The subject of a formula

1. The subject of a formula Here is a formula you may know from physics: V = IR where V is voltage, I is current and R is resistance. V is called the subject of the formula. The subject of a formula always appears in front of the equals sign without any other numbers or operations. Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula. Suppose, for example, that we want to make I the subject of the formula V = IR.

2. Changing the subject of the formula V = IR ×R ÷R V I = R V is the subject of this formula The formula: can be written as a function diagram: I V The inverse of this is: I V I is now the subject of this formula So:

3. Matchstick pattern Look at this pattern made from matchsticks: Pattern Number, n 1 2 3 4 Number of Matches, m 3 5 7 9 The formula for the number of matches, m, in pattern number n is given by the formula: m = 2n + 1 Which pattern number will contain 47 matches?

4. Changing the subject of the formula × 2 + 1 ÷ 2 – 1 m – 1 n = 2 m is the subject of this formula The formula: m = 2n + 1 can be written as a function diagram: n m The inverse of this is: n m n is the subject of this formula or

5. Changing the subject of the formula 47 – 1 n = 2 46 n = 2 m – 1 n = 2 To find out which pattern will contain 47 matches, substitute 47 into the rearranged formula. n = 23 So, the 23rd pattern will contain 47 matches.

6. Changing the subject of the formula F = + 32 subtract 32: 9C 9C 5(F – 32) = 9C multiply by 5: 5(F – 32) 5(F – 32) 5 5 F – 32 = 9 9 divide by 9: = C C = We can also change the subject by performing the same operations on both sides of the equals sign. For example, to make C the subject of

7. Change the subject of the formula 1

8. Find the equivalent formulae

9. Formulae where the subject appears twice Sometimes the variable that we are making the subject of a formula appears twice. For example, S = 2lw + 2lh + 2hw where S is the surface area of a cuboid, l is its length, w is its width and h is its height. Make w the subject of the formula. To do this we must collect all terms containing w on the same side of the equals sign. We can then isolate w by factorizing.

10. Formulae where the subject appears twice 2lw + 2hw = S – 2lh subtract 2lh from both sides: w(2l + 2h)= S – 2lh factorize: S – 2lh w = 2l + 2h S = 2lw + 2lh + 2hw Let’s start by swapping the left-hand side and the right-hand side so that the terms with w’s are on the left. 2lw + 2lh + 2hw = S divide by 2l + 2h:

11. Formulae involving fractions For example, if two resistors with a resistance a and b ohms respectively, are arranged in parallel their total resistance R ohms can be found using the formula, = + 1 1 1 R a b aΩ bΩ When a formula involves fractions we usually remove these by multiplying before changing the subject. Make R the subject of the formula

12. Formulae involving fractions multiply through by Rab: ab = Rb + Ra simplify: = + 1 1 1 R ab = R(b + a) a b factorize: ab ab Rab Rab Rab = R divide both sides by a + b: = + a + b a + b R a b R =

13. Formulae involving powers and roots c = √a2 + b2 c2 = a2 + b2 square both sides: c2 – b2 = a2 subtract b2 from both sides: √c2 – b2 = a square root both sides: a = √c2 – b2 The length c of the hypotenuse of a right-angled triangle is given by where a and b are the lengths of the shorter sides. Make a the subject of the formula

14. Formulae involving powers and roots T = 2π l g The time T needed for a pendulum to make a complete swing is where l is the length of the pendulum and g is acceleration due to gravity. Make l the subject of the formula When the variable that we wish to make the subject appears under a square root sign, we should isolate it on one side of the equation and then square both sides.

15. Formulae involving powers and roots T = 2π l l l T divide both sides by 2π: = g g g 2π T2 = square both sides: 4π2 T2g T2g = l multiply both sides by g: 4π2 4π2 l =

16. Change the subject of the formula 2