Expert: Ahmed Salami - 1/9/2006

Question
Hello:

I want to thank you for your reply.  I think that I have a calculation that may work and that does not involve algebra. It is as follows:

Increase the hours traveled by 1/2. (1/2 X 6 hours) + 6 hours = 9 hours.
6 hours are what part of 9 hours?

6/9 = 2/3

2/3 of 6 hours equals 4 hours, the answer.  Is this calculation correct since it produces the answer of 4 hours? Furthermore, if it is correct, can you explain why?

I thank you for your follow-up reply.
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Followup To
Question -
Hello:

I want to thank you for your reply and answer.  However, your answer involves algebra and variables. Can you think of another solution?

I thank you for your assistance!
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Followup To
Question -
Hello:

Here is my question:

If it takes 6 hours to complete a trip, how many hours will it take to complete the same trip if the rate of speed is increased by 1/2 of itself?
The answer is 4 hours.

Please note: The question came from a chapter of an old textbook on arithmetic. The chapter makes no mention of algebra calculations. Can the answer be determined without solving for some unknown variable?

I thank you for your reply.
Answer -
Hi,
I'm sure it can.
We know that speed(v) = distance(d)/time(t)
so, v = d/t
Let the speed and time for the first and second situations be v1,t1 and v2,t2. Note that the distance is the same for both trips.
We therefore have
v1 = d/t1 which gives
d = v1.t1
and
v2 = d/t2
d = v2.t2
equating the d's gives
v1.t1 = v2.t2
but v2 = v1 + (1/2)v1
      = (3/2)v1
Therefore
v1.t1 = (3/2)v1.t2
hence
t2 = (2/3)t1
  = 2/3 x 6
  = 4
I've made the working long so you can get the full preview.
I hope i have helped.
Regards.
Answer -
Hi,
Not at the moment. You can only use a comparison which would still involve algebra.
Happy to always help out.
Regards.

Answer
Hi,
I think the answer was just a mere coincidence, it happens. You've actually increased the time by half, but it is the speed that was asked to be increased by the question. The confusion arises because both v and t are 'inversely proportional' i.e if v is multiplied by k, t is multiplied by 1/k.
Increasing v by 1/2 means that it changes to
v + (1/2)v = (3/2)v
t would then have to be multiplied by 1/(3/2) = 2/3
and thats how we get (2/3)x 6 = 4
Note this, your method might still work at other times, for instance if you increased v by 1/3, but then it has no meaningful mathematical explanation.
I hope you get it.
Regards.