Can You Solve This Challenging Math Problem?

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The Problem

“A train leaves New York for Chicago at 60 miles per hour. Two hours later, another train leaves Chicago for New York traveling at 90 miles per hour. The distance between New York and Chicago is 900 miles. At what distance from New York do the two trains meet?”

Step 1: Understand the setup

We are given:

• Train A leaves New York at 60 mph toward Chicago.

• Train B leaves Chicago 2 hours later at 90 mph toward New York.

• The distance between the cities is 900 miles.

• They are moving toward each other on the same track.

We want to find how far from New York the two trains meet.

⏱️ Step 2: Assign variables

Let’s let t be the amount of time (in hours) Train A has been traveling when they meet.

So:

• In that time, Train A will have traveled:
  Distance_A = 60 × t

• Since Train B leaves 2 hours later, it will have traveled for (t − 2) hours.
  Distance_B = 90 × (t − 2)

Since they’re moving toward each other, their combined distances must equal 900 miles when they meet.

Step 3: Set up the equation

\text{Distance_A} + \text{Distance_B} = 900 $$60t+90(t-2)=900$$

✏️ Step 4: Solve the equation

Expand:

$$60t+90t-180=900$$ $$150t-180=900$$

Add 180 to both sides:

$$150t=1080$$

Divide both sides by 150:

t=1080150=7.2t = \frac{1080}{150} = 7.2t=1501080​=7.2

So the two trains meet 7.2 hours after Train A leaves New York.

Step 5: Find the distance from New York

Train A traveled for 7.2 hours at 60 mph:

$$\text{Distance from New York}=60\times 7.2=432\text{ miles}$$

✅ Answer: The two trains meet 432 miles from New York.

Why this problem is useful

This type of relative motion problem is common in algebra and physics because it combines concepts like:

• Rates of speed

• Time and distance relationships

• Understanding how objects move toward each other

It also reinforces how to set up equations from word problems, a vital skill in standardized tests and real-world applications.

✨ Extra Challenge

What if both trains left at the same time? Then the equation would become:

$$60t+90t=900\Rightarrow 150t=900\Rightarrow t=6$$

They would meet in 6 hours, and the distance from New York would be:

$$60\times 6=360\text{ miles}$$

You can tweak numbers like speed, time delay, or total distance to create a whole family of similar but increasingly complex problems!