Distance/Rate Problems, Word Problems

[AAPL]

Two siblings Mark and Steve start from the same point and walk in the same direction at speeds of 5 kilometres per hour and 10 kilometres per hour respectively. After walking for an hour, Mark turns around and walks back along the same path to the starting point. Mark rests for half an hour at the starting point, and then hires a taxi driving at 30 kilometres per hour to catch up with Steve. If the taxi charges $10 for the first 2 kilometres and $2 for every subsequent 500 metres, how much in dollars does Mark pay for the taxi ride?

A. 92
B. 102
C. 142
D. 152
E. 160

The OA is D

Source: E-GMAT

[deloitte247]

Mark's walking speed = 5km/h
Steve's walking speed = 10km/h
Mark walked for 1 hour, turned around and walked back to his starting point (another 1 hour).
Mark rested for half an hour at the starting point.
Total time = 1 + 1 + 1 0.5 = 2.5 hours
Distance covered by Mark in this 2.5 hours = 5 * 2.5 = 12.5km
Distance covered by Steve in this 2.5 hours = 10 * 2.5 = 25km
The speed of taxi hired by Mark = 30km/hr
Charges for the first 2km = $10
Charges for every 500m = $2
Let the total distance covered by Steve in t hours = x km
i.e t = x/10
The total distance covered by taxi to catch up with Steve = distance covered by Steve in 2.5 hours + distance covered by Steve in t hours = (25 + x) km

Time taken by Taxi to cover (25 + x)km in t hours =>
$$=\frac{25+x}{30}$$
Equating the time taken by Steve and Taxi =>
$$\frac{x}{10}=\frac{25+x}{30}$$
$$30x=250+10x$$
$$30x-10x=250$$
$$20x=250$$
$$x=\frac{250}{20}=12.5km$$
Therefore, the total distance travelled by the taxi = 25 + 12.5 = 37.5km
Charges for the first 2km covered = $10
For the remaining (37.5 - 2 = 35.5km)=?
$2 per 500m or $4 per 1000m/1km
=>35.5 * 4 = $142
Total charges = $10 + $142 = $152
Answer = option D

[Scott@TargetTestPrep]

Two siblings Mark and Steve start from the same point and walk in the same direction at speeds of 5 kilometres per hour and 10 kilometres per hour respectively. After walking for an hour, Mark turns around and walks back along the same path to the starting point. Mark rests for half an hour at the starting point, and then hires a taxi driving at 30 kilometres per hour to catch up with Steve. If the taxi charges $10 for the first 2 kilometres and $2 for every subsequent 500 metres, how much in dollars does Mark pay for the taxi ride?

A. 92
B. 102
C. 142
D. 152
E. 160

The OA is D

Solution:

We see that it takes Mark 2 hours to walk back to the starting point, and since he takes a half- hour break, Steve has walked 2 and a half hours and a distance of 2.5 x 10 = 25 km at the moment when Mark hires a taxi to catch up to him. Letting t = the time, in hours, needed for Mark to catch up with Steve, we can create the equation:

30t = 25 + 10t

20t = 25

t = 1.25

That is, when Mark catches up with Steve, Steve has walked 25 + 10(1.25) = 37.5 km. Since the taxi also has to travel 37.5 km, Mark has to pay:

10 + 2 x (37.5 - 2)/0.5 = 10 + 2 x 35.5 x 2 = 10 + 142 = $152

Answer: D