Componendo-dividendo rule: A theorem based on the topic of proportions permits a faster way to carry out the calculations and cut down the count of expansions required. It is helpful in situations involving equations with fractions or rational functions. The componendo and dividendo rule are used while studying ratios and proportions. If the ratio of any two given numbers is the same as the ratio of the other 2 numbers, then the ratios of the sum total of the numerator and denominator to the difference of the numerator and denominator of the two rational numbers are equal in nature.
What is Componendo and Dividendo Rule?
In the concept of componendo, the fundamental rule is if a : b : : c : d then (a + b) : b : : (c + d) : d. It can be observed that it is required to add the denominator to the numerator in the given ratios and then they are equated. If the rule is used on the left side, then it should be used on the right side too.
In the dividendo rule, if a : b : : c : d then (a – b) : b : : (c – d) : d. It can be observed that it is required to subtract the denominator from the numerator in the given ratios. Everything else is the same as the componendo rule.
In the componendo and dividendo rule, if a : b : : c : d then (a + b) : (a – b) : : (c + d) : (c – d). If this rule is applied, then componendo and dividendo are used together.
Componendo Dividendo Rule Proof
Consider 4 quantities a, b, c and d such that a : b = c : d.
(a / b) = (c / d)
b / a = d / c   [by invertendo]
(a + b) / b = (c + d) / d  [by componendo]
(a – b) / b = (c – d) / d   [by dividendo]
(a + b) / (a – b) = (c + d) / (c – d)
(a + b) : (a – b) = (c + d) : (c – d).
So, it can be expressed as a : b = c : d and (a + b ) : (a – b) = (c + d) : (c – d).
It is true that if the ratio of a to b = c to d, then the ratio of (a + b) to (a – b) is the same as (c + d) to (c – d). This is termed the componendo and dividendo rule.
Componendo Dividendo Rule – Solved Examples
Problem 1: If 2a – 3b = 0, then what is the value of (a – b) : (a + b)?
Answer:
Convert the equation given to the proportion equation form.
2a – 3b = 0
2a = 3b
(a / b) = (3 / 2)
On adding 1 on both sides,
(a / b) + 1 = (3 / 2) + 1
(a + b) / b = (3 + 2) / 2
(a + b) / b = 5 / 2
On subtracting 1 from both sides,
(a – b) / b = (3 – 2) / 2 = 1 / 2
On dividing both the equations,
(a – b) / (a + b) = 1 / 5
Problem 2: If (a / b) = (16 / 3), then find the value of (a – b) / (a + b).
Answer:
If (a / b) = (c / d) then [(a + b) / (a – b)] = [(c + d) / (c – d)].
[(a + b) / (a – b)]= (16 + 3) / (16 – 3)
= 19 / 13
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