Maths-
General
Easy
Question
Convert the following repeating decimals to fractions
a) 1.233333333333333333
Hint:
Perform necessary multiplications to make it as a fraction.
The correct answer is: 37/30
Complete step by step solution:
(a) Let x = 0.23333333…(i)
Now multiply by 10 on both the sides in (i),
We get, 10x = 2.3333333
Now multiply by 100 on both the sides in (i),
100x = 23.3333333
On subtracting 10x from both the sides, we have
![table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 100 x minus 10 x equals 23.3333 minus 2.3333333 end cell row cell not stretchy rightwards double arrow 90 x equals 21 end cell row cell not stretchy rightwards double arrow x equals 21 over 90 end cell end table](data:image/png;base64,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)
On simplification, we get ![x equals 7 over 30](data:image/png;base64,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)
Now we have a fraction for 0.2333333…. On adding it with 1, we have
![not stretchy rightwards double arrow 1 plus 7 over 30](data:image/png;base64,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)
as fraction
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