Question

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

Answer 1

SOLUTION :

Given : The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8.  

Let the numbers x and (8 - x) .

A.T.Q

15(1/x + 1/(8 - x) ) = 8

15[(8 - x)  + x]/ x(8 -x) = 8

[By taking LCM]

15[(8 - x)  + x]/ (8x - x²) = 8

15(8) = 8(8x - x²)  

120 = 64x - 8x²

8x² - 64x +120 = 0

8(x² - 8x + 15) = 0  

x² - 8x + 15 = 0  

x² - 5x  - 3x + 15 = 0  

[By middle term splitting method]

x(x - 5) - 3(x - 5) = 0

(x - 5) (x - 3) = 0

(x - 5)  = 0  or (x - 3) = 0

x = 5  or  x = 3

Hence, the two numbers are 5 & 3 .

HOPE THIS  ANSWER WILL HELP YOU..

Answer 2

Answer is ---

3, 5

Explanation:

Let's call the two numbers x and y.

We're told that x+y=8

We're also told that 15 times the sum of their reciprocal is also 8. I'll interpret what this says this way:

15(1x+1y)=8

We have two equations and two variables, so we should be able to solve this. Let's first solve the first equation for x:

x=8−y

And now substitute into the second equation:

15(18−y+1y)=8

18−y+1y=815

18−y(yy)+1y(8−y8−y)=815

yy(8−y)+8−yy(8−y)=815

8y(8−y)=815

Notice that with the numerators equal, we can say:

y(8−y)=15

8y−y2=15

y2−8y+15=0

(y−3)(y−5)=0⇒y=3,5

And by substituting these values back into our first equation, we get that x=5,3

Now let's check our answer:

15(1x+1y)=8

15(13+15)=8

15(515+315)=8

15(815)=8

8=8

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