1984 AMC 12 Problem 6

Attempt Problem 6 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters

b,

g,

t

to represent the number of boys, girls and teachers, respectively, then the total number of boys, girls and teachers can be represented by the expression

31b

\frac{37}{27}b

13g

\frac{37}{27}g

\frac{37}{27}t

Answer: B

Concepts:ratio and proportion algebraic manipulation

Difficulty rating: 1290

Small hint:

Express both

g

and

t

in terms of

b

Big hint:

The given ratios imply

g=\frac{b}{3}

and

t=\frac{b}{27}

Solution:

Because

b=3g

and

g=9t,

we have

g=\frac{b}{3}

and

t=\frac{b}{27}.

Hence

b+g+t=b+\frac b3+\frac b{27}=\frac{37}{27}b.

Therefore, the correct answer is B.