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# Componendo, and Dividendo

We have discussed the laws of proportion in our earlier articles and today we are going to discuss the other two laws of proportion, Componendo, and Dividendo. Questions based on these laws are very common in competitive exams and they are very easy to learn as well. Let us begin.

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## Componendo and Dividendo

We all know that ratio of two numbers is done to compare the numbers. These ratios become proportional. And to solve questions from proportion we require laws. We have discussed two laws Invertendo and Alternendo in our previous articles, and today the discussion is on Componendo and Dividendo.

### Componendo

In Componendo the basic rule that you need to remember is if a : b : : c : d than (a + b) : b : : (c + d) : d. Thus, in Componendo as you can see that you need to add the denominator to numerator given in the ratios and then equate them. If you are using the rule on the left-hand side then you also need to use the rule on the right-hand side.

### Example

Q. Suppose the weekly salaries of two women are in the ratio of 4: 7. If each of the ladies receives an increment of Rs. 25 in their respective salary, the ratio of their salaries will be changed to 3: 5. What is their respective salary?

A. 200 & 350                 B. 210 & 320
C. 220 & 350                 D. 210 & 350

Ans: Here we are given the ratio of the salaries of two women. After the increment in their salary, the ratio is changed. As the salary is increased in the given question, Componendo law will be applied on both the sides.

Let’s start by assuming that the initial salary of both the women is Rs. x. So, the ratio of their salaries will be 4x and 7x. Now, after an increase in their salaries, their final salary will be 4x + 25 and 7x + 25. This ratio will be equal to 3/5. Thus, the given equation will be written as,

4x + 25/7x + 25 = 3/5
=> 5(4x + 25) = 3(7x + 25)
=> 20x + 125 = 21x + 75 => x = 50

Thus, the initial salary was Rs. 50. Now, based on the ratio their salary will be 4(50) : 7(50) => 200 : 350. So, the correct answer is A.

### Dividendo

In the law of Dividendo, if a : b : : c : d then (a – b) : b : : (c – d) : d. In dividendo instead of addition, you are required to subtract the denominator from numerator in both the ratios. Rest everything in this law is similar to the componendo.

### Example

Q. The present ages of P and Q are represented in the ratio as 6: 4. Five years ago their ages were in the ratio 5 : 3. What are their present ages?

A. 25 and 15 years                   B. 26 and 16 years
C. 28 and 18 years                  D. 30 and 20 years

Suppose that P’s present age is 6x, so the current age of Q will be 4x years. Now, we are given the ratio of their age five years back. Thus, their age during that time would have been 6x – 5 and 4x – 5 respectively. It is given that ratio of their ages is 5 : 3. Thus, the equation will be, $$\frac{6x – 5}{4x – 5}$$ =  $$\frac{5}{3}$$
=> 3(6x – 5) = 5(4x – 5)
=> 18x – 15 = 20x – 25 => 2x = 10 => x = 5 years. Thus, the present ages of P and Q will be 30 and 20 respectively. So, the correct answer is D.

### Componendo and Dividendo

There is one more law where you can use both componendo and dividendo together. In this law if a : b : : c : d than (a + b) : (a – b) : : (c + d) : (c – d). Thus, when using together you have to apply components in the numerator and dividend in the numerator.

### Practice Questions

1. The total of Rs. 431 are to be divided among three workers P, Q, and R such that 8 times P’s share is equal to 12 times Q’s share which is equal to 6 times R’s share. How much did A make?

A. Rs. 122            B. Rs. 144              C. Rs. 150           D. Rs. 156

2. The monthly income of Ajay and Arbaaz is in the ratio of 4: 5. Their expenses are in the ratio of 5: 6. If Ajay saves Rs. 25 per month while Arbaaz saves Rs. 50/month, what was their Ajay’s income?

A. Rs. 400              B. Rs. 450            C. Rs. 420              D. Rs. 470

3. The income of A and B are in the ratio of 8: 11 and their expenditure are in the ratio of 7: 10. If each of them saved Rs. 500, what was their total expenditure?

A. Rs. 7500            B. Rs. 8000            C. Rs. 8500               D. Rs. 9000

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