creative thought-16

Difference of two numbers equals to its product

The difference between two numbers may be equal to its product.

Mathematically it is expressed as

a-b = d and ab = d

hence a – b = ab, where a > b .The dependency between these two variables becomes

a = b/(1-b) ; b = a/(a+l)

It shows that for ‘a’ to be positive, ’b’ must be less than unity but

greater than zero ,that is ‘b’ must be a fraction. ‘a’ may be a whole

number or a fraction ,but ‘b’ will always be a fraction. If ‘b’ is given integral value, ‘a’

becomes negative. Thus there is no solution with whole integral value for both ‘a’and ‘b’.

When ‘a’ takes a whole integral value n , ‘b’ becomes a fraction

n/(n+1) and when ‘a’ takes a fractional value x/y, ‘b’ becomes

x/(x+y).Few typical solutions are given in Table.1.

a is an integer a is a frac tion

a b a b

1 ½ 1/3 ¼

2 2/3 2/5 2/7

3 ¾ 3/8 3/11

… … … ….

n n/(n+1) N/n N/(N+n)

The pair of numbers whose difference d and product are equal can be related to d.

By eliminating one of the dependent variables (either ‘a’ or ‘b’) ,one can obtain

a quadratic equation

(axa) – d a – d = 0 . The positive root of the equation is

a = [d + √ (dxd) + 4d ]/2

which gives,

b = a – d = [ -d + √ (dxd) +4d]/2

The positive values of the pair of numbers under the given condition for a given d

are shown in Table.2.

d a or b

6 √ 15 ± 3

8 √ 24 ± 4

10 √ 35 ± 5

…. …….

2n √ n(n+2) ± n

Under the given condition ,the fractional pair of numbers may be with numerator or

the denominator identical. When the numerators are same, the pair of numbers is

assumed as x/b and x/a so that

(x/b) – (x/a) = (x/b).(x/a)

It gives an additional condition x = a-b. Thus the pair becomes [(a-b)/b, (a-b)/a].

The pair of numbers whose denominators are same can be derived directly from

the pair of numbers whose numerators are identical. By multiplying both the numerator

and the denominator of a fractional pair by the denominator of the other same . e.g.,

a b a-b/b a-b/a a(a-b)/ab b(a-b)/ab

7 3 4/3 4/7 28/21 12/21

3 5 2/3 2/5 10/15 6/15

7 11 4/7 4/11 44/77 28/77

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