Question

x = 1/12.13 + 1/13.14 +1/14.15 ......23.24 . Y= 1/36.37+1/37.38 + 1/38.39....1/71.72 then x/y ?
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Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{x = \frac{1}{12.13} + \frac{1}{13.14} + \frac{1}{14.15} + .. + \frac{1}{36.37} }$

$\displaystyle \sf{y = \frac{1}{36.37} + \frac{1}{37.38} + \frac{1}{38.39} + .. + \frac{1}{71.72} }$

TO DETERMINE

$\displaystyle \sf{ \frac{x}{y} }$

EVALUATION

$\displaystyle \sf{x = \frac{1}{12.13} + \frac{1}{13.14} + \frac{1}{14.15} + .. + \frac{1}{36.37} }$

$\displaystyle \sf{ \implies \: x = \frac{13 - 12}{12.13} + \frac{14 - 13}{13.14} + \frac{15 - 14}{14.15} + .. + \frac{37 - 36}{36.37} }$

$\displaystyle \sf{ \implies \: x = \frac{1}{12} - \frac{1}{13} + \frac{1}{13} - \frac{1}{14} + .. + \frac{1}{23} - \frac{1}{24} }$

$\displaystyle \sf{ \implies \: x = \frac{1}{12} - \frac{1}{24} }$

$\displaystyle \sf{ \implies \: x = \frac{24 - 12}{12.24} }$

$\displaystyle \sf{ \implies \: x = \frac{12}{12.24} }$

$\displaystyle \sf{ \implies \: x = \frac{1}{24} }$

Again

$\displaystyle \sf{y = \frac{1}{36.37} + \frac{1}{37.38} + \frac{1}{38.39} + .. + \frac{1}{71.72} }$

$\displaystyle \sf{ \implies \: y = \frac{37 - 36}{36.37} + \frac{38 - 37}{37.38} + \frac{39 - 38}{38.39} + .. + \frac{72 - 71}{71.72} }$

$\displaystyle \sf{ \implies \: y = \frac{1}{36} - \frac{1}{37} + \frac{1}{38} - \frac{1}{39} + .. + \frac{1}{71} - \frac{1}{72} }$

$\displaystyle \sf{ \implies \: y = \frac{1}{36} - \frac{1}{72} }$

$\displaystyle \sf{ \implies \: y = \frac{72 - 36}{36.72} }$

$\displaystyle \sf{ \implies \: y = \frac{ 36}{36.72} }$

$\displaystyle \sf{ \implies \: y = \frac{1}{72} }$

Hence

$\displaystyle \sf{ \frac{x}{y} }$

$\displaystyle \sf{ = \frac{ \frac{1}{24} }{ \frac{1}{72} } }$

$\displaystyle \sf{ = \frac{1}{24} \times \frac{72}{1} }$

$\displaystyle \sf{ = 3 }$

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