Rearranging Formulae

There are several methods for rearranging formulae that can be used within Physics, each requiring a different degree of understanding of Mathematics. 

In order to achieve highly at National 5 Level (or above) a full understanding of these methods is vital, and without it many calculations will not be possible. 

Language of Formulae

The diagram below shows the basic parts of a formula:-

In all calculations, the value that is to be found is known as the subject of the formula. When a formula is rearranged, the subject of the formula changes. 

Method 1 - The "Triangle Method"

The first method most students learn that allows the rearrangement of formulae is the Triangle Method.  

This method has the benefit of not requiring the student to understand any of the underlying Mathematics, but does have the limitation of only working for simple (three part) formulae. 

The diagram below shows the basic layout of the triangle method:-

In the triangle method, position of the letters determines the type of sign used in the calculation. If letters are on the bottom row, they multiply together. If the letters required include the top row, they divide together. 

In order to change the subject of the formula to a different letter, simply "cover over" the required subject, and whatever is left gives the rearranged formula.

Example 1 -

The circuit below shows a bulb within a series circuit. 

Use Ohm's Law (V  = I x R) to find the Resistance of the bulb. 

R  =  V / I 

R  =  12 / 2

R  =  6 Ω

Method 2 - "Crossing the Equals Sign" Method

As stated above, the triangle method will not work if a formula has more than three parts, and if that is the case, another method must be used. 

The next simple method is known as the "crossing the equals sign" method. In this method, the formula is changed by moving parts from one side of the equals sign to the other one at a time, and reversing the sign:-

Example 2 - 

The Acceleration of an object is given by the formula below:-

If a car starts at rest (u = 0) and accelerates at 2 ms-2  (a=2) for 5 seconds (t=5) , what is the car's final speed (v=?)? 

The above formula must be rearranged to make V the subject:-

Step 1 : Move the "divide by t " across the equals sign and change to "multiply by t":-

Step 2:  Move the "subtract by u" across the equals sign and change to "add by u":-

Step 3 : Rewrite with the subject on the left:-

Step 4 :  Substitute values into correct places:-

v  =  0 + (2 x 5) 

v  =  10 ms-1

Method 3 - "Same thing to each side" Method

The second method above is a simplified version of this next method. The "reverse the sign" process above is actually several mathematical stages condensed to one step. 

These "missing" stages are based on the following three mathematical rules:-

In Mathematics, you can change or add or remove from a formula at will, as long as you do the same thing to both sides of the formula. 

For example:-

2 + 2  = 4 

2 + 2 - 2 = 4 - 2

( 2 + 2 ) x 3 =  4 x 3 

( 2 + 2 ) / 19863549  =  4 / 19863549

Also In Mathematics, any value multiplied by 1 is equal to itself. 

For Example - 

2 x 1  =  2

14 x 1  =  14

187354 x 1  =  187354

A x 1  =  A 

Finally, also in Mathematics, any value subtracted by itself is equal to zero.

For example - 

2 - 2  =  0

14 - 14  =  0

187354 - 187354  =  0

A - A  =  0

By combining these concepts together, we can change the subject of a formula. 

Therefore in order to "move" a part from one side of the Equals sign to the other, the following steps are actually performed:-

Step 1 : Multiply each side by " t ":-

Step 2 : Simplify any sections of "divide by itself = 1":-

Step 3 :  Simplify any sections of "multiplied by 1 = itself":-

As can be seen, step 3 here is identical  to step 1 in method 2 above. 

In order to make "v" the subject, the process is then repeated a second time with "u":-

Step 4 : Add "u" to each side :-

Step 5 :  Simplify any sections of " value subtracted by itself = 0":-

Step 6 : Rewrite with the subject on the left:-