Difference between revisions of "2019 AMC 8 Problems/Problem 1"
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<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== | + | ==Solution 1== |
We know that there sandwiches cost <math>4.50</math> dollars. We can multiply <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 sodas cost <math>1.00</math> dollar each, they can buy 3 sodas, which makes them spend <math>30.00</math> Since they bought 6 sandwiches and 3 sodas, they bought a total of <math>9</math> items. Therefore, the answer is <math>\boxed{D = 9 }</math> | We know that there sandwiches cost <math>4.50</math> dollars. We can multiply <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 sodas cost <math>1.00</math> dollar each, they can buy 3 sodas, which makes them spend <math>30.00</math> Since they bought 6 sandwiches and 3 sodas, they bought a total of <math>9</math> items. Therefore, the answer is <math>\boxed{D = 9 }</math> | ||
+ | |||
+ | ==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>4.50s=30</cmath> | ||
+ | <cmath>s=\frac{30}{4.5}</cmath> | ||
+ | <cmath>s=6R30</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{D = 9}</math> |
Revision as of 17:04, 15 October 2020
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 there sandwiches cost dollars. We can multiply
by 6, which gives us
. Since they can spend
they have
dollars left. Since sodas cost
dollar each, they can buy 3 sodas, which makes them spend
Since they bought 6 sandwiches and 3 sodas, they bought a total of
items. Therefore, the answer is
Solution 2 (Using Algebra)
Let be the number of sandwiches and
be the number of sodas. We have to satisfy the equation of
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