<p>If 0 ≤ x ≤ y and (x + y)^ 2 − (x − y)^ 2 ≥ 25, what
is the least possible value of y ?</p>
<p>I guess its not entirely interesting, but is there a mathematical way to solve this problem. I sort of guestimated by just intuitively saying that if both are 2.5, then it would work.</p>
<p>Well if you simplify the expression you get xy ≥ 25/4 and if you know that x ≤ y then the lowest possible value of y is attained when x = y. 2.5 is right.</p>
<p>okay, thanks. also.
A store discounts merchandise by 10 percent of the
original price at the end of each week and stops when
the merchandise is priced at 50 percent of the original
price. Which of the following graphs could correctly
represent the price of an article of merchandise over
an eight-week period?</p>
<p>I got the question right, but this is for the future. If the original price is 500 dollars, then at the end of week one, the price is 450. Then for week 2, is it 20 percent of the original price-500? or is it 10 percent of 450? thanks I just wanted to clarify the meaning</p>
<p>To be more clear about the mathematical explanation on your first question:
(x + y)^ 2 − (x − y)^ 2 ≥ 25
x^2 + 2xy + y^2 − x^2 + 2xy − y^2 ≥ 25
4xy ≥ 25
y ≥ 25 / (4x)</p>
<p>Let’s look at it graphically. So we know y ≥ x and y ≥ 25 / (4x) and we’re in the 1st Quadrant (x and y are both ≥ 0). Graph y = x and y = 25 / (4x). Here it is in wolfram alpha: [Plot[{25/(4</a> x), x}, {x, 0,4}] - Wolfram|Alpha](<a href=“http://■■■■■■/97G8Dj]Plot[{25/(4”>http://■■■■■■/97G8Dj). For both of these inequalities to hold, we must be ABOVE BOTH of these lines. The lowest value of y occurs where the 2 lines intersect. Substituting one equation in the other, we get:
y = 25/4y
y^2 = 25/4
y = 5/2 or 2.5</p>
<p>Is this question from the SAT? That’s a tough one!</p>
<p>^^It’s 10% of 450.The store further dicounts the merchandise by 10% every succesive week.</p>
<p>Is the 1st question an official BB one?It is indeed tougher than usual SAT questions.</p>
<p>^Yes, it appeared on the March 2005 exam, last question in a math section.</p>
<p>No, it’s 10% of the original price of 500. Re-read the question.</p>
<p>No its 10 percent of 450.</p>
<p>10% of 450 is 45 but 20% of 500 is 100 which is 5 more. Hope this helps.</p>
<p>@fignewton-
“10 percent of the
original price at the end of each week”
Every succesive week,the ‘new’ original price shall be the price of the previous week.</p>
<p>While most problems which involve multiple percent discounts require you to use the most recent value to find your next percent change, this problem clearly refers you back to the original value. There is no such thing as “the most recent original value”! If they meant for you to use the last value, they would have said “the price is reduced by 10% of the previous week’s value”. And again, that’s what more expreienced test-takers would expect. But not this time.</p>
<p>Uh.Ohhh.
I think, fignewton it shall be better if you tell us which method you applied to get the correct answer.We’re all confused here now!(i know I am!)</p>
<p>^Hmmm, I wasn’t really involved in the problem-solving aspect of this thread. :)</p>
<p>Is the confusion about the percents problem? I think pckeller is correct: by “original price”, the question means the unchanging, very first price.</p>
<p>So if the first price is 500, the sequence of prices is 500-450-400-350-300-250. In this way, the percents add, so the 2nd week’s price is 20% off the (unchanging) original price, the third week’s price is 30% off, etc. </p>
<p>Note that if, instead, you take 10% off each successive price (i.e., you interpret “original price” to mean the “previous price”), you will <em>never</em> have a price that is 50% of the original price, whether you mean the unchanging version of original or the changing version (the reason a math teacher would use: (1.0-0.1)^n will never equal 0.5 for integer n).</p>
<p>
</p>
<p>Rule Number One:</p>
<p>To answer official SAT questions, you always need the FULL context. In this case, there are a number of graphs to choose from. Hence, there is a POE available. </p>
<p>This said, this problem is not confusing, as long as you do NOT introduce more “logic” than needed. Simply follow the instructions of the problem. </p>
<p>Minus 10 percent of the original price =
<p>So that means successive discounting of 50 each week. And the weeks will be</p>
<p>w1 500
w2 450
w3 400
w4 350
w5 300
w6 250 </p>
<p>Note you can start with Week 0 and decide to label your weeks at the start of the week or at the end. This makes no difference for the final answer, which consists of selecting the right graph.</p>
<p>Thanks for clearing that up fignewton,xiggi!</p>