# 2019 AMC 8 Problems/Problem 23

## Problem 23

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members?

$\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

## Solution 1

Given the information above, we start with the equation $\frac{t}{4}+\frac{2t}{7} + 15 + x = t$,where $t$ is the total number of points scored and $x\le 14$ is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation $x+15 = \frac{13}{28}t$, or $28x+28\cdot 15=13t$. Since $t$ is necessarily divisible by 28, let $t=28u$ where $u \ge 0$ and divide by 28 to obtain $x + 15 = 13u$. Then, it is easy to see $u=2$ ($t=56$) is the only candidate remaining, giving $x=\boxed{\textbf{(B)} 11}$.

-scrabbler94

## Solution 2

We first start by setting the total number of points as $28$, since $\text{LCM}(4,7) = 28$. However, we see that this does not work since we surpass the number of points just with the information given ($28\cdot\frac{1}{4}+28\cdot\frac{2}{7} + 15 = 30$ $(> 28)$ ). Next, we can see that the total number of points scored is $56$ as, if it is more than or equal to $84$, at least one of the others will score more than 2 points. With this, we have that Alexa, Brittany, and Chelsea score: $56\cdot\frac{1}{4}+56\cdot\frac{2}{7} + 15 = 45$, and thus, the other seven players would have scored a total of $56-45 = \boxed{\textbf{(B)} 11}$. (We see that this works since we could have $4$ of them score $2$ points, and the other $3$ of them score $1$ point.)

-aops5234 -Edited by Penguin_Spellcaster

## Solution 3 — Modular Arithmetic

Adding together Alexa's and Brittany's fractions, we get $\frac{15}{28}$ as the fraction of the total number of points they scored together. However, this is just a ratio, so we can introduce a variable: $\frac{15x}{28x}$ where $x$ is the common ratio. Let $y$ and $z$ and $w$ be the number of people who scored 1, 2, and 0 points, respectively. Writing an equation, we have $\frac{13x}{28x} = 15 + y + 2z + 0w.$ We want all of our variables to be integers. Thus, we want $15 + y + 2z = 0 \pmod {13}.$ Simplifying, $y+2z = 11 \pmod {13}.$ The only possible value, as this integer sum has to be less than $7 \cdot 2 + 1 = 15,$ must be 11. Therefore, $y+2z = 11,$ and the answer is $\boxed{ \textbf{(B) 11}}$.

- ab2024

## Solution 4: Answer choices

We can rewrite the question as an algebraic equation: $\frac{1}{4} x + \frac{2}{7} x + 15 + y$, where $x$ represents the total amount of points and $y$ the amount of points the $7$ other players scored. From there, we add the two fractions to get $\frac{15}{28} x + 15 = x$. Subtracting $\frac{15}{28} x$ from both sides, we get $\frac{13}{28} x = y + 15$. We multiply each side by $28$ to get rid of the denominator, in which we get $13x = 420 + 28y$. Now let’s think of this logically. This equation is telling us that if you add $420$ and $28$ times the amount of points scored by the extra $7$ players, you get $13$ times the amount of points total. And since we have to have a whole number of points total, this means that $420 + 28y$ must be divisible by $13$. Plugging in all the answer choices for $y$, we find that the only answer that makes $420 + 28y$ divisible by $13$ is $\boxed{ \textbf{(B) 11}}$.

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