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Difficult Rates question

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Difficult Rates question

by gustavo.taucei » Wed Dec 08, 2010 1:00 pm
16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
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by Bharat » Wed Dec 08, 2010 3:10 pm
Hi Gustavo,
Here is an explanation, let me know if it is not clear:
Average speed = distance/time
from time condition: t2 - t1 = 2 & t2 + t1 = 18; solve for t1: t1 = 8
from distance condition: s1 + s2 = 64; i.e., s2 = 64 - s1
from av. speed condition: v2 - v1 = 1, i.e., (s2/t2) - (s1/t1) = 1
replace value of t2, t1, s2 in this equation: (64-s1)/10 - (s1/8) = 1

solve for s1: s1 = 24
hence v1 = 24/8 = 3 mph ANSWER.
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by VivianKerr » Fri Dec 31, 2010 8:21 pm
Hey Gustavo,

I thought of it like this. First, let's consider Day 1 and Day 2's hours.

If x = hours on Day 1, then x + 2 = hours on Day 2. The question said he walked 18 hours total, so we can set up a simple equation:

x + (x + 2) = 18
2x + 2 = 18
2x = 16
x = 8

Therefore he walked 8 hours on Day 1 and 10 hours on Day 2.

We are told he went 1mph FASTER on Day 2. So if Day 1's mph is y, then Day 2's mph is y + 1.

Let's look at the D = R x T formula.

D1 = R1 x T1

D2 = R2 x T2

If we plug in what we know:

D1 = (y) x 8 hrs

D2 = (y + 1) x 10 hrs

We know that D1 + D2 must equal 64, so we can sum the two equations and set them equal to 64.

(y) x 8hrs + (y + 1) x 10hrs = 64

Simplifying...

64 = 8y + 10y + 10

64 = 18y + 10

54 = 18y

3 = y

Hope this is clear!

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by GMATGuruNY » Thu Jan 06, 2011 4:28 pm
gustavo.taucei wrote:16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
Total time = 18 hours.
Since the hiker walked 2 hours longer on the second day, time on the first day = 8 hours, time on the second day = 10 hours.

Now we can plug in the answer choices, which represent the speed on the first day.

Answer choice C: Rate on the first day = 4 mph
Distance on first day = r*t = 4*8 = 32 miles.
Rate on the second day = 4+1 = 5 mph.
Distance on the second day = r*t = 5*10 = 50 miles.
Total distance = 32+50 = 82 miles.
The rate is too fast. Eliminate C, D and E.

Answer choice B: Rate on the first day = 3 mph
Distance on first day = r*t = 3*8 = 24 miles.
Rate on the second day = 3+1 = 4 mph.
Distance on the second day = r*t = 4*10 = 40 miles.
Total distance = 24+40 = 64 miles. Success!

The correct answer is B.
Last edited by GMATGuruNY on Wed Sep 07, 2011 4:40 pm, edited 1 time in total.
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by bblast » Sat Jan 08, 2011 5:57 am
IMO B

Are these retired test code papers good practice ?
I have a couple with me.
Cheers !!

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by Jayanth2689 » Mon Sep 03, 2012 6:55 am
GMATGuruNY wrote:
gustavo.taucei wrote:16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
Total time = 18 hours.
Since the hiker walked 2 hours longer on the second day, time on the first day = 8 hours, time on the second day = 10 hours.

Now we can plug in the answer choices, which represent the speed on the first day.

Answer choice C: Rate on the first day = 4 mph
Distance on first day = r*t = 4*8 = 32 miles.
Rate on the second day = 4+1 = 5 mph.
Distance on the second day = r*t = 5*10 = 50 miles.
Total distance = 32+50 = 82 miles.
The rate is too fast. Eliminate C, D and E.

Answer choice B: Rate on the first day = 3 mph
Distance on first day = r*t = 3*8 = 24 miles.
Rate on the second day = 3+1 = 4 mph.
Distance on the second day = r*t = 4*10 = 40 miles.
Total distance = 24+40 = 64 miles. Success!

The correct answer is B.
Mitch, would a weighted average approach help in tackling such problems in case of time crunch?

Realising that the question by itself provides TOTAL DISTANCE and TOTAL TIME made me treat this as a WA problem without using a Chart.

Here the WA is 3.55. Given that the answer choices are only 1 mile/hour apart, this WA falls between 3 mph and 4 mph. Also, since the weight here is time, Day 2 has more weight as the hiker walked 2 hours longer.

Thus, the answer is 4 mph? Can i do it this way?
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by Jayanth2689 » Mon Sep 03, 2012 7:44 am
Jayanth2689 wrote:
GMATGuruNY wrote:
gustavo.taucei wrote:16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
Total time = 18 hours.
Since the hiker walked 2 hours longer on the second day, time on the first day = 8 hours, time on the second day = 10 hours.

Now we can plug in the answer choices, which represent the speed on the first day.

Answer choice C: Rate on the first day = 4 mph
Distance on first day = r*t = 4*8 = 32 miles.
Rate on the second day = 4+1 = 5 mph.
Distance on the second day = r*t = 5*10 = 50 miles.
Total distance = 32+50 = 82 miles.
The rate is too fast. Eliminate C, D and E.

Answer choice B: Rate on the first day = 3 mph
Distance on first day = r*t = 3*8 = 24 miles.
Rate on the second day = 3+1 = 4 mph.
Distance on the second day = r*t = 4*10 = 40 miles.
Total distance = 24+40 = 64 miles. Success!

The correct answer is B.
Mitch, would a weighted average approach help in tackling such problems in case of time crunch?

Realising that the question by itself provides TOTAL DISTANCE and TOTAL TIME made me treat this as a WA problem without using a Chart.

Here the WA is 3.55. Given that the answer choices are only 1 mile/hour apart, this WA falls between 3 mph and 4 mph. Also, since the weight here is time, Day 2 has more weight as the hiker walked 2 hours longer.

Thus, the answer is 4 mph? Can i do it this way?
Sorry TYPO error! the answer is 3 mph as the weight of the 1st day is lesser!
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by GMATGuruNY » Tue Sep 04, 2012 8:34 am
Jayanth2689 wrote:
Jayanth2689 wrote:
GMATGuruNY wrote:
gustavo.taucei wrote:16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
Total time = 18 hours.
Since the hiker walked 2 hours longer on the second day, time on the first day = 8 hours, time on the second day = 10 hours.

Now we can plug in the answer choices, which represent the speed on the first day.

Answer choice C: Rate on the first day = 4 mph
Distance on first day = r*t = 4*8 = 32 miles.
Rate on the second day = 4+1 = 5 mph.
Distance on the second day = r*t = 5*10 = 50 miles.
Total distance = 32+50 = 82 miles.
The rate is too fast. Eliminate C, D and E.

Answer choice B: Rate on the first day = 3 mph
Distance on first day = r*t = 3*8 = 24 miles.
Rate on the second day = 3+1 = 4 mph.
Distance on the second day = r*t = 4*10 = 40 miles.
Total distance = 24+40 = 64 miles. Success!

The correct answer is B.
Mitch, would a weighted average approach help in tackling such problems in case of time crunch?

Realising that the question by itself provides TOTAL DISTANCE and TOTAL TIME made me treat this as a WA problem without using a Chart.

Here the WA is 3.55. Given that the answer choices are only 1 mile/hour apart, this WA falls between 3 mph and 4 mph. Also, since the weight here is time, Day 2 has more weight as the hiker walked 2 hours longer.

Thus, the answer is 4 mph? Can i do it this way?
Sorry TYPO error! the answer is 3 mph as the weight of the 1st day is lesser!
Nice work!
The answer choices represent the slower speed.
The average speed for the entire trip = 64/18 ≈ 3.5.
Thus, the slower speed must be LESS than 3.5 miles per hour and the faster speed must be GREATER than 3.5 miles per hour.
The problem states that the difference between the two speeds is 1 mile per hour.
Thus, only answer choice B-- which implies a slower speed of 3 miles per hour and a faster speed of 4 miles per hour -- is viable.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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by Jayanth2689 » Tue Sep 04, 2012 8:41 am
GMATGuruNY wrote:
Jayanth2689 wrote:
Jayanth2689 wrote:
GMATGuruNY wrote:
gustavo.taucei wrote:16. A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?
(A) 2 mph
(B) 3 mph
(C) 4 mph
(D) 5 mph
(E) 6 mph

Answer is (B)
Source: GMAT test code 14
I still have problems grasping the answer
Total time = 18 hours.
Since the hiker walked 2 hours longer on the second day, time on the first day = 8 hours, time on the second day = 10 hours.

Now we can plug in the answer choices, which represent the speed on the first day.

Answer choice C: Rate on the first day = 4 mph
Distance on first day = r*t = 4*8 = 32 miles.
Rate on the second day = 4+1 = 5 mph.
Distance on the second day = r*t = 5*10 = 50 miles.
Total distance = 32+50 = 82 miles.
The rate is too fast. Eliminate C, D and E.

Answer choice B: Rate on the first day = 3 mph
Distance on first day = r*t = 3*8 = 24 miles.
Rate on the second day = 3+1 = 4 mph.
Distance on the second day = r*t = 4*10 = 40 miles.
Total distance = 24+40 = 64 miles. Success!

The correct answer is B.
Mitch, would a weighted average approach help in tackling such problems in case of time crunch?

Realising that the question by itself provides TOTAL DISTANCE and TOTAL TIME made me treat this as a WA problem without using a Chart.

Here the WA is 3.55. Given that the answer choices are only 1 mile/hour apart, this WA falls between 3 mph and 4 mph. Also, since the weight here is time, Day 2 has more weight as the hiker walked 2 hours longer.

Thus, the answer is 4 mph? Can i do it this way?
Sorry TYPO error! the answer is 3 mph as the weight of the 1st day is lesser!
Nice work!
The answer choices represent the slower speed.
The average speed for the entire trip = 64/18 ≈ 3.5.
Thus, the slower speed must be LESS than 3.5 miles per hour and the faster speed must be GREATER than 3.5 miles per hour.
The problem states that the difference between the two speeds is 1 mile per hour.
Thus, only answer choice B-- which implies a slower speed of 3 miles per hour and a faster speed of 4 miles per hour -- is viable.
Thanks Mitch!!
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