<p>1) In an art class, there were just enough staplers, rulers
and glue bottles so that every 2 students had to share a
stapler, every 3 students had to share a ruler, and every
4 students had to share a glue bottle. If the sum of the
number of staplers, rulers, and glue bottles used by the
class was 65, how many students were in the class?</p>
<p>2) If a and b are integers such that a + b < 1000 and a/b = 0.625, what is the greatest possible value of b?</p>
<p>3) For all values of y, let y@ be defined by y@ = y^2 - 1.Which of the ff is equal to (y@)@ ?
A) y^4 - 1
B) y^4 - y^2 - 1
c) y^4 + y^2 - 1
D) y^4 - 2y^2
E) y^4 - 2y^2 + 1</p>
<p>4) According to the formula F= 9/5*c + 32, if the degrees in fahrenheit(F) increases by 27
,by how much does the temperature in degrees Celcius increase?
A) 9
B) 15
C) 47
D) 243/5
E) 59</p>
<p>2) If a and b are integers such that a + b < 1000 and a/b = 0.625, what is the greatest possible value of b?</p>
<p>From the second equation a=0.625b. So a + b = 0.625b+b = 1.625b = 13/8 b.</p>
<p>So a+b = c (where c is less than 1000) becomes b = 8/13 c. So we want to choose c to be divisible by 13 and as close to 1000 as possible. So c = 988. Thus b = 8/13 * 988 = 608.</p>
<p>(1) The number of students must be divisible by 2, 3 and 4. Since lcm(2,3,4)=12, the number of students must be a multiple of 12. Let’s take a guess. If we guess there are 60 students, then there are 60/2=30 staplers, 60/3=20 rulers, and 60/4=15 glue bottles, for a total of 30+20+15=65. It works! So there are 60 students in the class.</p>
<p>An algebraic solution:</p>
<p>If there are x students in the class, then there are x/2 staplers, x/3 rulers, and x/4 glue bottles.</p>
<p>So x/2 + x/3 + x/4 = 65.</p>
<p>Multiplying each side by 12 (the lcd) gives:</p>
<p>3) The radius of the circle is the length of the diagonal of a little square (why?), and therefore r=3sqrt(2) (again, why?). So the area of the circle is 18pi.</p>
<p>1) Here’s one way: x+y-x=(x+y)/2-(x+y). So 2y=x+y-2x-2y, and thus x=-3y. Replace x by -3y in the picture, and it’s easy to see that the answer is E</p>
<p>I can see 2 other ways to do this. One by using an average, and another by picking numbers.</p>
I’d like to point to a common mistake people make answering this question.</p>
<p>a = 0.625b
0.625b + b <1000
1.625b < 1000
b < 615.38…
so the greatest possible (integer) value of b is 615.</p>
<p>This answer is wrong because for b = 615
a = 0.625b
a = 384.375 - not an integer.</p>
<p>+++++++++++++++++++
Here’s an approach a bit different from DrSteve’s:
a/b = 0.625
a/b = 5/8
a = 5x, b = 8x,where x is an integer.
a + b < 1000
5x + 8x < 1000
13x < 1000
x < 76.9…
The greatest possible integer value of x is 76, therefore
the greatest possible integer value of b is (8)76 = 608.</p>
<p>4) F = 9/5C (ignore constant) - note that this is a linear relationship.
if F = 0, C = 0
if F = 27, C = 5/9(27)
Therefore, for every increase of 27 in F, C increases by 15 (D)
15 (B)</p>