# The Sum Of Their Reciprocals Is 7/24, Can You Find Two Consecutive Even Integers?

The answer is 6 and 8.

The numbers are consecutive even numbers.  So, if one number is x then the other number, call it y, is x+2.  So, y = x + 2.

The reciprocals of x and y are 1/x and 1/y and the sum is equal 7/24.  So, 1/x + 1/y = 7/24.

Because you have two equations you can solve for the two unknown numbers.

We know
1/x + 1/y = 7/24

multiply both sides of the equation by x to get
1 + x/y = 7x/24

multiply both sides of the equation by y to get
y + x = 7xy/24

multiply both sides of the equation by 24 to get
24y + 24x = 7xy

we know y = x + 2, so change y to (x + 2) to get
24(x + 2) + 24x = 7x(x + 2)

use the distributive property of multiplication to get
24x + 48 +24x = 7x^2 + 14x

add the x terms together to simplify the equation to get
48x + 48 = 7x^2 + 14x

subtract 48x from both sides of the equation to get
48 = 7x^2 – 34x

subtract 48 from both sides of the equation to get
0 = 7x^2 – 34x - 48

this in the form of a quadratic equation which is ax^2 + bx + c = 0
the solution to a quadractic equation is
x = (-b + the square root of (b^2 – 4ac)) / 2a
or
x = (-b - the square root of (b^2 – 4ac)) / 2a
(which can be derived, but that will be the answer to another question)

in this case a = 7, b = -34, and c = -48

start with the first solution and substitute the values of a, b, and c to get
x = (-(-34) + the square root of (-34^2 - 4*7*(-48)) / 2*7

simplify the terms to get
x = (34 + the square root of (1156 + 1344)) / 14
or
x = (34 + the square root of (2500)) / 14

the square root of (2500) =  50 so
x = (34 + 50) / 14
x = 84/14
x = 6
and y = x + 2 = 6 + 2 = 8

you can try the second solution to see if it also works
x = (-(-34) - the square root of (-34^2 - 4*7*(-48)) / 2*7
x = (34 - 50) / 14
x = -16 / 14 = -8/7 which is not an even number, so this solution cannot satisfy the conditions of the stated problem

Check your work by substituting the values into the original equations
1/6 + 1/8 = 8/48 + 6/48 = 14/48 = 7/24
thanked the writer. 