Question 74282
One-tenth of a rod is colored red, one-twentieth orange, one-thirieth yellow, one-fortieth green, one-fiftieth blue, one-sixtieth indigo, and the remainder, which is 302 in. long, violet. Find the length of the rod. 
Let the length of the rod be: x
Then the red part is: 1/10x
the orange part is: 1/20x
The yellow part is: 1/30x
the green part is:1/40x
the blue part is:1/50x
the indigo part is:1/60x
The rest is: 302 in
Problem to solve is:
{{{(1/10)x+(1/20)x+(1/30)x+(1/40)x+(1/50)x+1/60)x+302=x}}}
The LCD is 600, multiply all of the terms by 600 to clear the fractions.
{{{600(1/10)x+600(1/20)x+600(1/30)x+600(1/40)x+600(1/50)x+600(1/60)x+600(302)=600(x)}}}
{{{60x+30x+20x+15x+12x+10x+181200=600x}}}
{{{147x+181200=600x}}}
{{{147x-147x+181200=600x-147x}}}
{{{181200=453x}}}
{{{181200/453=453x/453}}}
{{{400=x}}}
The rod is 400 in long.
Sanity check:
{{{(1/10)(400)+(1/20)(400)+(1/30)(400)+(1/40)(400)+(1/50)(400)+(1/60)(400)+302=400}}}
{{{40+20+40/3+10+8+20/3+302=400}}}
{{{40+20+10+8+302+(40+20)/3=400}}}
{{{40+20+10+8+302+60/3=400}}}
{{{40+20+10+8+302+20=400}}}
{{{400=400}}}
If you weren't insane before messing with all those fractions, you probably are now.  Any way we seem to be right.
Happy Calculating!!!!