If x, y, and k are positive numbers such that x/(x + y)*10 + y/(x + y)

Bunuel, I didn't understand how you got the 15 and 20 at the extreme ends in the equation. Could you please explain? thanks!
15<10*(1+(y/x+y)<20

main funda is

y/(x+y) is equal to 1/2 if x=y but when x<y then x+y<2y i.e. y/x+y >1/2 (x,y>0)

also the maximum value is x tends to 0 i.e. as x>0 say x be 0.0000...(infinite times )1 then the expression x+y tends to y i.e. y/ (x+y) <1

hence 0.5 < y/(x+y) < 1
=> 1+0.5 < 1+y/(x+y) < 1+1
=> 10*1.5 < 10*(1+(y/x+y) <2 *10
=> 15<10*(1+(y/x+y)<20

hope this helps..!!

Since \(x<y\) then \(\frac{y}{x+y}\) is more than \(0.5\) and less than \(1\): \(0.5<\frac{y}{x+y}<1\).

In this case for \(1+\frac{y}{x+y}\) is more than 1.5 and less than 2, so \(10*(1+\frac{y}{x+y})\) is more than 15 and less than 20: \(15<10*(1+\frac{y}{x+y})<20\).

Hope it's clear.

\(\frac{10x}{x+y}+\frac{20y}{x+y}=\frac{10(x+y)+10y}{x+y}=10+\frac{10y}{x+y}\).

\(\frac{10y}{x+y}>\frac{10y}{2y}=5\) and \(\,\,\frac{10y}{x+y}<\frac{10y}{y}=10\), therefore \(10+5<k<10+10\) or \(15<k<20\).

Answer D.

Hi Bunnel ,
Can you pls Explain how these Steps Came in your Solution

10*\frac{x}{x+y}+20*\frac{y}{x+y}=k

10*\frac{x+2y}{x+y}=k

10*(\frac{x+y}{x+y}+\frac{y}{x+y})=k

Finally we get: 10*(1+\frac{y}{x+y})=k

I Didn't Understood how The Steps 3 and 4 Came?

\(\frac{x+2y}{x+y}=\frac{x+y}{x+y}+\frac{y}{x+y}=1+\frac{y}{x+y}\).

Hope it's clear.

Hi All,

You can take advantage of the answer choices and some math "logic" to get to the correct answer. Let's TEST THE ANSWERS....

We're told that X, Y and K are POSITIVE and that X < Y.

We're also told that 10X/(X+Y) + 20Y/(X+Y) = K. We're asked what COULD be the value of K, which means that there's more than one possible answer. Since the answers are NUMBERS, one of the them MUST be a possible answer.

We can manipulate the given equation into:

10X + 20Y = K(X+Y)

Since all of the variables are POSITIVE, K MUST be between 10 and 20. Here's the proof:

IF.....K=10, then the equation becomes...
10X + 20Y = 10X + 10Y
20Y = 10Y
Since Y is positive, 20Y = 10Y is NOT possible.
Eliminate Answer A

In that same way, K can't be 20 (or greater) because the end equation would be an impossibility.

With the remaining 3 answers, we can TEST the possibilities...

IF...K = 12, then the equation becomes...
10X + 20Y = 12X + 12Y
8Y = 2X
4Y = X
In this scenario, X > Y which is the OPPOSITE of what we were told. This is NOT the answer.
Eliminate B.

IF....K=15, then the equation becomes...
10X + 20Y = 15X + 15Y
5Y = 5X
Y = X
In this scenario, X = Y, which is NOT a match for what we were told. This is NOT the answer.
Eliminate C.

IF....K=18, then the equation becomes....
10X + 20Y = 18X + 18Y
2Y = 8X
Y = 4X
Here, X < Y, which IS a match for what we were told. This IS a POSSIBLE answer.

Final Answer:


GMAT assassins aren't born, they're made,
Rich

what will happen when x>y in this question? since we dont know the value of x and y they can be 1:100 or something like that. what will be the ans then?

Hi anik19890,

You ask a couple of different questions.

First, the prompt tells us that X < Y, so we CAN'T have a situation in which X > Y (or even X = Y). THAT specific inequality helps to define the correct answer.

Second, the prompt asks for which of the following COULD be the value of K, meaning that there is more than one possible answer (but only one of the 5 answer choices is correct). Based on the inequality we're given, we COULD have X = 1, Y = 100 (as you described), which would give us...

(1/101)(10) + (100/101)(20) = K

10/101 + 2000/101 = K
2010/101 = K

But that answer is NOT among the answer choices.

GMAT assassins aren't born, they're made,
Rich

We are given:

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

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

We can combine the two fractions on the left side of the equation because they have the same denominator, (x + y).

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

We see that we have a weighted average equation in which x items have an average of 10, and another y items have an average of 20 and a weighted average of k. In this case, the value of k must be between 10 and 20. However, since x is less than y, the weighted average (or k) must be closer to 20 than to 10. Thus k must be 18.

Answer: D

Hi, Bunuel! I have a silly question. Can you please explain how you arrived at the following?
We know that \(x<y\)

Hence \(\frac{y}{x+y}\) is more than \(0.5\) and less than \(1\)

\(\frac{y}{x+y}\)

Since both x and y are positive, then the denominator is greater than numerator. So, \(\frac{y}{x+y}<1\).

Next, if x were equal to y, then we'd have y/(2y) = 1/2 but x < y, so denominator is less than 2y, which makes the fraction greater than 1/2.

Hope it's clear.

This one's all about pattern recognition.

Notice that the equation is giving us a weighted average. Because y is bigger than x, the average must be weighted towards y. Visually:

x---------avg---y
10--------avg---20

Thus, the answer must be between 15 and 20, exclusive.

Among the answer choices, only 18 fits the bill.

Choose D.

I suspect that many test-takers would not recognize that this is a weighted average problem. For those test-takers, a straightforward and efficient approach would be to plug in the answer choices, which represent the value of k.

Answer choice C: k= 15
(10x + 20y)/(x+y) = 15
10x + 20y = 15x + 15y
5y = 5x
y = x
Doesn't work because the problem states that x<y.

We need y to be larger, so let's try answer choice D: k=18
10x + 20y = 18x + 18y
2y = 8x
y/x = 8/2
Success! x<y.

The correct answer is D.

The process would be very quick if you recognized right away that since k = one of the answer choices, you could simply plug them in for k.

Bunuel

We know that x<y

Hence y/(x+y) is more than 0.5 and less than 1

how we know its between 0.5 and less than 1? x can be 3 and y can be 4..there will be many possibilities.

please clear my confusion.

Thanks.

Hi swadhakamal,

We're told that X and Y are both POSITIVE and X < Y.

This means that if you take the sum of X and Y, then the Y will make up MORE than 1/2 of that sum. The reason why can be explained with a comparison:

(X + X) vs. (X + Y)

If we add two Xs together, then each variable represents EXACTLY HALF of the total. However, since Y is GREATER than X, if we take (X + Y), then the Y would have to be MORE than half of that total.

If you don't immediately recognize that pattern, then that's okay - you might find it helpful to come up with a few examples, and you'll see that that is ALWAYS the case. We can use your first example (X=3 and Y=4):

X+Y = 7 and 4 is more than half of 7.

Thus, when we're thinking about Y/(X+Y), we know that we'll have a positive fraction AND since Y is MORE than half of (X+Y), that the fraction must be GREATER than 1/2 (but less than 1).

GMAT assassins aren't born, they're made,
Rich

Given: \(\frac{x}{x + y}*10 + \frac{y}{x + y}*20 = k\)

Rewrite as follows: \(\frac{10x}{x + y} + \frac{20y}{x + y} = k\)

Rewrite as follows: \((\frac{10x}{x + y} + \frac{10y}{x + y}) + \frac{10y}{x + y}= k\)

Simplify: \((\frac{10x + 10y}{x + y}) + \frac{10y}{x + y}= k\)

Simplify: \(10 + \frac{10y}{x + y}= k\)

Rewrite as follows: \(10 + 10(\frac{y}{x + y})= k\)

IMPORTANT: If \( x = y\), then \(\frac{y}{x + y} = \frac{y}{y + y} = \frac{y}{2y} = \frac{1}{2}\)

Since it's actually the case that \( x < y\), then we know that \(\frac{1}{2} < \frac{y}{x + y} < 1 \), which means \(10(\frac{y}{x + y})\) is greater than 5 but less than 10.

So, we know that \(15 < k < 20\)

Answer: D