Question

The difference of two numbers is 4 and the difference of their reciprocal is 4/21. Find the numbers

Answer 1

Just be seeing the denominator the answer is -7 and -3 or +7 and +3. Now for the analysis:

Let the two numbers be x and y. Then we have two equations as follows:

x - y = 4 … (1)

1/y-1/x=4/21 … (2).

From (2)

x - y = 4xy/21, or

21x - 21y = 4xy …(3). Multiply (1) by 21 to give

21x - 21y = 84 …(1a)

Subtract (1a) and (3) to get

4xy - 84 = 0, or

x = 21/y

Eq. (1) becomes 21/y - y = 4, or

21 - y^2 = 4y, or

y^2 +4y -21 = 0

(y+7)(y-3) = 0

Hence from Eq.(1), y = -7 and x = -3 or

y = +3 and x = +7.

Check: x-y = -3-(-7) = 4.

x-y = 7- 3 = 4