Question

In base $R_1$ the expanded fraction $F_1$ becomes $0.373737\ldots$ and the expanded fraction $F_2$ becomes $0.737373\ldots.$ In base $R_2$ fraction $F_1,$ when expanded, becomes $0.252525\ldots,$ while fraction $F_2$ becomes $0.525252\ldots.$ The sum of $R_1$ and $R_2,$ each written in base ten, is:

$24$

$22$

$21$

$20$

$19$

Answer: E

Concepts:number base repeating decimal system of equations

Difficulty rating: 1990

Small Hint:

A repeating pair $ab$ in base $R$ equals $\frac{aR+b}{R^2-1}$

Big Hint:

Add the equations for $F_1$ and $F_2,$ then subtract them

Solution:

The two descriptions give $\begin{aligned} F_1&=\frac{3R_1+7}{R_1^2-1}\\ &=\frac{2R_2+5}{R_2^2-1},\\ F_2&=\frac{7R_1+3}{R_1^2-1}\\ &=\frac{5R_2+2}{R_2^2-1}. \end{aligned}$ Adding yields $\frac{10}{R_1-1}=\frac{7}{R_2-1},$ or $10R_2-7R_1=3.$ Subtracting yields $\frac{4}{R_1+1}=\frac{3}{R_2+1},$ or $4R_2-3R_1=-1.$ Solving gives $R_1=11,$ $R_2=8,$ whose sum is $19.$

Therefore, the correct answer is E.

1966 AMC 12 Problem 39

Answer: E

Difficulty rating: 1990

Small Hint:

Big Hint:

Solution:

Problem 39 in Other Years