Help! Hard math questions!

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1

<ol>
<li><p>Marbles are to be removed from a jar that contains 12 red marbles and 12 black marbles. What is the least number of marbles that could be removed so that the ratio of red marbles to black marbles left in the jar will be 4:3?
(If someone can extent further with this that will be great, like how to tackle ratio problems.. I have hard time with ratios...)</p></li>
<li><p>If 0 <= (less than or equal to) x <= (less than or equal to) y and (x+y)^2 - (x-y)^2 >= 25, what is the least possible value of y?</p></li>
<li><p>If (p+1)(t-1)=0 and p is positive, what is the value of t?
a) -3
b) -1
c) 0
d) 1
e) 3</p></li>
<li><p>A stamp collecting club calculated that the average (arithmetic mean) number of stamps in its members' 10 collections was 88. However, it was discovered that 2 numbers in the calculations were entered incorrectly. The number 55 was entered as 75 and the number 78 as 88. What is the correct average number of stamps in the 10 collections?</p></li>
</ol>

<p>a) 91
b) 89
c) 87
d) 86
e) 85</p>

<ol>
<li>R is the midpoint of line segment PT, and Q is the midpoint of line segment PR. If S is a point between R and T such that the length of segment QS is 10 and the length of segment PS is 19, what is the length of segment ST?</li>
</ol>

<p>a) 13
b) 14
c) 15
d) 16
e) 17</p>

<ol>
<li>The interior dimensions of a rectangular fish tank are 4 feet long, 3 feet wide, and 2 feet high. The water level in the tank is 1 foot high. All of the water in this tank is poured into an empty second tank. If the interior dimensions of the second tank are 3 feet long, 2 feet wide, and 4 feet high, what is the height of the water in the second tank?</li>
</ol>

<p>a) 0.5 ft
b) 1 ft
c) 1.5 ft
d) 2 ft
2) 4 ft</p>

<ol>
<li>Each of the following inequalities is true for some values of x EXCEPT?</li>
</ol>

<p>a) x< x^2 < x^3
b) x< x^3 < x^2
c) x^2 < x^3 < x
d) x^3 < x < x^2
e) x^3 < x^2 < x</p>

<p>Please explain thoroughly (I just don't want the answers, because I have the answers ... duh), step by step. Thank you!</p>

2

<p>1.) 12/x = 4/3
x = 9. So we want to remove 3 marbles to get this ratio. </p>

<p>2.) Expand and you’ll see that the squared terms go away and you’re left with 4xy > 25
xy = 25/4
Least possible value of y occurs at the equality of x and y, so xy = 25/4 = y^2; y = ±5/2, but since they want a positive value only 5/2 works. </p>

<p>3.) 1. Since the expression needs to multiply to zero, and since the term with p in it can’t be 0, the t term needs to equal 0. t - 1 = 0 -> t = 1</p>

<p>4.) Incorrect sum of stamps: 88*10 = 880. Subtract 20 and 10 to correct for this to yield a sum of 850 for the 10 stamps. To average, 850/10 -> 85. </p>

<p>5.) PQ = QR; Since QS is 10 and PS in 19, we know that adding on PQ adds an additional 9 units, which means that QR is also 9. We also now know that RS is 1. Since PR = RT, we know that 18 = ST + 1, which means ST = 17. </p>

<p>6.) Volume of tank 1 is 4<em>3</em>2 = 24, but since it’s half full we know the water volume is 12 ft^3 . so now we take the volume of the new container, 3<em>2</em>4 = 24, which is the same before, but the distribution along the axes is different, so we must halve the height of the new box to get 2 ft.</p>

<p>7.) Analyze it case by case. </p>

<p>(a) True for any integer n, where n > 1.
(b) True a negative number in the interval -1 < x < 0
(c) Fails.
(d) True for any negative integer.
(e) True for any number on the interval 0 < x < 1</p>

<p>So your answer is choice (c)</p>