Hey guys,

I've posted a basic Algebra problem below that I've solved however I'm curious if there is a better way to solve it. While my solution works it is a 2 part solution so for every answer I get from the first equation I then need to do a second calculation to see if it's correct.

Since I was doing this by hand (I was away from my computer at the time) it was a rather time consuming process of trial and error and I'm just curious if there is a better way of doing this.

The problem is:

The only hint was to let x = the total number of votes.The results of an election for a sporting club is as follows... "Fifty votes ahead of her closest competition, Amanda will be the new sports captain, obtaining 35% of all votes. Belinda came in second place, with Cameron not far behind her obtaining 134 more votes than Daniel. Daniel obtained 12% of all votes."

If these 4 people were the only candidates, find the number of votes each candidate obtained.

So to solve this I simply created an equation where x = A + B + C + D where A = Amanda's votes, B = Belinda's votes, C = Cameron's votes and D = Daniel's votes.

Then I chose D as the main variable and created an equation to link all 4 candidates votes back to Daniel's. Using the above information the initial equation I came up with was:

$\displaystyle {\rm{Total}}\,{\rm{Votes (x) = }}A + B + C + D\\$

$\displaystyle {\rm{Total}}\,{\rm{Votes (x) = }}(D*\frac{{35}}{{12}}) + (D*\frac{{35}}{{12}} - 50) + (D + 134) + D$

I then simplified this to:

$\displaystyle {{\rm{x = }}D*\frac{{35}}{{12}} + D*\frac{{35}}{{12}} - 50 + D + 134 + D}$

$\displaystyle {{\rm{x = }}D(\frac{{35}}{{12}} + \frac{{35}}{{12}}) - 50 + 134 + D + D}$

$\displaystyle {{\rm{x = }}\frac{{70}}{{12}}D + 84 + 2D}$

$\displaystyle {x = (5\frac{{10}}{{12}}D + 2D) + 84}$

$\displaystyle {x = 7\frac{5}{6}D + 84}$

So now for any value of D I can get the total number of votes that would go along with it. However the main rule that needs to be satisfied is $\displaystyle \frac{d}{x} = 12\% $.

So basically what I ended up doing was trying different values of D with the above equation to get the total number of votes and then doing a second calculation to divide D by x to see if it equaled 12%.

Eventually after 11 attempts I came up with:

D = 168 so...

$\displaystyle x = 7\frac{5}{6}*168 + 84\\$

$\displaystyle x = 1400$

and then confirmed it by:

$\displaystyle \frac{D}{x} = \frac{{168}}{{1400}} = 0.12{\mkern 1mu} \,{\mkern 1mu} {\rm{and}}{\mkern 1mu} {\mkern 1mu} \,\frac{A}{x} = \frac{{(168*\frac{{35}}{{12}}\,)}}{{1400}} = 0.35{\mkern 1mu} \,(also{\mkern 1mu} \,matches)$

which gives the final answer of:

Amanda's Votes: $\displaystyle 168*\frac{{35}}{{12}} = 490$

Belinda's Votes: $\displaystyle 168*\frac{{35}}{{12}} - 50 = 440$

Cameron's Votes: $\displaystyle 168 + 134 = 302$

Daniel's Votes: $\displaystyle 168$

So finally I managed to get the correct answer. However not having access to a computer meant it was a rather slow process of trial and error with 2 calculations per attempt.

As a result I was just wondering if any of you guys can see a better way of doing this??? Is there anyway to express this as one equation so by solving that one equation you could tell if your answer was correct instead of having to do the 2nd step I did by checking if $\displaystyle \frac{D}{x} = 0.12$??? Any advice would be greatly appreciated.

Thanks in advance.