A boy rode his bicycle northward, then turned left and rode one km and again turned left and rode 2 kms. He found himself exactly one km west of his starting point. How far did he ride northwards initially ?

Explanation

To solve the problem step-by-step, let's break down the movements of the boy on his bicycle:

### Step 1: Understanding the Directions
- The boy starts at a point, which we will call point O (the starting point).
- He rides his bicycle northward. Let's denote the distance he rides north as \( x \) kilometers.

### Step 2: First Turn
- After riding north for \( x \) kilometers, he turns left. Since he was heading north, turning left means he is now heading west.
- He rides 1 kilometer to the west. Let's denote the new position after this ride as point A.

### Step 3: Second Turn
- From point A, he again turns left. Now, since he was heading west, turning left means he is now heading south.
- He rides 2 kilometers south. Let's denote the new position after this ride as point B.

### Step 4: Analyzing the Final Position
- According to the problem, after these movements, he finds himself exactly 1 kilometer west of his starting point O.
- This means that the horizontal distance from point O to point B is 1 kilometer to the west.

### Step 5: Establishing the Coordinates
- If we assume point O is at the origin (0,0):
- After riding north \( x \) kilometers, the coordinates of point A would be (0, x).
- After riding 1 kilometer west, the coordinates of point A become (-1, x).
- After riding 2 kilometers south, the coordinates of point B become (-1, x - 2).

### Step 6: Setting Up the Equation
- Since point B is 1 kilometer west of point O, its coordinates should be (-1, 0).
- Therefore, we can set up the equation:
\[ x - 2 = 0 \]

### Step 7: Solving for \( x \)
- Solving the equation gives:
\[ x = 2 \]

### Conclusion
- The boy initially rode northward for **2 kilometers**.