Given:

• Amit cycled 15 km south.
• Then turned right (west) and cycled 7 km.
• Turned right again (north) and cycled 10 km.
• Finally, turned left (west) and cycled 5 km.

Calculation:

Vertical displacement: $15 - 10 = 5 \, \text{km (south)}$
Horizontal displacement: $7 + 5 = 12 \, \text{km (west)}$
Using Pythagoras theorem:
Distance = √(5² + 12²) = √(25 + 144) = √169 = 13 \, \text{km}

Answer:

Amit needs to cycle 13 km to return home.