The word "proportion" comes from a Latin root and means "proportion". People often use it in daily life. They talk, for example, about the proportions of the human body or about proportions in cooking. Today we will find out what mathematicians mean by this word.
Let's consider two relationships. We remember that a ratio is the quotient of two numbers.
Note that in both the first and second cases the value of the quotient is three. Before us are two equal relationships. Let's write down the equality.
Fifteen is to five as twenty-four is to eight. This equality is called proportion. Sometimes this equality is written as an equality of ordinary fractions.
Let's formulate a definition: The equality of two ratios is called proportion.
Using letters, the proportion can be written:
Attitude a To b equal to the ratio c To d. Sometimes the proportion is read differently: “ a this applies to b, How c refers to d». The numbers involved in a proportion are called terms of the proportion. All terms are assumed to be different from zero.
Numbers a And d are called the extreme terms of the proportion, and the numbers b And c- average members. Indeed, in the first variant of writing the number b And c are in the middle, and the numbers a And d on the edge.
In the proportion discussed earlier 
Let's find the product of its middle and extreme terms.
Note that the two resulting products are equal.
Let us formulate the basic property of proportion in general form.
In the correct proportion, the product of the extreme terms is equal to the product of the middle terms.
The converse is also true.
If the product of the extreme terms is equal to the product of the middle terms of the proportion, then the proportiontrue.
Let's find the unknown term of the proportion, that is, solve the proportion.
The numbers 0.5 and 13 are extreme terms; numbers a and 2 are the middle terms. Let's use the basic property of proportion.
Let's solve the proportion.
Using the basic property of proportion, we get:
To get rid of the decimal in the denominator, multiply both the numerator and denominator of the fraction by 10. Reduce the resulting fraction by 4, and then again by 4.
Check if these proportions are correct:
In this task, you need to check whether equality between relations actually holds.
Let's find the product of the averages and the product of the extremes for each proportion. If the resulting products are equal, then the proportion is correct. Otherwise, the proportion is incorrect.
the correct proportion, because
incorrect proportion, because .
If the middle or extreme terms are swapped in the correct proportion, then the resulting new proportions are also correct.
This is so because with such a rearrangement the product of the extreme and middle terms does not change.
Let's look at an example. From this proportion, obtain two new ones by rearranging the extreme and middle terms. First, let's rearrange the middle terms (Fig. 1).
Rice. 1. Rearrangement of middle terms
Indeed, the product of the average and extreme has not changed, which means that the resulting proportion is correct. Let us rearrange the extreme terms (Fig. 2).
Rice. 2. Rearrangement of extreme members
And in this case, the product of the average and extreme has not changed. We got the correct proportion.
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. - M.: Education, Mathematics Teacher Library, 1989.
- Mathematics ().
- Internet portal Math-portal.ru ().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012: No. 762 (a, d, d), No. 765, No. 777.
- Other tasks: No. 767, No. 775.
(from lat. rgorortio- “commensurability”).
If the ratio A: b equal to the ratio With:d, then the identity A:b= s:d called proportion.
If , then equality will remain in the following cases:
(increase in proportion),
(decrease in proportion).
(composing proportions by addition),
(composing proportions by subtraction).
Please note that drawing up proportions is another way to solve problems involving percentages.
For example:
Tin is made from a mineral called cassiterite. How many tons of tin will be obtained from 25 tons of cassiterite if it contains 78% tin?
Solution. Let them get some tin. Taking the mass of the mineral as 100%, we write:
Solving 25.78 = 100x we find that x = 19.5t.
The concept of proportion is closely related to proportionality. Proportionality- this is a constant ratio of two quantities to each other. For example, the more we press the gas pedal in a car, the faster it will go.
Proportionality can be direct or inverse.
Direct proportionality - the growth of one value entails the growth of another.
Inverse proportionality exists when an increase in one value several times decreases another by the same amount. Continuing from the previous one example- inverse proportionality between pressing the brake pedal and the speed of the car - the more we press on the brake, the lower the speed.
3.6:1.2 and 6.3:2.1 are equal, since the values of the quotients are equal to 3. Therefore, we can write the equality 3.6:1.2 = 6.3:2.1, or
The equality of two ratios is called proportion.
Using letters, the proportion is written like this: a:b = c:d or
These entries read: “The ratio of a to b is equal to the ratio of c to d >>
or “a is to b as c is to d >>
.
In proportion, or a:b=c:d,
The numbers a and d are called extreme terms, and the numbers b and c are called middle terms. In what follows we will assume that all terms of the proportion are different from zero: .
In a proportion we find the product of its extreme terms and the product of its middle terms.
We get 3.6 2.1 = 7.56; 1.2 6.3 = 7.56. So, 3.6 2.1 = 1.2 6.3.
In the correct proportion, the product of the extreme terms is equal to the product of the middle terms. The converse statement is also true: if the product of the extreme terms is equal to the product of the middle terms of the proportion, then the proportion is correct.
This property is called the basic property of proportion.
The proportion 20:16 = 5:4 is correct, since 20 4 = 16 5 = 80. Let's swap the middle terms in this proportion.
We get a new proportion: 20:5 = 16:4. It is also correct, since with such a rearrangement the product of the extreme and the product of the middle terms did not change. These products will not change if the extreme terms in the proportion 20:5 = 16:4 are swapped.
If the middle members or extreme members are swapped in the correct proportion, then the resulting new proportions are also correct.
748. By rearranging the middle or extreme terms of the proportion, create three new correct proportions from the proportion:
749. Using the correct equality 4 9 = 0.2 180, create four correct proportions.
P 750. Calculate verbally:
751. What action sign must be substituted instead of * to get the correct equality:
752. Find the ratio of quantities:
a) 1.5 m and 30 cm;
b) 1 kg and 250 g;
c) 1 hour and 15 minutes;
d) 50 cm 2 and 1 dm 2.
753. numbers are equal to this number. What is this number?
754. What number must be added to the numerator and denominator of a fraction to get a fraction?
M 755. Which of the figures (Fig. 33) are developments:
a) quadrangular prism; b) triangular prism; c) a triangular pyramid?
756. 50 shots were fired from a gun, with 5 bullets flying past the target. Define.
757.Angle A is 30° and angle B is 50°. What part of angle A is from angle B? How many times is angle B larger than angle A?
758. The brigade was given the task of collecting 280 centners of grapes. She collected 350 quintals. By what percentage did the team exceed the task? To what percentage did the team complete the task?
759.Maple and oak trees were planted in the park, with one oak for every 4 maples. What percentage of all trees planted are maples? How many trees were planted in the park if 480 maples were planted?
D 760. Is the proportion correct:
a) 2.04:0.6 = 2.72:0.8; b) 0.0112:0.28=0.204:0.51?
761. Solve the equation:
762. From 225 kg of ore, 34.2 kg of copper were obtained. What is the percentage of copper in the ore?
763. 2 hours after leaving station A, the diesel locomotive increased its speed by 12 km/h and 5 hours after the start of movement arrived at destination B. What was the speed of the diesel locomotive at the beginning of the journey, if the distance from A to B is 261 km?
764. If you add 0.8 to an unknown number, you get 1.2. Find the unknown number.
765. Follow these steps:
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I.Zhokhov, Mathematics for 6th grade, Textbook for high school
Calendar-thematic planning in mathematics, video on mathematics online, Mathematics at school download
