So we run into a second problem, how can we figure out how long it took the trains to get those 243 miles apart? Well, we already know that the northbound train has been traveling for an hour more than the southbound train. We can either think of this in terms of finding the time of the northbound train OR the southbound train. If the time of the northbound train were ‘x’, then the time of the southbound train would be ‘x-1’ since the southbound train is one hour behind the northbound one.

Now that we have a basis for the time that the trains traveled, all that’s needed now is to solve the equation for x. Remember: total distance = (speed of north train * time of north train) + (speed of south train * time of south train)

total distance = 243 miles

speed of north train = 63 mph

time of north train = x hrs

speed of south train = 57 mph

time of south train = x-1 hrs

243 = 63x + 57(x-1)

Solve for x to find how long the northbound train traveled and then subtract 1 to find out how long the southbound train traveled.

2) I’m extreeeemely not sure about this one, but I think it should be correct. It’s a problem of mixing ingredients of different qualities. This is how I set the problem up:

1 kg at .5 = .5

x kg at 4 = 4x

You can ignore the fact that these are in percents since that’s irrelevant to our calculations. When combining the two different solutions, you take the sum of the qualities of the two solutions you’re mixing (.5 + 4x) and divide that by the number of total kilograms you’ll get by combining the solutions (1 + x). This will give you the quality of the final solution that you want (the 2%). If you then solve: 2 = (.5 + 4x)/(1 + x), then x = 0.75, which means you’ll need .75kg of the 4% solution, plus 1kg of the .5% solution in order to make a 2% solution.

63x + 57(x-1) = 243

Solve for x.

2) .04x + 0.005(1) = 0.02