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  • (i)Express the following in the form p by q , where p and q are integers and q is not equal to 0. 0.2

(i) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0. 0.2
(ii) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0.

0.888…….
(iii)  Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0.5.\bar{2}   
(iv) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0.0\cdot \overline{001}
(v) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0.0.2555……
(vii) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0..00323232…..
(viii) Express the following in the form \frac{p}{q}, where p and q are integers and q\neq 0..404040……..

 

 


       

 

 

Answers (1)

(i) Answer.     \frac{1}{5}
Solution.   We know that
0.2 can be written as \frac{2}{10}
Now,
  \frac{2}{10}= \frac{1}{5}
Hence the answer is \frac{1}{5}

(ii) Answer.    \frac{8}{9}
Solution.    Let x = 0.888…..        .…(i)
Multiply RHS and LHS by 10
10 x = 8.88…….         …(ii)
Subtracting equation (i) from (ii)
We get
10x – x = 8.8 – 0.8 
\Rightarrow 9x= 8
\Rightarrow x= \frac{8}{9}
Hence answer is  \frac{8}{9}

(iii) Answer.  \frac{47}{9}
Solution.  Let x = 5\cdot \bar{2}          …eq. (1)
Multiply by 10 on both sides
10x = 52\cdot \bar{2}                  …eq (2)

Subtracting equation (1) from (2)
We get
10x – x = 52\cdot \bar{2} – 5\cdot \bar{2}
\Rightarrow 9x = 47
\Rightarrow x = \frac{47}{9}
Hence the answer is \frac{47}{9}

(iv) Answer.   \frac{1}{999}
Solution.    Let x = 0\cdot \overline{001}              …. Eq. (1)
Multiply by 1000 on both sides
1000 x = 1\cdot \overline{001}            …eq.(2)
Subtracting eq. (1) from (2)
We get
1000 x – x = 1\cdot \overline{001} – 1\cdot \overline{001}

\Rightarrow 999x = 1
  \Rightarrow x = \frac{1}{999}
Hence the answer is \frac{1}{999} 

(v) Answer.   \frac{23}{90}

Solution. Let x = 0.2555 …..    …eq.(1)
Multiply by 10 on both sides
10x = 2.555…                         …eq.(2)
Multiply by 100 on both sides
100x = 25.55…                       …eq. (3)
Subtracting eq. (2) from (3),
We get
100x – 10x = 25.555… – 2.555…

\Rightarrow 90x = 23
\Rightarrow x = \frac{23}{90} 
  Hence the answer is \frac{23}{90}

(vii) Answer.    \frac{8}{2475}
Solution.   Let    x = 0.00323232…..      …eq.(1)
Multiply with 100 on both sides.
We get
100x = 0.3232…                …eq(2)
Multiply again with 100 on both sides,
10000x = 32.3232…                           …eq(3)
Equation (3) – (2)
we get,
10000 x – 100x = 32.3232… – 0.3232…
\Rightarrow  9900x = 32
 \Rightarrow x = \frac{32}{9900}
 x = \frac{8}{2475}
Hence the answer is  \frac{8}{2475}

(viii) Answer.     \frac{40}{99}
Solution. Let x = 0.404040…….           …(1)
Multiplying by 100 on both sides
we get
100x = 40.40…                       …(2)
Subtracting equation (1) from (2)
we get
100x – x = 40.40 – 0.40
\Rightarrow99x = 40
 \Rightarrow  x = \frac{40}{99} 
Hence the answer is \frac{40}{99}

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