The chair of the board of directors for a certain company is planning

Question

The chair of the board of directors for a certain company is planning to call a meeting of the board to vote on a measure. Passing a measure requires at least 75% of the members of the board to vote for the measure. The chair has taken an informal poll of the board members, and exactly x members (including the chair) said they would vote for the measure, while the remaining y members said they would vote against it. The chair correctly determined that the measure would not pass if all members voted as they said they would, but the measure would pass if the chair could convince just one member who said she would vote against the measure to instead vote for the measure,
assuming all other members vote as they said they would.

Select a value for x and a value for y that are jointly consistent with the information provided. Make only two selections, one in each column.

HarshavardhanR · Accepted Answer

"The chair correctly determined that the measure would not pass if all members voted as they said they would"

x < (3/4)*(x + y)
4x < 3x + 3y
x < 3y ..............(1)

"but the measure would pass if the chair could convince just one member who said she would vote against the measure to instead vote for the measure"

x + 1 >= (3/4)*(x + y)
4x + 4 >= 3x + 3y
x + 4 >= 3y .................(2)

From (1) and (2),

x < 3y < x + 4

For 3 times y to still be less than x + 4, y should be small. For instance, if y = 19, 3 x 19 = 57. No value of x from the choices will give an x + 4 that is greater than 3y.

So,
for y, consider the smaller values and see what happens.

(1) y = 4
Then 3y = 12
x<3y => x can only be 4 or 7. But in both these cases, x + 4 will be < 3y (8, 11 are both < 12). We need x+4 to be greater than 3y.
Hence y= 4 can be rejected.

(2) y = 7.
Then 3y = 21.
All the choices here will work for x (all are < 21). But there is exactly one case where x + 4 is > 21. x = 19.

When x = 19 and y = 7,
x < 3y < x + 4 (19 < 21 < 23).

This must be the answer (multiple answers satisfying the given conditions are not possible here!).

x = 19; y = 7.

___
Harsha

Gemmie · Answer

\frac{x}{x+y} < 75% = \frac{3}{4} 4x < 3x + 3y x < 3y \frac{x+1}{x+y} >= \frac{3}{4} 4x + 4 >= 3x + 3y x + 4 >= 3y x < 3y <= x + 4 Then x must be among big numbers: 12, 15, 19 y be among small number: 4, 7 +) y = 4 => ey = 12 => There is no answer for x < 12 except 7 but then x+4 = 11 < 12 +) y = 7 => 3y = 21 => Try x = 19 < 21; x + 4 = 23 > 21 Then x = 19; y = 7

mandilwar · Answer

I think it's better to look at the option here and make a guess using trial and error:

So, as evident, we need to take one from Y and add to X, means Y is smaller. By adding 1 to X, X must become 75% of total and Y must become 25% of total. So this mean the difference between the 2 must be significant after adding 1 to X.

Keeping this in mind, let's test  19&4 , so 19+1=20, 4-1=3, 20+3=23, so 19 was already higher than the req. 75%, rejected.

So, we need  lesser gap , let's test  19&7 . 19+1=20 and 7-1=6. Total= 20+6 = 26, now let's check if the purpose is fulfilled, 75% of 26=19.5=20 (as no. of people cant be in decimal). Yes, condition fulfilled. Y will be automatically 25%.

Be careful in these kinds of trial and error as if you don't eliminate options smartly it will take a lot of time and wouldn't be worth solving at the cost of much time.

KarishmaB · Answer

Here is the explanation of a similar question and how you can solve such questions very quickly using percentages: https://youtu.be/enQLnAMy_XQ

Goldenfuture · Answer

Go by options
x+1 >= 3/4(y+x)
x+4>= 3y

Look at options of Y and convert them into 3y
12 so x can be min 8
21.... can be min 19 ----- bingo answer--- pair found
24----x can be min 20

KevoK · Answer

Here's my take on this:

Mus be over 75% (3/4) to pass, thus according to the poll X/Y < 3

And according to the chair, (X=1) / (Y-1) >=3

From here on I just test the number, making sure the proportion between the X and Y that I test are under 3.

Adit_ · Answer

Interesting problem.
So the best way to do it is to visualise such that there exists s value ins 1st column such that second column*3>First column.
But once First+1 happens then Second-1 happens and the ratio increases over 3x times.

19,7  is the perfect option for this as adding +1 = 20
7-1 = 6
Ratio = 20/6 >3x or >75%
Taking 12,4 would mean its already 75% so that is not valid.
Taking 12,7 also not valid as 13/6 is still less than 3x.

The chair of the board of directors for a certain company is planning

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