Formulas & Rearranging

$C = 2\pi r$ , $A = \frac{1}{2}bh$ ... expressions like these are formulas: relationships that a whole family of problems uses again and again, written down once and kept in your pocket. A formula does two jobs: substituting — cooking the recipe forward; rearranging — running the recipe backwards.

A formula is a recipe

Every formula is an ingredients list:

Circumference of a circle: $C = 2\pi r$ — radius $r$ goes in, circumference $C$ comes out;
Area of a triangle: $A = \frac{1}{2}bh$ — base $b$ and height $h$ go in, area $A$ comes out.

Letters are just labels for the ingredients. The recipe stays fixed, different ingredients give different dishes — that is why a formula works once and forever.

Substituting: cook the recipe

For a triangle with base 8 cm and height 5 cm, fill the numbers into the recipe:

$A = \frac{1}{2} \times 8 \times 5 = 20 \ \mathrm{cm^2}$

Replace, compute, attach the unit — three steps and done. This drill is already well trained in substitution.

Rearranging: hand the lead role to another letter

Formulas often get asked backwards: given distance $d$ and speed $v$ , what is the time $t$ ? Then $t$ must stay alone on one side of the equals sign while everything else moves to the other. The tool is a balance scale: do the same thing to both sides at once, and the scale never tips.

To solve $d = vt$ for $t$ : divide both sides by $v$ , leaving only $t$ on the right:

$t = \frac{d}{v}$

Likewise, solve $C = 2\pi r$ for the radius: divide both sides by $2\pi$ :

$r = \frac{C}{2\pi}$

Whoever you want in the lead role stays alone on one side — that is all there is to it.

The balance never tips — both sides move together

An equation is a balance scale: divide the left by $v$ and the right must be divided by $v$ too; touch one side alone and the scale tips at once. That single rule is the whole discipline of rearranging. Once this hand is steady, harder challenges wait in linear equations.

Check the rearranged formula with numbers

Rearranging is easy to get backwards, but cheap to verify: pick a number, walk forward through the original, then walk back through the new one. Take $r = 2$ : the circumference is $C = 2\pi \times 2 = 4\pi$ . Feed $4\pi$ into the rearranged formula:

$r = \frac{4\pi}{2\pi} = 2$

A full loop returns to 2 — the rearrangement is sound. Same for the distance one: at $d = 100$ , $v = 20$ , we get $t = 100 \div 20 = 5$ , and checking, $20 \times 5 = 100$ . When forward and backward both work, the formula is truly yours.

InteractiveFormula Transformer

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

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Picture the formula as a machine: feeding it forward is substitution; thinking it backwards is rearranging. Knowing the output and recovering the input is exactly this machine's favourite game.

Check yourself

Quick quiz

1. d = vt with d = 150 and v = 30: what is t?
2. After rearranging C = 2πr, what does r equal?
3. A = ½bh: what is A when b = 6 and h = 4?