, 2022

Each hit adds 7 points to Shane’s score, and so 4 hits adds $7\times4=28$ points to his score. For each miss, 3 points are subtracted from Shane’s score, and so 2 misses subtracts $3\times2=6$ points from his score. A player’s score begins at 0, and so after these 6 throws, Shane’s score is $28-6=22$. Solution 1 After exactly $h$ hits and 6 misses, Susan’s score is given by the expression $7\times h-3\times6$ or $7h-18$. After these throws, Susan’s score is 59, and so $7h-18=59$. Solving this equation, we get $7h=59+18$ or $7h=77$, and so $h=11$. Solution 2 Susan’s 6 misses decrease her score by $3\times6=18$ points. Since her score is 59, then Susan must have scored $59+18=77$ points from hits. Since each hit is worth 7 points, then the value of $h$ is $77\div7=11$. Solution 1 We begin by making an initial guess at the value of $m$, and then systematically adjust the value upward or downward as needed. If we begin with $m=3$ for example, then the number of hits is $20-3=17$ and Souresh’s score is $7\times17-3\times3=119-9=110$. Since this score is greater than 105, then the value of $m$ is greater than 3 (more misses means fewer hits and a lower score). When $m=4$, the number of hits is $20-4=16$ and Souresh’s score is $7\times16-3\times4=112-12=100$. Since this score is greater than 85 and less than 105, then 4 is a possible value of $m$. When $m=5$, the number of hits is $20-5=15$ and Souresh’s score is $7\times15-3\times5=105-15=90$. This score is also greater than 85 and less than 105, and so 5 is a possible value of $m$. When $m=6$, the number of hits is 14 and Souresh’s score is $7\times14-3\times6=98-18=80$. This score is less than 85 and so 6 is not a possible value of $m$. Continuing to increase the number of misses will further decrease Souresh’s score, and thus the only possible values are $m=4$ and $m=5$. Solution 2 Since Souresh makes 20 throws and $m$ of these throws are misses, then the remaining $20-m$ throws are hits. After exactly $20-m$ hits and $m$ misses, Souresh’s score is given by the expression $7(20-m)-3m$ or $140-10m$. Since Souresh’s score ($140-10m$) is greater than 85, then $10m$ must be less than 55 (note that $140 - 85 =
55$) and so$m < $\frac{55}{10}\$.$

Since $\frac{55}{10}=5\frac12$ and $m$ is a positive integer, then $m\leq 5$. Since Souresh’s score is less than 105, then $10m$ must be greater than 35 (note that $140 - 105 = 35$) and so $m>\frac{35}{10}$. Since $\frac{35}{10}=3\frac12$ and $m$ is a positive integer, then $m\geq 4$. Thus, $m$ is a positive integer and $4\leq m \leq 5$, and so $m=4$ or $m=5$. (We can check that when $m=4$, Souresh’s score is $7\times16-3\times4=112-12=100$, and when $m=5$, Souresh’s score is $7\times15-3\times5=105-15=90$.)