Question

a,b,c are three positive numbers. The second number is greater than the first by the amount the third number is greater than the second. The product of the two smaller numbers is 85 and that of the two bigger numbers if 115. Then the value of (2012a – 1006c) is __________

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

a,b,c are three positive numbers. The second number is greater than the first by the amount the third number is greater than the second.

It means,0 < a < b < c and

$\rm \: b - a = c - b$

$\rm \: b + b = a + c$

$\rm\implies \:\boxed{ \rm{ \:2b = a + c}} - - - (1)$

Further given that, product of the two smaller numbers is 85.

$\rm \: ab = 85$

$\rm\implies \:a \: = \: \dfrac{85}{b} - - - (2)$

Also, given that, product of the two bigger numbers is 115.

$\rm \: bc = 115$

$\rm\implies \:c \: = \: \dfrac{115}{b} - - - (3)$

On substituting the values of a and c from equation (2) and (3), in equation (1), we get

$\rm \: 2b = \dfrac{85}{b} + \dfrac{115}{b}$

$\rm \: 2b = \dfrac{85 + 115}{b}$

$\rm \: 2b = \dfrac{200}{b}$

$\rm \: b = \dfrac{100}{b}$

$\rm \: {b}^{2} = 100$

$\rm\implies \:b = 10 \: \: \: \{ \: as \: b > 0 \}$

On substituting the value of b in equation (2) and (3), we get

$\rm\implies \:a = 8.5 \:$

and

$\rm\implies \:c = 11.5 \:$

Now, Consider

$\rm \: 2012a - 1006c$

$\rm \: = \: 1006(2a - c)$

$\rm \: = \: 1006(2 \times 8.5 - 11.5)$

$\rm \: = \: 1006(17 - 11.5)$

$\rm \: = \: 1006(5.5)$

$\rm \: = \: 5533$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\rm \: 2012a - 1006c = 5533 \: \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$