1961 AHSME Problems/Problem 37

Problem

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:

$\textbf{(A)}\ \text{Not determined by the given information} \qquad  \textbf{(B)}\ 58\qquad \textbf{(C)}\ 100\qquad \textbf{(D)}\ 116\qquad \textbf{(E)}\ 120$

Solution

Let $a$ be speed of $A$, $b$ be speed of $B$, and $c$ be speed of $C$.

Person $A$ finished the track in $\frac{d}{a}$ minutes, so $B$ traveled $\frac{db}{a}$ yards at the same time. Since $B$ is $20$ yards from the finish line, the first equation is \[\frac{db}{a} + 20 = d\] Using similar steps, the second and third equation are, respectively, \[\frac{dc}{b} + 10 = d\] \[\frac{dc}{a} + 28 = d\] Get rid of the denominator in all three equations by multiplying both sides by the denominator in each equation. \[db + 20a = da\] \[dc + 10b = db\] \[dc + 28a = da\] These equations can be rearranged to get the following. \[(d-20)a = db\] \[(d-10)b = dc\] \[(d-28)a = dc\]

Solving for $b$ in the first equation yields $\frac{(d-20)a}{d} = b$. Substituting for $b$ and solving for $c$ in the second equation yields $\frac{(d-10)(d-20)a}{d^2} = c$. Substituting for $c$ in the third equation yields \[(d-28)a = \frac{(d-10)(d-20)a}{d}\] \[d^2 - 28d = d^2 - 30d + 200\] Solve the equation to get $d = 100$. Thus, the track is $100$ yards long, which is answer choice $\boxed{\textbf{(C)}}$.


Solution 2

Let $s_a$, $s_b$, $s_c$ be the speeds of $A$, $B$, $C$, respectively and let $t_a$, $t_b$, $t_c$ be their respective times needed, such that \[d=s_at_a=s_bt_b=s_ct_b\] Putting each pair of them into a race gets the $3$ equations \[s_at_a-s_bt_a=20\] \[s_bt_b-s_ct_b=10\] \[s_at_a-s_ct_a=28\] Subtract the first equation from the third yields \[s_bt_a-s_ct_a=t_a(s_b-s_c)=8\] Divide the equation by the second original equation yields \[\frac{t_a}{t_b}=\frac{4}{5}\] Since $s_at_a=s_bt_b$ we have \[\frac{s_a}{s_b}=\frac{5}{4}\] Thus from the first equation we have \[s_at_a-s_bt_a=s_at_a-\frac{4}{5}s_at_a=20\Longrightarrow d=s_at_a=100 \boxed{C}\]

~ Nafer

See Also

1961 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 36
Followed by
Problem 38
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