Difference between revisions of "2012 AMC 12B Problems/Problem 7"
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== Problem == | == Problem == | ||
− | + | Small lights are hung on a string <math>6</math> inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of <math>2</math> red lights followed by <math>3</math> green lights. How many feet separate the 3rd red light and the 21st red light? | |
− | Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the 3rd red light and the 21st red light? | ||
− | '''Note:''' 1 foot is equal to 12 inches. | + | '''Note:''' <math>1</math> foot is equal to <math>12</math> inches. |
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 18.5\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 20.5\qquad\textbf{(E)}\ 22.5 </math> | <math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 18.5\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 20.5\qquad\textbf{(E)}\ 22.5 </math> | ||
== Solution == | == Solution == | ||
+ | We know the repeating section is made of <math>2</math> red lights and <math>3</math> green lights. The 3rd red light would appear in the 2nd section of this pattern, and the 21st red light would appear in the 11th section. There would then be a total of <math>44</math> lights in between the 3rd and 21st red light, translating to <math>45</math> <math>6</math>-inch gaps. Since the question asks for the answer in feet, the answer is <math>\frac{45*6}{12} \rightarrow \boxed{\textbf{(E)}\ 22.5}</math>. | ||
− | We know the | + | == Solution 2 == |
+ | We know that both <math>3</math> and <math>21</math> are odd. This means that they both start their respective patterns of <math>rrggg</math> as the first <math>r</math> value. | ||
+ | |||
+ | Let's take a look at one full gap of two reds <math>rrgggr</math>, in this gap there is a total of <math>2.5</math> feet of gap (considering that <math>6</math> inches is half a foot). So for each gap of two reds there is a <math>2.5</math> feet gap. | ||
+ | |||
+ | The total amount of red gaps of lights hung is <math>\frac{21-3}{2} = 9</math> gaps. Multiplying <math>9 * 2.5 = 22.5</math> inches. This gives <math>(E)</math>. | ||
+ | |||
+ | ~PeterDoesPhysics | ||
== See Also == | == See Also == | ||
− | |||
{{AMC12 box|year=2012|ab=B|num-b=6|num-a=8}} | {{AMC12 box|year=2012|ab=B|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 01:46, 10 August 2024
Contents
Problem
Small lights are hung on a string inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of
red lights followed by
green lights. How many feet separate the 3rd red light and the 21st red light?
Note: foot is equal to
inches.
Solution
We know the repeating section is made of red lights and
green lights. The 3rd red light would appear in the 2nd section of this pattern, and the 21st red light would appear in the 11th section. There would then be a total of
lights in between the 3rd and 21st red light, translating to
-inch gaps. Since the question asks for the answer in feet, the answer is
.
Solution 2
We know that both and
are odd. This means that they both start their respective patterns of
as the first
value.
Let's take a look at one full gap of two reds , in this gap there is a total of
feet of gap (considering that
inches is half a foot). So for each gap of two reds there is a
feet gap.
The total amount of red gaps of lights hung is gaps. Multiplying
inches. This gives
.
~PeterDoesPhysics
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 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 AMC 12 Problems and Solutions |
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