Question

express 0.66...+ 0.77...+0.477... in the form of p/q​

Answer 1

$\bf{\large{\underline{\underline{Answer:-}}}}$

p/q form of 0.66.. + 0.77.. + 0.477.. = 173/90

$\bf{\large{\underline{\underline{Explanation:-}}}}$

0.66... + 0.77.. + 0.477...

To add the given terms in the expression first express every term in p/q form.

Consider 0.66..

Let x = 0.66... --(1)

Here periodicity = 1

So, multiply eq(1) with 10

10 * x = 0.66.. * 10

10x = 6.66.... --(2)

Subtract (2) - (1)

10x = 6.666....

- x = 0.666....

________________

9x = 6.0

_________________

9x = 6

x = 6/9

So, 0.666... = 6/9

Consider 0.77...

Let x = 0.77... --(1)

Here periodicity = 1

So, multiply eq(1) with 10

10 * x = 0.77.. * 10

10x = 7.77.... --(2)

Subtract (2) - (1)

10x = 7.77....

- x = 0.77....

________________

9x = 7.0

_________________

9x = 7

x = 7/9

So, 0.77.... = 7/9

Consider 0.477...

Let x = 0.477... --(1)

Here periodicity = 1

So, multiply eq(1) with 10

10 * x = 0.477.. * 10

10x = 4.77.... --(2)

Subtract (2) - (1)

10x = 4.777....

- x = 0.477....

________________

9x = 4.3

_________________

9x = 4.3

x = 4.3/9

x = 43/90 [To eliminate decimal I multiplied both numerator and denominator by 10 i.e, 4.3 * 10 = 43, 9 * 10 = 90]

So, 0.477.. = 43/90

Now Coming addition of terms of expression (0.66.. + 0.77... + 0.477..)

$0.66... + 0.77.. + 0.477..$

We know that 0.77.. = 7/9 , 0.66... = 6/9, 0.477.. = 43/90

So substitute these rational numbers according to their decimal value to get the value of expression in p/q form.

$= \dfrac{6}{9} + \dfrac{7}{9} + \dfrac{49}{90}$

Here to add first the fractions should be proper fractions

To make them as proper fractions we need to know Least Commom Multiple of denominators

Least Common Factor of 9 and 90 = 90

Now multiply both numerators and denominators by a same number such that all denomiator should be same.

$= \dfrac{6 \times 10}{9 \times 10} + \dfrac{7 \times 10}{9 \times 10} + \dfrac{43 \times 1}{90 \times 1}$

$= \dfrac{60}{90} + \dfrac{70}{90} + \dfrac{43}{90}$

$= \dfrac{60 + 70 + 43}{90}$

$= \dfrac{130 + 43}{90}$

$= \dfrac{173}{90}$

So, p/q form of 0.66.. + 0.77.. + 0.477.. = 173/90

Answer 2

$\mathfrak{\underline{\underline{\green{Answer:-}}}}$

$\dfrac{173}{90}$

$\mathfrak{\underline{\underline{\green{Explanation:-}}}}$

Given:

0.66... + 0.77... + 0.477

To Find:

$\dfrac{p}{q}$ form of the sum of the given decimal forms

Solution:

First, let us find the $\dfrac{p}{q}$ form of the each term seperately-

p/q form of 0.66....

let, x = 0.66... ------(1)

As the periodictiy is one, so Multiply with '10' on both sides.

10 × x = 0.66... × 10

10x = 6.66... ------(2)

Now, eq. (2) - (1)

10x = 6.66

- x = 0.66

___________________

9x = 6

___________________

x = $\dfrac{6}{9}$

$\boxed {x = \dfrac{2}{3}}$

p/q form of 0.77...

let, x = 0.77... ------(3)

As the periodicity is one, so multiply with '10' on both sides

10 × y = 0.77... × 10

10y = 7.77... -------(4)

Now, eq. (4) - (3)

10y = 7.77...

- y = 0.77...

__________________

9y = 7

__________________

$\boxed {</strong><strong>y</strong><strong> = \dfrac{</strong><strong>7</strong><strong>}{</strong><strong>9</strong><strong>}}$

p/q form of 0.477...

let, z = 0.477... -------(5)

As the periodicity is one, so multiply with '10' on both sides

10 × z = 0.477... × 10

10z = 4.77.... --------(6)

Now, eq. (6) - (5)

10z = 4.77...

- z = 0.47...

____________________

9z = 4.3

____________________

By multiplying with 10 on both sides,

90z = 43

$\boxed {z = \dfrac{43}{90}}$

Let us, add the p/q form of the three terms

$= x+y+z$

$= \dfrac{2}{3} + \dfrac{7}{9}+ \dfrac{43}{90}$

LCM = 90

$= \dfrac{60+70+43}{90}$

$= \dfrac{173}{90}$

Hence,

p/q form of 0.66... + 0.77... + 0.477.. =

$</strong><strong> = \dfrac{173}{90}$