Question

Пропорция, основное свойство

Answer 1

Произведение крайних членов пропорции равно произведению ее средних членов.

Для пропорции a:b=c:d или a/b=c/d основное свойство записывается так: a·d=b·c.

Для нашей пропорции 12 : 3 = 16 : 4 основное свойство запишется так: 12·4=3·16. Получается верное равенство: 48=48. 

Answer 2

Proportion and its main property

A proportion is a mathematical relationship between four numbers, where the ratio of the first two numbers is equal to the ratio of the last two numbers. It can be represented as a : b = c : d. The main property of a proportion is that the product of the means (the middle terms) is equal to the product of the extremes (the outer terms).

For example, if we have the proportion 3 : 2 = 12 : 8, we can verify its property by multiplying the means and the extremes:

- Means: 3 * 8 = 24 - Extremes: 2 * 12 = 24

Since the product of the means is equal to the product of the extremes (both equal to 24), the proportion is valid.

This property holds true for any valid proportion. It allows us to solve various types of problems involving proportions, such as finding missing values or verifying the equality of ratios.

Let's look at a few examples to illustrate this:

1. Example 1: Solve the proportion 6/2 = 12/4.

- Multiply the means: 6 * 4 = 24. - Multiply the extremes: 2 * 12 = 24. - Since the product of the means is equal to the product of the extremes (both equal to 24), the proportion is valid.

2. Example 2: Solve the proportion 10/2 = 16/4.

- Multiply the means: 10 * 4 = 40. - Multiply the extremes: 2 * 16 = 32. - Since the product of the means is not equal to the product of the extremes (40 ≠ 32), the proportion is not valid.

These examples demonstrate the main property of proportions and how it can be used to determine their validity.

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