Problem Solving

Correct answer D

You can start by re-writing the equation as:

(10x + 20y) / (x+y) = k

Then split it up like this:

(10x + 10y) / (x+y) + (10y) / (x+y) = k

Cancel x+y from the left term:

10 + (10y) / (x+y) = k

At this point, you can start looking at the answer choices.

It can't be A) 10, because 10y would have to be zero, and we know y is positive.

It can't be B) 12, because then (10y) / (x+y) = 2, which would mean 10y = 2x+2y, which would mean x=4y. But x and y are both positive, and x<y, so x=4y is not possible.

It can't be C) 15, because then (10y) / (x+y) = 5, which would mean 10y = 5x+5y, which would mean x=y. But x and y are both positive, and x<y, so x=y is not possible.

It can't be E) 30, because then (10y) / (x+y) = 20, which would mean 10y = 20x+20y, which would mean -2x=y. But x and y are both positive, and therefore -2x=y is not possible.

Answer D) 18 works, because then (10y) / (x+y) = 8, which would mean 10y = 8x+8y, which would mean y=4x. Since x and y are both positive, and since x<y, y=4x is definitely possible.

Another approach:

k = 10 + 10y/(x+y)

which can be written as

k = 10 + 10/(x/y+1)

now if I look at the boundary conditions - say x is very small ~0, then k = 10 + 10 = 20, similarly if x is approaching y, k = 10 + 10/(1+1) = 15 hence the answer should be any number between 20 and 15 which is 18.

raz1024 wrote:You can start by re-writing the equation as:

(10x + 20y) / (x+y) = k

Then split it up like this:

(10x + 10y) / (x+y) + (10y) / (x+y) = k

Cancel x+y from the left term:

10 + (10y) / (x+y) = k

At this point, you can start looking at the answer choices.

It can't be A) 10, because 10y would have to be zero, and we know y is positive.

It can't be B) 12, because then (10y) / (x+y) = 2, which would mean 10y = 2x+2y, which would mean x=4y. But x and y are both positive, and x<y, so x=4y is not possible.

It can't be C) 15, because then (10y) / (x+y) = 5, which would mean 10y = 5x+5y, which would mean x=y. But x and y are both positive, and x<y, so x=y is not possible.

It can't be E) 30, because then (10y) / (x+y) = 20, which would mean 10y = 20x+20y, which would mean -2x=y. But x and y are both positive, and therefore -2x=y is not possible.

Answer D) 18 works, because then (10y) / (x+y) = 8, which would mean 10y = 8x+8y, which would mean y=4x. Since x and y are both positive, and since x<y, y=4x is definitely possible.

excellent explanation from both of you..

Also, if you had x female friends, who, on average, smoked 10 cigarettes a day, and y male friends, who smoked 20 cigarettes a day on average, the number average number of cigarettes your friends would smoke on average would be
(10x + 20y)/(x + y). If y > x, this average will be closer to 20 than to 10 (i.e. between 15 and 20).