Two trains start from two cities 200 miles apart and travel toward each other at constant speeds. One goes 40 mph, and the other goes 60 mph. How long does it take for them to meet?

To solve this problem, we need to find the time it takes for the two trains to cover a combined distance of 200 miles. Since they are traveling toward each other, we’ll add their speeds together to find their relative speed.

Let the time taken for the two trains to meet be represented by the variable t.

The distance covered by the first train (going at 40 mph) in time t is:

D₁ = V₁ × t

Where D₁ is the distance covered by the first train, V₁ is its speed (40 mph), and t is the time.

Similarly, the distance covered by the second train (going at 60 mph) in time t is:

D₂ = V₂ × t

Where D₂ is the distance covered by the second train, V₂ is its speed (60 mph), and t is the time.

Since the two trains are traveling toward each other and they meet when they cover a combined distance of 200 miles, we can write the equation:

D₁ + D₂ = 200

Substitute the expressions for D₁ and D₂ we derived above:

(V₁ × t) + (V₂ × t) = 200

Now, plug in the speeds of the trains (40 mph and 60 mph):

(40t) + (60t) = 200

Combine the terms with t:

100t = 200

Now, solve for t:

$t = \frac{200}{100}$

t = 2

So it takes 2 hours for the two trains to meet.