Difference between revisions of "2003 AMC 10B Problems/Problem 16"
m (Formatting) |
|||
Line 5: | Line 5: | ||
<math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math> | <math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math> | ||
− | + | ==Solution 1== | |
− | |||
Let <math>m</math> be the number of main courses the restaurant serves, so <math>2m</math> is the number of appetizers. Then the number of dinner combinations is <math>2m\times m\times3=6m^2</math>. Since the customer wants to eat a different dinner in all <math>365</math> days of <math>2003</math>, we must have | Let <math>m</math> be the number of main courses the restaurant serves, so <math>2m</math> is the number of appetizers. Then the number of dinner combinations is <math>2m\times m\times3=6m^2</math>. Since the customer wants to eat a different dinner in all <math>365</math> days of <math>2003</math>, we must have | ||
Line 16: | Line 15: | ||
The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>. | The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>. | ||
− | + | ==Solution 2== | |
Let <math>m</math> denote the number of main courses needed to meet the requirement. Then the number of dinners available is <math>3\cdot m \cdot 2m = 6m^2</math>. Thus <math>m^2</math> must be at least <math>365/6 \approx 61</math>. Since <math>7^2 = 49<61<64 = 8^2</math>, <math>\boxed{8}</math> main courses is enough, but 7 is not. The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>. | Let <math>m</math> denote the number of main courses needed to meet the requirement. Then the number of dinners available is <math>3\cdot m \cdot 2m = 6m^2</math>. Thus <math>m^2</math> must be at least <math>365/6 \approx 61</math>. Since <math>7^2 = 49<61<64 = 8^2</math>, <math>\boxed{8}</math> main courses is enough, but 7 is not. The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>. | ||
− | + | ==Solution 3== | |
Let <math>x</math> be the number of main courses and let <math>2x</math> be the number of appetizers. | Let <math>x</math> be the number of main courses and let <math>2x</math> be the number of appetizers. |
Latest revision as of 08:06, 6 September 2022
Problem
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year ?
Solution 1
Let be the number of main courses the restaurant serves, so
is the number of appetizers. Then the number of dinner combinations is
. Since the customer wants to eat a different dinner in all
days of
, we must have
Also, year 2003 is not a leap year, because 2003 divided by 4 does not equal an integer.
The smallest integer value that satisfies this is
.
Solution 2
Let denote the number of main courses needed to meet the requirement. Then the number of dinners available is
. Thus
must be at least
. Since
,
main courses is enough, but 7 is not. The smallest integer value that satisfies this is
.
Solution 3
Let be the number of main courses and let
be the number of appetizers.
Since there are 3 desserts, the number of possible dinner choices would be
for any number
. Since a year has
days, we can assume that:
The least option that is greater than
is
, so the answer is
.
~ Alfi06
Video Solution by WhyMath
~savannahsolver
See Also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.