Question

In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d,$ in yards, equals:

not determined by the given information

$58$

$100$

$116$

$120$

Answer: C

Concepts:distance rate and time ratio and proportion rational equation

Difficulty rating: 1550

Small Hint:

Translate each winning margin into a ratio of speeds

Big Hint:

Multiply the ratios $\frac{v_B}{v_A}$ and $\frac{v_C}{v_B}$ to get $\frac{v_C}{v_A}$

Solution:

The margins give $\begin{aligned} \frac{v_B}{v_A}&=\frac{d-20}{d},\\ \frac{v_C}{v_B}&=\frac{d-10}{d},\\ \frac{v_C}{v_A}&=\frac{d-28}{d}. \end{aligned}$ Therefore $\frac{(d-20)(d-10)}{d^2}=\frac{d-28}{d}.$ Expanding and simplifying yields $2d=200,$ so $d=100.$

Thus, the correct answer is C.

1961 AMC 12 Problem 37

Answer: C

Difficulty rating: 1550

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years