Problem Solving

Have been doing speed drills all morning...now having a major brain freeze. Can anyone point me in the right direction on these? thx!

14. Mr. Kramer, the losing candidate in a two-candidate election, received 942,568 votes, which was exactly 40 percent of all the votes cast. Approximately what percent of the remaining votes would he need to have received in order to have won at least 50 percent of all the votes cast?
(A) 10%
(B) 12%
(C) 15%
(D) 17%
(E) 20%

I need an equation of sorts here to limit the amount of hairy long division that I would otherwise have to do...


15. Which of the following inequalities is equivalent to –2 < x < 4 ?
(A) | x – 2 | < 4
(B) | x – 1 | < 3
(C) | x + 1 | < 3
(D) | x + 2 | < 4
(E) None of the above

16. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
(A) 6x
(B) 4x
(C) (5x)/3
(D) (3x)/2
(E) (3x)/5

1. he recieved 942568 votes=x
total votes x/0.4
60 % votes is 3x/2 this is the denominator
the numerator term will be

50% votes is 5x/4
he should recieve 5x/4-x votes ie x/4 votes additional
the percent ans is x/4 divided by 3x/2
ans is 16.67 ie 17%

2. this is a plug in type of problem
a. let x=5 so a is true but violates the problem Q
b. seems correct
c.d at x=-2 c is true but violates the Q

another way of solving is by removing the mod and solving ie for B we have x-1<3 ie x<4 and -x+1<3 ie x>-2

16. i am not sure of this one

14. Mr. Kramer, the losing candidate in a two-candidate election, received 942,568 votes, which was exactly 40 percent of all the votes cast. Approximately what percent of the remaining votes would he need to have received in order to have won at least 50 percent of all the votes cast?
(A) 10%
(B) 12%
(C) 15%
(D) 17%
(E) 20%

He needs 10% of total votes to make it an even 50%
remainder votes = 60%
....Now simply find 10% of 60% = 10/100 * 100/60 * 100 = 16.67% = Approx 17%

I hope I am correct

OA is D, or about 17% for #14.

16. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
(A) 6x
(B) 4x
(C) (5x)/3
(D) (3x)/2
(E) (3x)/5

Assuming that all temperatures are the same then each value would be x/5....Hence sum of the 3 largest = x/5 * 3 = 3x/5....but since this question says 3 greatest I am still wondering?

15. Which of the following inequalities is equivalent to –2 < x < 4 ?
(A) | x – 2 | < 4
(B) | x – 1 | < 3
(C) | x + 1 | < 3
(D) | x + 2 | < 4

Work with substitution

Start with C --- Ix+1I < 3
Now x can be 1,0,-1,-2,-3
Doesn't satisfy

D -- Ix+2I < 4
x can be 0,1,-1,-2,-3,-4,-5
Doesn't satisfy

B-- Ix-1I < 3
x can be 3,2,1,0,-1,

It does satisfy...hence B

The OA for #16 is B.

(obviously) not sure how to approach this one...i randomly assigned 10, 20, 30, 40, and 50 as the five positive temperatures. average is 30, sum of largest 3 is 120. B suffices.

but then if you put in, say 20, 25, 30, 35 and 50...none of the choices suffice. clearly this isn't the way to solve it.

bww wrote:Have been doing speed drills all morning...now having a major brain freeze. Can anyone point me in the right direction on these? thx!

14. Mr. Kramer, the losing candidate in a two-candidate election, received 942,568 votes, which was exactly 40 percent of all the votes cast. Approximately what percent of the remaining votes would he need to have received in order to have won at least 50 percent of all the votes cast?
(A) 10%
(B) 12%
(C) 15%
(D) 17%
(E) 20%

I need an equation of sorts here to limit the amount of hairy long division that I would otherwise have to do...

Just to simplify things, take x = 942568

Total votes = x/0.4

0.5 of total votes = 0.5x/0.4

Difference in votes needed = 0.5x/0.4 - x = 0.25x

Remaining votes = 0.6*x/0.4

Percentage = 0.25x / (0.6x/0.4) * 100 = 1/6*100 = 16.67%

bww wrote: 15. Which of the following inequalities is equivalent to –2 < x < 4 ?
(A) | x – 2 | < 4
(B) | x – 1 | < 3
(C) | x + 1 | < 3
(D) | x + 2 | < 4
(E) None of the above

To narrow down on the options, take the positive value inside the
absolute value

A - x-2 < 4 implies x < 6. Does not satisfy x<4. Discard

B - x-1 < 3 implies x < 4. Keep this

C - x+1 < 3 implies x < 2. Discard

D - x+2 < 4 implies x < 2. Discard

So, we are down to B and E.

For B, take the negative factor as well

-(x-1) < 3
-x + 1 < 3
-x < 2
x > -2

So, together B yields -2 < x < 4. Sufficient.

bww wrote:
16. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
(A) 6x
(B) 4x
(C) (5x)/3
(D) (3x)/2
(E) (3x)/5

If we assume all 5 temp to be equal, the highest 3 will yield a sum 3x.

If we assume 2 of the least temp to be 1 and 1, the sum of the rest
will be 5x-2

So, sum has to be > 3x and less than 5x-2

Only B fits the bill.

bww wrote:
... 16. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
(A) 6x
(B) 4x
(C) (5x)/3
(D) (3x)/2
(E) (3x)/5

a different approach ... let the sum of the 3 greatest be "a " and the other 2 be "b" ... so we have (a+b)/5=x ... so a+b= 5x ... now if the additon of the 3 greatest is 6x (a) .. then we have 6x+b=5x .. b= -x .. that is the additon of the 2 smallest will be equal to -x which is not possible since the tempratures are given to be positive ...

.. if a = 5x/3 then b wuld be equal to 10x/3 .. so that wuld mean the addition of 3 greatest is less than the sum of the 2 smallest which is not possible ..

similarly if a = 3x/2 .. then b wuld be equal to 7x/2 ..again a<b which is not possible

so on ..

now if u solve for a = 4x .. then we get b= x which is the culd be a possible value for b .. so answer is 4x

16. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
(A) 6x
(B) 4x
(C) (5x)/3
(D) (3x)/2
(E) (3x)/5


total = 5x

Since its a maxmizing question start with 4x

2x x x .5x .5x

possible

choose b

I chose 5 numbers : 1, 2, 3, 4, 5

(1 + 2 + 3 + 4 + 5)/5 = 3

Now x = 3

Substitute in the answer choices. The answer choice that gets us 12 (sum of the greatest three numbers) is the answer.

(B)