Wikipedia:Reference desk/Archives/Mathematics/2018 March 1

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March 1

Math homework question.

From a tenth grade math book.

A man and his wife walk up a moving escalator. The man walks twice as fast as his wife. When he reaches the top he has taken 28 steps. When she reaches the top she has taken 21 steps. How many steps are visible in the escalator at any one time?

Hint: It was in the "Fractional equations" section. — Preceding unsigned comment added by 2600:1700:2ED1:4020:AD71:72DF:DA0A:A11E (talk) 15:34, 1 March 2018 (UTC)

I'll give you a starting point. After the man gets to the top, he's covered 28 steps, but x1 steps have been skipped over because the escalator is moving. after the wife gets to the top, she's covered 21 steps, but x2 steps have been skipped over because the escalator is moving. As the escalator doesn't change, 28+x1 = 21+x2. Figure out the equation linking x1 and x2 together from the speeds of the 2 people, and the rest is simple algebra. Iffy★Chat -- 16:39, 1 March 2018 (UTC)

Here is a worked answer using lots of algebra.

A      answer aka number of steps visible on the elevator
V      velocity of elevator  (independent of speed of man or speed of wife)
L      Length of elevator
TM     time for man
TW     time for wife
SM     Speed for man
SW     Speed for wife

SM = 2 * SW

28 == A - V * TM
TM == L  / SM

21 == A - V * TW
TW == L / SW    

Putting it together

28 == A - V * L  / SM      next substitute SM with 2 * SW 
28 == A - V * L  /  ( 2 * SW )     call this AAA

21 == A - V *  L / SW      call this BBB

Next we calc AAA - BBB

7 == - V * L  /  ( 2 * SW ) -  ( - V *  L / SW )     Next we pull out the common factors  (V * L  / SW )
7 ==  (V * L  / SW ) * (-1/2 + 1 )
7 ==  (V * L  / SW ) * (1/2)                Next we multiply both sides by TWO
14 ==  (V * L  / SW ) 

recall BBB
21 == A - V *  L / SW        Next we substitute (V * L  / SW )  with 14
21 == A - 14                 Next we add 14 to BOTHSIDES
35 == A

110.22.20.252 (talk) 06:50, 2 March 2018 (UTC)

Unfortunately this answer is wrong. The time taken by the man to get to the top is 28/SM = 14/SW, and the time taken by the woman is 21/SW. putting these in to the equation I derived above gives us 28+14/SW = 21+21/SW. These are equal when SW=1, and gives us our answer of 42 steps. Iffy★Chat -- 09:52, 2 March 2018 (UTC)

That's what I got. (Insert Hitchhiker's joke here.) I think the problem is ambiguously worded (not to mention a bit sexist) when it says that the man walks twice as his wife. To me that means the man walks twice as fast on solid ground. But it could be interpreted to mean that the man moves twice as fast while on the escalator. --RDBury (talk) 10:12, 2 March 2018 (UTC)

PS. The problem seems to be originally from Essentials of Mathematics by Russell V. Person (1961). There is a worked solution here and here. --RDBury (talk) 10:28, 2 March 2018 (UTC)

The mistake in 110.22.20.252's solution is that when walking up the escalator, we must add the escalator's speed V to the man's speed SM or the woman's speed SW. So we have TM = L / (V + SM) and TW = L / (V + SW). Gandalf61 (talk) 10:40, 2 March 2018 (UTC)

The difficult bit here is turning the words into equations (which may then be solved fairly mechanically).

It was initially suggested that if the escalator moves $1x$ steps during the man's journey then it must move $2x$ steps during the women's - or going back one step in the thinking, that the women is on the escalator twice as long as is the man. So is this correct?

One way to test it is to consider extreme (limit) cases:

1. Is it correct when the escalator is moving much (1,000,000 times) faster than either (when they are both effectively standing)?
2. Is it correct when they are both walking much (1,000,000 times) faster than escalator is moving (when the escalator is stopped)?
3. If the women was walking at 1/1,000,000 instead of 1/2 the speed of the man (effectively standing), would she be on the escalator 1,000,000 times longer (for ever)?

The idea fails 2 out of 3 of these tests, so we must think again. It is often possible to use these sort of checks on the final answer too. --catslash (talk) 12:53, 2 March 2018 (UTC)

Assuming v is the speed of the woman, V is the speed of the escalator and k is total number of steps, you can write the share of the steps which she covered as

${\frac {v}{v+V}}={\frac {21}{k}}$

As for the man,

${\frac {2v}{2v+V}}={\frac {28}{k}}$

You then get a trivial quadratic equation which has only one solution as the other is eliminated be the requirement that v ≠ 0 ≠ V, and that solution is indeed k = 42. 93.142.83.134 (talk) 20:27, 2 March 2018 (UTC)