What is the smallest possible value of a + bIf a b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, what is the smallest possible value of a + b?I don't see any strategies here for solving this problem, any help? Thanks in advance!

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What is the smallest possible value of   a  +  bIf       a    b   rounded to the nearest trillionth is   0.008012018027, where   a and   b are positive integers, what is the smallest possible value of   a  +  b?I don&#039;t see any strategies here for solving this problem, any help? Thanks in advance!

rideonthebussp · Accepted Answer

The continued fraction representations of the limits of the interval are  0.0080120180265  =  [  0  ;  124  ,  1  ,  4  ,  2  ,  1  ,  463872  ,  1  ,  1  ,  12  ,  1  ,  1  ,  41  ]    0.0080120180275  =  [  0  ;  124  ,  1  ,  4  ,  3  ,  545777  ,  2  ,  13  ,  1  ,  1  ,  1  ,  1  ,  2  ]The simplest continued fraction (and therefore also the simplest ordinary fraction!) in that interval is  [  0  ;  124  ,  1  ,  4  ,  3  ]  =      16    1997    =  0.00801201802704056084  …and the sum of its numerator and denominator is   2013(I used Wolfram Alpha to expand the continued fractions fully. For a pencil-and-paper solution one only needs to carry out the expansion until they start differing, which requires only a handful of long divisions with remainder.)

hawwend8u · Accepted Answer

1    0.008012018027    =  124.8125000006    0.8125  =      13    16      124.8125  =      1997    16      a  +  b  =  1997  +  16  =  2013

What is the smallest possible value of a + b If a b </mfrac> rounde

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