The sum of two natural numbers is 8. determine the numbers, if the sum of their reciprocals is

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

Thus, the two numbers are 3 and 5 respectively.

Answer

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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

Given Data:

Sum of two natural number is 8.

The sum of their reciprocals is 8/15.

Concept Used:

(a + b)2 = (a – b)2 + 4ab

Calculation:

Let the number be a and b.

a + b = 8  …(i)

1/a + 1/b = 8/15

⇒ (b + a)/ab = 8/15

⇒ 8/ab = 8/15

⇒ ab = 15

(a + b)2 = (a – b)2 + 4ab

⇒ 82 = (a – b)2 + 4 × 15

⇒ 64 = (a – b)2 + 60

⇒ (a – b)2 = 4

⇒ a – b = 2    ….(ii)

Adding (i) and (ii), 2a = 10

⇒ a = 5 and b = 3

∴ The two natural numbers are 5 and 3.

Shortcut:

Directly we can get the value from option 4, as a + b = 8 and ab = 15.

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