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If a motorist had driven 1 hour longer on a certain day and [#permalink]

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24 Mar 2008, 13:37

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If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day?

70 additional miles came from 2 sources - 5 miles additional distance traveled for t hours (5t) + distance traveled in the additional hour. The later one is x+5 where in x is distance traveled in an hour with the original speed and 5 is the distance came from the additional speed (making total speed x+5 mph)

If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day? (A) 100 (B) 120 (C) 140 (D) 150 (E) 160

Interesting problem.

first we have (r+5)(t+1)=d+70 ---> rt+r+5t+5=d+70. Also notice that rt=d

thus we now have r+5t=65.

Second equation. (r+10)(t+2)=d+x

rt+2r+10t+20=d+x --> plug in rt and 2(r+5t)=2(65) ---->130 ---> rt+150=rt+x ---> x=150.

If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day? (A) 100 (B) 120 (C) 140 (D) 150 (E) 160

The guy covered 70 miles more in an hour by driving 5 miles/hour faster. Therefore his speed current is 70 Miles/hr and his original speed was 65 Miles /Hr (since he is traveling 5Miles/Hr faster). Therefore his new speed is 75Miles/Hr (65+10) and he will travel 150 (75x2) Miles more in 2 Hours.

my approach was similar too than the posts above ... but i'll share it anyway

avg rate of 5mph faster means that the motorist drove 5miles more in each hour. since he drove 5miles in the last (extra) hour too, so he drove 70-5=65 miles more in each of the hour earlier 65/5=13hrs this is the actual time he drove

avg rate of 10mph more means that he drove 10*13=130miles more in the first 13 hrs he covered 2*10=20miles more in the last 2 hours 130+20=150 more miles covered than he actually did

so D it is _________________

press kudos, if you like the explanation, appreciate the effort or encourage people to respond.

Re: Problem related to time and distance [#permalink]

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08 Jun 2010, 02:37

padmaranganathan wrote:

20. If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day? (A) 100 (B) 120 (C) 140 (D) 150 (E) 160

Re: Problem related to time and distance [#permalink]

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08 Jun 2010, 03:33

Hi sondenso, My approach was very similar to yours except for a minor change in the 3rd step;

"...How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day?" can be rephrased as;

20. If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day? (A) 100 (B) 120 (C) 140 (D) 150 (E) 160

Let \(t\) be the actual time and \(r\) be the actual rate.

"If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did" --> \((t+1)(r+5)-70=tr\) --> \(tr+5t+r+5-70=tr\) --> \(5t+r=65\);

"How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day?" --> \((t+2)(r+10)-x=tr\) --> \(tr+10t+2r+20-x=tr\) --> \(2(5t+r)+20=x\) --> as from above \(5t+r=65\), then \(2(5t+r)+20=2*65+20=150=x\) --> so \(x=150\).

Answer: D.

OR another way:

70 miles of surplus in distance is composed of driving at 5 miles per hour faster for \(t\) hoursplusdriving for \(r+5\) miles per hour for additional 1 hour --> \(70=5t+(r+5)*1\) --> \(5t+r=65\);

With the same logic, surplus in distance generated by driving at 10 miles per hour faster for 2 hours longer will be composed of driving at 10 miles per hour faster for \(t\) hoursplusdriving for \(r+10\) miles per hour for additional 2 hour --> \(surplus=x=10t+(r+10)*2\) --> \(x=2(5t+r)+20\) --> as from above \(5t+r=65\), then \(x=2(5t+r)+20=150\).

Answer: D.

Note that the solutions proposed by dushver and dimitri92 are not correct (though correct answer was obtained). For this question we can not calculate neither \(t\) not \(r\) of the motorist.

The motorist covers 70 miles in an hour when he drives 5 miles/hr faster => He is currently driving at 65 miles/hr

If he drives 10 miles/hr faster for 2 more hours he would cover 75*2 = 150 miles

(D) it is.

despite the correct answer, the logic here is flawed, isn't it ?

Quote:

The motorist covers 70 miles in an hour when he drives 5 miles/hr faster

M drives less than 70miles in one hour. Because M drove more than the original distance in the time before the extra hour starts. So that delta + another hour's worth of driving (at the elevated speed) is 70miles.

despite the correct answer, the logic here is flawed, isn't it ?

Quote:

The motorist covers 70 miles in an hour when he drives 5 miles/hr faster

M drives less than 70miles in one hour. Because M drove more than the original distance in the time before the extra hour starts. So that delta + another hour's worth of driving (at the elevated speed) is 70miles.

Yes, 70 miles is the sum of extra distance covered by increasing speed + extra distance covered in an hour.

But remember, this is a PS question. You do have the correct answer in the options and there is only one correct option. I will just assume any case and whatever I get will be my answer.

Say the original speed was 60 miles/hr. The speed increased to 65. So the motorist would have traveled 5 miles + 65 miles (= 70) extra. (He travels for only one hour initially.) If he instead increases speed by 10 miles/hr, his speed becomes 70. In the initial 1 hr, he will travel an extra 10 miles and then in additional 2 hrs he will travel an extra 70*2 = 140 miles. So he will travel an extra 150 miles.

I could have just as well assumed the original speed to be 55 miles/hr. If the speed increases to 60 and extra distance covered is 70, it means that the motorist travels for 2 hrs initially. 70 = 5 + 5 + 60 If instead, the speed increases by 10 miles/hr, the speed becomes 65. Extra distance covered = 10+10+2*65 = 150

In any case, the answer has to be the same since it is a PS question.

I could also have assumed the original speed to be 65 miles/hr (as done above) If the speed increases to 70 and I cover 70 miles extra, it means I didn't travel at all before. If the speed instead becomes 75, I travel 2*75 = 150 miles extra.
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