$0. 4 ¯$ as a rational number in simplest form is ____.

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Detailed Solution

0. 4 ¯

as a rational number in simplest form is

49

.
To express

0. 4 ¯

in the form of

p q

, assume this number as, say x, and it can be written in the expanded form as follows,

x = 0 ⋅ 444444.......

... (i)
So, it can be observed that the repeating part of the decimal has one digit.
So, let us first shift it by one place towards the left of the decimal, by multiplying each side of equation (1) by 10 as follows,

10 x = 4 ⋅ 44444.....

... (ii)
Now, to get the required simplest form of the given decimal, we subtract equation (ii) from (i) and simplify as follows,

10 x − x = 4.44444..... − 0.44444..... 9 x = 4 x = 49

From the above result, we can conclude that,

0. 4 ¯ = 49

Hence, the simplest form of

0. 4 ¯

49

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