Question

At one point in a game the shooting team has a ratio of hits to misses of \[5:1\]. After the team misses the next three shots, which are the last in the game, its ratio of hits to misses is \[5:2\]. What is the total number of shots taken by the team in the game?

Answer 1

Hint: First we will assume for Case 1 that the number of hits is \[5x\] and misses be \[x\] at one point in the game where ratio is \[5:1\] and for Case 2 that the number of hits is \[5y\] and misses be \[2y\] at one point in the game where ratio is \[5:2\]. Then we will form the equation for the cases according to the cases to find the value of \[y\]. Then we will add the number of hits and misses at the end to find the total number of shots taken by team.

Complete step by step answer:

We are given that at one point in a game the shooting team has a ratio of hits to misses of \[5:1\].
Let us assume for Case 1 that the number of hits is \[5x\] and misses be \[x\] at one point in the game where the ratio is \[5:1\].
Let us also assume for Case 2 that the number of hits is \[5y\] and misses be \[2y\] at one point in the game where the ratio is \[5:2\].
Since we are given that number of misses increased by 3 from the assumptions, so we have
\[ \Rightarrow 2y - x = 3{\text{ ......eq.(1)}}\]
But since we have the number of hits remain the same, so we get
\[ \Rightarrow 5x = 5y\]
Dividing the above equation by 5 on both sides, we get
\[
   \Rightarrow \dfrac{{5x}}{5} = \dfrac{{5y}}{5} \\
   \Rightarrow x = y \\
 \]
Substituting this value of \[x\] in the equation (1), we get
\[
   \Rightarrow 2y - y = 3 \\
   \Rightarrow y = 3 \\
 \]
Then, we have that \[x = 3\].

Adding the number of hits and misses at the end to find the total number of shots taken by the team, we get
\[
   \Rightarrow 5y + 2y \\
   \Rightarrow 7y \\
 \]

Substituting the above value of \[y\] in the above equation, we get
\[
   \Rightarrow 7\left( 3 \right) \\
   \Rightarrow 21 \\
 \]
This implies that the total number of shots taken by the team in the game is 21.

Note: In solving these types of questions, you need to know the basic properties and meaning of the equations and their solutions. One should know that a ratio shows the relative size of two or more values. Students take the equation of ratio for case 1 at the start of the shots taken by the team in the game, which is wrong.