Infinite series
0 . 7 1 2 7 1 2 7 1 2 7 1 2 7 1 2 … If we convert the non-terminating decimal number above in fraction, b a , where a and b are coprime positive integers, calculate the value of a + b .
The answer is 1711.
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6 solutions
Waw bro very nice solution
However i solved it by converting the number into an infinite series and then calculated limit of sum when number of terms approch to infinity
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Yeah Even I Did it the way.. @Callum Maddox did..
taught to me when we encountered our first c h a p t e r in M a t h s back in Cass IX
BTW @Aman Sharma Why don't you try converting 0 . 7 2 8 2 in to b a and find a + b . The result is really ...
Arya :)
why do you subtract and x from 1000x?
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Reason behind subtraction is to get a rational number (a/b form) from irrational number.
l e t m = ( x = 0 . 7 1 2 )
l e t n = ( 1 0 0 0 x = 7 1 2 . 7 1 2 )
n − m :
9 9 9 x = 7 1 2
x = 9 9 9 7 1 2
a n s = 7 1 2 + 9 9 9 = 1 7 1 1
712 is not a prime
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That is correct but the question said "coprime positive integers." This just means that they are prime in relation to each other. Essentially, just that the only factor the numbers share is 1, which is indeed the case with 712 and 999.
Let x = b a .Then: x = 0 . 7 1 2 7 1 2 7 1 2 ⋯ Multiplying both sides of the equation by 1 0 0 0 ,we get: 1 0 0 0 x = 7 1 2 . 7 1 2 7 1 2 7 1 2 ⋯ Now here,s the trick: Subtract the first equation from the second equation: 1 0 0 0 x − x = 7 1 2 . 7 1 2 7 1 2 7 1 2 ⋯ − 0 . 7 1 2 7 1 2 7 1 2 ⋯ Observe the the infinitely repeating 7 1 2 gets cancelled out and we are left with: 9 9 9 x = 7 1 2 → x = b a = 9 9 9 7 1 2 So a = 7 1 2 , b = 9 9 9 and a + b = 7 1 2 + 9 9 9 = 1 7 1 1
The easiest solution to this problem involves a bit of prior knowledge. As it turns out, fractions have a property where you can put any number over the same quantity of "9"s and that expression will equate to a repeating decimal of the top number. For example 23/99 = 0.23232323... Or 123/999 = 0.123123123123...
Furthermore, coprime simply means they are prime in relation to each other or in other words the only factor they share is 1.
Knowing this, figuring out the values of a and b is as easy as taking the repeating decimal "712" and putting it over 999. This does indeed equate to 0.712712712712.... And the numbers 712 and 999 are coprime as they will not reduce to be a simpler fraction.
The final step is just adding a + b or 712 + 999 which equals 1711.
Let x/y=.712712712……which is 7 ̇1 ̇2 ̇ that is 712/999 Hence x=712 and y=999 then x+y=712+999=1711
take the right hand side and let it be x. 1000x=712.712712712712. If you then subtract x from 1000x you end up with 999x=712. Therefore 712/999=x. This means a/b=712/999. Since this is the most simplified fraction a+b=712+999=1711. This rule can apply to integers of any number of digits. the number of digits/the same number of 9 will equal a recurring decimal of the first digits. eg 23/99=0.2323232323....