Difference between revisions of "2008 AMC 8 Problems"
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− | + | {{AMC8 Problems|year=2008}} | |
+ | |||
+ | ==Problem 1== | ||
+ | Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend? | ||
+ | |||
+ | <math>\textbf{(A)}\ 12 \qquad | ||
+ | \textbf{(B)}\ 14 \qquad | ||
+ | \textbf{(C)}\ 26 \qquad | ||
+ | \textbf{(D)}\ 38 \qquad | ||
+ | \textbf{(E)}\ 50</math> | ||
+ | |||
+ | [[2008 AMC 8 Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
Line 47: | Line 58: | ||
[[2008 AMC 8 Problems/Problem 5|Solution]] | [[2008 AMC 8 Problems/Problem 5|Solution]] | ||
− | + | ==Problem 6== | |
+ | In the figure, what is the ratio of the area of the gray squares to the area of the white squares? | ||
+ | <asy> | ||
+ | size((70)); | ||
+ | draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0)); | ||
+ | draw((-2.5,-7.5)--(7.5,2.5)); | ||
+ | draw((-5,-5)--(5,5)); | ||
+ | draw((-7.5,-2.5)--(2.5,7.5)); | ||
+ | draw((-7.5,2.5)--(2.5,-7.5)); | ||
+ | draw((-5,5)--(5,-5)); | ||
+ | draw((-2.5,7.5)--(7.5,-2.5)); | ||
+ | fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray); | ||
+ | fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray); | ||
+ | fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray); | ||
+ | </asy> | ||
+ | <math> \textbf{(A)}\ 3:10 \qquad\textbf{(B)}\ 3:8 \qquad\textbf{(C)}\ 3:7 \qquad\textbf{(D)}\ 3:5 \qquad\textbf{(E)}\ 1:1 </math> | ||
+ | |||
+ | [[2008 AMC 8 Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
Line 86: | Line 114: | ||
==Problem 9== | ==Problem 9== | ||
− | In <math>2005</math> Tycoon Tammy invested <math>100</math> dollars for two years. During | + | In <math>2005</math> Tycoon Tammy invested <math>100</math> dollars for two years. During the first year |
her investment suffered a <math>15\%</math> loss, but during the second year the remaining | her investment suffered a <math>15\%</math> loss, but during the second year the remaining | ||
investment showed a <math>20\%</math> gain. Over the two-year period, what was the change | investment showed a <math>20\%</math> gain. Over the two-year period, what was the change | ||
Line 111: | Line 139: | ||
==Problem 11== | ==Problem 11== | ||
+ | Each of the <math>39</math> students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and <math>26</math> students have a cat. How many students have both a dog and a cat? | ||
+ | <math>\textbf{(A)}\ 7\qquad | ||
+ | \textbf{(B)}\ 13\qquad | ||
+ | \textbf{(C)}\ 19\qquad | ||
+ | \textbf{(D)}\ 39\qquad | ||
+ | \textbf{(E)}\ 46</math> | ||
− | + | [[2008 AMC 8 Problems/Problem 11|Solution]] | |
− | |||
− | |||
− | |||
==Problem 12== | ==Problem 12== | ||
− | + | A ball is dropped from a height of <math>3</math> meters. On its first bounce it rises to a height of <math>2</math> meters. It keeps falling and bouncing to <math>\frac{2}{3}</math> of the height it reached in the previous bounce. On which bounce will it rise to a height less than <math>0.5</math> meters? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 3 \qquad |
− | \textbf{(B)}\ | + | \textbf{(B)}\ 4 \qquad |
\textbf{(C)}\ 5 \qquad | \textbf{(C)}\ 5 \qquad | ||
− | \textbf{(D)}\ | + | \textbf{(D)}\ 6 \qquad |
− | \textbf{(E)}\ | + | \textbf{(E)}\ 7</math> |
[[2008 AMC 8 Problems/Problem 12|Solution]] | [[2008 AMC 8 Problems/Problem 12|Solution]] | ||
Line 198: | Line 229: | ||
==Problem 18== | ==Problem 18== | ||
− | + | Two circles that share the same center have radii <math>10</math> meters and <math>20</math> meters. An aardvark runs along the path shown, starting at <math>A</math> and ending at <math>K</math>. How many meters does the aardvark run? | |
− | + | <asy> | |
− | (A) | + | size((150)); |
+ | draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle); | ||
+ | draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle); | ||
+ | draw((20,0)--(-20,0)); | ||
+ | draw((0,20)--(0,-20)); | ||
+ | draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow); | ||
+ | draw((-18,1)--(-12, 1), EndArrow); | ||
+ | draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow); | ||
+ | draw((1,-9)--(1,9), EndArrow); | ||
+ | draw((0,12)..(8.3, 8.3)..(12,0), EndArrow); | ||
+ | draw((12,-1)--(18,-1), EndArrow); | ||
+ | label("$A$", (0,20), N); | ||
+ | label("$K$", (20,0), E); | ||
+ | </asy> | ||
+ | <math> \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \\ \textbf{(E)}\ 20\pi+40</math> | ||
[[2008 AMC 8 Problems/Problem 18|Solution]] | [[2008 AMC 8 Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | 5 | + | Eight points are spaced around at intervals of one unit around a <math>2 \times 2</math> square, as shown. Two of the <math>8</math> points are chosen at random. What is the probability that the two points are one unit apart? |
+ | <asy> | ||
+ | size((50)); | ||
+ | dot((5,0)); | ||
+ | dot((5,5)); | ||
+ | dot((0,5)); | ||
+ | dot((-5,5)); | ||
+ | dot((-5,0)); | ||
+ | dot((-5,-5)); | ||
+ | dot((0,-5)); | ||
+ | dot((5,-5)); | ||
+ | </asy> | ||
+ | <math> \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} </math> | ||
+ | |||
+ | [[2008 AMC 8 Problems/Problem 19|Solution]] | ||
− | + | ==Problem 20== | |
+ | The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and <math>\frac{3}{4}</math> of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | ||
− | + | <math>\textbf{(A)}\ 12\qquad | |
+ | \textbf{(B)}\ 17\qquad | ||
+ | \textbf{(C)}\ 24\qquad | ||
+ | \textbf{(D)}\ 27\qquad | ||
+ | \textbf{(E)}\ 36</math> | ||
+ | |||
+ | [[2008 AMC 8 Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
Line 239: | Line 305: | ||
[[2008 AMC 8 Problems/Problem 21|Solution]] | [[2008 AMC 8 Problems/Problem 21|Solution]] | ||
− | 8 | + | ==Problem 22== |
+ | For how many positive integer values of <math>n</math> are both <math>\frac{n}{3}</math> and <math>3n</math> three-digit whole numbers? | ||
+ | |||
+ | <math>\textbf{(A)}\ 12\qquad | ||
+ | \textbf{(B)}\ 21\qquad | ||
+ | \textbf{(C)}\ 27\qquad | ||
+ | \textbf{(D)}\ 33\qquad | ||
+ | \textbf{(E)}\ 34</math> | ||
+ | |||
+ | [[2008 AMC 8 Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | In square <math>ABCE</math>, <math>AF=2FE</math> and <math>CD=2DE</math>. What is the ratio of the area of <math>\triangle BFD</math> to the area of square <math>ABCE</math>? | + | In square <math>ABCE</math>, <math>AF=2FE</math> and <math>CD=2DE</math>. What is the ratio of the area of <math>\triangle BFD</math> to the area of square <math>ABCE</math>? |
<asy> | <asy> | ||
size((100)); | size((100)); | ||
Line 295: | Line 370: | ||
filldraw(circle(A5, 1), black, black); | filldraw(circle(A5, 1), black, black); | ||
</asy> | </asy> | ||
− | <math> \textbf{(A)}\ 42\qquad \textbf{(B)}\ 44\qquad \textbf{( | + | |
+ | <math> \textbf{(A)}\ 42\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 46\qquad \textbf{(D)}\ 47\qquad \textbf{(E)}\ 49\qquad</math> | ||
[[2008 AMC 8 Problems/Problem 25|Solution]] | [[2008 AMC 8 Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2008|before=[[2007 AMC 8 Problems|2007 AMC 8]]|after=[[2009 AMC 8 Problems|2009 AMC 8]]}} | ||
+ | * [[AMC 8]] | ||
+ | * [[AMC 8 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 15:47, 28 June 2024
2008 AMC 8 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?
Problem 2
The ten-letter code represents the ten digits
, in order. What 4-digit number is represented by the code word
?
Problem 3
If February is a month that contains Friday the , what day of the week is February 1?
Problem 4
In the figure, the outer equilateral triangle has area , the inner equilateral triangle has area
, and the three trapezoids are congruent. What is the area of one of the trapezoids?
Problem 5
Barney Schwinn notices that the odometer on his bicycle reads , a palindrome, because it reads the same forward and backward. After riding
more hours that day and
the next, he notices that the odometer shows another palindrome,
. What was his average speed in miles per hour?
Problem 6
In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
Problem 7
If , what is
?
Problem 8
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
Problem 9
In Tycoon Tammy invested
dollars for two years. During the first year
her investment suffered a
loss, but during the second year the remaining
investment showed a
gain. Over the two-year period, what was the change
in Tammy's investment?
Problem 10
The average age of the people in Room A is
. The average age of the
people in Room B is
. If the two groups are combined, what is the average age of all the people?
Problem 11
Each of the students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and
students have a cat. How many students have both a dog and a cat?
Problem 12
A ball is dropped from a height of meters. On its first bounce it rises to a height of
meters. It keeps falling and bouncing to
of the height it reached in the previous bounce. On which bounce will it rise to a height less than
meters?
Problem 13
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than pounds or more than
pounds. So the boxes are weighed in pairs in every possible way. The results are
,
and
pounds. What is the combined weight in pounds of the three boxes?
Problem 14
Three , three
, and three
are placed in the nine spaces so that each row and column contain one of each letter. If
is placed in the upper left corner, how many arrangements are possible?
Problem 15
In Theresa's first basketball games, she scored
and
points. In her ninth game, she scored fewer than
points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than
points and her points-per-game average for the
games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
Problem 16
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
Problem 17
Ms. Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
Problem 18
Two circles that share the same center have radii meters and
meters. An aardvark runs along the path shown, starting at
and ending at
. How many meters does the aardvark run?
Problem 19
Eight points are spaced around at intervals of one unit around a square, as shown. Two of the
points are chosen at random. What is the probability that the two points are one unit apart?
Problem 20
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
Problem 21
Jerry cuts a wedge from a -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
Problem 22
For how many positive integer values of are both
and
three-digit whole numbers?
Problem 23
In square ,
and
. What is the ratio of the area of
to the area of square
?
Problem 24
Ten tiles numbered through
are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
Problem 25
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?
See Also
2008 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2007 AMC 8 |
Followed by 2009 AMC 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.