Problem 1. The thickness of 300 sheets of printer paper is 3.3 cm. What thickness will a pack of 500 sheets of the same paper have?
Solution. Let x cm be the thickness of a stack of paper of 500 sheets. There are two ways to find the thickness of one sheet of paper:
3,3: 300 or x : 500.
Since the sheets of paper are the same, these two ratios are equal. We get the proportion ( reminder: proportion is the equality of two ratios):
x=(3.3 · 500): 300;
x=5.5. Answer: pack 500 sheets of paper have a thickness 5.5 cm.
This is a classic reasoning and design of a solution to a problem. Such problems are often included in test tasks for graduates, who usually write the solution in the following form:
or they decide orally, reasoning like this: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times less. Divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100-sheet pack of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and get the answer: 5.5 cm.
Of course, this is justified, since the time for testing graduates and applicants is limited. However, in this lesson we will reason and write down the solution as it should be done in 6 class.
Task 2. How much water is contained in 5 kg of watermelon, if it is known that watermelon consists of 98% water?
Solution.
The entire mass of the watermelon (5 kg) is 100%. Water will be x kg or 98%. There are two ways to find how many kg are in 1% of the mass.
5: 100 or x : 98. We get the proportion:
5: 100 = x : 98.
x=(5 · 98): 100;
x=4.9 Answer: 5kg watermelon contains 4.9 kg water.
The mass of 21 liters of oil is 16.8 kg. What is the mass of 35 liters of oil?
Solution.
Let the mass of 35 liters of oil be x kg. Then you can find the mass of 1 liter of oil in two ways:
16,8: 21 or x : 35. We get the proportion:
16,8: 21=x : 35.
Find the middle term of the proportion. To do this, we multiply the extreme terms of the proportion ( 16,8 And 35 ) and divide by the known average term ( 21 ). Let's reduce the fraction by 7 .
Multiply the numerator and denominator of the fraction by 10 so that the numerator and denominator contain only natural numbers. We reduce the fraction by 5 (5 and 10) and on 3 (168 and 3).
Answer: 35 liters of oil have mass 28 kg.
After 82% of the entire field had been plowed, there was still 9 hectares left to plow. What is the area of the entire field?
Solution.
Let the area of the entire field be x hectares, which is 100%. There are 9 hectares left to plow, which is 100% - 82% = 18% of the entire field. We can express 1% of the field area in two ways. This:
X : 100 or 9 : 18. We make up the proportion:
X : 100 = 9: 18.
We find the unknown extreme term of the proportion. To do this, multiply the average terms of the proportion ( 100
And 9
) and divide by the known extreme term ( 18
). We reduce the fraction.
Answer: area of the entire field 50 hectares.
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§ 125. The concept of proportion.
Proportion is the equality of two ratios. Here are examples of equalities called proportions:
Note. The names of the quantities in the proportions are not indicated.
Proportions are usually read as follows: 2 is to 1 (unit) as 10 is to 5 (the first proportion). You can read it differently, for example: 2 is as many times more than 1, how many times is 10 more than 5. The third proportion can be read like this: - 0.5 is as many times less than 2, how many times 0.75 is less than 3.
The numbers included in the proportion are called members of the proportion. This means that the proportion consists of four terms. The first and last members, i.e. the members standing at the edges, are called extreme, and the terms of the proportion located in the middle are called average members. This means that in the first proportion the numbers 2 and 5 will be the extreme terms, and the numbers 1 and 10 will be the middle terms of the proportion.
§ 126. The main property of proportion.
Consider the proportion:
Let us multiply its extreme and middle terms separately. The product of the extremes is 6 4 = 24, the product of the middle ones is 3 8 = 24.
Let's consider another proportion: 10: 5 = 12: 6. Let's multiply the extreme and middle terms separately here too.
The product of the extremes is 10 6 = 60, the product of the middle ones is 5 12 = 60.
The main property of proportion: the product of the extreme terms of a proportion is equal to the product of its middle terms.
In general, the main property of proportion is written as follows: ad = bc .
Let's check it on several proportions:
1) 12: 4 = 30: 10.
This proportion is correct, since the ratios from which it is composed are equal. At the same time, taking the product of the extreme terms of the proportion (12 10) and the product of its middle terms (4 30), we will see that they are equal to each other, i.e.
12 10 = 4 30.
2) 1 / 2: 1 / 48 = 20: 5 / 6
The proportion is correct, which is easy to verify by simplifying the first and second ratios. The main property of proportion will take the form:
1 / 2 5 / 6 = 1 / 48 20
It is not difficult to verify that if we write an equality in which on the left side there is the product of two numbers, and on the right side the product of two other numbers, then a proportion can be made from these four numbers.
Let us have an equality that includes four numbers multiplied in pairs:
these four numbers can be terms of a proportion, which is not difficult to write if we take the first product as the product of the extreme terms, and the second as the product of the middle terms. The published equality can be compiled, for example, into the following proportion:
In general, from equality ad = bc the following proportions can be obtained:
Do the following exercise yourself. Given the product of two pairs of numbers, write the proportion corresponding to each equality:
a) 1 6 = 2 3;
b) 2 15 = b 5.
§ 127. Calculation of unknown terms of proportion.
The basic property of proportion allows you to calculate any of the terms of the proportion if it is unknown. Let's take the proportion:
X : 4 = 15: 3.
In this proportion one extreme member is unknown. We know that in any proportion the product of the extreme terms is equal to the product of the middle terms. On this basis we can write:
x 3 = 4 15.
After multiplying 4 by 15, we can rewrite this equation as follows:
X 3 = 60.
Let's consider this equality. In it, the first factor is unknown, the second factor is known, and the product is known. We know that to find an unknown factor, it is enough to divide the product by another (known) factor. Then it will turn out:
X = 60:3, or X = 20.
Let's check the result found by substituting the number 20 instead of X in this proportion:
The proportion is correct.
Let's think about what actions we had to perform to calculate the unknown extreme term of the proportion. Of the four terms of the proportion, only the extreme one was unknown to us; the middle two and the second extreme were known. To find the extreme term of the proportion, we first multiplied the middle terms (4 and 15), and then divided the found product by the known extreme term. Now we will show that the actions would not change if the desired extreme term of the proportion were not in the first place, but in the last. Let's take the proportion:
70: 10 = 21: X .
Let's write down the main property of proportion: 70 X = 10 21.
Multiplying the numbers 10 and 21, we rewrite the equality as follows:
70 X = 210.
Here one factor is unknown; to calculate it, it is enough to divide the product (210) by another factor (70),
X = 210: 70; X = 3.
So we can say that each extreme term of the proportion is equal to the product of the averages divided by the other extreme.
Let us now move on to calculating the unknown average term. Let's take the proportion:
30: X = 27: 9.
Let's write the main property of proportion:
30 9 = X 27.
Let's calculate the product of 30 by 9 and rearrange the parts of the last equality:
X 27 = 270.
Let's find the unknown factor:
X = 270:27, or X = 10.
Let's check with substitution:
30:10 = 27:9. The proportion is correct.
Let's take another proportion:
12: b = X : 8. Let's write the main property of proportion:
12 . 8 = 6 X . Multiplying 12 and 8 and rearranging the parts of the equality, we get:
6 X = 96. Find the unknown factor:
X = 96:6, or X = 16.
Thus, each middle term of the proportion is equal to the product of the extremes divided by the other middle.
Find the unknown terms of the following proportions:
1) A : 3= 10:5; 3) 2: 1 / 2 = x : 5;
2) 8: b = 16: 4; 4) 4: 1 / 3 = 24: X .
The last two rules can be written in general form as follows:
1) If the proportion looks like:
x: a = b: c , That
2) If the proportion looks like:
a: x = b: c , That
§ 128. Simplification of proportion and rearrangement of its terms.
In this section we will derive rules that allow us to simplify the proportion in the case when it includes large numbers or fractional terms. The transformations that do not violate the proportion include the following:
1. Simultaneous increase or decrease of both terms of any ratio by the same number of times.
EXAMPLE 40:10 = 60:15.
Multiplying both terms of the first ratio by 3 times, we get:
120:30 = 60: 15.
The proportion was not violated.
Reducing both terms of the second relation by 5 times, we get:
We got the correct proportion again.
2. Simultaneous increase or decrease of both previous or both subsequent terms by the same number of times.
Example. 16:8 = 40:20.
Let us double the previous terms of both relations:
We got the correct proportion.
Let us decrease the subsequent terms of both relations by 4 times:
The proportion was not violated.
The two conclusions obtained can be briefly stated as follows: The proportion will not be violated if we simultaneously increase or decrease by the same number of times any extreme term of the proportion and any middle one.
For example, reducing by 4 times the 1st extreme and 2nd middle terms of the proportion 16:8 = 40:20, we get:
3. Simultaneous increase or decrease of all terms of the proportion by the same number of times. Example. 36:12 = 60:20. Let's increase all four numbers by 2 times:
The proportion was not violated. Let's decrease all four numbers by 4 times:
The proportion is correct.
The listed transformations make it possible, firstly, to simplify proportions, and secondly, to free them from fractional terms. Let's give examples.
1) Let there be a proportion:
200: 25 = 56: x .
In it, the members of the first ratio are relatively large numbers, and if we wanted to find the value X , then we would have to perform calculations on these numbers; but we know that the proportion will not be violated if both terms of the ratio are divided by the same number. Let's divide each of them by 25. The proportion will take the form:
8:1 = 56: x .
We have thus obtained a more convenient proportion, from which X can be found in the mind:
2) Let's take the proportion:
2: 1 / 2 = 20: 5.
In this proportion there is a fractional term (1/2), from which you can get rid of. To do this, you will have to multiply this term, for example, by 2. But we do not have the right to increase one middle term of the proportion; it is necessary to increase one of the extreme members along with it; then the proportion will not be violated (based on the first two points). Let's increase the first of the extreme terms
(2 2) : (2 1/2) = 20:5, or 4:1 = 20:5.
Let's increase the second extreme member:
2: (2 1/2) = 20: (2 5), or 2: 1 = 20: 10.
Let's look at three more examples of freeing proportions from fractional terms.
Example 1. 1 / 4: 3 / 8 = 20:30.
Let's bring the fractions to a common denominator:
2 / 8: 3 / 8 = 20: 30.
Multiplying both terms of the first ratio by 8, we get:
Example 2. 12: 15 / 14 = 16: 10 / 7. Let's bring the fractions to a common denominator:
12: 15 / 14 = 16: 20 / 14
Let's multiply both subsequent terms by 14, we get: 12:15 = 16:20.
Example 3. 1 / 2: 1 / 48 = 20: 5 / 6.
Let's multiply all terms of the proportion by 48:
24: 1 = 960: 40.
When solving problems in which some proportions occur, it is often necessary to rearrange the terms of the proportion for different purposes. Let's consider which permutations are legal, i.e., do not violate the proportions. Let's take the proportion:
3: 5 = 12: 20. (1)
Rearranging the extreme terms in it, we get:
20: 5 = 12:3. (2)
Let us now rearrange the middle terms:
3:12 = 5: 20. (3)
Let us rearrange both the extreme and middle terms at the same time:
20: 12 = 5: 3. (4)
All these proportions are correct. Now let's put the first relation in the place of the second, and the second in the place of the first. You get the proportion:
12: 20 = 3: 5. (5)
In this proportion we will make the same rearrangements as we did before, that is, we will first rearrange the extreme terms, then the middle ones, and finally, both the extremes and the middle ones at the same time. You will get three more proportions, which will also be fair:
5: 20 = 3: 12. (6)
12: 3 = 20: 5. (7)
5: 3 = 20: 12. (8)
So, from one given proportion, by rearranging, you can get 7 more proportions, which together with this one makes 8 proportions.
The validity of all these proportions is especially easy to discover when writing in letters. The 8 proportions obtained above take the form:
a: b = c: d; c: d = a: b ;
d: b = c: a; b:d = a:c;
a: c = b: d; c: a = d: b;
d: c = b: a; b: a = d: c.
It is easy to see that in each of these proportions the main property takes the form:
ad = bc.
Thus, these permutations do not violate the fairness of the proportion and can be used if necessary.
Solving most problems in high school mathematics requires knowledge of formulating proportions. This simple skill will help you not only perform complex exercises from the textbook, but also delve into the very essence of mathematical science. How to make a proportion? Let's figure it out now.
The simplest 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.
From a mathematical point of view, a proportion is the equality of two ratios. Interdependence is characteristic of all parts of the proportion, as well as their unchanging result. You can understand how to create a proportion by familiarizing yourself with the properties and formula of proportion. To understand the principle of solving proportions, it will be sufficient to consider one example. Only by directly solving proportions can you quickly and easily learn these skills. And this article will help the reader with this.
Properties of proportion and formula
- Reversal of proportion. In the case when the given equality looks like 1a: 2b = 3c: 4d, write 2b: 1a = 4d: 3c. (And 1a, 2b, 3c and 4d are prime numbers other than 0).
- Multiplying the given terms of the proportion crosswise. In literal expression it looks like this: 1a: 2b = 3c: 4d, and writing 1a4d = 2b3c will be equivalent to it. Thus, the product of the extreme parts of any proportion (the numbers at the edges of the equality) is always equal to the product of the middle parts (the numbers located in the middle of the equality).
- When drawing up a proportion, its property of rearranging the extreme and middle terms can also be useful. The formula of equality 1a: 2b = 3c: 4d can be displayed in the following ways:
- 1a: 3c = 2b: 4d (when the middle terms of the proportion are rearranged).
- 4d: 2b = 3c: 1a (when the extreme terms of the proportion are rearranged).
- Its property of increasing and decreasing helps perfectly in solving proportions. When 1a: 2b = 3c: 4d, write:
- (1a + 2b) : 2b = (3c + 4d) : 4d (equality by increasing proportion).
- (1a – 2b) : 2b = (3c – 4d) : 4d (equality by decreasing proportion).
- You can create a proportion by adding and subtracting. When the proportion is written as 1a:2b = 3c:4d, then:
- (1a + 3c) : (2b + 4d) = 1a: 2b = 3c: 4d (the proportion is made by addition).
- (1a – 3c) : (2b – 4d) = 1a: 2b = 3c: 4d (the proportion is calculated by subtraction).
- Also, when solving a proportion containing fractional or large numbers, you can divide or multiply both of its terms by the same number. For example, the components of the proportion 70:40=320:60 can be written as follows: 10*(7:4=32:6).
- An option for solving proportions with percentages looks like this. For example, write down 30=100%, 12=x. Now you should multiply the middle terms (12*100) and divide by the known extreme (30). Thus, the answer is: x=40%. In a similar way, if necessary, you can multiply the known extreme terms and divide them by a given average number, obtaining the desired result.
If you are interested in a specific proportion formula, then in the simplest and most common version, the proportion is the following equality (formula): a/b = c/d, in which a, b, c and d are four non-zero numbers.
In the last video lesson we looked at solving problems involving percentages using proportions. Then, according to the conditions of the problem, we needed to find the value of one or another quantity.
This time the initial and final values have already been given to us. Therefore, the problems will require you to find percentages. More precisely, by how many percent has this or that value changed. Let's try.
Task. The sneakers cost 3,200 rubles. After the price increase, they began to cost 4,000 rubles. By what percentage was the price of sneakers increased?
So, we solve through proportion. The first step - the original price was 3,200 rubles. Therefore, 3200 rubles is 100%.
In addition, we were given the final price - 4000 rubles. This is an unknown percentage, so let's call it x. We get the following construction:
3200 — 100%
4000 - x%
Well, the condition of the problem is written down. Let's make a proportion:
The fraction on the left cancels perfectly by 100: 3200: 100 = 32; 4000: 100 = 40. Alternatively, you can shorten it by 4: 32: 4 = 8; 40: 4 = 10. We get the following proportion:
Let's use the basic property of proportion: the product of the extreme terms is equal to the product of the middle terms. We get:
8 x = 100 10;
8x = 1000.
This is an ordinary linear equation. From here we find x:
x = 1000: 8 = 125
So, we got the final percentage x = 125. But is the number 125 a solution to the problem? No way! Because the task requires finding out by how many percent the price of sneakers was increased.
By what percentage - this means that we need to find the change:
∆ = 125 − 100 = 25
We received 25% - that’s how much the original price was increased. This is the answer: 25.
Problem B2 on percentages No. 2
Let's move on to the second task.
Task. The shirt cost 1800 rubles. After the price was reduced, it began to cost 1,530 rubles. By what percentage was the price of the shirt reduced?
Let's translate the condition into mathematical language. The original price is 1800 rubles - this is 100%. And the final price is 1,530 rubles - we know it, but we don’t know what percentage it is of the original value. Therefore, we denote it by x. We get the following construction:
1800 — 100%
1530 - x%
Based on the received record, we create a proportion:
To simplify further calculations, let's divide both sides of this equation by 100. In other words, we will cross out two zeros from the numerator of the left and right fractions. We get:
Now let's use the basic property of proportion again: the product of the extreme terms is equal to the product of the middle terms.
18 x = 1530 1;
18x = 1530.
All that remains is to find x:
x = 1530: 18 = (765 2) : (9 2) = 765: 9 = (720 + 45) : 9 = 720: 9 + 45: 9 = 80 + 5 = 85
We got that x = 85. But, as in the previous problem, this number in itself is not the answer. Let's go back to our condition. Now we know that the new price obtained after the reduction is 85% of the old one. And in order to find changes, you need from the old price, i.e. 100%, subtract the new price, i.e. 85%. We get:
∆ = 100 − 85 = 15
This number will be the answer: Please note: exactly 15, and in no case 85. That's all! The problem is solved.
Attentive students will probably ask: why in the first problem, when finding the difference, did we subtract the initial number from the final number, and in the second problem did exactly the opposite: from the initial 100% we subtracted the final 85%?
Let's be clear on this point. Formally, in mathematics, a change in a quantity is always the difference between the final value and the initial value. In other words, in the second problem we should have gotten not 15, but −15.
However, this minus should under no circumstances be included in the answer, because it is already taken into account in the conditions of the original problem. It says directly about the price reduction. And a price reduction of 15% is the same as a price increase of −15%. That is why in the solution and answer to the problem it is enough to simply write 15 - without any minuses.
That's it, I hope we have sorted this out. This concludes our lesson for today. See you again!