Suppose the first number is x

Since sum of two numbers = 8

=> Other number = (8 - x)

We know that for any number x,





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

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

=> 8/(8x - x^2) = 8/15

=> 8x - x^2 = 15

=> x^2 - 8x + 15 = 0

=> x^2 - 5x - 3x + 15 = 0

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

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

=> x = 3 or x = 5

=> 8 - x = 8 - 3 = 5 (if x = 3)
=> 8 - x = 8 - 5 = 3 (if x = 5)

Problem Statement: The sum of two numbers is 8. Determine the numbers, if the sum of their reciprocals is 8/15.

Solution:

Let's assume that the two numbers are x and y. Then we have:

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

We need to solve these two equations to find the values of x and y.

Step 1: Simplify the second equation

To solve the second equation, we need to simplify it. We can do this by finding the common denominator (xy) and then multiplying both sides by xy. This gives us:

- y/x + x/y = 8/15
- (y^2 + x^2) / xy = 8/15
- 15(y^2 + x^2) = 8xy

Step 2: Use the first equation to eliminate one variable

We can eliminate one variable (y) by substituting it with (8 - x) from the first equation:

- y = 8 - x

Substituting this in the second equation gives us:

- 15((8-x)^2 + x^2) = 8x(8-x)
- 15(64 - 16x + x^2 + x^2) = 64x - 8x^2
- 15(2x^2 - 16x + 64) = 8x(8-x)
- 30x^2 - 240x + 960 = 64x^2 - 8x^3
- 8x^3 - 34x^2 + 240x - 960 = 0

Step 3: Solve for x

We can solve this cubic equation using various methods (such as synthetic division or using a calculator). The solutions are x = 2, x = 6, and x = 10.

Step 4: Find the values of y

We can find the values of y by using the first equation:

- x + y = 8

Substituting the values of x, we get:

- If x = 2, then y = 6
- If x = 6, then y = 2
- If x = 10, then y = -2 (which is not a valid solution)

Step 5: Check the solutions

We need to check if the solutions satisfy both equations. For example:

- If x = 2 and y = 6, then:

- 1/2 + 1/6 = 8/15 (satisfied)
- 2 + 6 = 8 (satisfied)

Therefore, the solution is x = 2 and y = 6.

In conclusion, the two numbers are 2 and 6, and their reciprocals add up to 8/15.