M07-24

Question

A cyclist traveled for two days. On the second day, she rode 4 hours longer and at an average (arithmetic mean) speed 10 miles per hour slower than on the first day. If over these two days, she covered a total of 280 miles in 12 hours, what was her average speed on the second day?

A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph

A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph

Bunuel · Accepted Answer

Official Solution:

A cyclist traveled for two days. On the second day, she rode 4 hours longer and at an average (arithmetic mean) speed 10 miles per hour slower than on the first day. If over these two days, she covered a total of 280 miles in 12 hours, what was her average speed on the second day?

A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph

A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph

Approach 1 - Algebra: 
 
Given that on the second day the cyclist traveled for 4 hours more than on the first day and the total travel time over two days was 12 hours, we can set up the equation  t + (t + 4) = 12 . Solving for  t , we get  t = 4 . This means she traveled for 4 hours on the first day and 8 hours on the second day.
 
If we let the  rate on the second day  be  r  miles per hour, then the equation for the total distance covered on both days is:  4(r + 10) + 8r = 280 . Solving for  r , we find  r = 20  miles per hour.
 
 Approach 2 - Logic: 
 
The average rate of the cyclist is  \frac{	ext{total distance}}{	ext{total time}}=\frac{280}{12}=23\frac{1}{3} . Now, since the weighted average of 2 individual averages ( r  and  r+10 ) must lie between these individual averages, we have  r \lt 23\frac{1}{3} \lt r+10 . Only answer choice C fits, as the rate from the correct answer choice must be less than  23\frac{1}{3} , and that rate plus 10 must be more than  23\frac{1}{3} .

Answer: C

PrayasT · Answer

I think this is a  high-quality  question and I agree with explanation.

patto · Answer

I think this is a  high-quality  question and I agree with explanation.

AndrewN · Answer

Another approach : You can also solve this one with another sort of logic and number sense. Focus first on the time. Namely, 12 hours across 2 days can be broken down readily enough to 6 hours per day  on average . Since we know there was a 4-hour difference between days one and two, we can take the 4 and divide it by 2 and skew our 6-hour split by going 2 hours in either direction:

6 - 2 hours = 4 hours (Day 1)
6 + 2 hours = 8 hours (Day 2)
--------------------------
12 hours altogether over two days, with the correct split

If the question asks about the  second day , pick a number in the middle, say, (B), 10 mph:

10 mph * 8 hours = 80 miles (Day 2)
20 mph * 4 hours = 80 miles (Day 1)
-------------------------------
12 hours altogether over two days, but only 160 total miles traveled-- TOO LOW . Out with (A) and (B), onto (D). Why (D)? Because regardless of whether it ends up being the answer, we will know definitively whether it is too high or too low.

30 mph * 8 hours = 240 miles (Day 2)
40 mph * 4 hours = 160 miles (Day 1)
--------------------------------
12 hours altogether over two days, but with 400 miles traveled-- TOO HIGH . Thus, (D) and (E) are out, and  (C), 20 mph, must be the answer .

This mental math is simple enough and takes about a minute to work through, maybe a minute and a half in a Day 2-type situation.

Cheers,
Andrew

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Danny134 · Answer

Using the algebra approach: Why can't we take the first day's speed as r and second day's as r-10 making the equation as 
4r+ 8(r-10)=280?

Bunuel · Answer

That would yield r = 30, thus, the average speed on the second day, r - 10 = 20.

sindhureddyalla · Answer

I think this is a  high-quality  question and the explanation isn't clear enough, please elaborate. Why cannot we use 4r + 8(r-10) = 280 . Then the answer would be 30mph. I am little confused on this

Bunuel · Answer

There are two explanations, each quite detailed. You just need to read the question and the solution carefully enough. The question asks for the speed on the second day, while r in your equations stands for the speed on the first day, making the speed for the second day equal to r - 10 = 30 - 10 = 20.

SergejK · Answer

Why can't we use s+s-10=280/12 to calculate the speed? Is it only possible when the distance is equal in both instances?

Bunuel · Answer

The average rate of the cyclist is  \frac{	ext{total distance}}{	ext{total time}}=\frac{280}{12}=23\frac{1}{3} . But why would it be equal to r + r - 10? Did you mean (r + r - 10)/2?

If the distances covered at the rate of x and the rate of y are equal, say d, then the average rate is:

\frac{	ext{total distance}}{	ext{total time}}=\frac{d + d}{d/x + d/x}= \frac{2}{1/x + 1/y}=\frac{2xy}{x + y}  .

For example, if 100 miles were covered at 20 miles per hour and another 100 miles were covered at 50 miles per hour, the average rate for the total 200 miles would be:

\frac{	ext{total distance}}{	ext{total time}}=\frac{100 + 100}{100/20 + 100/50}= \frac{200}{5 + 2}=\frac{200}{7}   or directly:  \frac{2xy}{x + y}=\frac{200}{7}\approx 29 .

Hope it's clear.

SergejK · Answer

Thank you for the explanation. I conflated working together on a specific project with 2 different entities with distance problems, where you can't apply the addition of 2 rates as you are not driving the same distance with 2 different entities so to speak.

Premkungg · Answer

I like the solution - it’s helpful.

saynchalk · Answer

This is a great question that’s helpful for learning.

anushree01 · Answer

I like the solution - it’s helpful.

SamCodes · Answer

Hello  Bunuel ,

Thanks for the question.
Can you please explain in detail about the Logical solution using weighted average?
I seem to be missing some conceptual clarity here, would really appreciate your help.

Bunuel · Answer

Because you’re combining two speeds, the total average must lie between them. That’s always true for any average of two values. Here, the two speeds are r and r + 10, so the total average must be between them.

SamCodes · Answer

Got it. This clarifies everything, thanks!

AdarshSambare · Answer

Speed = Distance / Time

Since time required is 12 hours:
1st time = t
so Given:
t+t+4 = 12
2t = 8
t=4

Now Distance = Speed * Time, 
speed = s
and time from above, Given
4 (s) + 8(s-10) = 280
s = 30

So, speed for 2nd trip will be: 
30-10 = 20 
option

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