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The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.
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Let the numbers be x and 8 - x
Given that the sum of these numbers is 8
And 15 times the sum of their reciprocals as 8
`rArr15(1/x+1/(8-x))=8`
`rArr15(((8-x)+x)/(x(8-x)))=8`
⇒ 15 ((8 - ЁЭСе) + ЁЭСе) = 8(ЁЭСе(8 - ЁЭСе))
⇒ 15 [8 - ЁЭСе + ЁЭСе] = 8ЁЭСе(8 - ЁЭСе)
⇒ 120 = 64x - 8x2
⇒ 8ЁЭСе2 - 64ЁЭСе + 120 = 0
⇒ 8[ЁЭСе2 - 8ЁЭСе + 15] = 0
⇒ ЁЭСе2 - 5ЁЭСе - 3ЁЭСе + 15 = 0
⇒ (ЁЭСе - 5)(ЁЭСе - 3) = 0
⇒ x = 5 or x = 3
∴ The two numbers are 5 and 3.
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