Hoozan wrote:
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
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.
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