Question: Alice and Bob traveled in the same direction along the same route at their respective constant speeds of 12 km per hour and 6 km per hour. They each started to travel from their own houses. Bob’s house is halfway between Alice’s house and the destination. After passing Bob, Alice took 10 minutes to reach the destination. How many minutes did it take Bob to reach the destination after Alice passed him?
- 5 min
- 6 min
- 8 min
- 10 min
- 20 min
“Alice and Bob traveled in the same direction along the same route at”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide of GMAT Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
Convert rates to minutes.
A's rate in minutes:
12 km/1 hr = 12 km/60 mins = 1 km/5 mins
B's rate in minutes:
6 km/1 hr = 6 km/60 mins = 1 km/10 mins
Information to ignore or avoid:
1) Bob's house as a halfway point
2) Relative speed
3) Whatever happens before the passing point
As soon as Alice passes Bob, the chase is over. Both travel the same distance to the end.*
We need only two pieces of information:
1) the distance from pass point to end, which we can derive from Alice's time; and
2) the time it takes for Bob to travel that distance, which we can derive from his rate
Distance to end
In 10 minutes, the distance Alice covers is
D=R∗T, so
D = 1 km/5 mins∗10 mins =2 kms
Time Bob takes to cover distance of 2km
R∗T=D, so T= D/R
T= 2 km/1 km/10 mins = (2 km∗10 mins/1 km) = 20 minutes
Correct Answer: E
Approach Solution 2:
Alice does travel 2d, but we don't care about that because we don't know start times.
After she passes him, both travel the same distance.
We find distance from her rate and time. In minutes. (Or a fraction of an hour.)
Then find Bob's time from distance and rate.
All that matters: separate rates and times.
She travels at 12 km per hour.
One hour = 60 minutes.
Rate: She travels 12 km in 60 minutes:
D= (12 km/60 mins∗10 minutes) = 2
Bob's speed = 6 km/hr, OR 6 km in 60 minutes.
Bob's time?
T = D/R
T= 2/(6/60) = (2∗60/6) = 20
OR get time from Bob's unit rate = 6 km/60 mins = 1 km/10 mins
To go 2 km, at a rate of 1 km per 10 minutes, it will take him 2 * 10 = 20 minutes.
Correct Answer: E
Approach Solution 3:
Given:
1. Alice and Bob travelled in the same direction along the same route at their respective constant speeds of 12 km per hour and 6 km per hour.
2. They each started to travel from their own houses.
3. Bob’s house is halfway between Alice’s house and the destination.
4. After passing Bob, Alice took 10 minutes to reach the destination.
Asked: How many minutes did it take Bob to reach the destination after Alice passed him?
- Bob’s house is halfway between Alice’s house and the destination.
Alice House -----------------x km -----------------Bob's House ---------------------x km ------------------------Destination
- After passing Bob, Alice took 10 minutes to reach the destination.
Distance travelled by Alice = (10 /60) * 12 = 2 km
Time taken by Bob to cover 2 km = 2/6 hr = 20 mins
Correct Answer: E
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