Express the following in the form `P/q` where p and q are integers and `q ne 0.` (i) 0.2 , (ii) 0.888…. , (iii) `5.bar2` (iv) `0. bar(001)` , ( v) 0.2555…. ,(vi) `0.1bar(34)` (vii) 0.00323232….. ,(vii) 0.404040…

Explanation

Let `x=0.2 =2/10 -1/5`
(ii) Let x=0.888…….
On multiplying both sides of Eq.(i) by 10 we, get
10 x. 8.888…….
On subtracting Eq. (i) from Eq. (ii),we get
10x-x=(8.88) - (0.888)
9x=8
`x=8/9`
(iii) Let `x=5.bar2=5.222.... `
On multiplying both sides of Eq. (i) by 10, we get
10x=52.222......
On subtracting Eq. (i) from Eq. (ii) , we get
10x-x=(52.222.....)-(5.222......)
9x =47
`x=47/9`
(v) Let `x=0.bar(001)`
`x=0.bar(001) = 0.001001`
On multiply both sides of Eq. (i) by 1000, we get
1000x=001.001.....
On subtrating Eq. (i) from Eq. (ii) we, get
`1000x-x=001.001 .... -(0.001001....)`
999x = 001
`x=1/999`
Let x= 0.2555......
On multplying both sides of Eq. (i) by 10, we get
10x=2.555....
On multiplying both sides of Eq. (ii) by 10, we get
100x=25.55......
On subtracting Eq. (ii) from Eq. (iii) , we get
100x-10x=25.55...-(2.555...)
90x=23
`x=23/90`
(vi) Let `x= bar(134)`
`x=0 bar(134)= 0.13434......`
On multiplying both sides of Eq. (i) by 10, we get
10x=13434.........
On multplying both sides of Eq. (ii) by 100 , we get
1000x= 134.3434.........
On subtracting Eq. (ii) from Eq. (iii) we, get
100x-10x= 134.34..... -(1.3434....)
990x=133
`x=133/990`
(vii) Let x= 0.00323232
On Multiplying both sides of Eq. (i) by 100 we get
100x = 0.3232......
On multiplying both sides of Eq. (ii) by 100 , we get
1000x=32.3232.........
On subtracting Eq. (ii) [from Eq. (iii) we get
1000x-100x=32.32 ......-0.3232.......
9900x=32
x=`32/9900 = 8/2475` [dividing numerator and denominator by 4]
(viii) Let x=0.404040.
On multiplying both sides of eq. (i) by 100, we get
100x = 40.4040
On subtracting Eq. (i) from (ii),we get
100x -x =40.4040 ...... - 0.404040.......
99x =40
`x= 40/99`