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Difference between revisions of "2007 AMC 10A Problems"

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== Problem 1 ==
 
== Problem 1 ==
 +
One ticket to a show costs <math>20</math> at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?
 +
 +
<math>\mathrm{(A)}\ 2\qquad \mathrm{(B)}\ 5\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 15\qquad \mathrm{(E)}\ 20</math>
  
 
[[2007 AMC 10A Problems/Problem 1|Solution]]
 
[[2007 AMC 10A Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
Define <math>a@b = ab - b^{2}</math> and <math>a\#b = a + b - ab^{2}</math>. What is <math>\frac {6@2}{6\#2}</math>?
 +
 +
<math>\text{(A)}\ - \frac {1}{2}\qquad \text{(B)}\ - \frac {1}{4}\qquad \text{(C)}\ \frac {1}{8}\qquad \text{(D)}\ \frac {1}{4}\qquad \text{(E)}\ \frac {1}{2}</math>
  
 
[[2007 AMC 10A Problems/Problem 2|Solution]]
 
[[2007 AMC 10A Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
 +
 +
<math>\text{(A)}\ 0.5 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 1.5 \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 2.5</math>
  
 
[[2007 AMC 10A Problems/Problem 3|Solution]]
 
[[2007 AMC 10A Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
A school store sells 7 pencils and 8 notebooks for <math>\</math>4.15<math>. It also sells 5 pencils and 3 notebooks for </math>\<math>1.77</math>. How much do 16 pencils and 10 notebooks cost?
 +
 +
<math>\text{(A)}\ \</math> <math>1.76 \qquad \text{(B)}\ \</math> <math>5.84 \qquad \text{(C)}\ \</math> <math>6.00 \qquad \text{(D)}\ \</math> <math>6.16 \qquad \text{(E)}\ \</math> <math>6.32</math>
  
 
[[2007 AMC 10A Problems/Problem 4|Solution]]
 
[[2007 AMC 10A Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
The larger of two consecutive odd integers is three times the smaller. What is their sum?
 +
 +
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20</math>
  
 
[[2007 AMC 10A Problems/Problem 5|Solution]]
 
[[2007 AMC 10A Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
At Euclid High School, the number of students taking the [[AMC 10]] was <math>60</math> in 2002, <math>66</math> in 2003, <math>70</math> in 2004, <math>76</math> in 2005, <math>78</math> and 2006, and is <math>85</math> in 2007. Between what two consecutive years was there the largest percentage increase?
 +
 +
<math>\text{(A)}\ 2002\ \text{and}\ 2003 \qquad \text{(B)}\ 2003\ \text{and}\ 2004 \qquad \text{(C)}\ 2004\ \text{and}\ 2005 \qquad \text{(D)}\ 2005\ \text{and}\ 2006 \qquad \text{(E)}\ 2006\ \text{and}\ 2007</math>
  
 
[[2007 AMC 10A Problems/Problem 6|Solution]]
 
[[2007 AMC 10A Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
Last year Mr. Jon Q. Public received an inheritance. He paid <math>20\%</math> in federal taxes on the inheritance, and paid <math>10\%</math> of what he had left in state taxes. He paid a total of <math>\</math>10500 for both taxes. How many dollars was his inheritance?
 +
 +
<math>(\mathrm {A})\ 30000 \qquad (\mathrm {B})\ 32500 \qquad(\mathrm {C})\ 35000 \qquad(\mathrm {D})\ 37500 \qquad(\mathrm {E})\ 40000</math>
  
 
[[2007 AMC 10A Problems/Problem 7|Solution]]
 
[[2007 AMC 10A Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Triangles <math>ABC</math> and <math>ADC</math> are isosceles with <math>AB=BC</math> and <math>AD=DC</math>. Point <math>D</math> is inside triangle <math>ABC</math>, angle <math>ABC</math> measures 40 degrees, and angle <math>ADC</math> measures 140 degrees. What is the degree measure of angle <math>BAD</math>?
 +
 +
<math>\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E)}\ 60</math>
  
 
[[2007 AMC 10A Problems/Problem 8|Solution]]
 
[[2007 AMC 10A Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Real numbers <math>a</math> and <math>b</math> satisfy the equations <math>3^{a} = 81^{b + 2}</math> and <math>125^{b} = 5^{a - 3}</math>. What is <math>ab</math>?
 +
 +
<math>\text{(A)}\ -60 \qquad \text{(B)}\ -17 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 60</math>
  
 
[[2007 AMC 10A Problems/Problem 9|Solution]]
 
[[2007 AMC 10A Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is <math>20</math>, the father is <math>48</math> years old, and the average age of the mother and children is <math>16</math>. How many children are in the family?
 +
 +
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math>
  
 
[[2007 AMC 10A Problems/Problem 10|Solution]]
 
[[2007 AMC 10A Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
The numbers from <math>1</math> to <math>8</math> are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
 +
 +
<math>\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 24</math>
  
 
[[2007 AMC 10A Problems/Problem 11|Solution]]
 
[[2007 AMC 10A Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
 +
 +
<math>\text{(A)}\ 56 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 64</math>
  
 
[[2007 AMC 10A Problems/Problem 12|Solution]]
 
[[2007 AMC 10A Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from the stadium?
 +
 +
<math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac 78</math>
  
 
[[2007 AMC 10A Problems/Problem 13|Solution]]
 
[[2007 AMC 10A Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
A triangle with side lengths in the ratio <math>3 : 4 : 5</math> is inscribed in a circle with radius <math>3</math>. What is the area of the triangle?
 +
 +
<math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math>
  
 
[[2007 AMC 10A Problems/Problem 14|Solution]]
 
[[2007 AMC 10A Problems/Problem 14|Solution]]

Revision as of 15:43, 6 January 2008

Problem 1

One ticket to a show costs $20$ at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?

$\mathrm{(A)}\ 2\qquad \mathrm{(B)}\ 5\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 15\qquad \mathrm{(E)}\ 20$

Solution

Problem 2

Define $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$. What is $\frac {6@2}{6\#2}$?

$\text{(A)}\ - \frac {1}{2}\qquad \text{(B)}\ - \frac {1}{4}\qquad \text{(C)}\ \frac {1}{8}\qquad \text{(D)}\ \frac {1}{4}\qquad \text{(E)}\ \frac {1}{2}$

Solution

Problem 3

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?

$\text{(A)}\ 0.5 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 1.5 \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 2.5$

Solution

Problem 4

A school store sells 7 pencils and 8 notebooks for $$4.15$. It also sells 5 pencils and 3 notebooks for$\$1.77$. How much do 16 pencils and 10 notebooks cost?

$\text{(A)}\ $ $1.76 \qquad \text{(B)}\ $ $5.84 \qquad \text{(C)}\ $ $6.00 \qquad \text{(D)}\ $ $6.16 \qquad \text{(E)}\ $ $6.32$

Solution

Problem 5

The larger of two consecutive odd integers is three times the smaller. What is their sum?

$\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

Solution

Problem 6

At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ and 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase?

$\text{(A)}\ 2002\ \text{and}\ 2003 \qquad \text{(B)}\ 2003\ \text{and}\ 2004 \qquad \text{(C)}\ 2004\ \text{and}\ 2005 \qquad \text{(D)}\ 2005\ \text{and}\ 2006 \qquad \text{(E)}\ 2006\ \text{and}\ 2007$

Solution

Problem 7

Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $$10500 for both taxes. How many dollars was his inheritance?

$(\mathrm {A})\ 30000 \qquad (\mathrm {B})\ 32500 \qquad(\mathrm {C})\ 35000 \qquad(\mathrm {D})\ 37500 \qquad(\mathrm {E})\ 40000$

Solution

Problem 8

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside triangle $ABC$, angle $ABC$ measures 40 degrees, and angle $ADC$ measures 140 degrees. What is the degree measure of angle $BAD$?

$\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E)}\ 60$

Solution

Problem 9

Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$. What is $ab$?

$\text{(A)}\ -60 \qquad \text{(B)}\ -17 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 60$

Solution

Problem 10

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

Solution

Problem 11

The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

$\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 24$

Solution

Problem 12

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

$\text{(A)}\ 56 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 64$

Solution

Problem 13

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?

$\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac 78$

Solution

Problem 14

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius $3$. What is the area of the triangle?

$\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?

$\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 12$

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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