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Q1. Show that 0.235353535 can be expressed in the form p/q, where p and q are integers and q is not equal to zero.

Solution.

Let x=0.235353535. Over here, note that 2 does not repeat, but the block 35 repeats.Since two digits are repeating, we multiply x by 100 to get

100x = 23.535353...

100x = 23.3 + 0.23535353... = 23.3 + x

99x = 23.3

99x = 233/10, which gives x = 233/990

so, the form p/q is x = 233/990
Solution

Q2. Express 1.4393939... in the form p/q, where p and q are integers and q is not equal to zero.

Solution.

Let x = 1.4393939.

Multiplying both sides by 10, we get

10x = 14.393939 ------------------(1)

Here, two digits are repeated continuously, therefore again multiplying both sides by 100, we get
1000x = 1439.393939 --------------(2)

Now, subtract equation 2 from equ. 1

1000x - 10x = 1425

990x = 1425

x=1425/990 x=285/198

Solution

Q3. Express 48.484848... in the form p/q, where p and q are integers and q is not equal to zero.

Solution.

Let x = 48.484848. -----------(1)

Here two digits are repeating, so multipying both sides by 100,

we get,

x = 48.484848

100x = 4848.484848 ------------(2)

subtracting equ. (1) from equ. (2), we get

100x - x = 4848.484848-48.484848

99x = 4800

x = 4800/99

x = 1600/33

Solution

Q4. Express 9.32222...+ 0.454545..... as a fraction in simplest form.

Solution.

Let x = 9.322.... and y = 0.454545....
(i) Consider x = 9.322.... = 9.3222--------(1)
multiply equ.,(1) by 10(because one digit is not repeated)
10x = 93.2222--------(2)
again multiply equation (2) by 10 (because one digit is repeat)
100x = 932.222--------(3)
Now, subtract equation (2) by equation (3)
100x - 10x = 839
90x = 839
x = 839/90
y = 0.4545-----(1)
two digit are repeating. So, multiplied both side by 100.
100y = 45.4545------(2)
Now subtract equation (1) by equation (2)
100y = 45.454545
y = 0.454545
99y = 45
y = 45/99
Therefore,x + y = 9.32222.... + 0.454544...
= 839/90 + 45/99
= 839/90 + 5/11
= 9679/990

Solution

Q5. Express 12.383838....+ 0.453333... as a fraction in simplest form.

Solution.

Let x = 12.383838.... and y = o.4533333....
(i) Consider x = 12.383838---------(1)
multiply equation (1) by 100(becausee two digit are repeating)
100x = 1238.3838---------(2)
Subtract equation (2) by equation (1)
100x = 1238.383838
x = 12.383838
99x = 1226
x =1226/99
(ii) Consider y = 0.453333--------(1).
multiply equation (1) by 100(because two digit are non repeating)
100y = 45.333------(2)
again muliply equation (2) by 10 (because one digit is repeating)
1000y = 453.333------(3)
Now, subtract equation (3) by equation (2)
1000y = 453.3333
100y = 45.3333
900y = 408
y = 408/900 , = 136/300
= 68/150 , = 34/75
Therefore, x + y = 12.383838 + 0.453333
= 1226/99 + 34/75

Solution