https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Rguan&feedformat=atomAoPS Wiki - User contributions [en]2022-01-20T08:05:47ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_11&diff=451122012 AMC 10B Problems/Problem 112012-02-24T21:53:56Z<p>Rguan: Created page with "== Problem 11 == A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same desse..."</p>
<hr />
<div>== Problem 11 ==<br />
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?<br />
<br />
<math> \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\2304 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 11|Solution]]<br />
<br />
<br />
<br />
== Solutions ==<br />
We have 4 choices for desserts.<br />
<br />
However, the same dessert cannot be served for 2 straight days, meaning that you only have 3 choices for a dessert for the next day. <br />
It is also given that there must be cake on Friday.<br />
<br />
So,<br />
<br />
<math>4*3*3*3*3*1*3=\boxed{972}</math><br />
<br />
OR<br />
<br />
<br />
<math> \textbf{(B)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_10&diff=451112012 AMC 10B Problems/Problem 102012-02-24T21:30:42Z<p>Rguan: Created page with "== Problem 10 == How many ordered pairs of positive integers (M,N) satisfy the equation <math>\frac {M}{6}</math> = <math>\frac{6}{N}</math> <math> \textbf{(A)}\ 6\qquad..."</p>
<hr />
<div>== Problem 10 ==<br />
How many ordered pairs of positive integers (M,N) satisfy the equation <math>\frac {M}{6}</math> = <math>\frac{6}{N}</math><br />
<br />
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\10 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
<br />
<br />
== Solution ==<br />
<br />
<math>\frac {M}{6}</math> = <math>\frac{6}{N}</math><br />
<br />
is a ratio; therefore, you can cross-multiply.<br />
<br />
<math>MN=36</math><br />
<br />
Now you find all the factors of 36:<br />
<br />
1*36=36<br />
<br />
2*18=36<br />
<br />
3*12=36<br />
<br />
4*9=36<br />
<br />
6*6=36.<br />
<br />
Now you can reverse the order of the factors for all of the ones listed above, because they are ordered pairs except for 6*6 since it is the same back if you reverse the order.<br />
<br />
<math>4*2+1=\boxed{9}</math> <br />
<br />
OR<br />
<br />
<math> \textbf{(D)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_9&diff=451102012 AMC 10B Problems/Problem 92012-02-24T21:23:37Z<p>Rguan: Created page with "== Problem 9 == Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of th..."</p>
<hr />
<div>== Problem 9 ==<br />
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
<br />
<br />
== Solutions ==<br />
<br />
Lets say that all 6 integers added are : a,b,c,d,e, and f.<br />
<br />
If a+b=26<br />
<br />
and a+b+c+d=41<br />
<br />
Then,<br />
c+d=15<br />
<br />
Also, <br />
<br />
a+b+c+d+e+f=57<br />
<br />
a+b+c+d=41<br />
<br />
Then,<br />
e+f=16<br />
<br />
<br />
So<br />
<br />
a+b=26<br />
<br />
c+d=15<br />
<br />
e+f=16<br />
<br />
a,b,e,f can be all odd since odd + odd= even. And the sum of the two respective pairs are even.<br />
<br />
However, either c or d has to be even to get a odd sum.<br />
<br />
Therefore, there is <math>\boxed{1}</math> even integer<br />
<br />
OR<br />
<br />
<math> \textbf{(A)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_8&diff=451092012 AMC 10B Problems/Problem 82012-02-24T20:50:06Z<p>Rguan: Created page with "== Problem 8 == What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>? <math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19..."</p>
<hr />
<div>== Problem 8 ==<br />
<br />
What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\25 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
<br />
<br />
<br />
== Solutions ==<br />
<br />
<math>(x-2)^2</math> = perfect square.<br />
<br />
1< perfect square< 25<br />
<br />
Perfect square can equal: 4, 9, or 16<br />
<br />
Solve for x:<br />
<br />
<math>(x-2)^2=4</math> <br />
<br />
<math>x=4</math><br />
<br />
and<br />
<br />
<math>(x-2)^2=9</math><br />
<br />
<math>x=5</math><br />
<br />
and<br />
<br />
<math>(x-2)^2=16</math><br />
<br />
<math>x=6</math><br />
<br />
''What is the sum of all integer solutions''<br />
<br />
<math>4+5+6=\boxed{15}</math><br />
<br />
OR<br />
<br />
<math> \textbf{(C)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_7&diff=450822012 AMC 10B Problems/Problem 72012-02-24T04:02:35Z<p>Rguan: Created page with "== Problem 7 == For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid..."</p>
<hr />
<div>== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
<br />
== Solutions ==<br />
x=number of acorns that both animals had.<br />
<br />
<math>\frac{x}{3}=(\frac{x}{4})+4</math><br />
<math>\frac{x}{12}=4</math><br />
<math>\boxed{x=48}</math><br />
<br />
OR<br />
<br />
<math> \textbf{(D)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_6&diff=450812012 AMC 10B Problems/Problem 62012-02-24T03:44:34Z<p>Rguan: Created page with "== Problem 6 == In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, an..."</p>
<hr />
<div>== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
<br />
<br />
== Solutions ==<br />
<br />
Say Z=is the amount rounded up by and down by. <br />
<br />
''Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values''.<br />
<br />
Which translates to:<br />
<br />
<math>(X+Z)-(Y-Z)</math>=<math>X+Z-Y+Z</math>=<math>X+2Z-Y</math><br />
<br />
This is 2Z bigger than the original amount of <math>X-Y</math>.<br />
<br />
Therefore, her estimate is larger than <math>X-Y</math><br />
<br />
<br />
or<br />
<br />
<br />
<math> \textbf{(A)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_5&diff=450802012 AMC 10B Problems/Problem 52012-02-24T03:34:42Z<p>Rguan: Created page with "== Problem 5 == Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax i..."</p>
<hr />
<div>== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
<br />
== Solutions ==<br />
<br />
Let X be the cost of her dinner. <br />
<br />
<math>27.50=X+\frac{1}{10}*X+\frac{3}{20}*X</math><br />
<br />
<math>27+\frac{1}{2}=\frac{5}{4}*X</math><br />
<br />
<math>\frac{55}{2}=\frac{5}{4}X</math><br />
<br />
<math>\frac{55}{2}*\frac{4}{5}=X</math><br />
<br />
<math>\boxed{22=X}</math><br />
<br />
'''OR'''<br />
<br />
<math> \textbf{(D)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_4&diff=450792012 AMC 10B Problems/Problem 42012-02-24T03:27:17Z<p>Rguan: </p>
<hr />
<div>== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
<br />
== Solution ==<br />
<br />
<br />
In total, there were <math>3+4=7</math> marbles left from both Ringo and Paul. <math>7/6</math>=1R1. <math>\text{This means that that there is}</math> <math> \boxed{1}</math> <math>\text{marbles left}</math> or <br />
<br />
<math> \textbf{(A)}</math></div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_4&diff=450782012 AMC 10B Problems/Problem 42012-02-24T03:26:14Z<p>Rguan: </p>
<hr />
<div>== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
<br />
In total, there were <math>3+4=7</math> marbles left from both Ringo and Paul. <math>7/6</math>=1R1. <math>\text{This means that that there is}</math> <math> \boxed{1}</math> <math>\text{marbles left}</math> or <br />
<br />
['''A''']</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450772012 AMC 10B Problems2012-02-24T03:23:31Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
[[2012 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
<br />
== Problem 8 ==<br />
<br />
What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\25 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
<br />
== Problem 10 ==<br />
How many ordered pairs of positive integers (M,N) satisfy the equation <math>\frac {M}{6}</math> = <math>\frac{6}{N}</math><br />
<br />
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\10 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
<br />
<br />
== Problem 11 ==<br />
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?<br />
<br />
<math> \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\2304 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 11|Solution]]<br />
<br />
<br />
<br />
== Problem 12 ==<br />
<br />
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is <math>10\sqrt 2</math>, and <math>\angle BAC</math>= 45 degrees. Point D is 20 meters due North of point C. The distance Ad is between which two integers?<br />
<br />
<br />
('''A''') 30 and 31 ('''B''') 31 and 32 ('''C''') 32 and 33 ('''D''')33 and 34 ('''E''')34 and 35 <br />
<br />
[[2012 AMC 10B Problems/Problem 12|Solution]]<br />
<br />
<br />
<br />
== Problem 13 ==<br />
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?<br />
<br />
<math> \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\52 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 12|Solution]]<br />
<br />
<br />
<br />
== Problem 14 ==<br />
Two equilateral triangles are contained in square whose side length is <math>2\sqrt 3</math>. The bases of these triangles are the opposite side of the square, and their intersection is a rhombus. What is the area of the rhombus?<br />
<br />
<math><br />
\text{(A) } \frac{3}{2}<br />
\qquad<br />
\text{(B) } \sqrt 3<br />
\qquad<br />
\text{(C) } 2\sqrt 2 - 1 <br />
\qquad<br />
\text{(D) } 8\sqrt 3 - 12<br />
\qquad<br />
\text{(E)} \frac{4\sqrt 3}{3} <br />
</math><br />
<br />
[[2012 AMC 10B Problems/Problem 14|Solution]]<br />
<br />
<br />
<br />
==Problem 15==<br />
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?<br />
<br />
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\6 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 15|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450712012 AMC 10B Problems2012-02-24T02:51:36Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
[[2012 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
<br />
== Problem 8 ==<br />
<br />
What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\25 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
<br />
== Problem 10 ==<br />
How many ordered pairs of positive integers (M,N) satisfy the equation <math>\frac {M}{6}</math> = <math>\frac{6}{N}</math><br />
<br />
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\10 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450652012 AMC 10B Problems2012-02-24T02:48:15Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
[[2012 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
<br />
== Problem 8 ==<br />
<br />
What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\25 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450582012 AMC 10B Problems2012-02-24T02:43:20Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
[[2012 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
<br />
== Problem 8 ==<br />
<br />
What is the sum of all integer solutions to <math>1<(x-2)^2<25</math>?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\25 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 8|Solution]]<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450572012 AMC 10B Problems2012-02-24T02:40:17Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
[[2012 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450552012 AMC 10B Problems2012-02-24T02:37:58Z<p>Rguan: /* Problem 6 */</p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0<br />
<br />
== Problem 7 ==<br />
<br />
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
<br />
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450542012 AMC 10B Problems2012-02-24T02:35:00Z<p>Rguan: </p>
<hr />
<div>== Problem 1 ==<br />
<br />
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
<br />
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
<br />
<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
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[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
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== Problem 4 ==<br />
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When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
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<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
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[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
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== Problem 5 ==<br />
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Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
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<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
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[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
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== Problem 6 ==<br />
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In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
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A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0.<br />
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== Problem 7 ==<br />
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For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?<br />
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<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54 </math><br />
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[[2012 AMC 10B Problems/Problem 7|Solution]]<br />
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== Problem 23 ==<br />
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A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
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<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
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[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
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== Problem 24 ==<br />
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Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
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<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
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[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems&diff=450532012 AMC 10B Problems2012-02-24T02:31:02Z<p>Rguan: </p>
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<div>== Problem 1 ==<br />
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Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?<br />
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<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80 </math><br />
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[[2012 AMC 10B Problems/Problem 1|Solution]]<br />
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== Problem 2 ==<br />
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A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?<br />
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<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math><br />
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[[2012 AMC 10B Problems/Problem 2|Solution]]<br />
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== Problem 3 ==<br />
<br />
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?<br />
<br />
<math> \textbf{(A)}\ (998,2012)\qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) </math><br />
<br />
[[2012 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?<br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? <br />
<br />
A) Her estimate is larger than x-y B) Her estimate is smaller than x-y C) Her estimate equals x-y D) Her estimate equals y - x E) Her estimate is 0.<br />
== Problem 23 ==<br />
<br />
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?<br />
<br />
<math> \textbf{(A)}\ \frac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} </math><br />
<br />
[[2012 AMC 10B Problems/Problem 23|Solution]]<br />
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== Problem 24 ==<br />
<br />
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?<br />
<br />
<math> \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 </math><br />
<br />
[[2012 AMC 10B Problems/Problem 24|Solution]]</div>Rguanhttps://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10B_Problems/Problem_4&diff=450472012 AMC 10B Problems/Problem 42012-02-24T02:18:37Z<p>Rguan: Created page with "In total, there were <math>3+4=7</math> marbles left from both Ringo and Paul. <math>7/6</math>=1R1. <math>\text{This means that that there is}</math> <math> \boxe..."</p>
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<div>In total, there were <math>3+4=7</math> marbles left from both Ringo and Paul. <math>7/6</math>=1R1. <math>\text{This means that that there is}</math> <math> \boxed{1}</math> <math>\text{marbles left}</math> or <br />
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['''A''']</div>Rguan