Difference between revisions of "1975 AHSME Problems/Problem 17"
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\textbf{(E)}\ \text{ not enough information given to solve the problem} | \textbf{(E)}\ \text{ not enough information given to solve the problem} | ||
</math> | </math> | ||
| + | |||
| + | |||
| + | == Solution == | ||
| + | |||
| + | The man has three possible combinations of transportation: | ||
| + | <cmath>\text{Morning train, Afternoon bus (m.t., a.b.)}</cmath> | ||
| + | <cmath>\text{Morning bus, Afternoon train (m.b., a.t.)}</cmath> | ||
| + | <cmath>\text{Morning bus, Afternoon bus (m.b, a.b.)}</cmath>. | ||
| + | |||
| + | |||
| + | Let <math>y</math> be the number of times the man takes the <math>\text{a.t.}</math>. Then, <math>9-y</math> is the number of times he takes the <math>\text{m.t.}</math>. Keep in mind that <math>\text{m.b.}=y</math> and <math>\text{a.b.}=9-y</math>. | ||
| + | |||
| + | |||
| + | Let <math>z</math> be the number of times the man takes the <math>\text{m.b.}</math> and <math>\text{a.b.}</math>. Now, we get the two equations <cmath>y+z=8</cmath> and <cmath>9-y+z=15.</cmath> | ||
| + | |||
| + | Solving the system of equations, we get <math>y=1</math> and <math>z=7</math>. | ||
| + | |||
| + | |||
| + | So during the <math>x</math> working days, the man took the <math>\text{(m.t., a.b.)}</math> on <math>9-1=8</math> days, the <math>\text{(m.b., a.t.)}</math> on <math>1</math> day, and the <math>\text{(m.b., a.b.)}</math> on <math>7</math> days. | ||
| + | |||
| + | |||
| + | Therefore, <math>x=8+1+7= \boxed{\textbf{(D)}\ 16}</math>. ~[[User:Jiang147369|jiang147369]] | ||
| + | |||
==See Also== | ==See Also== | ||
{{AHSME box|year=1975|num-b=16|num-a=18}} | {{AHSME box|year=1975|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 22:15, 21 July 2021
Problem
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of 
working days, the man took the bus to work in the morning 
times, came home by bus in the afternoon 
times, and commuted by train (either morning or afternoon) 
times. Find .
Solution
The man has three possible combinations of transportation:
![\[\text{Morning train, Afternoon bus (m.t., a.b.)}\]](http://latex.artofproblemsolving.com/3/b/1/3b193e023d5afb3aa920650a0b72a07079508dd3.png)
![\[\text{Morning bus, Afternoon train (m.b., a.t.)}\]](http://latex.artofproblemsolving.com/0/1/5/015f70bc8b7b384f9650a0aa8623449fce0bff70.png)
![\[\text{Morning bus, Afternoon bus (m.b, a.b.)}\]](http://latex.artofproblemsolving.com/2/f/9/2f9bb495eac9ba3cbbf60f1b8006b9cc635c376a.png)
.
Let 
be the number of times the man takes the 
. Then, 
is the number of times he takes the 
. Keep in mind that 
and 
.
Let 
be the number of times the man takes the 
and 
. Now, we get the two equations ![\[y+z=8\]](http://latex.artofproblemsolving.com/9/9/0/99037dced914b21ed981b091b03e91f7323881d3.png)
and ![\[9-y+z=15.\]](http://latex.artofproblemsolving.com/4/6/e/46ed1c514853fa32ccee600139c761e0d235f190.png)
Solving the system of equations, we get 
and 
.
So during the working days, the man took the 
on 
days, the 
on 
day, and the 
on 
days.
Therefore, 
. ~jiang147369
See Also
| 1975 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
