Source: e-GMAT
A straight highway connects two cities P and Q, and goes via two checkpoints M and N, in that order. A bus started from P and moved to M at an average speed of 30 mph. After crossing M, it increased its speed to 50 mph and moved towards N. When it reached N, it further increased its speed to 60 mph and continued to move at this speed till it reached Q. What is the average speed of the bus throughout the whole journey?
1) The ratio of time the bus took to cover the distances PM, MN, and NQ respectively is 2:1:3.
2) Out of the three distances, NQ is 3 times the distance PM and more than 3 times the distance MN.
The OA is A.
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A straight highway connects two cities P and Q, and goes
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From P to M, speed = 30mph
From M to N, speed = 50mph
From N to Q, speed = 60mph
Question=> What is the average speed of the bus throughout the whole journey?
Let distance P to M = x, M to N = y and N to Q =z.
$$Average\ speed\ of\ the\ whole\ journey\ =\frac{\left(x+y+z\right)}{\left(\frac{x}{30}+\frac{y}{50}+\frac{z}{60}\right)}$$
In statement 1: The ratio of time the bus took to cover the distance PM, MN and NQ is respectively 2 : 1 : 3.
Let the time taken for individual parts be 't'. Hence, time will be in ratio 2t : 1t : 3t.
Distances are x, y and z respectively. Therefore,
$$\frac{x}{30}=2t;\ \ \ \ \ \ x=60t$$
$$\frac{y}{50}=t;\ \ \ \ \ \ y=50t$$
$$\frac{z}{60}=3t;\ \ \ \ \ \ z=180t$$
$$Average\ speed=\frac{\left(total\ dist.\ travelled\right)}{total\ time\ taken}=\frac{\left(x+y+z\right)}{\left(\frac{x}{30}+\frac{y}{50}+\frac{z}{60}\right)}$$ $$=\frac{\left(60t+50t+180t\right)}{\left(\frac{60t}{30}+\frac{50t}{50}+\frac{180t}{60}\right)}=\frac{290t}{6t}$$
$$=\frac{145}{3}=48.33mph$$
$$Hence,\ statement\ 1\ is\ SUFFICIENT$$
For statement 2: Out of the 3 distances, WQ is 3 times the distance PM and more than 3 times the distance MN. The information provided here does not give us the exact distance of MN. Thereby, makes 'y' an unknown variable.
Hence, statement 2 is INSUFFICIENT
Option A is the correct answer because statement 1 alone is sufficient
From M to N, speed = 50mph
From N to Q, speed = 60mph
Question=> What is the average speed of the bus throughout the whole journey?
Let distance P to M = x, M to N = y and N to Q =z.
$$Average\ speed\ of\ the\ whole\ journey\ =\frac{\left(x+y+z\right)}{\left(\frac{x}{30}+\frac{y}{50}+\frac{z}{60}\right)}$$
In statement 1: The ratio of time the bus took to cover the distance PM, MN and NQ is respectively 2 : 1 : 3.
Let the time taken for individual parts be 't'. Hence, time will be in ratio 2t : 1t : 3t.
Distances are x, y and z respectively. Therefore,
$$\frac{x}{30}=2t;\ \ \ \ \ \ x=60t$$
$$\frac{y}{50}=t;\ \ \ \ \ \ y=50t$$
$$\frac{z}{60}=3t;\ \ \ \ \ \ z=180t$$
$$Average\ speed=\frac{\left(total\ dist.\ travelled\right)}{total\ time\ taken}=\frac{\left(x+y+z\right)}{\left(\frac{x}{30}+\frac{y}{50}+\frac{z}{60}\right)}$$ $$=\frac{\left(60t+50t+180t\right)}{\left(\frac{60t}{30}+\frac{50t}{50}+\frac{180t}{60}\right)}=\frac{290t}{6t}$$
$$=\frac{145}{3}=48.33mph$$
$$Hence,\ statement\ 1\ is\ SUFFICIENT$$
For statement 2: Out of the 3 distances, WQ is 3 times the distance PM and more than 3 times the distance MN. The information provided here does not give us the exact distance of MN. Thereby, makes 'y' an unknown variable.
Hence, statement 2 is INSUFFICIENT
Option A is the correct answer because statement 1 alone is sufficient
