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 big 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 common 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?
We translate the condition to 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:
Let's separate both parts to simplify further calculations given 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 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!
A proportion is a mathematical expression that compares two or more numbers to each other. Proportions can compare absolute values and quantities or parts of a larger whole. Proportions can be written and calculated in several different ways, but the basic principle is the same.
Steps
Part 1
What is proportion-
Familiarize yourself with what proportion means. As noted above, proportions allow us to determine the relationship between two or more quantities. For example, if you need 2 cups of flour and 1 cup of sugar to make cookies, we say that there is a 2 to 1 ratio between the amount of flour and sugar.
- Proportions can be used to show how different quantities relate to each other, even if they are not directly related (unlike a recipe). For example, if there are five girls and ten boys in a class, the ratio of girls to boys is 5 to 10. In this case, one number is not dependent on or directly related to the other: the proportion may change if someone leaves the class or vice versa , new students will come to it. A proportion simply allows you to compare two quantities.
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pay attention to various ways expressions of proportions. Proportions can be written in words or using mathematical symbols.
- In everyday life, proportions are more often expressed in words (as above). Proportions are used in a variety of fields, and unless your profession is related to mathematics or other science, this is the way you will most often come across this way of writing proportions.
- Proportions are often written using a colon. When comparing two numbers using a proportion, they can be written with a colon, for example 7:13. If more than two numbers are being compared, a colon is placed consecutively between each two numbers, for example 10:2:23. In the above example for a class, we are comparing the number of girls and boys, with 5 girls: 10 boys. Thus, in this case the proportion can be written as 5:10.
- Sometimes a fraction sign is used when writing proportions. In our class example, the ratio of 5 girls to 10 boys would be written as 5/10. In this case, you should not read the “divide” sign and you must remember that this is not a fraction, but a ratio of two different numbers.
Part 2
Operations with proportions-
Reduce the proportion to its simplest form. Proportions can be simplified, like fractions, by reducing their members by a common divisor. To simplify a proportion, divide all numbers included in it by common divisors. However, we should not forget about the initial values that led to this proportion.
- In the example above with a class of 5 girls and 10 boys (5:10), both sides of the proportion have a common factor of 5. Dividing both quantities by 5 (the greatest common factor) gives a ratio of 1 girl to 2 boys (i.e. 1:2) . However, when using a simplified proportion, you should remember the original numbers: there are not 3 students in the class, but 15. The reduced proportion only shows the ratio between the number of girls and boys. For every girl there are two boys, but this does not mean that there is 1 girl and 2 boys in the class.
- Some proportions cannot be simplified. For example, the ratio 3:56 cannot be reduced, since the quantities included in the proportion do not have a common divisor: 3 is prime number, and 56 is not divisible by 3.
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To “scale” proportions can be multiplied or divided. Proportions are often used to increase or decrease numbers in proportion to each other. Multiplying or dividing all quantities included in a proportion by the same number keeps the relationship between them unchanged. Thus, the proportions can be multiplied or divided by the “scale” factor.
- Let's say a baker needs to triple the amount of cookies he bakes. If flour and sugar are taken in a ratio of 2 to 1 (2:1), to triple the amount of cookies, this proportion should be multiplied by 3. The result will be 6 cups of flour to 3 cups of sugar (6:3).
- You can do the opposite. If the baker needs to reduce the amount of cookies by half, both parts of the proportion should be divided by 2 (or multiplied by 1/2). The result is 1 cup of flour per half cup (1/2, or 0.5 cup) of sugar.
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Learn to find an unknown quantity using two equivalent proportions. Another common problem for which proportions are widely used is finding an unknown quantity in one of the proportions if a second proportion similar to it is given. The rule for multiplying fractions greatly simplifies this task. Write each proportion as a fraction, then equate these fractions to each other and find the required quantity.
- Suppose we have no large group students of 2 boys and 5 girls. If we want to maintain the ratio between boys and girls, how many boys should there be in a class of 20 girls? First, let's create both proportions, one of which contains the unknown quantity: 2 boys: 5 girls = x boys: 20 girls. If we write the proportions as fractions, we get 2/5 and x/20. After multiplying both sides of the equality by the denominators, we obtain the equation 5x=40; divide 40 by 5 and ultimately find x=8.
Part 3
Troubleshooting-
When operating with proportions, avoid addition and subtraction. Many problems with proportions sound like the following: “To prepare a dish you need 4 potatoes and 5 carrots. If you want to use 8 potatoes, how many carrots will you need?” Many people make the mistake of trying to simply add up the corresponding values. However, to maintain the same proportion, you should multiply rather than add. This is wrong and correct solution of this task:
- Incorrect method: “8 - 4 = 4, that is, 4 potatoes were added to the recipe. This means that you need to take the previous 5 carrots and add 4 to them so that... something is wrong! Proportions work differently. Let's try again".
- Correct method: “8/4 = 2, that is, the number of potatoes has doubled. This means that the number of carrots should be multiplied by 2. 5 x 2 = 10, that is, 10 carrots must be used in the new recipe.”
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Convert all values to the same units. Sometimes the problem occurs because quantities have different units. Before writing down the proportion, convert all quantities into the same units. For example:
- The dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in dragon hoards?
- Grams and kilograms are different units of measurement, so they should be unified. 1 kilogram = 1,000 grams, that is, 10 kilograms = 10 kilograms x 1,000 grams/1 kilogram = 10 x 1,000 grams = 10,000 grams.
- So the dragon has 500 grams of gold and 10,000 grams of silver.
- The ratio of the mass of gold to the mass of silver is 500 grams of gold/10,000 grams of silver = 5/100 = 1/20.
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Write down the units of measurement in the solution to the problem. In problems with proportions, it is much easier to find an error if you write down its units of measurement after each value. Remember that if the numerator and denominator have the same units, they cancel. After all possible abbreviations, your answer should have the correct units of measurement.
- For example: given 6 boxes, and in every three boxes there are 9 balls; how many balls are there in total?
- Incorrect method: 6 boxes x 3 boxes/9 marbles = ... Hmm, nothing is reduced, and the answer comes out to be “boxes x boxes / marbles“. It does not make sense.
- Correct method: 6 boxes x 9 balls/3 boxes = 6 boxes x 3 balls/1 box = 6 x 3 balls/1= 18 balls.
Find out what proportions are for. Proportions are used as in scientific research, and in Everyday life to compare different values and quantities. In the simplest case, two numbers are compared, but a proportion can include any number of quantities. When comparing two or more proportions can always be applied. Knowing how quantities relate to each other allows, for example, to write chemical formulas or recipes for various dishes. Proportions will be useful to you for a variety of purposes.
Proportion – equality of two relations, i.e. equality of the form a: b = c: d , or, in other notations, equality
If a : b = c : d, That a And d called extreme, A b And c - averagemembers proportions.
There is no escape from “proportion”; many tasks cannot be done without it. There is only one way out - to deal with this relationship and use proportion as a lifesaver.
Before we begin to consider proportion problems, it is important to remember the basic rule of proportion:
In proportion
the product of the extreme terms is equal to the product of the middle terms
If some quantity in a proportion is unknown, it will be easy to find it based on this rule.
For example,
That is, the unknown value of the proportion - the value of the fraction, in the denominator
which is the number that stands opposite the unknown quantity
, in the numerator – the product of the remaining terms of the proportion
(regardless of where this unknown quantity stands
). 
Task 1.
From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
Solution:
We understand that a decrease in the weight of the seed by a certain factor entails a decrease in the weight of the resulting oil by the same amount. That is, the quantities are directly related.
Let's fill out the table: 
An unknown quantity is the value of a fraction, in the denominator of which - 21 - the value opposite the unknown in the table, in the numerator - the product of the remaining members of the proportion table.
Therefore, we find that 1.7 kg of oil will come out of 7 kg of seed.
To Right When filling out the table, it is important to remember the rule:
Identical names must be written below each other. We write percentages under percentages, kilograms under kilograms, etc.
Task 2.
Convert to radians.
Solution:
We know that . Let's fill out the table:
Task 3.
A circle is depicted on checkered paper. What is the area of the circle if the area of the shaded sector is 27?
Solution:
It is clearly seen that the unshaded sector corresponds to the angle in (for example, because the sides of the sector are formed by the bisectors of two adjacent right angles). And since the entire circle is , then the shaded sector accounts for .
Let's make a table:
Where does the area of a circle come from?
Task 4. 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:
The entire field is 100%, and since 82% is plowed, then 100%-82%=18% of the field remains to be plowed.
Fill out the table: 
From where we get that the entire field is (ha).
And the next task is an ambush.
Task 5.
A passenger train covered the distance between two cities at a speed of 80 km/h in 3 hours. How many hours will it take a freight train to cover the same distance at a speed of 60? km/h?
If you solve this problem similarly to the previous one, you will get the following:
the time it takes for a freight train to travel the same distance as a passenger train is hours. That is, it turns out that walking at a lower speed, he covers (in the same time) the distance faster than a train with a higher speed.
What is the error in reasoning?
So far we have considered problems where the quantities were directly proportional to each other , that is height of the same value several times, gives height the second quantity associated with it by the same amount (similarly with a decrease, of course). And here we have a different situation: the speed of a passenger train more the speed of a freight train is several times higher, but the time required to cover the same distance is required by a passenger train smaller as many times as a freight train. That is, values to each other inversely proportional .
The scheme that we have used so far needs to be slightly changed in this case.
Solution:
We reason like this:
A passenger train traveled for 3 hours at a speed of 80 km/h, therefore it traveled km. This means that a freight train will cover the same distance in an hour.
That is, if we were making a proportion, we should have swapped the cells of the right column first. Would get: h.
That's why, please be careful when drawing up the proportions. First, figure out what kind of dependence you are dealing with - direct or inverse.
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