# Unable to solve challenging Q: "A train goes 50km from one station to another; it takes the train 2 hours 12 minutes...."

#### Jonathan897867

##### New member
A train has to move for 50km from one station to another it takes the train 2 hours and 12 minutes. The train can move at a speed of 50km/hr along flat land and at 10km/hr on a slope. How much of the journey was on flat land and how much of the journey was on sloped land? In advance thank you for any comments.

#### tkhunny

##### Moderator
Staff member
Distance = Rate * Time

That is almost all you need.

Rule #1 - If you don't know something, and you need to talk about it, give it a name!

Level Trip
F km = Trip DISTANCE on Flat Land -- This is what we need to answer the first question. Don't get distracted from this goal. Focus!
Distance = Rate * Time
F km = 50 km/h * Hmmm... We'll need another name!
T hr = Trip TIME on Flat Land
F km = 50 km/h * T hr

UpSlope Trip
Distance = Rate * Time
(50 km - F km) = 10 km/h * (2.2 hr - T hr)

Realization: IF you solve for the Average Rate, it had better be somewhere between 10 km/h and 50 km/h. 50 km / 2.2 hr = 22.72 km/h -- Nice.

• topsquark

#### HallsofIvy

##### Elite Member
I would have done this in a slightly different way. Let x be the TIME, in hours, spent at 50 km/h on flat land. Then using TKHunny's "Distance = Rate * Time" the train will have gone a distance of 50x km on the flat land. Let y be the TIME, in hours, spent at 10 km/h on the slope. The train will have gone 10y km on the slope. We are told two things: that the total distance covered was 50 km, so 50x+ 10y= 50, which is equivalent to 5x+ y= 5, and that it took "2 hours and 12 min"= 2+ 12/60= 2+ 1/5= 2.2 hours so x+ y= 2.2.

So the problem becomes to solve the two equations, x+ y= 2.2 and 5x+ y= 5. There are many different ways to solve "two equations in two unknowns" but here the simplest, perhaps, is to solve for y, y= 2.2- x, and replace the "y" in the second equation with that: 5x+ 2.2- x= 4x+ 2.2= 5. 4x= 5- 2.2= 2.8. x= 2.8/4= 0.7. Since y= 2.2- x, y= 2.2- 0.7= 1.5.

Now, the question was "how much of the journey was on the flat land and how much of the journey was on the slope". In my mind that is ambiguous. Does "how much of the journey" refer to time or distance? In terms of time, x= 0.7 hrs, 0.7/2.2= 31.8% of the journey was on the flat land, while y= 1.5 hrs, 1.5/2.2= 68.2%, was on the slope. In terms of distance, 0.7(50)= 35 km, 35/50= 70% of the journey was on the flat land, while 1.5(10)= 15 km, 15/50= 30% of the journey was on the slope.

• topsquark