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?

# 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?

1. A
79.5
2. B
81.5
3. C
83.5
4. D
85.5

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### Solution:

The average score of class X = 83
The average score of class Y = 76
The average score of class Z = 85
The average score of class X and Y = 79
The average score of class Y and Z = 81
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.
$\frac{p}{a}$ = 83, $\frac{q}{b}$ = 76, $\frac{r}{c}$ = 85
$\frac{p+q}{a+b}=79$ and $\frac{q+r}{b+c}=81$
$⇒p+q=79a+79b$
$⇒83a+76b=79a+79b$
$⇒4a=3b$
$⇒b=\frac{4a}{3}$
And $q+r=81b+81c$
$⇒76b+85c=81b+81c$
$⇒4c=5b$
$⇒c=\frac{5b}{4}=\frac{5}{4}×\frac{4a}{3}=\frac{5a}{3}$
Average score of the combined class X, Y, and Z = $\frac{p+q+r}{a+b+c}=\frac{83a+76b+85c}{a+b+c}=81.5$

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