Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1961 AHSME Problems/Problem 37"

(→‎Solution 2)
(→‎Solution 2)
Line 37: Line 37:
  
 
Let <math>s_a</math>, <math>s_b</math>, <math>s_c</math> be the speeds of <math>A</math>, <math>B</math>, <math>C</math>, respectively and let <math>t_a</math>, <math>t_b</math>, <math>t_c</math> be their respective times needed, such that  
 
Let <math>s_a</math>, <math>s_b</math>, <math>s_c</math> be the speeds of <math>A</math>, <math>B</math>, <math>C</math>, respectively and let <math>t_a</math>, <math>t_b</math>, <math>t_c</math> be their respective times needed, such that  
−
<math>d=s_at_a=s_bt_b=s_ct_b</math>.
+
<cmath>d=s_at_a=s_bt_b=s_ct_b</cmath>
 +
Putting each pair of them into a race gets the <math>3</math> equations
 +
<cmath>s_at_a-s_bt_a=20</cmath>
 +
<cmath>s_bt_b-s_ct_b=10</cmath>
 +
<cmath>s_at_a-s_ct_a=8</cmath>
  
 
==See Also==
 
==See Also==

Revision as of 17:20, 23 December 2019

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 \[s_at_a-s_bt_a=20\] \[s_bt_b-s_ct_b=10\] \[s_at_a-s_ct_a=8\]

See Also

1961 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 36 		Followed by
Problem 38
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1961_AHSME_Problems/Problem_37&oldid=113289"