$3 \cdot 31$ In a race of $d$ yards run at a constant speed, $A$ beats $B$ by 20 yards, $B$ beats $C$ by 10 yards, and $A$ beats $C$ by 28 yards. What is the value of $d$ in yards?
(A) Cannot be determined.
(B) 58.
(C) 100.
(D) 116.
(E) 120.
(12th American High School Mathematics Examination, 1961)
[Solution] Let the speeds of $A, B, C$ be $v_{A}, v_{B}, v_{C}$. According to the problem, we have
$$
\left\{\begin{array}{l}
\frac{d}{v_{A}}=\frac{d-20}{v_{B}} \\
\frac{d}{v_{B}}=\frac{d-10}{v_{C}} \\
\frac{d}{v_{A}}=\frac{d-28}{v_{C}} .
\end{array}\right.
$$

From (1) and (3), we have $\frac{d-20}{v_{B}}=\frac{d-28}{v_{C}}$
Dividing (4) by (2) gives $\frac{d-20}{d}=\frac{d-28}{d-10}$, which yields $d=100$. Therefore, the answer is $(C)$.