Question

show that the following numbers are rational 0.325 where 25 has a bar​

Answer 1

Given:

A number -

• $\bf : \longmapsto 0.3\overline{25}$

What To Find:

We have to -

• Show the given number as a rational number [p/q].

Solution:

• First Method/Way:

Let's take -

$\bf : \longmapsto x = 0.3\overline{25}$

Can be written as,

$\bf : \longmapsto x = 0.3252525 \dots$

Multiply both sides with 10,

$\bf : \longmapsto 10x = 3.252525 \dots$

Multiply both sides with 1000,

$\bf : \longmapsto 1000x = 325.2525 \dots$

Here we have formed 2 equations -

• $\bf : \longmapsto 10x = 3.252525 \dots -[1^{st} eq.]$

• $\bf : \longmapsto 1000x = 325.2525 \dots - [2^{nd} eq.]$

Subtract the first eq. from second eq.,

$\bf : \longmapsto 1000x - 10x = 325.2525\dots \: - 3.252525 \dots$

Solve both sides,

$\bf : \longmapsto 990x = 322$

Take 990 to RHS,

$\bf : \longmapsto x = \dfrac{322}{990}$

Simplify the RHS,

$\bf : \longmapsto x = \dfrac{161}{495}$

∴ Thus, we have shown that $\bf 0.3\overline{25}$ is a rational number.

• Second Method/Way:

We know that -

$\bf :\longmapsto \dfrac{p}{q} \: form = \dfrac{Complete \: no. - No. \: formed \: by \: NRD}{No. \: of \: 9 \: as \: RD \: after \: that \: write \: as\: 0 \: as \: NRD \: after \: decimal}$

Where -

• $\bf :\longmapsto NRD = Non - Repe{a}ting \: Digits$

• $\bf :\longmapsto RD =Repe{a}ting \: Digits$

Substitute,

$\bf :\longmapsto \dfrac{p}{q} \: form = \dfrac{325 - 3}{990}$

Solve the numerator,

$\bf :\longmapsto \dfrac{p}{q} \: form = \dfrac{322}{990}$

Simplify the RHS,

$\bf : \longmapsto \dfrac{p}{q} \: form = \dfrac{161}{495}$

∴ Thus, we have shown that $\bf 0.3\overline{25}$ is a rational number.