What Is The Componendo And Dividendo Rule?

Proportion is mostly taught using ratios and fractions. When a fraction is expressed as a/b, and a ratio is expressed as a:b, a proportion asserts that two ratios are equal. In this case, a and b can be any two numbers. The ideas of ratio and proportion are important foundations for understanding different concepts in mathematics and science. Proportion may be used to solve a variety of difficulties in everyday life, such as in business when dealing with transactions or in the kitchen, for example. It creates a relationship between two or more variables, making comparison easier.

Componendo-Dividendo, also known as componendo and dividendo, is a rule for comparing and analysing ratios and proportions. This rule will be stated in the next part, let's see what happens.

Componendo-Dividendo

Componendo and dividendo is a proportionality theorem that allows for rapid computations while reducing the amount of expansions required. It's very beneficial when working with fractions or rational functions in mathematics Olympiads, especially when fractions are involved. As a result, if the ratio of any two numbers equals the ratio of another two numbers, the sum of numerator and denominator to the difference of numerator and denominator of both rational numbers is equal. Let's begin by defining a few key concepts.

Ratios and Proportions

The ratio is a method of dividing two quantities of the same sort to compare them. a:b or a/b is the ratio formula for two integers a and b. The ratio is unaffected by multiplying and dividing each term of a ratio by the same number (non-zero). Two or more such ratios are said to be in proportion when they are equal.

Proportional fourth, third, and mean. If a : b = c : d, the following is true:

• The fourth proportionate to a, b, and c is known as d.
• The third proportionate to a and b is known as c.
• Between a and b, the mean proportional is (ab).

Proportion Tips & Tricks:

a/b = c/d ⇒ ad = bc; a/b = c/d ⇒ b/a = d/c; a/b = c/d ⇒ a/c = b/d; a/b = c/d ⇒ (a + b)/b = (c + d)/d; a/b = c/d ⇒ (a - b/b = (c - d)/d; a/ (b + c) = b/ (c + a) = c/ (a + b) and a + b + c ≠0, then a = b = c; The componendo -dividendo rule states that (a + b)/ (a - b) Equals (c + d)/ (c - d); In the ratio a:b, if both the integers a and b are multiplied or divided by the same number, the outcome is the same as the original ratio.

Properties of Proportion

Proportion is a mathematical concept that provides an equal relationship between two ratios. The proportional characteristics that this relationship follows are:

• Addendo – the value of each ratio is a + c : b + d, If a : b = c : d.
• Subtrahendo – the value of each ratio is a – c : b – d, If a : b = c : d.
• Dividendo – a – b : b = c – d : d, If a : b = c : d.
• Componendo – a + b : b = c + d : d, If a : b = c : d.
• Alternendo –a: c = b: d, If a : b = c : d.
• Invertendo – b: a = d: c, If a: b = c : d.
• Componendo and dividendo – a + b: a – b = c + d : c – d, If a : b = c : d.

Direct Proportion

If the rise (or decrease) of one quantity causes the other to increase (or decrease) to the same amount, the two are said to be directly proportional. Example 1: The cost is proportionate to the number of articles. (More articles, higher price). Example 2: The amount of work done is proportionate to the number of guys who work (More Men, More Work).

Indirect Proportion

If one quantity increases, the other decreases to the same amount, and vice versa, two quantities are said to be indirectly proportional. Example 1: The time it takes an automobile to travel a given distance is inversely related to its speed. (The time it takes to cross a distance, decreases as speed increases.) Example 2: The length of time it takes to complete a task is inversely related to the number of people who work on it. (As the number of people increases, the time it takes to do a task decreases). Remarks: When using the chain rule to solve an issue, we compare each item to the word to be discovered.

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