Componendo and Dividendo Rule
Shivam Singh31 Jul, 2025
The Componendo and Dividendo Rule is a theorem based on proportions that offers a quicker way to perform calculations and reduces the number of steps needed. It is especially useful in solving equations involving fractions or rational expressions. It states that if two ratios are equal, then the ratio of the sum of the numerator and denominator to their difference will also be equal for both. HereIn this blog, we learn about componendo and dividendo Rule in detail with examples and proof.
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What is the Componendo and Dividendo Rule
The Componendo and Dividendo Rule is a useful algebraic method for simplifying problems that involve two equal ratios. When two ratios are equal, like a/b = c/d, this rule allows us to transform the equation into a new, simpler form, i.e.
a+ b/ a − b = c + d/ c − d.
This helps reduce calculation steps, especially when dealing with fractions or rational expressions. Instead of using long multiplication or expanding terms, the rule quickly rewrites the problem using just addition and subtraction.
It is especially useful in topics like ratios, proportions, and linear equations. While it may not be taught in early classes, it is often applied in higher-level math and competitive exams.
What is the Componendo Rule?
The Componendo Rule is used to simplify an equation that compares two ratios. According to this rule, if two ratios are equal, then the ratio of the sum of the numerator and denominator to the denominator will also be equal.
Equation:
If
a/b = c/d
then, by Componendo,
a+ b/b = c + d/d
This rule is particularly useful in converting ratios into forms where cross-multiplication or comparison becomes easier.
Componendo Rule Proof
We start with the given proportion:
a/b = c/d
Step 1: Add 1 to both sides
Adding 1 to both sides of an equation preserves equality:
a/b + 1 = c/d+1
Step 2: Write 1 as the denominator over itself
Note that:
1 = b/ b and 1= d/d
So we can rewrite the equation as:
a/b + b/b = c/d + d/d
Step 3: Add the fractions
Since both terms on each side have the same denominator, we can combine them:
a + b/b = c + d/d
Thus, we have proved that:
a + b/b = c + d/d
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What is the Dividendo Rule?
The Dividendo Rule is the counterpart to the Componendo Rule. It states that if two ratios are equal, then the ratio of the difference of the numerator and denominator to the denominator will also be equal.
Equation
If
ab = cd
then, by Dividendo,
a − b/b = c − d/d
It is often used in algebra to simplify equations involving proportions, especially where subtraction is needed to isolate terms.
Proof of the Dividendo Rule
We begin with the given proportion:
ab = cd
Step 1: Subtract 1 from both sides
Just like we added 1 in the Componendo Rule, here we subtract 1:
a/b − 1 = c/d −1
Step 2: Rewrite 1 as a fraction
1 = b/b and 1= d/d
So we can write:
a/b − b/b = c/d - d/d
Step 3: Subtract the fractions
Since both terms on each side have the same denominator, we subtract the numerators:
a − b/b = c − d/d
This is the exact expression stated in the Dividendo Rule.
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Proof of the Componendo and Dividendo Rule
To prove this rule, we start with a basic proportion involving four numbers. Let:
a/b = c/d
This tells us that the ratio of a to b is the same as the ratio of c to d. We now aim to show that:
a + b/a − b = c + d/c − d
We will prove this step by step using two known ratio rules: Componendo and Dividendo.
Step 1: Apply the Componendo Rule
According to the Componendo Rule, if two ratios are equal, then the ratio of the sum of the numerator and denominator to the denominator is also equal. That means:
a+b/b = c + d/d
Step 2: Apply the Dividendo Rule
According to the Dividendo Rule, the ratio of the difference of the numerator and denominator to the denominator is also equal. So:
a−b/b = c− d/d
Step 3: Divide Equation 1 by Equation 2
a + b/b/a − b/b = c + d/d/c - d/d
Since both sides have the same denominator in numerator and denominator (i.e., b and d), they cancel out.
a + b/a − b = c + d/c − d
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Componendo and Dividendo Rules Solved Examples
Example 1. If 3a − 4b = 0, find the ratio (a−b):(a+b).
Step 1: Rewrite the equation as a ratio
We are given:
3a = 4b
Divide both sides by b:
a/b = 4/3
Step 2: Apply the Componendo and Dividendo Rule
The rule says:
a + b/a − b = 4 + 3 / 4−3=7
Now take reciprocal to find (a−b):(a+b)
a−b/a+b = 1/7
Therefore,
(a−b): (a+b)=1:7
Example 2: If a/b = 2/5. find the ratio (a+b):(a−b)
Solution:
Step 1: Identify the ratio
We are given:
a/b = 2/5
Step 2: Apply the Componendo and Dividendo Rule
a + b/ a − b = 2 + 5/2 − 5 = 7/-3
(a+b) : (a−b) = −7:3
Note: Since the denominator is negative, this ratio is also negative.
Example 3: If (a + b):(a − b) = 5:2 then find the value of a:b
Solution:
We are given:
a + b/a − b = 5/2
Now apply the reverse Componendo and Dividendo Rule:
a/b = 5 + 2/5 − 2 = 7/3
Hence,
a : b = 7:3
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