Pascal has 96 miles remaining to complete his cycling trip.

Given:

\(\frac{96}{s - 4} - \frac{96}{3s/2} = 16\)
\(\frac{6}{s - 4} - \frac{4}{s} = 1\)

Don't try to solve it now. Using the options, we can easily see that s = 8 will satisfy this equation.
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Here Plugin helps....I always start with C and it gives me a good indication whether to move up or down in answer choices. Note that answer choices are in ascending or descending order always....

So let's start with C, If Current speed is 10 then time take with speed of 10-4 =6 m/hr is 16 hours
Time take at (10+5) m/hr 96/15 ~ 6...hrs difference is 10 hrs

So speed will be lower....

try current speed as 8 and we get at speed of (8-4), time taken 96/4= 24 hrs
and then time taken at (8+4) m/hr is 8 hrs...difference is 16 hrs...

That is what is given in the answer stem. So ans is B ie. 8

It took me around 2.5 minutes to solve this question.

Please see my solution below:

Let the current speed be x miles per hour.
Time taken if speed is 50% faster (i.e. 3x/2 = 1.5x) = 96/1.5x
Time taken if speed is reduced by 4 miles/hr (i.e. (x-4)) = 96/(x-4)

As per question, 96/(x-4) - 96/1.5x = 16
Solving this we get x = 8.

So, the correct answer is .

Thank you for your explanations throughout these forums.. it has helped me grasp a number of concepts..

In this case, I attempted this question using the methods in Manhattan's book by setting up a chart.. although it is not ideal for this question, I can not see where I am going wrong.. Below is my workings out:

(1) faster speed: 1.5r x t = 96
(2) slower speed: (r-4)(t+16) = 96

Is there a way to proceed if you foil it out? It got messy when I did so and in case I do not see the more elegant method, I would like to see if foiling out will give the correct answer, regardless of the extra time it would take. The main issue I have is how do you know how to set up the equations to get the answer once you are able to set up the given information. Thank you in advance.

Always try to keep minimum variables. More variables means messier calculations.

1.5rt = 96
rt = 96/(3/2) = 64
t = 64/r

(r-4)(t+16) = 96
rt + 16r - 4t - 64 = 96
64 + 16r - 4t - 64 = 96
4r - t = 24
4r - 64/r = 24

Try options which are divisible by 64. Try r = 8. It fits.
4*8 - 64/8 = 24

Answer (B)
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If we let Pascal’s current rate = r, then his time to complete the remainder of the trip = 96/r.

If we let Pascal’s reduced rate = r - 4, then his new time is 96/(r - 4).

If we let Pascal’s increased rate = 1.5r, then his new time is 96/1.5r = 960/15r = 64/r,

Since the remainder of the trip would take him 16 hours longer with his reduced rate than his increased rate:

64/r = 96/(r - 4) - 16

Multiplying the entire equation by r(r - 4), we have:

64(r - 4) = 96r - 16r(r - 4)

64r - 256 = 96r - 16r^2 + 64r

16r^2 - 96r - 256 = 0

r^2 - 6r - 16 = 0

(r - 8)(r + 2) = 0

r = 8 or r = -2

Since the rate can’t be negative, r = 8. Pascal’s current rate is 8 miles per hour.

Answer: B
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Hi all, I think I found a quick way to solve this problem using RELATIVE RATES. I get the correct answer, but can someone take a look and validate it for me?

Call Pascal's regular rate "r". The lower rate (r-4) is called L. The higher rate (1.5r) is called H. Let us set up the formula for the relative rate between H and L.

Speed: H - L = 1.5r-(r-4) = 0.5r+4
Time: 16 hrs (diff between his time with high rate and low rate)
Distance: 0 miles, because he covers the same amount of miles in both speeds.

speed x time = distance
(H - L) x 16 = 0 miles
(0.5r + 4) x 16 = 0
8r + 64 = 0
r = -8. Since r can't be negative, r = 8.

Is this method correct, or is it just giving me the right answer as a fluke? Thanks!

This is not correct. Difference in the two speeds is 0.5r + 4 and difference in time taken in the two cases is 16 hrs but the distance covered cannot be 0.

Note that
Distance1 - Distance2 = Speed1*TIme1 - Speed2*Time2

(Distance1 - Distance2) is NOT (Speed1 - Speed2)*(Time1 - Time2) (which is what you have done)

Also, you get r as -8 i.e. rate as -8. You cannot just drop the negative sign. If you have got rate as a negative number, you need to go back to your equations to check what you have done wrong.
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I thought of a pretty need way to do this

Because he reduced his speed by 4 miles per hours. He has to travel 16 hrs extra. That's 16*4 = 64 miles extra.

And this 64 miles extra would be "canceled" if he was to travel 50% faster (1.5 speed). So it's the 0.5 * speed * 16 hrs is what cancel the 64 miles

Hence: 0.5 * speed * 16 hrs = 64 --> speed = 8 miles/hr

Can anyone please help me in understandig "1.5 r"
If we let Pascal’s increased rate = 1.5r, then his new time is 96/1.5r = 960/15r = 64/r,
how do we calculate 1.5r??
 

The "1.5r" represents Pascal's increased speed, where his original speed is increased by 50%. If we denote his current speed as r miles per hour, increasing this speed by 50% means you're getting the speed of 1.5r.
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