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Difference between revisions of "2006 AMC 12B Problems"

m (Changing links to subpages)
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== Problem 1 ==
 
== Problem 1 ==
 +
What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>?
 +
\[
 +
\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 1|Solution]]
 
[[2006 AMC 12B Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
For real numbers <math>x</math> and <math>y</math>, define <math>x\spadesuit y = (x + y)(x - y)</math>.  What is <math>3\spadesuit(4\spadesuit 5)</math>?
 +
\[
 +
\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) } 24 \qquad \text {(E) } 72
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 2|Solution]]
 
[[2006 AMC 12B Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
A football game was played between two teams, the Cougars and the Panthers.  The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points.  How many points did the Panthers score?
 +
\[
 +
\text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 3|Solution]]
 
[[2006 AMC 12B Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Mary is about to pay for five items at the grocery store.  The prices of the items are <math>\</math><math>7.99, </math>\<math></math> 4.99, <math>\</math><math>2.99, </math>\<math></math>1.99, and <math>\</math><math>0.99.  Mary will pay with a twenty-dollar bill.  Which of the following is closest to the percentage of the </math>\<math></math>20.00 that she will receive in change?
 +
\[
 +
\text {(A) } 5 \qquad \text {(B) } 10 \qquad \text {(C) } 15 \qquad \text {(D) } 20 \qquad \text {(E) } 25
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 4|Solution]]
 
[[2006 AMC 12B Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour.  If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
 +
\[
 +
\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) } 120
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 5|Solution]]
 
[[2006 AMC 12B Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade.  There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar.  Water contains no calories.  How many calories are in 200 grams of her lemonade.
 +
\[
 +
\text {(A) } 129 \qquad \text {(B) } 137 \qquad \text {(C) } 174 \qquad \text {(D) } 223 \qquad \text {(E) } 411
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 6|Solution]]
 
[[2006 AMC 12B Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
Mr. and Mrs. Lopez have two children.  When they get into their family car, two people sit in the front, and the other two sit in the back.  Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat.  How many seating arrangements are possible?
 +
\[
 +
\text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text {(D) } 24 \qquad \text {(E) } 48
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 7|Solution]]
 
[[2006 AMC 12B Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
The lines <math>x = \frac 14y + a</math> and <math>y = \frac 14x + b</math> intersect at the point <math>(1,2)</math>.  What is <math>a + b</math>?
 +
\[
 +
\text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94
 +
\]
  
 
[[2006 AMC 12B Problems/Problem 8|Solution]]
 
[[2006 AMC 12B Problems/Problem 8|Solution]]

Revision as of 12:43, 13 November 2007

Problem 1

What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$? \[ \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006 \]

Solution

Problem 2

For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$? \[ \text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) } 24 \qquad \text {(E) } 72 \]

Solution

Problem 3

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? \[ \text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24 \]

Solution

Problem 4

Mary is about to pay for five items at the grocery store. The prices of the items are $$$7.99,$\$$ (Error compiling LaTeX. Unknown error_msg) 4.99, $$$2.99,$\$$ (Error compiling LaTeX. Unknown error_msg)1.99, and $$$0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the$\$$ (Error compiling LaTeX. Unknown error_msg)20.00 that she will receive in change? \[ \text {(A) } 5 \qquad \text {(B) } 10 \qquad \text {(C) } 15 \qquad \text {(D) } 20 \qquad \text {(E) } 25 \]

Solution

Problem 5

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? \[ \text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) } 120 \]

Solution

Problem 6

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade. \[ \text {(A) } 129 \qquad \text {(B) } 137 \qquad \text {(C) } 174 \qquad \text {(D) } 223 \qquad \text {(E) } 411 \]

Solution

Problem 7

Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? \[ \text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text {(D) } 24 \qquad \text {(E) } 48 \]

Solution

Problem 8

The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$? \[ \text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94 \]

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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