Question

The sum of two numbers is 10. The su of their reciprocals is 5/12. What are the numbers?

In my answer sheet it says the answers are 4 and 6.

Answer 1

Note: A reciprocal is the number turned over...

The reciprocal of 2/3 = 3/2, and so forth

Since x = x/1, the reciprocal of x is 1/x....

The sum of 2 numbers is 10:

Equation 1) x + y = 10

The sum of their reciprocals is 5/12:

Equation 2) 1/x + 1/y = 5/12

From equation 1 we can solve for x in terms of y

(or vice versa...)

x = 10-y

Plug 10-y in place of x in equation 2 and solve for y

1/(10-y) + 1/y = 5/12

We need to find a common denominator on the left.

That would be y(10-y)

(y/y)(1/(10-y)) + [(10-y)/(10-y)](1/y)

= y/(y(10-y)) + (10-y)/(y(10-y))

= (y+10-y)/(y(10-y))

= 10/(10y-y2)

This is the left side of the equal sign combined

to a single fraction. It is still equal to 5/12

10/(10y-y2) = 5/12

Cross multiply:

10(12) = 5(10y-y2)

120 = 50y - 5y2

Move everything to the left side to create a quadratic by

adding 5y2 and subtracting 50y from both sides

5y2 - 50y +120 = 0

Note that each term in the polynomial is divisible by 5

so factor out a 5

5(y2 - 10y + 24) = 0

Divide both sides by 5

y2 - 10y + 24 = 0

The factors of 24 that add to -10 are (-6)(-4) so this factors to:

(y-6)(y-4) = 0

Thus either y-6 = 0 and y=6

OR

y-4 = 0 and y=4

If y = 4 then x=6

If y = 6 then x=4

Your numbers are 4, 6

Check Answer: x+y=10 yes

1/x + 1/y = 5/12

1/6 + 1/4 = 5/12

2/12 + 3/12 = 5/12

5/12 = 5/12

Answer checks as correct

Explanation:

Answer 2

The numbers are 4 and 6.

Explanation:

It helps to start with the right equation. The one you wrote is the difference of the reciprocals. The problem statement tells you their sum is 5/12.

1/x +1/y = 5/12

Multiply by 12xy to clear fractions:

12y +12x = 5xy

12(x+y) = 5xy

Now, you can substitute 10 for x+y, and (10-x) for y. This results in a quadratic equation that can be solved by any of the usual methods.

12(10) = 5x(10-x)

Divide by 5

24 = x(10 -x)

This can be rearranged to ...

x^2 -10x = -24

Add 25 to complete the square:

x^2 -10x +25 = 1

(x -5)^2 = 1

Take the square root:

x -5 = ±√1 = ±1

x = 5 ± 1 . . . . . . . . add 5

The numbers are 4 and 6 or 6 and 4. Either way, one of them is 4 and the other is 6.