Posted on : 10-09-2018 Posted by : Admin

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

- Direct Proportionality
- Inverse Proportionality or Indirect Proportionality

Let us study each of them in some detail.

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 value is called Proportionality constant. It is denoted by the letter ‘C’. Here if one of the value (here Y ) increases, then the other corresponding value (here X ) increases automatically as, the two values are interdependent.

Direct proportionality is expressed as $Y\mathit{}\alpha \mathit{}X$

This equation can be written as $y\mathit{=}cx\mathit{}\mathit{\left(}or\mathit{\right)}\mathit{}\frac{y}{x}\mathit{=}c$

Where C is the proportionality constant.

For example,

If Y_{1} is directly proportional to X_{1} and Y_{2} is directly proportional to X_{2} then we can write,

${Y}_{\mathit{1}}\mathit{}\alpha \mathit{}{X}_{\mathit{1}}\mathit{}and\mathit{}{Y}_{\mathit{2}}\mathit{}\alpha \mathit{}{X}_{\mathit{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{Y}_{\mathit{1}}={C}_{\mathit{1}}{X}_{\mathit{1}}\mathit{}and\mathit{}{Y}_{\mathit{2}}={C}_{\mathit{2}}{X}_{\mathit{2}}\phantom{\rule{0ex}{0ex}}\frac{{y}_{\mathit{1}}}{{x}_{\mathit{1}}}\mathit{=}{C}_{\mathit{1}}\mathit{}and\mathit{}\frac{{y}_{\mathit{2}}}{{x}_{\mathit{2}}}\mathit{=}{C}_{\mathit{2}}$

If C1 = C_{2} =C then,

we can write,

$\frac{{y}_{\mathit{1}}}{{x}_{\mathit{1}}}\mathit{=}\frac{{y}_{\mathit{2}}}{{x}_{\mathit{2}}}\mathit{=}C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{\left(}OR\mathit{\right)}\phantom{\rule{0ex}{0ex}}\frac{{y}_{\mathit{1}}}{{y}_{\mathit{2}}}\mathit{=}\frac{{x}_{\mathit{1}}}{{x}_{\mathit{2}}}\mathit{=}C\left(OR\right)\phantom{\rule{0ex}{0ex}}{y}_{1}{x}_{2}={y}_{2}{x}_{1}=C$

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 $y\mathit{}\alpha \mathit{}\frac{\mathit{1}}{\mathit{x}}$

This equation can be written as $y\mathit{}\alpha \mathit{}\frac{\mathit{C}}{\mathit{x}}\mathit{}or\mathit{}yx\mathit{=}C$

C is the proportionality constant.

For example,

If Y_{1} is inversely proportional to X_{1} & Y_{2} is inversely proportional to X_{2} then we can write,

${Y}_{\mathit{1}}\mathit{}\alpha \mathit{}\frac{\mathit{1}}{{x}_{\mathit{1}}}\mathit{}and\mathit{}{Y}_{\mathit{2}}\mathit{}\alpha \mathit{}\frac{\mathit{1}}{{x}_{\mathit{2}}}\phantom{\rule{0ex}{0ex}}{Y}_{\mathit{1}}\mathit{}\alpha \mathit{}\frac{{C}_{\mathit{1}}}{{x}_{\mathit{1}}}\mathit{}and\mathit{}{Y}_{\mathit{2}}\mathit{}\alpha \mathit{}\frac{{C}_{\mathit{2}}}{{x}_{\mathit{2}}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{Y}_{\mathit{1}}{X}_{\mathit{1}}\mathit{=}{C}_{\mathit{1}}\mathit{}and\mathit{}{Y}_{\mathit{2}}{X}_{\mathit{2}}\mathit{=}{C}_{\mathit{2}}$

If C_{1} = C_{2} =C, then we can write

${y}_{1}{x}_{1}={y}_{2}{x}_{2}=C\left(OR\right)\phantom{\rule{0ex}{0ex}}\frac{{y}_{1}}{{x}_{2}}=\frac{{y}_{2}}{{x}_{1}}=C\left(OR\right)\phantom{\rule{0ex}{0ex}}\frac{{y}_{\mathit{1}}}{{y}_{2}}\mathit{=}\frac{{x}_{\mathit{2}}}{{x}_{1}}\mathit{=}\mathit{}C$

**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

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

Hence, **T _{1}M_{1} = T_{2}M_{2}**

**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.

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

Hence,** E _{1}T_{1} = E_{2}T_{2}**

**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.

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

Hence, **E _{1}M_{1} = E_{2}M_{2}**

**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,

**W _{1}T_{2}=W_{2}T_{1}**

**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,

**W _{1}D_{2} =W_{2}D_{1}**

**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,** **

**W _{1}M_{2}=W_{2}M_{1}**

**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,** **

**W _{1}E_{2}=W_{2}E_{1}**

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

**W _{1}T_{2}D_{2}M_{2}E_{2 }= W_{2}T_{1}D_{1}M_{1}E_{1}**

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