Electric Christmas Tree

The principles of Thevenin,s theorem can be applied in order to simplify the calculation for the total resistance across the entire circuit between A and B. By considering only a portion of the circuit at a time and then using the calculated equivalent resistance between selected points in subsequent calculations the circuit can be dealt with in manageable pieces.

The resulting total resistance can be calculated by applying these expressions

Resistors in Series ${ R }_{ Total }={ R }_{ 1 }+{ R }_{ 2 }+{ R }_{ 3 }+...$

Resistors in Parallel
${ R }_{ Total }={ R }_{ 1 }||{ R }_{ 2 }||{ R }_{ 3 }||...$ Using || to indicate parallel resistances.

${ R }_{ Total }=\frac { { R }_{ 1 }{ R }_{ 2 }... }{ { R }_{ 1 }+{ R }_{ 2 }+... }$

Beginning at the top of the diagram.

${ R }_{ 2 }||{ R }_{ 3 }=\frac { { R }_{ 2 }{ R }_{ 3 } }{ { R }_{ 2 }+{ R }_{ 3 } } =\frac { 1*1 }{ 1+1 } =\frac { 1 }{ 2 } \Omega$

${ R }_{ top }={ R }_{ R2||R3 }+R1+R4=\frac { 1 }{ 2 } +1+1=\frac { 5 }{ 2 } \Omega$

Total equivalent resistance between points E and F considering only resistors. R1,R2,R3,R4,R5

${ R }_{ EF }={ R }_{ top }||{ R }_{ 5 }=\frac { \frac { 5 }{ 2 } }{ \frac { 5 }{ 2 } +1 } =\frac { \frac { 5 }{ 2 } }{ \frac { 7 }{ 2 } } =\frac { 5 }{ 7 } \Omega$

${ R }_{ DG }=({ R }_{ EF }+{ R }_{ 6 })||{ R }_{ 7 }=\frac { { (R }_{ EF }+{ R }_{ 7 })*{ R }_{ 7 } }{ { (R }_{ EF }+{ R }_{ 7 })+{ R }_{ 7 } } =\frac { { 5 }/{ 7 }+1 }{ { 5 }/{ 7 }+1+1 } =\frac { { 12 }/{ 7 } }{ { 19 }/{ 7 } } =\frac { 12 }{ 19 } \Omega$

${ R }_{ 12 }+{ R }_{ 13 }=2\Omega$

${ R }_{ Underleft }=({ R }_{ 12 }+{ R }_{ 13 })||{ R }_{ 14 }=\frac { 2 }{ 3 } \Omega$

${ R }_{ Underright }=({ R }_{ 15}+{ R }_{ 16 })||{ R }_{ 17 }=\frac { 2 }{ 3 } \Omega$

${ R }_{ 8 }||{ R }_{ 9 }=\frac { 1 }{ 2 } \Omega$

${ R }_{ AB }={ R }_{ underleft }+{ R }_{ 10 }+{ R }_{ DG }+{ R }_{ R8||R9 }+{ R }_{ 11 }+{ R }_{ underright }=\frac { 2 }{ 3 } +1+\frac { 12 }{ 19 } +\frac { 1 }{ 2 } +1+\frac { 2 }{ 3 }$

$LCD=3\ast19\ast2=114$

${ R }_{ AB }=\frac { 76 }{ 114 } +\frac { 114 }{ 114 } +\frac { 72 }{ 114 } +\frac { 57 }{ 114 } +\frac { 114 }{ 114 } +\frac { 276 }{ 114 } =\frac { x }{ y } =\frac { 509 }{ 114 } \Omega$

answer x+y = 509+114 = 623