## Componendo dividendo to solve number system and ratio and proportion problems quickly

We have seen Componendo dividendo to simplify competitive math problems in topics as diverse as—number system, ratio and proportion, surds, trigonometry or algebra. Often, it plays a key role in solving the problems quickly.

In this session we will show how Componendo dividendo can be applied to quickly solve problems in number system and ratio and proportion.

You may refer to the earlier sessions on componendo dividendo to have a better awareness.

**Componendo divididendo explained**

**Componendo dividendo in Algebra.**

### Problem example 1. Componendo dividendo in Number system

If 1 is added to both the numerator and denominator of a fraction it become $\displaystyle\frac{1}{4}$. If 2 is added to both the numerator and denominator it becomes $\displaystyle\frac{1}{3}$. Sum of the numerator and denominator is,

- 13
- 8
- 27
- 22

** Solution** 1:** Problem analysis and execution:**

As the operations in both cases on the numerator and denominator are uniform, we decide to use Componendo dividendo.

Assuming the fraction to be, $\displaystyle\frac{x}{y}$, in the first case,

$\displaystyle\frac{x+1}{y+1}=\frac{1}{4}$.

Adding 1 to both sides,

$\displaystyle\frac{x+y+2}{y+1}=\frac{5}{4}$

Subtracting both sides from 1,

$\displaystyle\frac{y-x}{y+1}=\frac{3}{4}$.

Dividing second by first,

$\displaystyle\frac{y-x}{x+y+2}=\frac{3}{5}$.

Doing similar operations in the second case we get,

$\displaystyle\frac{y-x}{x+y+4}=\frac{1}{2}$.

Dividing first by second,

$\displaystyle\frac{x+y+2 +2}{x+y+2}=\frac{6}{5}$,

Or, $1 + \displaystyle\frac{2}{x+y+2}=1+\frac{1}{5}$,

Or, $x+y+2=10$,

Or, $x+y=8$.

**Answer:** Option b: 8.

**Key concepts used:** Pattern recognition and use -- **Componendo dividendo -- efficient simplification.**

The process of componendo dividendo is so easy and systematic that it can be carried out in mind. We do not need to write the steps as above.

#### Note: The problem expressions do not conform to componendo dividendo at the outset

It was not apparent from the problem description that componendo dividendo can be apllied directly, but we recognized the opportunity from the uniformity of operation in both numerator and denominator, that too in both cases. It needed some amount of goal-directed manipulation of expression to apply componendo dividendo, but overall, the ease of the individual steps, if you are able to visualize them, enables us to solve the problem quickly in mind.

We call this type of application as * Indirect Componendo dividendo*.

That is the * hallmark of componendo dividendo*—if you manage to apply it to a problem, you will certainly be able to solve it quickly and in mind.

#### Problem example 2. Componendo dividendo in ratio proportion problems

If 12 is added to two numbers, their ratio changes from $3:4$ to $13:16$. The larger number is,

- 48
- 144
- 52
- 36

#### Problem solving by componendo technique

Componendo dividendo is a powerful algebraic technique that frequently can be applied to simplify a particular form of expression in a few seconds. The form of expression it is applied to is,

$\displaystyle\frac{a+b}{a-b}=\frac{5}{2}$, say.

By adding 1 to both sides, then subtracting 1 from both sides, and finally taking the ratio of the two, we get a quick simple result of,

$\displaystyle\frac{a}{b}=\frac{5+2}{5-2}=\frac{7}{3}$.

Many times we apply the first part of the technique, that is componendo technique, to simplify the numerator,

$\displaystyle\frac{a+b}{a-b}+1=\frac{5}{2}+1$,

Or, $\displaystyle\frac{2a}{a-b}=\frac{7}{2}$.

We have simplified the numerator by eliminating $b$.

Using this technique on our problem subtracting both sides of the equation from 1, we get,

$\displaystyle\frac{3x+12}{4x+12}=\frac{13}{16}$, introducing cancelled out HCF as $x$ factor to both the numbers

Or, $\displaystyle\frac{x}{4x+12}=\frac{3}{16}$,

Inverting the LHS and simplifying,

$\displaystyle\frac{4x+12}{x}=\frac{16}{3}$,

Or, $4 + \displaystyle\frac{12}{x}=\frac{16}{3}$,

Or, $\displaystyle\frac{12}{x}=\frac{4}{3}$,

Or $x=9$.

These tasks of adding 1 to fractions, inverting the fraction and so on are simple processes and can be quickly done in mind without using pen and paper. If you are skilled in this type of mental manipulation, this approach should reach you to the solution pretty quick.

#### Note: We have used part of comnponendo dividendo

Observe that we have not applied all the three steps of componendo dividendo in this problem, we have just applied the first step to simplify the numerator to one variable expression so that we can reverse and simplify all in mind dealing with simpler expressions.