the-difference-of-two-number-is-3-if-the-sum-of-their-reciprocal-is-7-10-find-the-numbres-

Question Number 10157 by ketto last updated on 27/Jan/17

$${the}\:{difference}\:{of}\:{two}\:{number}\:{is}\:\mathrm{3}. \\ $$$${if}\:{the}\:{sum}\:{of}\:{their}\:{reciprocal}\:{is}\: \\ $$$$\frac{\mathrm{7}}{\mathrm{10}}\:.\:{find}\:{the}\:{numbres} \\ $$

Answered by FilupSmith last updated on 28/Jan/17

a−b=3 ⇒ a=3+b (1/a)+(1/b)=(7/(10)) ((a+b)/(ab))=(7/(10)) ((3+b+b)/((3+b)b))=(7/(10)) ((3+2b)/(3b+b^2 ))=(7/(10)) 10(3+2b)=7(3b+b^2 ) 30+20b=21b+7b^2 7b^2 +b−30=0 b=((−1±(√(1+840)))/(14)) b=((−1±29)/(14)) b = 2, −((15)/7) a=3+b ∴ a=5, (3−((15)/7))

$${a}−{b}=\mathrm{3}\:\:\:\:\Rightarrow\:\:\:\:{a}=\mathrm{3}+{b} \\ $$$$\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}=\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\frac{{a}+{b}}{{ab}}=\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\frac{\mathrm{3}+{b}+{b}}{\left(\mathrm{3}+{b}\right){b}}=\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\frac{\mathrm{3}+\mathrm{2}{b}}{\mathrm{3}{b}+{b}^{\mathrm{2}} }=\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\mathrm{10}\left(\mathrm{3}+\mathrm{2}{b}\right)=\mathrm{7}\left(\mathrm{3}{b}+{b}^{\mathrm{2}} \right) \\ $$$$\mathrm{30}+\mathrm{20}{b}=\mathrm{21}{b}+\mathrm{7}{b}^{\mathrm{2}} \\ $$$$\mathrm{7}{b}^{\mathrm{2}} +{b}−\mathrm{30}=\mathrm{0} \\ $$$${b}=\frac{−\mathrm{1}\pm\sqrt{\mathrm{1}+\mathrm{840}}}{\mathrm{14}} \\ $$$${b}=\frac{−\mathrm{1}\pm\mathrm{29}}{\mathrm{14}} \\ $$$${b}\:=\:\:\:\:\mathrm{2},\:\:\:\:−\frac{\mathrm{15}}{\mathrm{7}} \\ $$$$\: \\ $$$${a}=\mathrm{3}+{b} \\ $$$$\therefore\:{a}=\mathrm{5},\:\left(\mathrm{3}−\frac{\mathrm{15}}{\mathrm{7}}\right) \\ $$

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