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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. - Mathematics

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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.

Sum
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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 − ๐‘ฅ) + ๐‘ฅ) = 8(๐‘ฅ(8 − ๐‘ฅ))

⇒ 15 [8 − ๐‘ฅ + ๐‘ฅ] = 8๐‘ฅ(8 − ๐‘ฅ)

⇒ 120 = 64x − 8x2

⇒ 8๐‘ฅ2 − 64๐‘ฅ + 120 = 0

⇒ 8[๐‘ฅ2 − 8๐‘ฅ + 15] = 0

⇒ ๐‘ฅ2 − 5๐‘ฅ − 3๐‘ฅ + 15 = 0

⇒ (๐‘ฅ − 5) (๐‘ฅ − 3) = 0

⇒ x = 5 or x = 3

∴ The two numbers are 5 and 3.

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Chapter 4: Quadratic Equations - Exercise 4.7 [Page 51]

APPEARS IN

RD Sharma Mathematics [English] Class 10
Chapter 4 Quadratic Equations
Exercise 4.7 | Q 9 | Page 51
Nootan Mathematics [English] Class 10 ICSE
Chapter 5 Quadratic equations
Exercise 5E | Q 4. | Page 91

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