Leo+DadMade for Leo
Algebraic Expressions & Substitution
Rung 4 of 4 · Mastery

Formulas in the Real World

A formula is just an expression with a job. Costs, perimeters, fares — substitute the real number, and sometimes work backwards from the answer. This is where it all pays off.


BuildSubstitute forwards, then flip to the “reverse challenge” and hunt the input that hits the target.
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Every formula you'll meet — the perimeter of a rectangle, the cost of a phone plan, the fare in a taxi — is just an algebraic expression with the letters standing for real things. Substitution is the skill that makes them all work.

Substitute Real Numbers

The perimeter of a rectangle is P = 2(l + w). For a rectangle 5 long and 3 wide, substitute: P = 2(5 + 3) = 2 × 8 = 16. A skate-park costing cost = 5n + 20 for n visits? Eight visits gives 5 × 8 + 20 = 60 dollars. Same two moves as always — drop the numbers in, then evaluate in order.

Say it plainly: a formula is an expression with a name. Put the real numbers in where the letters are, evaluate, and you've got your answer in real units.

Working Backwards

Sometimes you know the answer and need the input. "I spent $45 at the skate-park — how many visits?" Start from 5n + 20 = 45, undo the +20 to get 5n = 25, then undo the ×5 to get n = 5. In the toy's reverse challenge, you slide the input until the formula lands exactly on the target — that's the same idea by feel. Undoing a formula like this is the doorway to solving equations next.

Why This Is the Finish Line

You started with a letter as a placeholder, learned to substitute and evaluate, dodged the 2x-versus-x² traps — and now you can run real formulas both forwards and backwards. That's the move that shows up costing a job, measuring a yard, or sitting an exam. That's mastery.

Us, Thinking Out Loud

Which formula in our everyday life could we write with a letter in it?

The backwards one — how do we “undo” a formula to find the input?

Of the four rungs, which should we re-visit in a fortnight?