<p>Can someone explain these?</p>
<p>1) For any positive integer a: ((a)) = 1/2^a+1</p>
<p>What is the ratio of ((a+3)) to ((a))</p>
<p>Answer: 1/8</p>
<p>2) Each of 100 cards has none, one, or two of the letters A and C written on it. If 75 cards have the letter A, 30 have the letter C, and fewer than 15 are blank, what is the largest possible number of cards that have both A and C written on them?</p>
<p>Answer: 19</p>
<p>Number of cards with A + number of cards with B +number of cards blank - number of cards containing both A and C = total number of cards</p>
<p>75+30+14(fewer than 15)-n=100
119-n=100
-n=-19
n=19</p>
<p>Here is a quick algebraic solution to the first one:</p>
<p>I assume you mean ((a)) = (1/2)^(a+1). Under that assumption we have:</p>
<p>((a+3))/((a)) = [(1/2)^(a+4)] / [(1/2)^(a+1)] = (1/2)^3 = 1/8.</p>
<p>Note: I have simply subtracted exponents: (a+4)-(a+1) = a+4-a-1 = 3</p>
<p>Alternative solution for number 1: just pick a value for a, say a=2. </p>
<p>Then ((a)) = ((2)) = (1/2)^3 = 1/8</p>
<p>((a+3)) = ((5)) = (1/2)^6 = 1/64</p>
<p>((a+3))/((a)) = (1/64)/(1/8) = 1/8 (use your calculator).</p>