1990 AMC 12 Problem 24

Question

All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?

	Adams	Baker	Adams & Baker
Boys:	$71$	$81$	$79$
Girls:	$76$	$90$	?
Boys & Girls:	$74$	$84$

81

82

83

84

85

Answer: D

Concepts:weighted average ratio system of equations

Difficulty rating: 1720

Small hint:

Use each school’s combined average to find its ratio of girls to boys

Big hint:

Use the combined boys’ average to relate the numbers of boys at the two schools

Solution:

Let Adams have

x

boys and

y

girls. From its average,

71x+76y=74(x+y),

so

y=\frac{3x}{2}.

If Baker has

u

boys and

v

girls, its average gives

u=2v.

The combined boys’ average gives

71x+81u=79(x+u),

so

u=4x

and

v=2x.

Hence the combined girls’ average is

\frac{76(\frac{3x}{2})+90(2x)}{\frac{3x}{2}+2x}=84.

Thus the correct answer is D.