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# Proportionality: Basic Concept, Formulae and Short tricksPosted on : 10-09-2018 Posted by : Admin

## Introduction

In mathematics there are two kinds of proportionalities as mentioned below,

1. Direct Proportionality
2. Inverse Proportionality or Indirect Proportionality

Let us study each of them in some detail.

## Direct Proportionality

Y is said to be directly proportional to X if and only if their ratio is always constant.

But what does a constant ratio mean?

When we divide Y by X we should get some constant value every time. This constant is called Proportionality constant. It is denoted by the letter ‘C’. Here is one of the value increases, the other corresponding value increases automatically as, the two values are interdependent.

Direct proportionality is expressed as

This equation can be written as

Where C is the proportionality constant.

### Direct proportionality

For example,

If Y1 is directly proportional to X1 and Y2 is directly proportional to X2 then we can write,

If C1  = C2   =C then,

we can write,

$\frac{{\mathit{y}}_{\mathit{1}}}{{\mathit{x}}_{\mathit{1}}}\mathit{=}\frac{{\mathit{y}}_{\mathit{2}}}{{\mathit{x}}_{\mathit{2}}}\mathit{=}\frac{{\mathit{y}}_{\mathit{1}}}{{\mathit{x}}_{\mathit{2}}}\mathit{=}\frac{{\mathit{y}}_{\mathit{2}}}{{\mathit{x}}_{\mathit{1}}}\mathit{=}C$

## Inverse Proportionality

Y is said to be inversely or indirectly proportional to X if their product is always constant.

And what does this constant ratio mean?

When we multiply Y with X we should get some constant value every time. This constant is called Proportionality constantT. This is also denoted by letter ‘C’. Here the increase or decrease in one of the value has no effect on the other value as, is one of the value increases, the other corresponding value increases automatically as, the two values are interdependent.

Inverse proportionality is expressed as

This equation can be written as

C is the proportionality constant.

### Inverse proportionality

For example,

If Y1 is inversely proportional to X1 & Y2 is inversely proportional to X2 then we can write,

If C1 = C2 =C, then we can write Y1X1 = Y2X2 = C

## Application of proportionality concept to Work and Time

TIME AND MEN

To complete the given work,

• If number of men is more, then time taken to complete the work is less
• If no of men are less, then the time taken to complete the work is more

So here, Time taken to complete the work (T) and number of men required to complete the work (M) are inversely proportional

$\mathbit{T}\mathbit{\alpha }\frac{\mathit{1}}{\mathit{M}}$

Hence, T1M1 = T2M2

EFFICIENCY AND TIME

To complete the given work,

• If the efficiency of workers is more, then the time taken to complete the work is less
• If the efficiency of the workers is less, then the time taken to complete the work is more

So here, efficiency of workers (E) and Time taken to complete the work (T) are inversely proportional.

$\mathbit{E}\mathbit{\alpha }\frac{\mathit{1}}{\mathit{T}}$

Hence, E1T1 = E2T2

EFFICIENCY AND MEN

To complete the given work,

• If the efficiency of workers is more, then the number of men required to complete the work are less
• If the efficiency of the workers is less, then the number of men required to complete the work are more.

So here, efficiency of workers (E) and Number of workers (M) are inversely proportional.

$\mathbit{E}\mathbit{\alpha }\frac{\mathit{1}}{\mathit{M}}$

Hence, E1M1 = E2M2

WORK AND TIME, DAYS, MEN AND EFFICIENCY

Work is directly proportional to Men (M), Efficiency (E), Time (T) and Days (D)

Equation 1: If work is more then, time required to complete the work is more or If more time is given then, more work can be done (W α T)

This can be represented as,

W1T2=W2T1

Equation 2: If work is more number of days required to do the work are more or if number of days given are more then, more work can be done (W α D)

This can be represented as,

W1D2 =W2D1

Equation 3: If work is more no of men are more or If Men are more then, more work can be done  (W α M)

This can be represented as,

W1M2=W2M1

Equation 4: If work is more then, efficiency should be more or If efficiency is more then, more work can be done (W α E)

This can be represented as,

W1E2=W2E1

Hence, on combining all the above four equation we can say,

W1T2D2M2E= W2T1D1M1E1

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