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

Question

Thanks for the helps this confuses me so much!

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