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Difference between revisions of "2009 AMC 10A Problems"

(→‎Problem 8)
Line 1: Line 1:
 
== Problem 1 ==
 
== Problem 1 ==
 
One can holds <math>12</math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
 
One can holds <math>12</math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
−
<math>a)\, 7\qquad
+
 
−
b)\, 8\qquad
+
<math>
−
c)\, 9\qquad
+
\mathrm{(A)}\ 7
−
d)\, 10\qquad
+
\qquad
−
e)\, 11</math>
+
\mathrm{(B)}\ 8
 +
\qquad
 +
\mathrm{(C)}\ 9
 +
\qquad
 +
\mathrm{(D)}\ 10
 +
\qquad
 +
\mathrm{(E)}\ 11
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 1|Solution]]
 
[[2009 AMC 10A Problems/Problem 1|Solution]]
Line 12: Line 19:
 
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could ''not'' be the total value of the four coins, in cents?
 
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could ''not'' be the total value of the four coins, in cents?
  
−
<math>a)\, 15\qquad
+
<math>
−
b)\, 25\qquad
+
\mathrm{(A)}\ 15
−
c)\, 35\qquad
+
\qquad
−
d)\, 45\qquad
+
\mathrm{(B)}\ 25
−
e)\, 55</math>
+
\qquad
 +
\mathrm{(C)}\ 35
 +
\qquad
 +
\mathrm{(D)}\ 45
 +
\qquad
 +
\mathrm{(E)}\ 55
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 2|Solution]]
 
[[2009 AMC 10A Problems/Problem 2|Solution]]
Line 23: Line 36:
 
Which of the following is equal to <math>1 + \frac{1}{1+\frac{1}{1+1}}</math>?
 
Which of the following is equal to <math>1 + \frac{1}{1+\frac{1}{1+1}}</math>?
  
−
<math>a)\, \frac{5}{4}\qquad
+
<math>
−
b)\, \frac{3}{2}\qquad
+
\mathrm{(A)}\ \frac{5}{4}
−
c)\, \frac{5}{3}\qquad
+
\qquad
−
d)\, 2\qquad
+
\mathrm{(B)}\ \frac{3}{2}
−
e)\, 3</math>
+
\qquad
 +
\mathrm{(C)}\ \frac{5}{3}
 +
\qquad
 +
\mathrm{(D)}\ 2
 +
\qquad
 +
\mathrm{(E)}\ 3
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 3|Solution]]
 
[[2009 AMC 10A Problems/Problem 3|Solution]]
Line 34: Line 53:
 
Eric plans to compete in a triathalon. He can average <math>2</math> miles per hour in the <math>\frac{1}{4}</math>-mile swim and <math>6</math> miles per hour in the <math>3</math>-mile run. His goal is to finish the triathlon in <math>2</math> hours. To accomplish his goal what must his average speed in miles per hour, be for the <math>15</math>-mile bicycle ride?
 
Eric plans to compete in a triathalon. He can average <math>2</math> miles per hour in the <math>\frac{1}{4}</math>-mile swim and <math>6</math> miles per hour in the <math>3</math>-mile run. His goal is to finish the triathlon in <math>2</math> hours. To accomplish his goal what must his average speed in miles per hour, be for the <math>15</math>-mile bicycle ride?
  
−
<math>a)\, \frac{120}{11}\qquad
+
<math>
−
b)\, 11\qquad
+
\mathrm{(A)}\ \frac{120}{11}
−
c)\, \frac{56}{5}\qquad
+
\qquad
−
d)\, \frac{45}{4}\qquad
+
\mathrm{(B)}\ 11
−
e)\, 12</math>
+
\qquad
 +
\mathrm{(C)}\ \frac{56}{5}
 +
\qquad
 +
\mathrm{(D)}\ \frac{45}{4}
 +
\qquad
 +
\mathrm{(E)}\ 12
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 4|Solution]]
 
[[2009 AMC 10A Problems/Problem 4|Solution]]
Line 45: Line 70:
 
What is the sum of the digits of the square of <math>111,111,111</math>?
 
What is the sum of the digits of the square of <math>111,111,111</math>?
  
−
<math>a)\, 18\qquad
+
<math>
−
b)\, 27\qquad
+
\mathrm{(A)}\ 18
−
c)\, 45\qquad
+
\qquad
−
d)\, 63\qquad
+
\mathrm{(B)}\ 27
−
e)\, 81</math>
+
\qquad
 +
\mathrm{(C)}\ 45
 +
\qquad
 +
\mathrm{(D)}\ 63
 +
\qquad
 +
\mathrm{(E)}\ 81
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 5|Solution]]
 
[[2009 AMC 10A Problems/Problem 5|Solution]]
Line 56: Line 87:
  
 
[[2009 AMC 10A Problems/Problem 6|Solution]]
 
[[2009 AMC 10A Problems/Problem 6|Solution]]
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
== Problem 7 ==
 
== Problem 7 ==
 
A carton contains milk that is <math>2</math>% fat, an amount that is <math>40</math>% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
 
A carton contains milk that is <math>2</math>% fat, an amount that is <math>40</math>% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
  
−
<math>a)\, \frac{12}{5}\qquad
+
<math>
−
b)\, \frac{10}{3}\qquad
+
\mathrm{(A)}\ \frac{12}{5}
−
c)\, 9\qquad
+
\qquad
−
d)\, 38\qquad
+
\mathrm{(B)}\ \frac{10}{3}
−
e)\, 42</math>
+
\qquad
 +
\mathrm{(C)}\ 9
 +
\qquad
 +
\mathrm{(D)}\ 38
 +
\qquad
 +
\mathrm{(E)}\ 42
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 7|Solution]]
 
[[2009 AMC 10A Problems/Problem 7|Solution]]
Line 71: Line 120:
 
Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a <math>50</math>% discount as children. The two members of the oldest generation receive a <math>25</math>% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <dollar/>6.00, is paying for everyone. How many dollars must he pay?
 
Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a <math>50</math>% discount as children. The two members of the oldest generation receive a <math>25</math>% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <dollar/>6.00, is paying for everyone. How many dollars must he pay?
  
−
<math>a)\, 34\qquad
+
<math>
−
b)\, 36\qquad
+
\mathrm{(A)}\ 34
−
c)\, 42\qquad
+
\qquad
−
d)\, 46\qquad
+
\mathrm{(B)}\ 36
−
e)\, 48</math>
+
\qquad
 +
\mathrm{(C)}\ 42
 +
\qquad
 +
\mathrm{(D)}\ 46
 +
\qquad
 +
\mathrm{(E)}\ 48
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 8|Solution]]
 
[[2009 AMC 10A Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
 +
Positive integers <math>a</math>, <math>b</math>, and <math>2009</math>, with <math>a<b<2009</math>, form a geometric sequence with an integer ratio. What is <math>a</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 7
 +
\qquad
 +
\mathrm{(B)}\ 41
 +
\qquad
 +
\mathrm{(C)}\ 49
 +
\qquad
 +
\mathrm{(D)}\ 289
 +
\qquad
 +
\mathrm{(E)}\ 2009
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 9|Solution]]
 
[[2009 AMC 10A Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 10|Solution]]
 
[[2009 AMC 10A Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 11|Solution]]
 
[[2009 AMC 10A Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 12|Solution]]
 
[[2009 AMC 10A Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 13|Solution]]
 
[[2009 AMC 10A Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 14|Solution]]
 
[[2009 AMC 10A Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 15|Solution]]
 
[[2009 AMC 10A Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 16|Solution]]
 
[[2009 AMC 10A Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 17|Solution]]
 
[[2009 AMC 10A Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 18|Solution]]
 
[[2009 AMC 10A Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 19|Solution]]
 
[[2009 AMC 10A Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 20|Solution]]
 
[[2009 AMC 10A Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 21|Solution]]
 
[[2009 AMC 10A Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 22|Solution]]
 
[[2009 AMC 10A Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 23|Solution]]
 
[[2009 AMC 10A Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 24|Solution]]
 
[[2009 AMC 10A Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
 +
<math>
 +
\mathrm{(A)}\
 +
\qquad
 +
\mathrm{(B)}\
 +
\qquad
 +
\mathrm{(C)}\
 +
\qquad
 +
\mathrm{(D)}\
 +
\qquad
 +
\mathrm{(E)}\
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 25|Solution]]
 
[[2009 AMC 10A Problems/Problem 25|Solution]]

Revision as of 03:54, 13 February 2009

Problem 1

One can holds $12$ ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?

$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 10 \qquad \mathrm{(E)}\ 11$

Solution

Problem 2

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?

$\mathrm{(A)}\ 15 \qquad \mathrm{(B)}\ 25 \qquad \mathrm{(C)}\ 35 \qquad \mathrm{(D)}\ 45 \qquad \mathrm{(E)}\ 55$

Solution

Problem 3

Which of the following is equal to $1 + \frac{1}{1+\frac{1}{1+1}}$?

$\mathrm{(A)}\ \frac{5}{4} \qquad \mathrm{(B)}\ \frac{3}{2} \qquad \mathrm{(C)}\ \frac{5}{3} \qquad \mathrm{(D)}\ 2 \qquad \mathrm{(E)}\ 3$

Solution

Problem 4

Eric plans to compete in a triathalon. He can average $2$ miles per hour in the $\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$-mile bicycle ride?

$\mathrm{(A)}\ \frac{120}{11} \qquad \mathrm{(B)}\ 11 \qquad \mathrm{(C)}\ \frac{56}{5} \qquad \mathrm{(D)}\ \frac{45}{4} \qquad \mathrm{(E)}\ 12$

Solution

Problem 5

What is the sum of the digits of the square of $111,111,111$?

$\mathrm{(A)}\ 18 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 45 \qquad \mathrm{(D)}\ 63 \qquad \mathrm{(E)}\ 81$

Solution

Problem 6

Solution

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Problem 7

A carton contains milk that is $2$% fat, an amount that is $40$% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?

$\mathrm{(A)}\ \frac{12}{5} \qquad \mathrm{(B)}\ \frac{10}{3} \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 38 \qquad \mathrm{(E)}\ 42$

Solution

Problem 8

Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25$% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <dollar/>6.00, is paying for everyone. How many dollars must he pay?

$\mathrm{(A)}\ 34 \qquad \mathrm{(B)}\ 36 \qquad \mathrm{(C)}\ 42 \qquad \mathrm{(D)}\ 46 \qquad \mathrm{(E)}\ 48$

Solution

Problem 9

Positive integers $a$, $b$, and $2009$, with $a<b<2009$, form a geometric sequence with an integer ratio. What is $a$?

$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 41 \qquad \mathrm{(C)}\ 49 \qquad \mathrm{(D)}\ 289 \qquad \mathrm{(E)}\ 2009$

Solution

Problem 10

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 11

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 12

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 13

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 14

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 15

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 16

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 17

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 18

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 19

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 20

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 21

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 22

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 23

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 24

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 25

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

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