**Introduction**

While rummaging through a dusty attic or exploring the depths of an old library, one might stumble upon ancient texts that hold timeless knowledge. Such was the case when this collection of formula rearrangements was discovered inside an old mathematics textbook. The yellowing pages and worn cover spoke of countless hours spent in study and contemplation by students of the past. Within these pages lay the exercise of manipulating formulas—a skill that remains as vital today as it was then. This serendipitous find serves as a reminder of the enduring nature of mathematical principles and the continuous journey of learning that spans generations. The carefully penned exercises, though aged, are as relevant now as they were when first inscribed, offering invaluable insights into the foundational techniques of mathematical problem-solving.

In the world of mathematics and physics, formula manipulation is a crucial skill. It allows us to solve for a desired variable and understand the relationships between different quantities. This guide will walk you through the process of changing the subject of various equations, illustrating the steps with clear explanations. Let’s dive into 24 different formulas and see how to rearrange each to isolate the desired variable.

**1.** **Volume in terms of height**

**2. Time (t) in terms of initial velocity (u), final velocity (V), and acceleration (f)**

**3. Velocity (v) in terms of weight (w), gravity (g), and tension (T)**

**4. Radius (r) in terms of time (t), velocity (v), and a constant (p)**

**5. Velocity (v) in terms of kinetic energy (K), weight (w), and gravity (g)**

**6. Number of turns (n) in terms of current (I), resistance (R), and other constants**

**7. Length (l) in terms of wire length (WL) and resistance (R)**

**8. Area (a) in terms of energy (E), time (T), mass (m), and constants**

**9. Height (h) in terms of (x) and (y)**

**10. ( y ) in terms of (R) and (x)**

**11. Height (h) in terms of area (A), radius (r), and (\pi)**

**12. Height (h) in terms of (H) and (a)**

**13. Radius (r) in terms of volume (V)**

**14. Radius (r) in terms of area (A)**

**15. Radius (r) in terms of area (A)**

**16. ( x ) in terms of (R) and angle (\theta)**

**17. ( x ) in terms of (D) and (h)**

**18. Length (L) in terms of pendulum period (T) and gravity (G)**

**19. Area (a) in terms of (R) and length (P)**

**20. Acceleration (a) in terms of resistance (R)**

**21. Diameter (d) in terms of area (a)**

**22. Current (I) in terms of force (F) and magnetic field (B)**

**23. Time (t) in terms of energy (E) and distance (d)**

**24. Distance (d) in terms of velocity (v) and energy (E)**

**Summary**

Mastering the art of rearranging formulas is essential for students and professionals in science and engineering fields. By understanding these steps and practicing, one can become adept at manipulating equations to solve for any variable of interest.