Tag: mental additions and subtractions

$50342135 - 46380651 = 3961484$.

$1 + 2 - 3 + 4 - 5 + 6 - 7 + 8$

    1 0

  5 0 0

 $99999$
$-10000$
______
$89999$

Let Y closed pipe after a min .
Then part filled by (X+Y) in a min +Part filled by X in (18-x) 
So $a(\frac{1}{24}+\frac{1}{32})+(18-a)\frac{1}{24}= 1$
Multy by 96 
Or 4a+3a+72-4a=96
Or 3x=24
Or x=8 min

$x=\frac{4\sqrt{2}}{\sqrt{2+1}}$
$\frac{1}{\sqrt{2}}\left ( \frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x-2\sqrt{2}} \right )$
Put the value of x
$\frac{1}{\sqrt{2}}\left ( \frac{\frac{4\sqrt{2}}{\sqrt{2+1}}+2}{\frac{4\sqrt{2}}{\sqrt{2+1}}-2}+\frac{\frac{4\sqrt{2}}{\sqrt{2+1}}+2\sqrt{2}}{\frac{4\sqrt{2}}{\sqrt{2+1}}-2\sqrt{2}} \right )$
=$\frac{1}{\sqrt{2}}\left ( \frac{4\sqrt{2}+2\sqrt{2}+2}{4\sqrt{2}-2\sqrt{2}-2} \right )+\left ( \frac{4\sqrt{2}+4+2\sqrt{2}}{4\sqrt{2}-4-2\sqrt{2}} \right )$
=$\frac{6\sqrt{2}+2}{2\sqrt{2-2}}+\frac{6\sqrt{2}+4}{2\sqrt{2}-4}$
=$\frac{1}{\sqrt{2}}\left ( \frac{32-24\sqrt{2}}{16-12\sqrt{2}} \right )$
=$\sqrt{2}$

$
:Q:R\quad =\quad 6:5:4\ Let\quad P\quad \quad =\quad 6x\ Q=\quad 5x\ R\quad =\quad 4x\ { P }^{ 2 }+{ Q }^{ 2 }{ +\quad R }^{ 2 }\quad =\quad 192500\ { (6x) }^{ 2 }+{ (5x) }^{ 2 }+(4x)^{ 2 }\quad =\quad 192500\ 77{ x }^{ 2 }\quad =\quad 192500\ { x }^{ 2 }\quad =\quad 2500\ x\quad =\quad 50\ \ \frac { P+Q-R }{ 2 } \quad =\quad \frac { 6x+5x-4x }{ 2 } \quad =\quad \frac { 7x }{ 2 } \quad =\quad \frac { 7\times 50 }{ 2 } \quad =\quad 175
$

  1  2
  5  A  9
  3  B  7
  2  C  8
1 1  1  4
$2+A+B+C=11$
$A+B+C=9$
Clearly max value of $C=9$