantity is the price of one unit of the 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 candy bars. Then we get that one candy bar costs 20 rubles

Comparison of values

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

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

To find out how many times the first value is greater than the second, you need to write the larger value into the numerator of the ratio and the smaller value into the denominator.

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

Consider the numbers 20 and 2. Let's find out how many times the number 20 is more number 2. To do this, 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. This means that 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 more girls there are than boys.

We record the attitude of girls towards boys. In the numerator of the ratio we write the number of girls, in the denominator of the ratio - the number of boys:

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

There is no longer a 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 the number 20?

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

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 fraction. This answer will be easier to understand:

This means that the number 2 of the number 20 is one tenth.

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

20: 10 = 2

2 × 1 = 2

We got the number 2. This means that one tenth of the number 20 is the number 2. From here we conclude that the problem was 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 record the ratio of boys to the total number of schoolchildren. We write five boys in the numerator of the ratio, and the total number of schoolchildren in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write the number 15 in the denominator of the ratio

To find the value of a given 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

Dividing 5 by 15 produces a periodic fraction. Let's convert this fraction to an ordinary fraction

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

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

If we find 15 schoolchildren to check, then we will get 5 boys

15: 3 = 5

5 × 1 = 5

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

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

The value of this ratio is 7. This means that the number 35 is seven times greater than 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 record the ratio of girls to the total number of schoolchildren. We write ten girls in the numerator of the ratio, and the total number of schoolchildren in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write the number 15 in the denominator of the ratio

To find the value of a given 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

Dividing 10 by 15 produces a periodic fraction. Let's convert this fraction to an ordinary fraction

Let's reduce the resulting fraction by 3

We received the final answer. This means girls make up two thirds of the entire class.

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

If we find 15 schoolchildren to check, we will 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 value of a given 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 into an ordinary fraction

Let's reduce the resulting fraction by 2

We received the final answer. So 10 cm is equal to 25 cm.

Example 8. How many times is 25 cm greater than 10 cm?

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

We received an answer of 2.5. This means 25 cm is 2.5 times greater than 10 cm (two and a half times)

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

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

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

First, let's make sure that both lengths are expressed in the same unit of measurement. To do this, 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

We received an answer of 1.5. This means 150 cm is 1.5 times greater than 100 cm (one and a half times).

And if we had not started converting meters into centimeters and immediately tried to find the ratio of 150 cm to one meter, then we would have gotten the following:

It would turn out that 150 cm is one hundred and fifty times more than one meter, but this is incorrect. 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 increased

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

We received an answer of 1.08. This means that the salary increased by 1.08 times. In the future, when we become familiar with percentages, we will express indicators such 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 width of the house greater 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 greater than its height.

Relationship property

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

This is one of the most important properties relations 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 a relation is nothing more than a division, the quotient property works for it too.

Let's return to the attitude of girls towards boys (10:5). This ratio showed that for every boy there are two girls. Let's check how the relation 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 terms of the relation by their greatest common divisor (GCD).

The gcd of the terms 10 and 5 is the number 5. Therefore, we can divide the terms of the relation by the number 5

We got a new attitude. This is a two to one ratio (2:1). This ratio, like the previous 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 previous ratio of 10: 5 for one boy there are 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 than boys in first grade? How many times are there more girls than boys in the second grade?

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

The relation property allows you to build various models, which have similar parameters to the real object. Let's assume that 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 30:10.

Let's 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 get a ratio of 3 cm: 1 cm. Now you 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 that is as similar to the real one as possible.

Attitude can be understood in another way. It was originally said that the real house was 30 meters wide and 10 meters high. The total is 30+10, that is, 40 meters.

These 40 meters can be understood as 40 parts. A ratio of 30:10 means that 30 parts are in width and 10 parts are in height.

Next, the terms 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 in width, one in height. In this case, you usually need to find out exactly how many meters there are in width and height.

In other words, you need to find out how many meters there are in 3 parts and how many meters there are in 1 part. First you need to find out how many meters there are per 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 the width is:

10 m × 3 = 30 m

Let's determine how many meters there are in 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. 18 apples purchased. These apples were divided between mother, father and daughter in a ratio of 2: 1: 3. How many apples did each person get?

The ratio 2: 1: 3 means that mom received 2 parts, dad - 1 part, daughter - 3 parts. In other words, each term in the 2:1:3 ratio is a specific portion of 18 apples:

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

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 part)

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

Let's find out how many apples mom got:

3 × 2 = 6 (apples)

Let's find out how many apples dad got:

3 × 1 = 3 (apples)

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

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 must be taken 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, let’s find out how many parts there will be in four kilograms of silver:

3 + 4 + 13 = 20 (parts)

Let's determine how many kilograms there will be per part:

4 kg: 20 = 0.2 kg

Let's determine how many kilograms of nickel will be contained in 4 kg of new silver. The 3:4:13 ratio indicates that three parts of the alloy 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. The 3:4:13 ratio indicates that four parts of the alloy contain zinc. Therefore, we multiply 0.2 by 4:

0.2 kg × 4 = 0.8 kg zinc

Now let’s determine how many kilograms of copper will be contained in 4 kg of new silver. The 3:4:13 ratio indicates that thirteen parts of the alloy 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 masses of which are in the ratio 3:2. To make a piece of brass, 120 g of copper is required. How much zinc is required to make this piece of brass?

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

120:3 = 40 grams per part

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 3:2 ratio it is indicated that two parts contain zinc:

40 g × 2 = 80 grams zinc

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

Solution

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

Let's determine the mass of one part. To do this, first add up 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 piece 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 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.