Lesson Another unusual Travel problem (Arnold's problem on two walking old women)

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<H2>Another unusual Travel problem (Arnold's problem on two walking old women)</H2>

(The first one is in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/travel/One-unusual-Travel-problem.lesson>One unusual Travel problem</A>&nbsp; in this site).

Two old women started at sunrise and each walked at a constant speed. &nbsp;One went from &nbsp;A&nbsp; to &nbsp;B&nbsp; and the other from &nbsp;B&nbsp; to &nbsp;A&nbsp; on the same road.

They met at noon and, &nbsp;continuing with no stop, &nbsp;arrived respectively at &nbsp;B&nbsp; at &nbsp;4 p.m.&nbsp; and at &nbsp;A&nbsp; at &nbsp;9 p.m.

At what time was the sunrise that day?

Solution

Similar to the first &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/travel/One-unusual-Travel-problem.lesson>unusual Travel problem</A>, &nbsp;the distance is not a given in this problem. &nbsp;Nevertheless, &nbsp;we can solve it by applying elementary logic.

<TABLE>
 <TR>
 <TD>
Figure 1 shows the road two old women were walking on. &nbsp;For simplicity, &nbsp;the road is shown as a straight line in the &nbsp;Figure 1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

Actually, &nbsp;it doesn't matter, &nbspand the road might be curved :-) &nbsp;The point &nbsp;C&nbsp; in the &nbsp;Figure 1&nbsp; represents the meeting point.

Let &nbsp;{{{v[a]}}}&nbsp; be the velocity of lady &nbsp;"A"&nbsp; moving from &nbsp;A&nbsp; to &nbsp;B, &nbsp;and let &nbsp;{{{v[b]}}}&nbsp; be the velocity of lady &nbsp;"B"&nbsp; moving from &nbsp;B&nbsp; to &nbsp;A.

Next, &nbsp;let &nbsp;t&nbsp; be the time &nbsp;(in hours)&nbsp; that both women spent walking from the starting moment &nbsp;("the sunrise")&nbsp; till noon.

</TD>
 <TD>
&nbsp;&nbsp;{{{drawing( 300, 100, 0, 7, 0.5, 3, 
 line( 0.5, 1.5, 6.5, 1.5),

circle(0.5, 1.5, 0.06),
            circle(6.5, 1.5, 0.06),
            circle(4.0, 1.5, 0.06),

locate(0.4,  1.4, A),
            locate(6.4,  1.4, B),
            locate(3.95, 1.4, C),

line (0.5, 2.0, 1.5, 2.0),
            line (1.2, 2.1, 1.5, 2.0),
            line (1.2, 1.9, 1.5, 2.0),
            locate(0.5, 2.6, Va),

line (5.5, 2.0, 6.5, 2.0),
            line (5.5, 2.0, 5.8, 2.1),
            line (5.5, 2.0, 5.8, 1.9),
            locate(6.1, 2.6, Vb)

)}}}
Figure 1. &nbsp;The road between &nbsp;A&nbsp; and &nbsp;B
 </TD>
 </TR>
</TABLE>

Lady &nbsp;"A"&nbsp; covered partial way &nbsp;|AC|&nbsp; in &nbsp;t&nbsp; hours and lady &nbsp;"B"&nbsp; covered it in &nbsp;9&nbsp; hours. &nbsp;It gives the equation &nbsp;|AC| = {{{v[a]*t}}} = {{{v[b]*9}}}, &nbsp;&nbsp;or simply &nbsp;&nbsp;{{{v[a]*t}}} = {{{v[b]*9}}}, &nbsp;&nbsp;for short.

Similarly, &nbsp;lady &nbsp;"B"&nbsp; covered partial way &nbsp;|BC|&nbsp; in &nbsp;t&nbsp; hours and lady &nbsp;"A"&nbsp; covered it in &nbsp;4&nbsp; hours. &nbsp;It gives the equation &nbsp;|BC| = {{{v[b]*t}}} = {{{v[a]*4}}}, &nbsp;&nbsp;or simply &nbsp;&nbsp;{{{v[b]*t}}} = {{{v[a]*4}}}.

From the last two equations &nbsp;&nbsp;{{{v[a]/v[b]}}} = {{{v[b]/v[a]}}}*{{{9/4}}}. &nbsp;&nbsp;It gives &nbsp;&nbsp;{{{v[a]^2/v[b]^2}}} = {{{9/4}}}, &nbsp;&nbsp;or&nbsp;&nbsp; {{{va/v[b]}}} = {{{3/2}}}.

Now, &nbsp;let us consider the entire walking journey from &nbsp;A&nbsp; to &nbsp;B&nbsp; and from &nbsp;B&nbsp; to &nbsp;A. &nbsp;&nbsp;Lady &nbsp;"A"&nbsp; covered it in &nbsp;t+4&nbsp; hours and lady &nbsp;"B"&nbsp; in &nbsp;t+9&nbsp; hours. &nbsp;It gives the equation

{{{v[a]*(t+4)}}} = {{{v[b]*(t+9)}}}, &nbsp;since the distance from &nbsp;A&nbsp; to &nbsp;B&nbsp; is the same as the distance from &nbsp;B&nbsp; to &nbsp;A.

Simplifying it step by step, &nbsp;you get  &nbsp;&nbsp;{{{(t+9)/(t+4)}}} = {{{v[a]/v[b]}}} = {{{3/2}}}, &nbsp;&nbsp;then&nbsp;&nbsp;{{{2*(t+9)}}} = {{{3*(t+4)}}}, &nbsp;&nbsp;{{{2*t + 18}}} = {{{3*t + 12}}}, &nbsp;&nbsp;and, &nbsp;at last, &nbsp;&nbsp;{{{t}}} = {{{6}}}.

Thus the women were walking &nbsp;6&nbsp; hours before noon. &nbsp;&nbsp;It means that the sunrise was at &nbsp;6:00 am&nbsp; that day. &nbsp;&nbsp;The problem is solved.

There is another way to complete the solution. &nbsp;From equations &nbsp;{{{v[a]*t}}} = {{{v[b]*9}}}&nbsp; and &nbsp;{{{v[b]*t}}} = {{{v[a]*4}}}&nbsp; you have &nbsp;{{{t/9}}} = {{{v[b]/v[a]}}}&nbsp; and &nbsp;{{{4/t}}} = {{{v[b]/v[a]}}}, &nbsp;i.e. &nbsp;{{{t/9}}} = {{{4/t}}}, &nbsp;or {{{t^2}}} = 36&nbsp; and &nbsp;t = {{{sqrt(36)}}} = 6.

Answer. &nbsp;The sunrise was at &nbsp;6:00 am&nbsp; that day.

<H3>The story</H3>This problem has its own &nbsp;story. &nbsp;It became known &nbsp;("widely known in narrow circles")&nbsp; after Vladimir Arnold.

<A HREF=http://en.wikipedia.org/wiki/Vladimir_Arnold>Vladimir Arnold</A>&nbsp; (1937 - 2010) &nbsp;was&nbsp; the famous soviet, &nbsp;russian and word mathematician. &nbsp;In one of his interviews he remembered that this problem was given in the classroom in the Moscow school by his mathematics teacher, &nbsp;when he was a &nbsp;6-th&nbsp; grade &nbsp;(12&nbsp; years old)&nbsp; student. &nbsp;(6-th&nbsp; grade in soviet school at that time approximately corresponded to the &nbsp;6-th - 7-th &nbsp; grade of American middle school). &nbsp;The problem was a big challenge for him. &nbsp;He spent the entire day trying to solve it, &nbsp;and was very happy when, &nbsp;finally, &nbsp;he found the solution. &nbsp;He memorized that impression, &nbsp;the problem itself and his teacher for all his life.

You can make a &nbsp;Google search&nbsp; about this story in the &nbsp;Internet&nbsp; with keywords &nbsp;"Vladimir Arnold problem on two walking women" &nbsp;if you want.

Recently this problem came in another form.

<H3>Problem 2</H3>Winnie and Piglet decided to visit one another and set off at the same time to each others house. &nbsp;When they met on the road 
joining their houses, &nbsp;they forgot that they wanted to see each other and continued walking. &nbsp;Winnie reached Piglet's house 
8&nbsp; minutes after the meeting and Piglet got to Winnie's house &nbsp;18&nbsp; minutes after the meeting. 
How long did it take Piglet to reach Pooh's house from the time he left his own house?

Solution

<pre>
Let "W" be Winnie's speed and "P" be Piglet's speed.

Let "t" be the time from the start to the meeting moment.

Then you have these equations

W*t = P*18,   (1)   and
P*t = W*8.    (2)

It implies

{{{W/P}}} = {{{18/t}}},         (1')   ( from (1) ) and
{{{W/P}}} = {{{t/8}}}.          (2')   ( from (2) ).

Hence,  {{{18/t}}} = {{{t/8}}},  and then  {{{t^2}}} = {{{18*8}}} = {{{144}}}.

Therefore, t = {{{sqrt(144)}}} = 12.

In other words, it took 12 minutes from the start to the meeting moment.

Then the total time for Piglet to reach Pooh's house was 12 + 18 minutes = 30 minutes.

Answer. The total time for Piglet to reach Pooh's house was 12 + 18 minutes = 30 minutes.
</pre>

My other lessons on &nbsp;Travel and Distance&nbsp; problems in this site are

<TABLE>
 <TR>
 <TD>
- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems.lesson>Travel and Distance problems</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems-for-two-bodies-moving-toward-each-other.lesson>Travel and Distance problems for two bodies moving in opposite directions</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-catching-up-Travel-and-Distance-problems.lesson>Travel and Distance problems for two bodies moving in the same direction (catching up)</A>

- <A HREF=http://www.algebra.com/algebra/homework/NumericFractions/Using-fractions-to-solve-Travel-problems.lesson>Using fractions to solve Travel problems</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems.lesson>Wind and Current problems</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/More-problems-on-upstream-and-downstream-round-trips.lesson>More problems on upstream and downstream round trips</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems-solvable-by-quadratic-equations.lesson>Wind and Current problems solvable by quadratic equations</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Unpowered-raft-moving-downstream-along-a-river.lesson>Unpowered raft floating downstream along a river</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-boat-floating-Upstream-and-Downstream.lesson>Selected problems from the archive on the boat floating Upstream and Downstream</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-plane-flying-with-and-against-the-wind.lesson>Selected problems from the archive on a plane flying with and against the wind</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-Travel-and-Distance-problems-from-the-archive.lesson>Selected Travel and Distance problems from the archive</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Had-a-car-move-faster-it-would-arrive-quicker.lesson>Had a car move faster it would arrive sooner</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/How-far-do-you-live-from-school.lesson>How far do you live from school?</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-unusual-Travel-problem.lesson>One unusual Travel problem</A>

- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-problem-on-a-messenger-moving-back-and-forth-along-the-marching-army-column.lesson>Travel problem on a messenger moving back and forth along the marching army's column</A>

- <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-person-walking-along-the-street-and-the-buses-traveling-in-the-same-and-opposite-directions.lesson>A person walking along the street and buses traveling in the same and opposite directions</A>

</TD>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Calculating-an-average-speed.lesson>Calculating an average speed: a train going from A to B and back</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-more-problem-on-calculating-an-average-speed.lesson>One more problem on calculating an average speed</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Clock-problems.lesson>Clock problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Advanced-clock-problems.lesson>Advanced clock problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Problems-on-bodies-moving-on-a-circle.lesson>Problems on bodies moving on a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-telegraph-post-and-passing-a-bridge.lesson>A train passing a telegraph post and passing a bridge</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-platform.lesson>A train passing a platform</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-tunnel.lesson>A train passing through a tunnel</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-light-rail-train-passing-a-walking-person.lesson>A light-rail train passing a walking person</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-another-train.lesson>A train passing another train</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-man-crossing-a-bridge-when-a-train-comes-from-behind.lesson>A man crossing a bridge and a train coming from behind</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/27-A-problem-on-a-rower-going-on-a-river-who-missed-the-bottle-of-whiskey-under-a-bridge.lesson>A rower going on a river who missed the bottle of whiskey under a bridge</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Non-traditional-Travel-and-Distance-problems.lesson>Non-traditional Travel and Distance problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-distance-covered-by-a-free-falling-body-during-last-second-of-the-fall.lesson>The distance covered by a free falling body during last second of the fall</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-Doppler-shift.lesson>The Doppler Shift</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Entertainment-Travel-and-Distance-problems.lesson>Entertainment Travel and Distance problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/OVERVIEW-of-lessons-on-Travel-and-Distance.lesson>OVERVIEW of lessons on Travel and Distance</A>

</TD>
 </TR>
</TABLE>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.