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ICSE Class X Math 2015 Paper

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Question : 50 of 52
Marks: +1, -0
Sum of two natural numbers is 8 and the difference of their reciprocal is   215\;\frac{2}{15}. Find the numbers.
Solution:  
Let 1 st no. be xx and 2 nd be 8−x8-x.
By the given condition,
    1x−  18−x=  215\;\; \frac{1}{x}-\; \frac{1}{8-x}=\; \frac{2}{15}
  ⇒      8−x−xx(8−x)=  215\; \Rightarrow \;\; \; \frac{8-x-x}{x(8-x)} = \; \frac{2}{15}
  ⇒      8−2x8x−x2=  215\; \Rightarrow \;\; \; \frac{8-2x}{8x-x^2} = \; \frac{2}{15}
⇒    120−30x  =16x−2x2\Rightarrow \;\; 120-30 x \;= 16 x-2 x^2
2x2−46x+120  =02 x^2-46 x+120\;=0
x2−23x+60  =0x^2-23 x+60\;=0
x2−20x−3x+60  =0x^2-20 x-3 x+60\;=0
x(x−20)−3(x−20)  =0x(x-20)-3(x-20)\;=0
(x−3)(x−20)  =0(x-3)(x-20)\;=0
x=3   or   x  =20   (not possible)   x=3 \; \text{ or } \; x \;=20 \; \text{ (not possible) } \;
1  st      no.   =31^{\;\text{st }\;} \; \text{ no. } \;=3    and   2  nd      no.     =8−3=5.\; \text{ and } \; 2^{\;\text{nd }\;} \; \text{ no. } \;\;=8-3=5 .
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