Question

Three maths classes: X, Y, and Z take an algebra test. The average score of class X is 83. The average score of class Y is 76. The average score of class Z is 85. The average score of class X and Y is 79 and the average score of class Y and Z is 81. What is the average score of the combined classes X, Y, and Z?

Answer 1

Hint: Assume that the number of students in class X, Y, and Z are a, b, and c respectively, and the sum of the scores of the students in class X, Y, and Z are p, q, and r. Now, get the average scores of the class X, Y, Z, Class X and Y, and the Class Y and Z. Use these and compare them with the information provided in the question and then get the value of p, q, r, b, and c in terms of a. For the combined class X, Y, and Z, the average score is \[\dfrac{p+q+r}{a+b+c}\] . Now, put the value of p, q, r, b, and c in terms of a. Solve it further and get the required answer.

Complete step-by-step answer:
According to the question, we have
The average score of class X = 83 …………………………………………(1)
The average score of class Y = 76 …………………………………………(2)
The average score of class Z = 85 …………………………………………(3)
The average score of class X and Y = 79 …………………………………………(4)
The average score of class Y and Z = 81 …………………………………………(5)
Let us assume that the number of students in class X, Y, and Z are a, b, and c respectively, and the sum of the scores of the students in class X, Y, and Z are p, q, and r.
Now, using the above, we get
The average score of class X = \[\dfrac{p}{a}\] …………………………………………..(6)
 The average score of class Y = \[\dfrac{q}{b}\] …………………………………………..(7)
The average score of class Z = \[\dfrac{r}{c}\] …………………………………………..(8)
The average score of Class X and Y = \[\dfrac{p+q}{a+b}\] ……………………………………………………..(9)
The average score of Class Y and Z = \[\dfrac{q+r}{b+c}\] ……………………………………………………..(10)
Now, from equation (1) and equation (6), we get
\[\Rightarrow \dfrac{p}{a}=83\]
\[\Rightarrow p=83a\] …………………………………………….(11)
Similarly, from equation (2) and equation (7), we get
 \[\Rightarrow \dfrac{q}{b}=76\]
\[\Rightarrow q=76b\] ………………………………………………(12)
 Similarly, from equation (3) and equation (8), we get
\[\Rightarrow \dfrac{r}{c}=85\]
\[\Rightarrow r=85c\] ……………………………………………………(13)
Similarly, from equation (4) and equation (9), we get
\[\Rightarrow \dfrac{p+q}{a+b}=79\]
\[\Rightarrow p+q=79a+79b\] ……………………………………………..(14)
From equation (11), equation (12), and equation (14), we get
 \[\begin{align}
  & \Rightarrow 83a+76b=79a+79b \\
 & \Rightarrow 83a-79a=79b-76b \\
 & \Rightarrow 4a=3b \\
\end{align}\]
\[\Rightarrow b=\dfrac{4a}{3}\] ………………………………………………………(15)
Now, from equation (5) and equation (10), we get
\[\Rightarrow \dfrac{q+r}{b+c}=81\]
\[\Rightarrow q+r=81b+81c\] ……………………………………………………………..(16)
From equation (12), equation (13), and equation (16), we get
\[\begin{align}
  & \Rightarrow 76b+85c=81b+81c \\
 & \Rightarrow 85c-81c=81b-76b \\
 & \Rightarrow 4c=5b \\
\end{align}\]
\[\Rightarrow c=\dfrac{5b}{4}\] …………………………………………………….(17)
Now, from equation (15) and equation (17), we get
 \[\Rightarrow c=\dfrac{5}{4}\times \dfrac{4a}{3}\]
\[\Rightarrow c=\dfrac{5a}{3}\] …………………………………………………………..(18)
We have to find the average score of the combined class X, Y, and Z.
The sum of total scores of students for the combined class X, Y, and Z = \[p+q+r\] ………………………………………………(19)
The total number of students for the combined class X, Y, and Z = \[a+b+c\] ………………………………………………(20)
Now, the average score of the combined class X, Y, and Z = \[\dfrac{p+q+r}{a+b+c}\]
………………………………………………(21)
Now, from equation (11), equation (12), equation(13), and equation (21), we get
\[\Rightarrow \dfrac{83a+76b+85c}{a+b+c}\] …………………………………………..(22)
On putting the value of b and c from equation (15) and equation (18) in equation (22), we get
\[\begin{align}
  & =\dfrac{83a+76\times \dfrac{4a}{3}+85\times \dfrac{5a}{3}}{a+\dfrac{4a}{3}+\dfrac{5a}{3}} \\
 & =\dfrac{83\times 3+76\times 4+85\times 5}{3+4+5} \\
 & =\dfrac{249+304+425}{12} \\
 & =\dfrac{978}{12} \\
 & =81.5 \\
\end{align}\]
Therefore, the average of the combined class X, Y, and Z is 81.5.

Note: In this question, one might get confused because neither the number of students nor the sum of the scores of any of the class is given. So, don’t get confused here, just assume them and at the end the variable will cancel each other and we will get the absolute numerical value.