"Spoooooooky" Russian rational expressions 5

Algebra Level 1

E = ( 2 m + 1 2 m − 1 − 2 m − 1 2 m + 1 ) ÷ 4 m 10 m − 5 \mathscr{E} = \left( \dfrac{2m+1}{2m-1} - \dfrac{2m-1}{2m+1}\right) \div \dfrac{4m}{10m-5}

Let m = 2016 m = 2016 . If the value of E \mathscr{E} can be represented in the form a b \dfrac{a}{b} , where a a and b b are coprime positive integers . Find a + b a+b .

Credit: My former Trig teacher's worksheet of Russian rational expressions


The answer is 4043.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

⇒ ( 2 m + 1 2 m − 1 − 2 m − 1 2 m + 1 ) ÷ 4 m 10 m − 5 \Rightarrow \left( \dfrac{2m+1}{2m-1} - \dfrac{2m-1}{2m+1}\right) \div \dfrac{4m}{10m-5}

= ( 2 m + 1 ) 2 − ( 2 m − 1 ) 2 ( 2 m + 1 ) ( 2 m − 1 ) × 10 m − 5 4 m =\dfrac{(2m+1)^2-(2m-1)^2}{(2m+1)(2m-1)}×\dfrac{10m-5}{4m}

= 4 m 2 + 1 + 4 m − 4 m 2 − 1 + 4 m ( 2 m + 1 ) ( 2 m − 1 ) × 5 ( 2 m − 1 ) 4 m = \dfrac{4m^2+1+4m-4m^2-1+4m}{(2m+1)(2m-1)}×\dfrac{5(2m-1)}{4m}

= 2 2 m + 1 × 5 =\dfrac{2}{2m+1}×5

= 10 2 m + 1 =\dfrac{10}{2m+1}

Pluging m = 2016 m=2016 .

= 10 4033 =\dfrac{10}{4033}

∴ a + b = 4043 \therefore a+b=\boxed{4043}

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