Difference between revisions of "2010 AMC 8 Problems/Problem 13"
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==Problem== | ==Problem== | ||
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The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is <math>30\%</math> of the perimeter. What is the length of the longest side? | The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is <math>30\%</math> of the perimeter. What is the length of the longest side? | ||
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math> | <math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math> | ||
| − | ==Solution 1== | + | ==Solution 1(algebra solution)== |
| − | Let <math>n</math>, <math>n+1</math>, and <math>n+2</math> be the lengths of the sides of the triangle. Then the perimeter of the triangle is <math>n + (n+1) + (n+2) = 3n+3</math>. Using the fact that the length of the | + | Let <math>n</math>, <math>n+1</math>, and <math>n+2</math> be the lengths of the sides of the triangle. Then the perimeter of the triangle is <math>n + (n+1) + (n+2) = 3n+3</math>. Using the fact that the length of the smallest side is <math>30\%</math> of the perimeter, it follows that: |
<math>n = 0.3(3n+3) \Rightarrow n = 0.9n+0.9 \Rightarrow 0.1n = 0.9 \Rightarrow n=9</math>. The longest side is then <math>n+2 = 11</math>. Thus, answer choice <math>\boxed{\textbf{(E)}\ 11}</math> is correct. | <math>n = 0.3(3n+3) \Rightarrow n = 0.9n+0.9 \Rightarrow 0.1n = 0.9 \Rightarrow n=9</math>. The longest side is then <math>n+2 = 11</math>. Thus, answer choice <math>\boxed{\textbf{(E)}\ 11}</math> is correct. | ||
==Solution 2== | ==Solution 2== | ||
| − | Since the length of the shortest side is a whole number and is equal to <math>\frac{3}{10}</math> of the perimeter, it follows that the perimeter must be a multiple of <math>10</math>. Adding the two previous integers to each answer choice, we see that <math>11+10+9= | + | Since the length of the shortest side is a whole number and is equal to <math>\frac{3}{10}</math> of the perimeter, it follows that the perimeter must be a multiple of <math>10</math>. Adding the two previous integers to each answer choice, we see that <math>11+10+9=30</math>. Thus, answer choice <math>\boxed{\textbf{(E)}\ 11}</math> is correct. |
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| + | ==Video by MathTalks== | ||
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| + | https://www.youtube.com/watch?v=6hRHZxSieKc | ||
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==See Also== | ==See Also== | ||
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{{AMC8 box|year=2010|num-b=12|num-a=14}} | {{AMC8 box|year=2010|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 19:54, 22 January 2023
Problem
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 
of the perimeter. What is the length of the longest side?
Solution 1(algebra solution)
Let 
, 
, and 
be the lengths of the sides of the triangle. Then the perimeter of the triangle is 
. Using the fact that the length of the smallest side is of the perimeter, it follows that:
. The longest side is then 
. Thus, answer choice 
is correct.
Solution 2
Since the length of the shortest side is a whole number and is equal to 
of the perimeter, it follows that the perimeter must be a multiple of 
. Adding the two previous integers to each answer choice, we see that 
. Thus, answer choice is correct.
Video by MathTalks
https://www.youtube.com/watch?v=6hRHZxSieKc
See Also
| 2010 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
