If a soccer team scores an average of x goals per game for n games, sc

Average goal for all games = \(\frac{TotalGoals}{TotalGames}\) i.e.,

Overall Average = \(\frac{(Ave)(Qty) + (Ave)(Qty) + (Ave)(Qty)}{(TotalQty)}\)

Average goals scored = \(Ave = A\)
Number of games = \(Qty = n\)

1) average of x goals per game for n games
\(A_1 = x\)
\(n_1 = n\)
\(A_1*n_1 = S_1\)
\(xn = S_1\)

2) average of z goals for n-1 games
\(A_2 = z\)
\(n_2 = (n-1)\)
\(A_2*n_2 = S_2\)
\(z(n-1) = zn - z = S_2\)

3) an average of y goals in one game
\(A_3 = y\)
\(n_3 = 1\)
\(A_3*n_3 = S_3\)
\(y*1 = y = S_3\)

Total Goals: \(S_1 + S_2 + S_3 =\)
\(xn + (zn-z) + y = (xn + zn -z + y)\)

Total Games: \(n_1 + n_2 + n_3 =\)
\(n + (n-1) + 1 = (2n)\)

Overall Average: \(\frac{xn + zn - z + y}{2n}\)

Answer

Assign values
This approach might save time if you have a good grasp of the algebra involved.
Let
n = 2
x = 1
z = 1
y = 1

1) "average of x goals per game for n games"
(1 * 2 ) = 2 goals
2) "average of z goals for (n-1) games"
(1 * 1) = 1 goal
3) "average of y goals for 1 game"
(1 * 1) = 1 goal

Overall total goals: (2 + 1 + 1) = 4
Overall total games: (2 + 1+ 1) = 4
Average goals per game: \(\frac{4}{4}=1\)

With assigned variables, find the answer that yields 1
If you check, the choice that yields 1 is

Answer

Took me 4 mins to do this. Is there any shorter way?

Avr goals=(Total goals)/(Total games)
Total goals=x*n + z*(n-1) + y = (xn+zn-z+y)
Total games = n+n-1+1=2n
So avr goals = (xn+y+sn-z)/(2n)
A is correct answer