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Difference between revisions of "1951 AHSME Problems/Problem 50"

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== Problem ==
 
== Problem ==
 
Tom, Dick and Harry started out on a <math>100</math>-mile journey. Tom and Harry went by automobile at the rate of <math>25</math> mph, while Dick walked at the rate of <math>5</math> mph. After a certain distance, Harry got off and walked on at <math>5</math> mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:  
 
Tom, Dick and Harry started out on a <math>100</math>-mile journey. Tom and Harry went by automobile at the rate of <math>25</math> mph, while Dick walked at the rate of <math>5</math> mph. After a certain distance, Harry got off and walked on at <math>5</math> mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:  
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<cmath>500-4d_1=100+2d_2=100+4d_1-4d_2</cmath>
 
<cmath>500-4d_1=100+2d_2=100+4d_1-4d_2</cmath>
  
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We have that <math>100+2d_2=100+4d_1-4d_2</math>, so <math>4d_1=6d_2\Rightarrow d_1=\frac{2}{3}d_2</math>. This then shows that <math>500-4d_1=100+\frac{4}{3}d_1\Rightarrow 400=\frac{16}{3}d_1\Rightarrow d_1=75</math>, which in turn gives that <math>d_2=50</math>. Now we only need to solve for <math>T</math>:
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We have that <math>100+2d_2=100+4d_1-4d_2</math>, so <math>4d_1=6d_2\Rightarrow d_1=\frac{3}{2}d_2</math>. This then shows that <math>500-4d_1=100+\frac{4}{3}d_1\Rightarrow 400=\frac{16}{3}d_1\Rightarrow d_1=75</math>, which in turn gives that <math>d_2=50</math>. Now we only need to solve for <math>T</math>:
  
 
<cmath>T=\frac{d_1}{25}+\frac{100-d_1}{5}=\frac{75}{25}+\frac{100-75}{5}=3+5=8</cmath>
 
<cmath>T=\frac{d_1}{25}+\frac{100-d_1}{5}=\frac{75}{25}+\frac{100-75}{5}=3+5=8</cmath>
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The journey took 8 hours, so the correct answer is <math>\boxed{\textbf{(D)}}</math>.
 
The journey took 8 hours, so the correct answer is <math>\boxed{\textbf{(D)}}</math>.
  
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== See also ==
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== See Also ==
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{{AHSME box|year=1951|num-b=49|after=Last Question}}
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{{AHSME 50p box|year=1951|num-b=49|after=Last Question}}
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[[Category:Intermediate Algebra Problems]]
 +
[[Category:Rate Problems]]
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{{MAA Notice}}

Latest revision as of 14:55, 6 April 2020

Problem

Tom, Dick and Harry started out on a $100$-mile journey. Tom and Harry went by automobile at the rate of $25$ mph, while Dick walked at the rate of $5$ mph. After a certain distance, Harry got off and walked on at $5$ mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ \text{none of these answers}$

Solution

Let $d_1$ be the distance (in miles) that Harry traveled on car, and let $d_2$ be the distance (in miles) that Tom backtracked to get Dick. Let $T$ be the time (in hours) that it took the three to complete the journey. We now examine Harry's journey, Tom's journey, and Dick's journey. These yield, respectively, the equations

\[\frac{d_1}{25}+\frac{100-d_1}{5}=T,\]

\[\frac{d_1}{25}+\frac{d_2}{25}+\frac{100-(d_1-d_2)}{25}=T,\]

\[\frac{d_1-d_2}{5}+\frac{100-(d_1-d_2)}{25}=T.\]

We combine these three equations:

\[\frac{d_1}{25}+\frac{100-d_1}{5}=\frac{d_1}{25}+\frac{d_2}{25}+\frac{100-(d_1-d_2)}{25}=\frac{d_1-d_2}{5}+\frac{100-(d_1-d_2)}{25}\]

After multiplying everything by 25 and simplifying, we get

\[500-4d_1=100+2d_2=100+4d_1-4d_2\]

We have that $100+2d_2=100+4d_1-4d_2$, so $4d_1=6d_2\Rightarrow d_1=\frac{3}{2}d_2$. This then shows that $500-4d_1=100+\frac{4}{3}d_1\Rightarrow 400=\frac{16}{3}d_1\Rightarrow d_1=75$, which in turn gives that $d_2=50$. Now we only need to solve for $T$:

\[T=\frac{d_1}{25}+\frac{100-d_1}{5}=\frac{75}{25}+\frac{100-75}{5}=3+5=8\]

The journey took 8 hours, so the correct answer is $\boxed{\textbf{(D)}}$.

See Also

1951 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 49 		Followed by
Last Question
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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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