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Question
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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Solution
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 − ๐ฅ) + ๐ฅ) = 8(๐ฅ(8 − ๐ฅ))
⇒ 15 [8 − ๐ฅ + ๐ฅ] = 8๐ฅ(8 − ๐ฅ)
⇒ 120 = 64x − 8x2
⇒ 8๐ฅ2 − 64๐ฅ + 120 = 0
⇒ 8[๐ฅ2 − 8๐ฅ + 15] = 0
⇒ ๐ฅ2 − 5๐ฅ − 3๐ฅ + 15 = 0
⇒ (๐ฅ − 5) (๐ฅ − 3) = 0
⇒ x = 5 or x = 3
∴ The two numbers are 5 and 3.
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