Difference between revisions of "2019 AMC 8 Problems/Problem 1"
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| − | ==Problem 1== | + | == Problem 1 == |
| − | Ike and Mike go into a sandwich shop with a total of <math>\$30.00</math> to spend. Sandwiches cost <math>\$4.50</math> each and soft drinks cost <math>\$1.00</math> each. Ike and Mike plan to buy as many sandwiches as they can and use | + | Ike and Mike go into a sandwich shop with a total of <math>\$30.00</math> to spend. Sandwiches cost <math>\$4.50</math> each and soft drinks cost <math>\$1.00</math> each. Ike and Mike plan to buy as many sandwiches as they can, |
| + | and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how | ||
| + | many items will they buy? | ||
<math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math> | <math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math> | ||
| − | ==Solution 1== | + | == Solution 1 == |
| + | We know that the sandwiches cost <math>4.50</math> dollars. Guessing will bring us to multiplying <math>4.50</math> by 6, which gives us <math>27.00</math>. Since they can spend <math>30.00</math> they have <math>3</math> dollars left. Since soft drinks cost <math>1.00</math> dollar each, they can buy 3 soft drinks, which makes them spend <math>30.00</math> Since they bought 6 sandwiches and 3 soft drinks, they bought a total of <math>9</math> items. Therefore, the answer is <math>\boxed{\textbf{(D) }9}</math>. | ||
| − | + | - SBose | |
| − | + | == Solution 2 (Using Algebra) == | |
| + | Let <math>s</math> be the number of sandwiches and <math>d</math> be the number of sodas. We have to satisfy the equation of | ||
| + | <cmath>4.50s+d=30</cmath> | ||
| + | In the question, it states that Ike and Mike buys as many sandwiches as possible. | ||
| + | So, we drop the number of sodas for a while. | ||
| + | We have: | ||
| + | <cmath>\begin{align*} | ||
| + | 4.50s&=30 \\ | ||
| + | s&=\frac{30}{4.5} \\ | ||
| + | s&=6R3 | ||
| + | \end{align*}</cmath> | ||
| + | We don't want a remainder so the maximum number of sandwiches is <math>6</math>. | ||
| + | The total money spent is <math>6\cdot 4.50=27</math>. | ||
| + | The number of dollar left to spent on sodas is <math>30-27=3</math> dollars. | ||
| + | <math>3</math> dollars can buy <math>3</math> sodas leading us to a total of | ||
| + | <math>6+3=9</math> items. | ||
| + | Hence, the answer is <math>\boxed{\textbf{(D) }9}</math>. | ||
| − | + | - by interactivemath | |
| − | + | == Video Solution == | |
| − | ==Solution | + | ==Video Solution by Math-X (First fully understand the problem!!!)== |
| − | + | https://youtu.be/IgpayYB48C4?si=2nxxlhOYNmBnEhj5 | |
| + | ~Math-X | ||
| − | ==See | + | |
| − | {{AMC8 box|year=2019|before=First Problem|num-a=2}} | + | The Learning Royal: https://youtu.be/IiFFDDITE6Q |
| + | |||
| + | == Video Solution 2 == | ||
| + | Solution detailing how to solve the problem: https://www.youtube.com/watch?v=Puzy1HAlAKk&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=2 | ||
| + | |||
| + | ==Video Solution 3== | ||
| + | https://youtu.be/Y7wzKYaSOhI | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution (CREATIVE THINKING!!!)== | ||
| + | https://youtu.be/WggTFT9FhNg | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)== | ||
| + | https://youtu.be/Xm4ZGND9WoY | ||
| + | |||
| + | ~Hayabusa1 | ||
| + | |||
| + | ==See also== | ||
| + | {{AMC8 box|year=2019|before = First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 17:56, 24 June 2024
Contents
Problem 1
Ike and Mike go into a sandwich shop with a total of 
to spend. Sandwiches cost 
each and soft drinks cost 
each. Ike and Mike plan to buy as many sandwiches as they can,
and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how
many items will they buy?
Solution 1
We know that the sandwiches cost 
dollars. Guessing will bring us to multiplying by 6, which gives us 
. Since they can spend 
they have 
dollars left. Since soft drinks cost 
dollar each, they can buy 3 soft drinks, which makes them spend Since they bought 6 sandwiches and 3 soft drinks, they bought a total of 
items. Therefore, the answer is 
.
- SBose
Solution 2 (Using Algebra)
Let 
be the number of sandwiches and 
be the number of sodas. We have to satisfy the equation of
![\[4.50s+d=30\]](http://latex.artofproblemsolving.com/9/c/8/9c8e776fe1543c6342e6785f6d11ae6e2cc34420.png)
In the question, it states that Ike and Mike buys as many sandwiches as possible.
So, we drop the number of sodas for a while.
We have:
We don't want a remainder so the maximum number of sandwiches is 
.
The total money spent is 
.
The number of dollar left to spent on sodas is 
dollars.
dollars can buy sodas leading us to a total of
items.
Hence, the answer is .
- by interactivemath
Video Solution
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/IgpayYB48C4?si=2nxxlhOYNmBnEhj5 ~Math-X
The Learning Royal: https://youtu.be/IiFFDDITE6Q
Video Solution 2
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=Puzy1HAlAKk&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=2
Video Solution 3
~savannahsolver
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)
~Hayabusa1
See also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 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. 
