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The sum of two numbers is 8 determine the numbers if the sum of their reciprocal is 8 by 15
Solution
Let the two natural numbers be x and y
X+y=8
X=8-y ---(Equation-1)
1/x+1/y=8/15 ---(Equation -2)
We get,
1/8-y+1/y=8/15
Y+8-y/-y2+8y=8/15
8/-y2+8y=8/15
120=-8y2+64y
-8y2+64y=120
-8y2+64y-120=0
8y2-64y+120=0
y2-8y+15=0
y2-5y-3y+15=0
(sum=-8,Product=15)
y(y-5)-3(y-5)=0
y-5=0 (or)y-3=0
y=5 (or) y=3
If y=5 then x=8-5
X=3
If y=3 then x=8-3
X=5
Therefore the two natural numbers are 3and 5
0
Let the two numbers be x and y respectively.Then,x + y = 8 ….(i)⇒ x = 8 - yAnd,
`1/x + 1/y = 8/15`
⇒ `[ y + x ]/[xy] = 8/15`
⇒ `8/(xy) = 8/15` .....[ From(1) ]⇒ xy = 15⇒ ( 8 - y )y = 15
⇒ 8y - y2 = 15
⇒ y2 - 8y + 15 = 0
⇒ y2 - 3y - 5y + 15 = 0⇒ y( y - 3 ) - 5( y - 3 ) = 0⇒ ( y - 3 )( y - 5 ) = 0⇒ y = 3 or y = 5⇒ x = 5 or x = 3
Thus, the two numbers are 3 and 5 respectively.