A Guide to Changing the Subject of Equations


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

\text{Original formula: } V = \frac{b^2h}{3}
\text{Multiply by (3), divide by} {(b^2).}
\text{Rearranged: } h = \frac{3V}{b^2}

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

\text{Original formula: } V = u + ft
\text{Subtract (u), divide by (f).}
\text{Rearranged: } t = \frac{V - u}{f}

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

\text{Original formula: } T = \frac{wv^2}{g}
\text{Multiply by (g), divide by (w), take the square root.}
\text{Rearranged: } v = \sqrt{\frac{Tg}{w}}

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

\text{Original formula: } t = \frac{2vr}{p+2}
\text{Multiply by (p+2), divide by (2v).}
\text{Rearranged: } r = \frac{t(p+2)}{2v}

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

\text{Original formula: } K = \frac{30wv^2}{2g}
\text{Multiply by (2g), divide by (30w), take the square root.}
\text{Rearranged: } v = \sqrt{\frac{2gK}{30w}}

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

\text{Original formula: } I = \frac{mne}{mR + nr}
\text{Cross-multiply, solve for (n).}
\text{Rearranged: } n = \frac{I(mR)}{me - Ir}

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

\text{Original formula: } WL = \sqrt{l^2 - R^2}
\text{Square both sides, solve for (l).}
\text{Rearranged: } l = \sqrt{WL^2 + R^2}

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

\text{Original formula: } E = \frac{4WT^3}{3mda^2}
\text{Multiply by } (3mE), \text{ divide by } (4WT^3), \text{ take the square root.}
\text{Rearranged: } a = \sqrt{\frac{4WT^3}{3mE}}

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

\text{Original formula: } x + y = 4\sqrt{x^2 + 4h^2}
\text{Square both sides, solve for (h).}
\text{Rearranged: } h = \sqrt{\frac{(x + y)^2 - 16x^2}{16}}

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

\text{Original formula: } R = \sqrt{x^2 + y^2}
\text{Square both sides, solve for (y).}
\text{Rearranged: } y = \sqrt{R^2 - x^2}

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

\text{Original formula: } A = \pi r \sqrt{h^2 + r^2}
\text{Divide by (\(\pi r\)), square both sides, solve for (h).}
\text{Rearranged: } h = \sqrt{\left(\frac{A}{\pi r}\right)^2 - r^2}

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

\text{Original formula: } H = \sqrt{\frac{a^2}{4} + h^2}
\text{Square both sides, solve for (h).}
\text{Rearranged: } h = \sqrt{H^2 - \frac{a^2}{4}}

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

\text{Original formula: } V = \frac{4}{3}\pi r^3
\text{Multiply by (\(\frac{3}{4\pi}\)), take the cube root.}
\text{Rearranged: } r = \sqrt[3]{\frac{3V}{4\pi}}

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

\text{Original formula: } A = 4\pi r^2
\text{Divide by (4\(\pi\)), take the square root.}
\text{Rearranged: } r = \sqrt{\frac{A}{4\pi}}

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

\text{Original formula: } A = \frac{6r^2}{2}
\text{Multiply by (2), divide by (6), take the square root.}
\text{Rearranged: } r = \sqrt{\frac{A}{3}}

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

\text{Original formula: } R = \frac{x}{2 \sin \frac{\theta}{2}}
\text{Multiply by (2 \(\sin \frac{\theta}{2}\)).}
\text{Rearranged: } x = 2R \sin \frac{\theta}{2}

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

\text{Original formula: } D = h + \frac{x^2}{4h}
\text{Subtract ( h ), multiply by ( 4h ), take the square root.}
\text{Rearranged: } x = \sqrt{4h(D - h)}

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

\text{Original formula: } T = 2\pi \sqrt{\frac{L}{G}}
\text{Divide by (2\(\pi\)), square, multiply by (G).}
\text{Rearranged: } L = \frac{T^2G}{4\pi^2}

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

\text{Original formula: } R = \frac{Pl}{a}
\text{Multiply by ( a ), divide by ( R ).}
\text{Rearranged: } a = \frac{Pl}{R}

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

\text{Original formula: } Ri = Ro(i + at)
\text{Expand, subtract ( Roi ), divide by ( Rot ).}
\text{Rearranged: } a = \frac{Ri - Roi}{Rot}

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

\text{Original formula: } a = \frac{\pi d^2}{4}
\text{Multiply by (4), divide by (\(\pi\)), take the square root.}
\text{Rearranged: } d = \sqrt{\frac{4a}{\pi}}

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

\text{Original formula: } F = BlI
\text{Divide by ( Bl ).}
\text{Rearranged: } I = \frac{F}{Bl}

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

\text{Original formula: } E = \frac{Bld}{t}
\text{Multiply by ( t ), divide by ( E ).}
\text{Rearranged: } t = \frac{Bld}{E}

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

\text{Original formula: } v = \frac{eV}{md}
\text{Multiply by ( md ), divide by ( v ).}
\text{Rearranged: } d = \frac{eV}{mv}


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.

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