Two Buses, P and Q, Started Simultaneously from Opposite Ends GMAT Problem Solving

Question: Two buses, P and Q, started simultaneously from opposite ends of a 140 mile route and traveled towards each other on parallel roads. Bus P, traveling at a constant rate, completed the 140 miles trip in 9 hours and Bus Q, traveling at constant speed, completed the 140 mile in 5 hours. How many miles had Bus P traveled when it first met Bus Q?

1. 50
2. 60
3. 70
4. 90
5. 100

Answer:
Solution with Explanation:
Approach Solution (1):

Total space =140

Speed of P =140/9 which can be approximated as (140/10) = 14
Speed of Q = 140/5= 28

Relative speed (sum since they are going in opposite direction) =14 + 28 = 42
Space to cover =140

Relative Time = Space/Relative speed = 140/42 = approximately 3 (time P and Q encounter each other)
For t=3 calculate Space travelled by P
Thus space= speed * relative time = 14 *3 = 42 (approximate) (distance travelled by P when it encounters Q)

Since we approximated the speed of Q by increasing the denominator to 10 the overall result is underestimated, it should be a little more than 14
In addition we estimated T = 3 but it should be a little more than 3

Thus the product 14*3 should be higher and the closed answer is A

Correct Option: A

Approach Solution (2):

Speed = Distance/Time
Speed (Bus P) = 140/9
Speed (Bus P) = 140/5
Relative speed: Total time = Distance between them/(Sum of speeds) = 140/(140/9 + 140/5) = 45/14 hours
Distance traveled by Bus P in 45/14hours = 140/9 * 45/14 = 50 kms

Correct Option: A

Approach Solution (3):

Speed of P =140/9 = 14 (approx)
Speed of Q = 140/5= 28

Relative speed (sum since they are going in opposite direction) =14 + 28 = 42
Space to cover =140
Relative Time = Space/Relative speed = 140/42 = 3 (approx)

For t=3, space= speed * relative time = 14 *3 = 42

Since we approximated the speed of Q by increasing the denominator to 10 the overall result is underestimated, it should be a little more than 14
In addition we estimated T=3 but it should be a little more than 3
Thus the product 14*3 should be higher

Correct Option: A

“Two buses, P and Q, started simultaneously from opposite ends of a 140 mile route and traveled towards each other on parallel roads. Bus P, traveling at a constant rate, completed the 140 miles trip in 9 hours and Bus Q, traveling at constant speed, completed the 140 mile in 5 hours. How many miles had Bus P traveled when it first met Bus Q?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative 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.

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