, 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 $7\times15-3\times5=105-15=90$.)