How to understand the proportion 1 to 3. Ratios

basis mathematical research is the ability to gain knowledge about certain quantities by comparing them with other quantities that are either equal, or more or less than those that are the subject of the study. This is usually done with 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 are comparing an unknown quantity with others that not equal her, but more or less of her. Here we need a different approach to data processing. We may need to know, for example, how much one value is greater than the other, or how many times one contains the other. To find answers to these questions, we will find out what is ratio two sizes. One ratio is called arithmetic, and another geometric. Although it is worth noting that both of these terms were not adopted by chance or just for the sake of distinction. Both arithmetic and geometric relations apply to both arithmetic and geometry.

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

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

339. If both members of an arithmetic relation multiply or divide by the same amount, then ratio, will eventually be multiplied or divided by that 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 the terms of an arithmetic ratio add to 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 ratios,
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 of 11 - 4 is 7
And the arithmetic ratio 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 members 6 - 2 is 4, - the difference of ratios.

341. geometric ratio is the relationship between quantities, which is expressed PRIVATE if one value is divided by another.
So 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 is basically 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. Geometric ratio is also written by placing two points one above the other between the compared values.
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 couple, in which the first term is called antecedent, and the last one is consequential.

343. This dotted notation and the other, in the form of a fraction, are interchangeable as necessary, with 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 relation are given any two, then the third one can be found.

Let a= antecedent, c= consequent, r= relation.
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.
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 the ratios and antecedents of two pairs are equal, then the consequents are equal, and if the ratios and consequents are equal, then the antecedents are equal.

345. If two compared quantities equal, then their ratio is equal to unity or equality. The ratio 3 * 6:18 is equal to one, since the quotient of any value 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 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 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 ratio is the ratio of two reciprocals.
So the ratio of the inverse of 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 cosequence b divided by the antecedent a.

Hence the inverse relation is expressed by inverting a fraction, which displays a direct relationship, or, when notation is done using dots, inverting the order of writing members.
Thus a is related to b in the opposite way that b is related to a.

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

Here a complex relation is obtained by multiplying together two antecedents and also two consequents of simple relations.
So the ratio is composed
From the ratio a:b
And c:d ratios
and the ratio h:y
This is the relation $ach:bdy=\frac(ach)(bdy)$.
A complex relationship does not differ in its nature from any other ratio. This term is used to show the origin of a relation 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 of 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 next 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 summed from the ratios a:b and b:c and c:d and d:h. So the complex relation 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 lower terms and in the remainder the complex relation 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 himself or to another equal ratio. These ratios are called double, triple, quadruple, and so on, according to the number of multiplications.

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

Made up of three, i.e, cube simple ratio is called triple, etc.

Similarly, the ratio square roots two quantities is called the ratio square root, and the ratio cube roots- ratio cube root, etc.
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, etc.
The ratio of 6 to 2 is 6:2 = 3
If we double this ratio, that is, the ratio twice, we get 12:2 = 6
We triple this ratio, that is, this ratio three times, we get 18: 2 = 9
BUT double ratio, that is square ratio is 6 2:2 2 = 9
AND triple the ratio, i.e. 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 it can be stated with certainty whether they are equal to each other, or whether one of them is greater or less. A foot is to an inch like 12 to 1: it is 12 times larger than an inch. But one cannot, for example, say that an hour is longer or shorter than a stick, or an acre is greater or less than a degree. However, if these values 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 is how the change in one or two terms that are compared with each other will affect the ratio itself. Recall that a direct ratio is expressed as a fraction, where antecedet couples are always numerator, but 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: the numerator divided by the denominator. (Art. 341.) It has now been shown that multiplying the numerator of a fraction by any value is the same as multiplying meaning by the same amount and that dividing the numerator is the same as dividing the values of a fraction. That's why,

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

Resp. 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 a pair by any value, as a result, we obtain 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 12:4 ratio is 3.
Here is the consequent of the second pair in twice more, but the ratio twice less than the first.
The ratio a:b is $\frac(a)(b)$
And the ratio a:nb is equal to $\frac(a)(nb)$.

Resp. For a given antecedent, the larger the consequent, the smaller the ratio. Conversely, the larger the ratio, the smaller the consequent.

354. It follows from the last two articles that multiplication antecedent pairs by any value will have the same effect on the ratio as division of the consequent by this amount, and antecedent division, will have the same effect as consequent multiplication.
So the 8:4 ratio is 2
Multiplying the antecedent by 2, the 16:4 ratio is 4
Dividing the antecedent by 2, the 8:2 ratio 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 relation.

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

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

355. As 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 two fractions, which have a common denominator, the same as the ratio of their numerators.
Thus 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 their reciprocal ratio denominators.

356. It is easy to determine the ratio of any two fractions from the article. 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 greater inequality ratio be given as 1+n:1
And any ratio a:b
A complex ratio will be (Art. 347,) a + na:b
What is greater than the ratio a:b (Art. 351 resp.)
But the ratio less inequality, added with another ratio, reduces his.
Let the ratio of the smaller difference 1-n:1
Any given ratio a:b
Complex ratio a - na:b
What 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 relation amounts antecedents to the sum of consequents, namely, a + c to b + d, is also the same.
That is, $\frac(a+c)(b+d)$ = $\frac(c)(d)$ = $\frac(a)(b)$.

Proof.

1. By assumption, $\frac(a)(b)$ = $\frac(c)(d)$
2. Multiply by b and by 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 difference antecedents to the difference of consequents are also the same.

358. If the ratios in several pairs are equal, then the sum of all antecedents is to the sum of all consequents as any antecedent is 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.

358b. Ratio greater inequalitydecreases, adding the same amount to both members.
Let a given relation 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. (Art. 351 resp.)

But the ratio less inequality increases, adding the same value to both terms.
Let the given relation 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 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 bigger: 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 made up 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? 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 you get: 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 ratio.

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

10. What is the ratio made up of 3:7, and triple the ratio of x:y, and extracting the root from the ratio of 49:9?
Rep. x3:y3.

Proportion Formula

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

ratio 1 : 10 is equal to the ratio of 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

Proportion inversion: if a:b=c:d , then b:a=d:c

1 10 7 70 10 1 = 70 7

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

1 10 7 70 1 7 = 10 70

Permutation of extreme members: 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

A task: you need to drink 1 tablet of activated charcoal per 10 kilograms of weight. How many tablets should be taken 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

A 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 was useful to me both in order to proportionally reduce pictures, and in the HTML layout of a web page, and in everyday situations.

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

In mathematics, the ratio is often used as "the relation of something to something". For example, the ratio of four apples and two pears, which we considered above, in mathematics will be 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 various ways:

a to b
a refers to b
attitude a to b

We write the ratio of four apples and two pears using the ratio symbol:

4: 2

If we swap 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 are equal to four apples" .

In what follows, we will refer to the relation as a relation.

Lesson content

What is an attitude?

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

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

The ratio allows you to find out how much of one entity is per 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 there are 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 the answer to the question of how many apples there are 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 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 every pear.

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

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

Reduce the resulting ordinary fraction by 5

Got an answer (half a pear). So two pears and four apples (2: 4) are correlated (interrelated with each other) so that one apple accounts for half a pear

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

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

Consider other examples of relationships. A recipe is made to prepare something. The recipe is built from the ratios between the products. For example, making oatmeal usually requires a glass of cereal to two glasses of milk or water. This results in a 1:2 ratio ("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. Calculating this fraction, we get 0.5. So one glass of cereal and two glasses of milk are correlated (correlated) so that there is half a glass of cereal for one glass of milk.

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

Example 2 There are 15 students in the class. Of these, 5 are boys, 10 are girls. It is possible to write down a ratio of girls to boys of 10:5 and convert this ratio to a fraction. Calculating this fraction, we get 2. That is, girls and boys are related to each other so 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 for every boy there are two girls.

It is not always possible to convert a ratio into a fraction and find a quotient. In some cases it will be illogical.

So, if you turn the ratio upside down, and this is the ratio of boys to girls. If you calculate this fraction, you get 0.5. It turns out that five boys are related to ten girls so 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 cannot just be taken and divided like a pear or an apple.

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

The distance is denoted by the variable S, time - through a 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 a car travels 100 kilometers in 2 hours. Then the ratio of 100 kilometers traveled to 2 hours will be the speed of the car:

Velocity is the distance traveled by a body per unit of time. The unit of time is 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 for one hour of movement. We see that for every hour of movement there are 50 kilometers

So 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 commodity to its quantity is the price of one unit of the commodity.

If we took 5 chocolate bars in 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 one bar accounts for 20 rubles

Comparison of values

Earlier we learned that the ratio between quantities of different nature form a new quantity. Thus, the ratio of 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 the commodity.

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

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

To find out what part 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. Write the number 20 in the numerator of the ratio, and the number 2 in the denominator

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 more girls are than boys.

Write down the attitude of girls to 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. It means that in a class of 15 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 number 2 is from 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 into an ordinary one. This answer will be easier to understand:

So 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 x 1 = 2

We got the number 2. So one tenth of the number 20 is the number 2. From this we conclude that the problem has been 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 students are boys.

We write down the ratio of boys to the total number of students. 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 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

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

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

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

15: 3 = 5

5 x 1 = 5

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

We write 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 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 write down the ratio of girls to the total number of students. 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 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

Let's reduce the resulting fraction by 3

Got the final answer. So girls make up two-thirds of the whole class

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

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

15: 3 = 5

5 x 2 = 10

Example 7 What part of 10 cm is 25 cm

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

To find the value 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

Let's reduce the resulting fraction by 2

Got the final answer. So 10 cm is 25 cm.

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

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

Got the answer 2.5. So 25 cm is 2.5 times more than 10 cm (two and a half times)

Important note. When finding the ratio of the same 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 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 sure that both lengths are expressed in the same unit. 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 relation

Got the answer 1.5. So 150 cm is more than 100 cm by 1.5 times (one and a half times).

And if we didn’t begin to convert meters into 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 one hundred and fifty times more than one meter, 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 relation. 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 much the salary has increased

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

Let's find the value of this relation

Got the answer 1.08. So the salary increased by 1.08 times. In the future, when we get acquainted with percentages, we will express such indicators as salary as a percentage.

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

We write 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.

relation property

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

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

Let us 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).

GCD of members 10 and 5 is the number 5. Therefore, you can divide the terms of the relation by the number 5

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

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

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

There are twice as many girls as boys in both classes, since the ratios of 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, like the previous ratio is 3

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 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 another way. Initially, it was 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 30 parts for the width and 10 parts 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 fall on the width, one on the height. In this case, you usually need to find out exactly how many meters per width and height.

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

Let's determine how many meters the width is:

10 m × 3 = 30 m

Let's determine how many meters fall on the height:

10 m × 1 = 10 m

Multiple members of a relation

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

Example 1. Bought 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 of 2: 1: 3 indicates that the mother received 2 parts, the father - 1 part, the daughter - 3 parts. In other words, each member of the 2:1:3 ratio is a certain fraction of 18 apples:

If you add the terms 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 fall on one part. To do this, divide 18 apples by 6

18:6 = 3 (apples per part)

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

Find out how many apples mom got:

3 × 2 = 6 (apples)

Find out how many apples dad got:

3 × 1 = 3 (apples)

Find out how many apples the 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 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, we find out how many parts there will be in four kilograms of silver:

3 + 4 + 13 = 20 (parts)

Determine how many kilograms will fall on one part:

4 kg: 20 = 0.2 kg

Let us determine how many kilograms of nickel will be contained in 4 kg of new silver. In the 3:4:13 ratio, three parts of the alloy are said to contain nickel. So 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 the 3:4:13 ratio, four parts of the alloy are said to contain zinc. So 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. In the 3:4:13 ratio, 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

So, 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 whose mass ratio is 3:2. It takes 120g of copper to make a piece of brass. How much zinc is required to make this piece of brass?

Let's determine how many grams of the alloy falls on one part. The condition says 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 the alloy are in one part:

120: 3 = 40 grams per piece

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

40 g × 2 = 80 grams of zinc

Example 4. They took two alloys of gold and silver. In one, the ratio of these metals is 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 related as 1: 4?

Solution

15 kg of a new alloy should be in a ratio of 1: 4. This ratio indicates that one part of the alloy will have gold, and four parts will have 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 all the 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 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 task - " How much to take each alloy? »

We will take 10 kg of the first alloy, since the ratio of gold and silver in it is 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.

Did you like the lesson?
Join our new Vkontakte group and start receiving notifications of new lessons

To solve most problems in high school mathematics, knowledge of proportioning is required. This simple skill will help not only perform complex exercises from the textbook, but also delve into the very essence of mathematical science. How to make a proportion? Now let's figure it out.

The simplest example is a problem where three parameters are known, and the fourth must be found. The proportions are, of course, different, but often you need to find some number by percentage. 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 make a proportion. The main thing is to do it. There were originally ten apples. Let it be 100%. This we marked all his apples. He gave one-fourth. 1/4=25/100. So, he has left: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the amount of fruit left over the amount of fruit that was available first. Now we have three numbers by which we can already solve the proportion. 10 apples - 100%, X apples - 75%, where x is the desired amount of fruit. How to make a proportion? It is necessary to understand what it is. Mathematically it looks like this. The equal sign is 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 more x, the more percent is this number from the original. We solve this proportion and get that x=7.5 apples. Why the boy decided to give a non-integer amount, we do not know. Now you know how to make a proportion. The main thing is to find two ratios, one of which contains the desired unknown.

Solving a proportion often comes down to simple multiplication and then division. Children are not taught in schools why this is so. While 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, it is often necessary to convert percentages to ordinary fractions. That is, a record of 95% will not work. And if you immediately write 95/100, then you can make solid reductions without starting the main count. It’s worth saying right away that if your proportion turned out with two unknowns, then it cannot be solved. No professor can help you here. And your task, most likely, has a more complex algorithm for correct actions.

Consider another example where there are no percentages. The 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, we denote by x the required amount of money. You can solve this problem yourself and then check the answer. If you have not yet figured out how to make a proportion, then look. 5 liters of gasoline is 150 rubles. As in the first example, let's write 5l - 150r. Now let's find the third number. Of course, it's 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;

We solve this proportion:

x = 900 rubles.

That's what 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. Also you can solve it. As you can see, there is nothing complicated in this.

A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute values 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 quantities. The simplest ratios relate only two numbers, but there are ratios that compare three or more values. In any situation in which more than one quantity is present, a ratio can be written. By linking some 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 relation is a relationship between two (or more) values of the same kind. For example, if a cake requires 2 cups of flour and 1 cup of sugar, then the ratio of flour to sugar is 2 to 1.
- Ratios can also be used when two quantities are not related to each other (as in the cake example). For example, if there are 5 girls and 10 boys in the class, then the ratio of girls to boys is 5 to 10. These quantities (the number of boys and the number of girls) do not depend on 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 values of quantities.
Notice the different ways in which ratios are represented. Relationships can be represented in words or with mathematical symbols.
- Very often ratios are expressed in words (as shown above). Especially this form of representation of ratios is used in everyday life, far from science.
- Also, ratios can be expressed through 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, put 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 like this: 5 girls: 10 boys. Or like this: 5:10.
- Less commonly, ratios 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, numbers are not part of a single 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 of 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. The 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 are not 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 the ratio. A common problem is to increase or decrease two values that are proportional to each other. If you are given a ratio and need to find a larger or smaller ratio that matches it, 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 the recipe says the ratio of flour to sugar is 2:1 (2:1), then the baker will multiply each term by 3 to get a ratio of 6:3 (6 cups of flour to 3 cups of sugar).
  - On the other hand, if the baker needs to halve the amount of ingredients given in the recipe, then the baker will divide each ratio term by 2 and get a ratio of 1:½ (1 cup flour to 1/2 cup sugar).
3. Search for an unknown value when two equivalent ratios 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 ratio as a 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 is preserved)? 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; hence x = 40/5 = 8.
Part 3
Common Mistakes
1. Avoid addition and subtraction in text ratio problems. Many word problems look something like this: “The recipe calls for 4 potato tubers and 5 root carrots. If you want to add 8 potatoes, how many carrots do you need to keep the ratio the same?” When solving such problems, students often make the mistake of adding the same amount of ingredients to the original number. However, to keep the ratio, you need to use multiplication. Here are examples of right and wrong decisions:
  - Incorrect: “8 - 4 = 4 - so we added 4 potatoes. So, you need to take 5 carrot roots and add 4 more to them ... Stop! Ratios don't work that way. Worth trying again."
  - Correct: “8 ÷ 4 = 2 - so we multiplied the number of potatoes by 2. Accordingly, 5 carrot roots also need to be multiplied by 2. 5 x 2 = 10 - 10 carrot roots need to be added to the recipe.”

total:0
In contact with 0
Google+ 0
OK 0
Facebook 0