Question

a,b,c are positive numbers such that a+b+ab= 8, b+c+bc=15 and c+a+ca = 35
What is the value of a+b+c+abc?

Answer 1

HELLO DEAR,

GIVEN:-

a + b + ab = 8

$\Rightarrow$ b + a(1 + b) = 8

$\bold{\Rightarrow a = \frac{8 - b}{1 + b}}$-------( 1 )

b + c + bc = 15

$\Rightarrow$ b + c(1 + b) = 15

$\bold{\Rightarrow c = \frac{15 - b}{1 + b}}$ ------( 2 )

a + c + ac = 35

$\bold{\Rightarrow \frac{8 - b}{1 + b} + \frac{15 - b}{1 + b} + \frac{8 - b}{1 + b}\times \frac{15 - b}{1 + b} = 35}$

From----------( 1 ) &-----------( 2 )

$\bold{\Rightarrow \frac{8 - b + 15 - b}{1 + b} + \frac{120 - 8b - 15b + b^2}{(1 + b)^2} = 35}$

$\Rightarrow$ $\bold{(23 - 2b)(1 + b) + (120 - 23b + b^2) = 35(1 + b)^2}$

$\Rightarrow$ $\bold{23 + 23b - 2b - 2b^2 + 120 - 23b + b^2 = 35 + 35b^2 + 70b}$

$\Rightarrow$ $\bold{143 - 35 - 2b - b^2 = 35b^2 +  70b}$

$\Rightarrow$ $\bold{108 - 72b - 36b^2 = 0}$

$\Rightarrow$ $\bold{b^2 + 2b - 3 = 0}$

$\Rightarrow$ $\bold{b^2 + 3b - b - 3 = 0}$

$\Rightarrow$ $\bold{b(b + 3) - 1(b + 3)}$

$\Rightarrow$ $\bold{(b + 3)(b - 1)}$

so, b = 1 , b = -3[neglect]

because a , b , c ,are positive,

now,

from----------( 1 )

a = (8 - b)(1 + b)

$\Rightarrow$ a = (8 - 1)(1 + 1)

$\Rightarrow$ a = 7/2

from-----------( 2 )

c = (15 - b)(1 + b)

$\Rightarrow$ c = (15 - 1)(1 + 1)

$\Rightarrow$ c = 14/2

$\Rightarrow$ c = 7

thus, the value of "a + b + c + abc" is

$\Rightarrow$ 7/2 + 1 + 7 + (7 * 7/2 * 1)

$\Rightarrow$ {(7 + 2 + 14 )/2 + 49/2}

$\Rightarrow$  (23/2 + 49/2)

$\Rightarrow$ (23 + 49)/2

$\Rightarrow$ 72/2

$\Rightarrow$ 36

HENCE, the value of (a + b + c + abc) = 36

I HOPE ITS HELP YOU DEAR,

THANKS

Answer 2

Given, a + b + ab = 8 ........(i)
b + c + bc = 15 ...........(ii)
c + a + ca = 35 ............(iii)

we have to find (a + b + c + abc) = ?

from equation (ii),
b + c + bc = 15
b + c(b + 1) = 15
c = (15 - b)/(b + 1)
put it in equation (iii),
(15 - b)/(b + 1) + a + (15 - b)a/(b+ 1) = 35
(15 - b) + a(b + 1) + (15 - b)a = 35(b + 1)
15 - b + ab + a + 15a - ab = 35b + 35
15 + a - b + 15a = 35b + 35
a - b + 15a - 35b = 35 - 15 = 20
16a - 36b = 20
4a - 9b = 5
4a = 5 + 9b
a = (5 + 9b)/4 ......(iv)

put equation (iv) in equation (i),
(5 + 9b)/4 + b + (5 + 9b)/4 × b = 8
(5 + 9b) + 4b + 5b + 9b² =8 × 4
18b + 9b² + 5 = 32
9b² + 18b - 27= 0
b² + 2b - 3 = 0
(b + 3)(b - 1) = 0.
b = -3 and 1 but a, b , c are positive numbers
so, b = 1

put equation (iv), a = (5 + 9)/4 = 7/2
c = (15 - b)/(b + 1) = (15 - 1)/(1 + 1) = 7

now, a + b + c + abc = 7/2 + 1 + 7 + 7/2 × 1 × 7
= 7/2 + 8 + 49/2
= 56/2 + 8
=28 + 8 = 36

hence, answer is 36