Which of the following is greatest?

Which of the following is greatest?

A. \(10\sqrt{3}=\sqrt{100*3}=\sqrt{300}\)

B. \(9\sqrt{4}=\sqrt{81*4}=\sqrt{324}\)

C. \(8\sqrt{5}=\sqrt{64*5}=\sqrt{320}\)

D. \(7\sqrt{6}=\sqrt{49*6}=\sqrt{294}\)

E. \(6\sqrt{7}=\sqrt{36*7}=\sqrt{252}\)

Answer: B.
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We can use a nice rule that says (√x)(√y) =√(xy)
Also, we'll rewrite each integer as a root.
For example 10 = √100

So, we get...
A) 10√3 = (√100)(√3) = √300
B) 9√4 = (√81)(√4) = √324
C) 8√5 = (√64)(√5) = √320
D) 7√6 = (√49)(√6) = √294
E) 6√7 = (√36)(√7) = √252

√324 is the greatest

Answer: B

Say you extend the list of numbers in the answer choices to go further in both directions:

13√0, 12√1, 11√2, 10√3, ..., 3√10, 2√11, 1√12, 0√13

Notice now that the first number in the list is zero, and the last number is zero. So these numbers can't constantly increase going from left to right, nor can they constantly decrease. They must first increase from zero, then decrease back to zero. If you want to see what this looks like graphically, you can plug the function "y = (13 - x)(√x)" into an online graphing calculator (the answer choices are the points on the graph where x equals the integers from 3 through 7). You'll see you get a curve resembling a downwards parabola which peaks somewhere around x = 4, so it peaks roughly around where our function equals 9√4.

More conceptually, if you take two positive numbers a and b, and find their product ab, and then you decide to add 1 to a, and subtract 1 from b, and find the new product (a+1)(b-1), there's no way to tell if the new product is larger or smaller than the old one, as you can see by first imagining a = 1 and b = 100 (then the new product 2*99 is almost double the old product of 1*100) and then by imagining a = 100 and b = 2 (then the new product 101*1 is roughly half of the old product of 100*2). But instead if I tell you we'll increase a by, say, 20%, and we'll decrease b by, say, 10%, then we can tell if our product will increase: our new product becomes 1.2a * 0.9b = 1.08ab, and our new product is 8% bigger than our old one. So that's what matters: we care about how we're changing each number in percent terms, or in ratio terms, and we don't care about how much we're adding or subtracting. In this question, going from say 4√9 to 5√8, we're increasing '4' by 25% to get to '5', and we're decreasing '√9' by only about 6% to go to '√8', so our product will grow (since 1.25*0.94 is greater than 1). But going from say 9√4 to 10√3, we're now only increasing the '9' by roughly 11%, and we're decreasing the √4 by about 13%, and that will lead to a small decrease.

And for interest only (irrelevant on the GMAT) -- finding precisely where this product (13 - x)(√x) is at a maximum is a calculus problem. In calculus, when we find the "derivative" of a function, we're finding a new function which tells us the slope at every point of our original function. Here we need to use the "product rule" to find the derivative of our function (if you haven't learned calculus, this will all seem very mysterious, but it's something you'll use a lot in an MBA, never on the GMAT though) to find the derivative of y = (13 - x)(√x), and doing that, we learn that the derivative is y' = [ (13 - x) / 2√x ] + (-1)(√x) = [ (13 - x)/2√x ] - √x. When a curve is at its maximum or minimum point, a tangent line to that curve is horizontal, so it has a slope of zero. So if we set the derivative equal to zero, we can locate the maximum point on our original function:

(13 - x)/2√x - √x = 0
(13 - x)/2√x = √x
13 - x = 2x
x = 13/3

So at x = 13/3, our function is at its maximum. Plugging x = 13/3 into our original function y = (13 - x)(√x), we find that function will be at its peak when y = (26/3)(√(13/3)) ~ (8.66)(√4.33). If you compute the numerical value of that product, it's roughly 18.06, so slightly larger than the OA to this question, 9√4 = 18.

A.
Other option have a simultaneous increase in denominator and decrease in denominator.

Sent from my Moto G (5) Plus using GMAT Club Forum mobile app

A. 10*1.732 = 17.32
B. 9*2 = 18
C. 8*2.XX = 16.XX
D. 7*2.XX = 14.XX
E. 6*2.XX = 12.XX

Hence, correct answer will be (B)
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i think you misunderstood the ques.
it is not division., it is square root.
integer* sq.root of an integer

B. is the answer

A.) square it = 300
B.) square it = 324
C.) square it = 320
D.) square it = 294
E.) square it = 252

Let’s estimate √3, √5, √6, and √7.

√3 ≈ 1.7

√5 ≈ 2.2

√6 ≈ 2.4

√7 ≈ 2.7

Thus:

A) 10 x 1.7 = 17

B) 9 x 2 = 18

C) 8 x 2.2 = 17.6

D) 7 x 2.4 = 16.8

E) 6 x 2.7 = 16.2

Alternate Solution:

Another way to get rid of the roots to compare each answer is to square each answer choice:

A) (10√3)^2 = 100 x 3 = 300

B) (9√4)^2 = 81 x 4 = 324

C) (8√5)^2 = 64 x 5 = 320

D) (7√6)^2 = 49 x 6 = 294

E) (6√7)^2 = 36 x 7 = 252

Answer: B
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HI Bunuel,

I squared all the answer choices and then reached my result. Is that method correct?

Yes, it is correct. Since all the numbers are greater than 1, the square of a greater number will be greater.

Best,
Gladi


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There is a bit of a problem with this method.

8 * 2.XX needn't necessarily be 16.YY

In fact, 8*2.3 = 18.4
8*2.6 = 20.8
8*2.9 = 23.2
and so on...

So check the method used by Bunuel above.

B is correct answer as 9*4^1/2=9*2=18

so OA is B

Move everything under square root:

A. 100*3 = 300
B. 81*4 = 324
C. 64* 5 = 320
D. 49*6 = 294
E. 36*7 = <300

ANSWER: B

Hi All,

We’re asked which of the 5 answer choices is GREATEST. Calculating all of those square roots would be tedious; thankfully, we can modify all 5 answer choices in the same way so that we can AVOID doing those specific calculations.

From the wording of the prompt, we know that one of the 5 answers is clearly largest, so if we multiplied all the answers by 2, then the largest answer would still be largest. If we added 1 to each answer, then the largest answer would still be greatest. Etc.

So what can we do to eliminate all of those square root signs? SQUARE each answer…

Answer A becomes (10)(10)(3) = 300
Answer B becomes (9)(9)(4) = 324
Answer C becomes (8)(8)(5) = 320
Answer D becomes (7)(7)(6) = 294
Answer E becomes (6)(6)(7) = 252

Final Answer:

GMAT Assassins aren’t born, they’re made,
Rich

Answer: Option B

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IanStewart KarishmaB avigutman BrentGMATPrepNow could you guys share your views on the below thought proces. I want to understand how and why the numbers are functioning in a certain way.


From (A) to (E) we see that the integer value keeps falling by 1 unit i.e. 10 - 9 - 8... while the value inside the root keeps increasing by 1 unit i.e. 3 - 4 - 5...

But still, we see that when 10 drops to 9 and when 3 increases to 4 the overall value of the expression increases BUT from C onwards as we continue to follow this pattern (i.e. drop the integer value by 1 unit AND increase the root value by 1) the overall expression keeps falling. Why?? I mean I get the entire calculation in order to solve this question but it was interesting to note that there is a pattern being followed in the answer choices, however, this pattern doesn't help much.

What I also realized is that from (A) to (B) the integer value drops by one i.e. 10 --> 9 while the value of the root increases by 0.3 i.e. √3 --> √4 But from (C) to (E) the value of the root is all most the same i.e. √4 ---> √5 is a 0.2 increase. From √5 to √6 is a 0.2 increase and from √6 to √7 is again a 0.2 increase

Hoozan

The decrease is of 1 each from 10 to 9 to 8 etc but from root 3 to root 4 to root 5 etc, the numbers are increasing by different amounts. Hence, it doesn't follow a defined increasing or decreasing pattern. Also note that even if the increase/decrease is by the same amount, the relation is a quadratic.

Say we compare

100*2
95*3
90*4
... and so on

f(x) = (100 - 5a)(2 + a)
a goes from 0 to 1 to 2 etc...

This is a quadratic. We know that a quadratic is a parabola. So whether the values will be increasing or decreasing or will reach a maximum and then decrease or reach a minimum and then increase depends on what part of the parabola you are dealing with.

But yes, keep thinking of patterns and numbers! Good job!

The simplest approach for me is to square everything.

A) 100*3 = 300
B) 81*4 = 324
C) 64*5 = 320
D) 49*6 = <300
E) 36*7 = <300

B is correct.
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Thank you so much for this Was a great read and a super cool conceptual understanding