Is 4^(x + y) = 8^(10) ? (1) x - y = 9 (2) y/x = 1/4

SOLUTION

Is \(4^{x+y}=8^{10}\) ?

Work with the same base: is \(4^{x+y}=8^{10}\) ? --> is \(2^{2(x+y)}=2^{30}\) ? --> is \(2(x+y)=30\)? is \(x+y=15\)?

(1) x - y = 9. Not sufficient.
(2) y/x = 1/4 --> \(x=4y\). Not sufficient.

(1)+(2) We have two distinct linear equation with two unknowns (\(x - y = 9\) and \(x=4y\)), hence we can solve for both of them and get whether \(x+y=15\) is true. Sufficient.

Answer: C.
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The question is simply is x + y = 15?
All we need is to know the value of the sum of x and y to get sufficiency.

(1) x -y= 9

10 - 1 is 9
11 - 2 is 9
24 - 9 is 15

INSUFFICIENT

(2) y/x = 1/4 or 4y = x many possibilities , INSUFFICENT

X - y = 9
(4y) - Y = 9
y = 3
x = 12

answer: c

the given eqn can be solved as:
2^2(x+y) = 2^3(10)

=> x+y = 15

Stmt 1) x - y = 9 can take many values for x and y and satisfy the eqn
Stmt 2) y/x = 1/4, here also x and y can take many values

combining both 1) and 2) the eqn can be solved to get values of x and y and can be verified if these x and y satisfies the given eqn

Hence C
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Hi, could you please help clarify? I'm a little bit confused.

Isn't the answer D?

(1) X-Y=9 ----> X=9+Y
We can just put "X=Y+9" in the initial equation X+Y=15 to solve the equation without any help from the 2nd equation, y/x = 1/4.

The same goes to the second equation
(2) y/x = 1/4 -------> X=4Y
Just put substitute the X in the initial equation, X+Y=15, with the 4Y and then solve the equation for both X and Y.

So, we can actually get the answer using only 1 of statement. So why is the answer C.

Billion thanks in advance!

Welcome to GMAT Club!

Actually the correct answer is C, not D.

From the stem the question became: is x + y = 15?

Now, (2) says that x - y = 9. Can we tell from this whether x + y = 15? No! Consider x = 10 and y = 1 for a NO answer and x = 12 and y = 3 for an YES answer. Hence, the first statement is NOT sufficient.

The same for the second statement.

The problem with your solution is that you assumed that we have two equations for each statement, whereas we have just one: x - y = 9 for (1) and y/x = 1/4 for (2). The second equation, x + y = 15 is not given to be true, we are asked to find whether it's true.

Hope it's clear.
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Here's the approach I did, it took me roughly 5 minutes to complete, would you be able to tell me how to get faster, and if i plugged Y into the correct question.

From the Q.Stem I simplified the Question down to, does "x+y=15" Yes/No.

Statement 1) x-y=9
x+y=9+y
X=9+Y

I used substation, plugged in X with the original equation, and I got 9+2y=15, reduced down to y=3.

So the question is reduced to "does Y=3, yes or no" from what I know, I'm Not sure. So therefore it's not sufficient.

Statement 2) y/x=1/4 is simplified to 4Y=X,

I Plugged in to the original equation X+Y=15
So now I have 4Y+Y=15
5Y=15, The question is now rephrased as does y equal 3? again, it's not sufficient.

(C) X+Y=15
From Statement 1 & Statement 2, Y=3, I plugged in the following

X+Y=15
X-3=-3
X=12

Banuel, If I went wrong anywhere, can you tell me where I went wrong. Should I test the answer choice to see if it's insufficient, instead of labeling the answer not sufficient?

For statement one, I found myself answering, yes it's sufficient to solve, but X is not given, so I should not assume that the number is automatically "x=12" correct? The simplification is reducing the question still" Does Y=3? and that changes it all, because I don't know if Y=3. So is that enough information to move on? Same reasoning on the second statement. Threw me off for a bit.

5 minutes is too much for this problem.

The question boils down to whether \(x+y=15\).

(1) says x - y = 9. Can we answer whether \(x+y=15\)? No, because infinitely many pairs of (x, y) satisfy x - y = 9, and only one of them yields the sum of 15, namely x=12 and y=3.

(2) says x=4y. Basically the same here: can we answer whether \(x+y=15\)? No, because infinitely many pairs of (x, y) satisfy x=4y, and only one of them yields the sum of 15.

When we combine the statements we have x-y=9 and x=4y. So, we have two two distinct linear equation with two unknowns, hence we can solve for both of them and get whether \(x+y=15\) is true.

As you can see we don't need to solve anything for this question to get the answer.

Hope it helps.
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Hi, could someone please help me evaluate my approach? I got stuck...

I did it this way:

\(4^x^+^y = 8^1^0\\
--> 4^x * 4^y = 2^1^0 * 4^1^0\\
--> 2^2^x * 4^y = 2^1^0 * 4^1^0\)

Now we have the same bases on the left and right side, so I can get rid of them:

\(2x * y = 10 * 10\)

By looking at the equation I concluded that x needs to be "5" (-> 2*5=10) and y needs to be "10". Then the left side would equal the right side.

From (1) we get that the difference of x and y is 9. I figured that given my equation, we need a difference of 5 (x=5 and y=10 -> difference between both is 5).


Can someone please help, if (i) my approach of simplifying the given equation was correct, and (ii) my train of thought is correct (apparently it is not)

Many thanks

Yes, x + y must be 15 for \(4^{x+y}=8^{10}\) to hold true but it's not necessary that x = 10 and y = 5, there are many other solutions.

For example, consider this if x = 0, then we'd have \(4^{y}=8^{10}\) --> \(2^{2y}=2^{30}\) --> y = 15. Or if x = 1, then we'd have \(4^{1+y}=8^{10}\) --> \(2^{2+2y}=2^{30}\) --> y = 14 ...

Hope it helps.
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We need to determine whether 4^(x+y) = 8^10. We start by breaking down our two bases into prime factors.

4^(x+y) = (2^2)^(x+y) = 2^(2x+2y)

8^10 = (2^3)^10 = 2^30

We can now rephrase the question as:

Is 2^(2x+2y) = 2^30 ?

Because the bases are the same, we can drop them and set the exponents equal to each other. The question becomes:

Is 2x+2y = 30 ?

Is x + y = 15 ?

After simplifying the equation, we see that we need to determine whether the sum of x and y is equal to 15.

Statement One Alone:

x – y = 9

Knowing the difference of x and y is not the same as knowing the sum of x and y; thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

y/x = ¼

When we cross multiply obtain:

4y = x

4y = x is not enough information to determine the value of x + y. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two we know the following:

x – y = 9 and 4y = x

Since 4y = x, we can substitute 4y for x into the equation x – y = 9 and we have:

4y – y = 9

3y = 9

y = 3

Since y = 3, x = 4(3) = 12.

Thus, x + y = 12 + 3 = 15. We can answer yes to the question. Both statements together are sufficient.

The answer is C.
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Target question: Is 4^(x + y) = 8^10?
This is a good candidate for rephrasing the target question.

Given equation: 4^(x + y) = 8^10
Rewrite 4 and 8 as powers of 2 to get: (2²)^(x + y) = (2³)^10
Apply power of a power law to get: 2^(2x +2y) = 2^30
This means that: 2x + 2y = 30
Divide both sides by 2 to get: x + y = 15
In other words, asking whether 4^(x + y) = 8^10 is the SAME as asking whether x + y = 15
REPHRASED target question: Is x + y = 15?

Statement 1: x - y = 9
Is this enough information to answer the REPHRASED target question? No.
Consider these two CONFLICTING cases:
Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15
Case b: x = 10 and y = 1. In this case, x + y = 10 + 1 = 11. So, x + y does NOT equal 15
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y/x = 1/4
Is this enough information to answer the REPHRASED target question? No.
Consider these two CONFLICTING cases:
Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15
Case b: x = 8 and y = 2. In this case, x + y = 8 + 2 = 10. So, x + y does NOT equal 15
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x - y = 9
Statement 2 tells us that y/x = 1/4
Since we have 2 different linear equations with 2 variables, we COULD solve the system for the individual values of x and y, which means we COULD answer the REPHRASED target question with certainty. Of course, we wouldn't waste precious time performing such calculations, since our sole goal is to determine the sufficiency of the combined statements.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

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Hey Guys,

Quick question from my side. Why are we allowed in Statement B to make this move:

y/x=1/4 -> 4y = x

I am confused, because we don`t know for sure that either x or y are really positive and hence we would not be allowed to divide or multiple a variable.

Thank you for your help!

We are concerned about the sign when dealing with inequalities: we should keep the sign if we multiply by a positive value and flip the sign when we multiply by a negative value.

For equations we can multiply/divide by a variable regardless of its sign (providing it's not 0).

Hope it's clear.
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For statement 1 to prove insufficient, shouldn't we be looking for a case that gives us 9 and adds to get 15?

None of the samples above give us x + y = 15 so only half the work is done.

24 - 9 is 15
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Hi Bunuel, QQ:

from stem I got, (4)^x+y= 8^10

((2)^2)^x+y= ((2)^3)^10

2x+2y=30
x+y=15 << stem

st1: x-y= 9 << 1

subtract x+y=15
x-y=9

2y=6
y=3, from stem equation, x+y=15, y=3, thus, x=12

4^x+y= 4^ 15= 8^10. where did I go wrong?

The stem does not say that x + y = 15, it asks whether x + y = 15. So, when considering the first statement we don't have the system of equations x + y = 15 and x - y = 9. We have only x - y = 9 and want to find whether x + y = 15, which is obviously impossible to do knowing only that x - y = 9.

Hope it's clear.

P.S. Please check thus Writing Mathematical Formulas on the Forum
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