solve this "level 5" SAT math question

analogous · October 10, 2009, 6:08pm

In an election, 2.8 million votes were cast and each vote was for either Candidate I or Candidate II. Candidate I received 28,000 more votes than Candidate II. What percent of the 2.8 million votes were for Candidate I?
(A) 50.05%
(B) 50.1%
(C) 50.5%
(D) 51%
(E) 55%

My attempt:

(1) 2.8M/2 = 1.4M
(2) 1.4M + 28k = 1,428,000 (# of votes for Candidate I)
(3) 1,428,000/2.8M = .51 = 51% = choice (D)

The answer is (C) and i can't figure out what I'm doing wrong. 
Thanks for the help.

(#8 in BB 2nd edition; first practice test section 7)

Tilgaham · October 10, 2009, 6:14pm

2,800,000 *.51= 1,428,000
1,428,000+1,400,000=2,828,000. =wrong.

lockn · October 10, 2009, 6:14pm

You can't just add 28k, becuase then there would be 2.828 million votes. You need to add 14k to one and subtract 14k from the other.

Or you could do algebra.
2x-28,000=2,800,000
x=1414000

njwhitekid · October 10, 2009, 6:17pm

You can't assume that each have at least half and then add from there.
You must assign each candidate a variable and calculate using algebra:

The solution:

Candidate 1: C1
Candidate 2: C2

C1 + C2 = 2,800,000
C1 = C2 + 28,000
Substitution method:
( C2 + 28,000 ) + C2 = 2,800,000
2(C2) + 28,000 = 2,800,000
2(C2) = 2,772,000
C2 = 1,386,000
Thus...
C1 = ( 1,386,000 ) +28,000
C1 = 1,414,000

C1 / total * 100 = his %
( 1,414,000 / 2,800,000 ) * 100 = 50.5%

Therefore, the answer is 50.5% of the vote... or C.

I deliberately wrote the long explanation to help you. Hope it does.

Yankee_Belle · October 10, 2009, 6:20pm

You should have split the difference of 28,000 votes in half. Candidate 1 went up 14,000 votes and candidate 2 went down 14,000 votes from 50%. Your solution has a vote separation of 56,000 votes.

analogous · October 10, 2009, 6:34pm

Oops, lol. Silly mistake. So Candidate I should actually have 1,414,000 votes.

1.414M/2.8M = .505 = 50.5%
=P silly me

Thanks guys!

187 · October 10, 2009, 6:53pm

here's the fastest way to do it:
x+y = 2.8E6
x = y + 2.8E4, which is x - y = 2.8E4

set it as a 2x3 matrix
[1 1 2.8E6]
[1 -1 2.8E4]

rref of the matrix; x = 1414E3
x/total = .505

analogous · October 10, 2009, 6:59pm

^LOOL

I haven't even used matrices in my life. I think once you understand what you're looking for, the fastest way to do it is whichever way "clicks" inside your head. If i wouldn't have made at dumb mistake the time it would have taken me to solve the problem is probably ~40 seconds.

187 · October 10, 2009, 7:20pm

Solve it however way you want. I'm just showing you a faster method for solving problems that require substitution. It's faster AND the only mistake you can possibly make would be putting in the numbers incorrectly.

The fastest way is NOT whatever way "clicks" in your head. Take, for instance, a rate problem. There's 3 types of people: those who don't get it, those who do it the long way, and those who use the formula to get the answer in less than 10 seconds. Which would you prefer? Faster methods offer more time to either go over the section again, or to spend more time on a problem you're stuck on.

analogous · October 10, 2009, 7:22pm

Woah, you sound very defensive 187, chillax dude. I'm going to PM you, you seem like you can offer me help in the math section!

silverturtle · December 21, 2009, 6:34pm

A matrix is not the fastest way to solve the problem. I solved it in about five or six seconds by just dividing by two, adding 14,000, and dividing by the total.

Sci-Fry · December 21, 2009, 6:50pm

No, here's the fastest way to do it. Took me about 10 seconds.

2800000/2 = 1400000

Since there's a difference of 28000 in the votes, candidate I has 1400000 + 14000 votes.

1414000/2800000 = 0.505 = 50.5%

silverturtle · December 21, 2009, 7:08pm

Sci-Fry, that's the method I was trying to convey.

Anonymous217 · December 21, 2009, 7:16pm

Nice job with the necro-post. :

Sci-Fry · December 21, 2009, 7:33pm

Ah, silverturtle, didn't see your post before! I was referring to one of the posts above yours. Sorry for the confusion.

silverturtle · December 21, 2009, 8:37pm

Mistake on my part, too. I didn't realize the age of this thread. :)

aznandrew · August 5, 2011, 5:37pm

i think you do 2x-28000=2,800,000
then that answer is 1,414,000.
Last guess and check till you find the right answer.
2,800,000*50.5=1,414,000

Dusterbug · August 5, 2011, 5:48pm

Lol necroed two different times.

Anyway the best way is definitely substitution. No chance for error and it takes like 30 seconds.

Aceventura74 · August 5, 2011, 6:31pm

Analogous,

Set up equations like you would in a beginner algebra class.

Let:
x = # of votes cast for candidate II
x + 28,000 = # of votes cast for candidate I (Candidate I received 28,000 more votes than Candidate II)

candidate II + candidate I = 2,800,000
x + (x + 28,000) = 2,800,000
x = 1,386,000

x = candidate II = 1,386,000 votes
x + 28,000 = candidate I = 1,386,000 + 28,000 = 1.414,000 votes

At this point you've found the # of votes each candidate got. Now we must convert candidate I's votes into a percent ( What percent of the 2.8 million votes were for Candidate I?)

(1,414,000/2,800,000) x 100 = 50.5% choice (c)

~Aceventura74

xiggi · August 5, 2011, 6:43pm

Hope you saw that this is an old thread. The OP is long gone.

Fwiw, this problem does not require fancy steps or equations.

How hard is it to compare 28,000 and 2,800,000? No calculator is needed to see that the difference is 1 percent. How can you represent a ONE percent difference on a 100 scale? The only way is ... 50.5 and 49.5.

Again, use your head before grabbing that TI-89 or focusing on basic HS math.