Short Answer

The fundamental rule of proportion states that in any proportion, the product of the extreme terms is equal to the product of the mean terms. In simple words, if a:b = c:d, then a × d = b × c.

This rule helps us check whether two ratios are equal and also helps in finding missing values. It is very useful in solving mathematical problems involving proportion.

Detailed Explanation:

Fundamental rule of proportion

The fundamental rule of proportion is a basic principle in mathematics that explains the relationship between the four terms of a proportion. A proportion is written in the form a:b = c:d, where a, b, c, and d are numbers. According to this rule, the product of the first and last terms (called extremes) is equal to the product of the middle terms (called means).

Terms in a proportion

In a proportion like a:b = c:d, there are four terms. The first term (a) and the last term (d) are called extremes. The second term (b) and the third term (c) are called means. For example, in the proportion 2:3 = 4:6, the extremes are 2 and 6, while the means are 3 and 4.

Cross multiplication rule

The fundamental rule is often explained using cross multiplication. It means multiplying the first term with the last term and the second term with the third term. For example, in 2:3 = 4:6, we multiply 2 × 6 and 3 × 4. Both give 12, so the proportion is correct. This shows that the ratios are equal.

Checking correctness of proportion

This rule helps us check whether a given proportion is correct or not. If the product of extremes is equal to the product of means, then the proportion is true. If the products are not equal, then the proportion is not correct. This makes it easy to verify proportions.

Finding missing value

The fundamental rule is also useful in finding unknown values in a proportion. For example, if 3:5 = x:10, we can apply the rule. Multiply 3 × 10 = 5 × x, which gives 30 = 5x. Solving this, we get x = 6. This method is simple and widely used.

Importance of the rule

This rule is very important because it simplifies calculations involving proportions. It helps in solving problems related to ratio, scaling, and comparison. It is widely used in mathematics, science, and real-life situations.

Use in daily life

In daily life, this rule is used in many situations like cooking, shopping, and measuring quantities. For example, if a recipe for 2 people uses 1 cup of rice, we can use proportion to find the quantity needed for more people. The rule helps maintain the correct balance.

Common mistakes

A common mistake is mixing up the order of terms. The position of numbers in a proportion must be correct. Another mistake is not multiplying correctly during cross multiplication. Care should be taken to apply the rule properly to get accurate results.

Conclusion

The fundamental rule of proportion states that the product of extremes is equal to the product of means. It is a simple and powerful rule used to check proportions and find unknown values. It plays an important role in solving mathematical and real-life problems.