Making a quantity the subject of a formula

How to rearrange o = 2(a + b) or s = v · t so another quantity is the subject, not the one it was written for.

Turn the formula your way

The formula s = v · t says: distance is speed times time. Great when you are looking for distance. But what if you know the distance and the speed — and you are looking for time?

You can substitute numbers every time and solve an equation. Or you do it once for all: you turn the formula so it says t = something.

s = v · t → divide both sides by v → t = s/v. Done. From this moment you have a formula for time.

This is called making a quantity the subject of a formula and it is exactly the same work as solving an equation. The only difference: instead of numbers you move letters. The scales rule holds without change — what you do to one side, do to the other too.

This skill is gold in physics and chemistry. Anyone who can turn formulae carries one formula in their head instead of three. Anyone who cannot crams three versions and still mixes them up.

Sofie is calculating in physics: a train travelled 210 km at 70 km/h, how long? Filip next to her is flipping through his notebook: “Where is the formula for time? We only have s = v · t…” Sofie: “So turn it. You need t on its own, so divide both sides by v.” She writes t = s/v, substitutes: t = 210/70 = 3 hours. Filip stares: “You made that formula.” — “I did not make it. I just treated it as an equation. A formula is an equation where nobody has substituted numbers yet.”

💡

Treat letters exactly like numbers. You want t on its own and “times v” is in the way? Divide both sides by v. The same move as when you do / :4 with the equation 4x = 20.

How to make a quantity the subject step by step

1. Underline what you want as the subject. Say b in the formula o = 2(a + b).

2. Clear everything else with opposite operations — from the furthest to the nearest, like with equations:

o = 2(a + b). The two is in the way (it multiplies the brackets): divide both sides by two → o/2 = a + b.

a is in the way (it is added): subtract it → o/2 − a = b.

3. Flip the writing so the unknown is on the left: b = o/2 − a.

4. Check with numbers. Invent simple ones: a = 3, b = 2 → perimeter o = 2 · 5 = 10. Now into the turned formula: b = 10/2 − 3 = 2. It fits — the turning is correct.

A check with numbers is as important with rearranging as a check with equations. Letters invite mechanical mistakes and numbers catch them.

Common shapes: s = v · t → v = s/t, t = s/v. Perimeter of a square o = 4a → a = o/4. Average mark p = (a + b)/2 → a = 2p − b.

Example 1: from s = v · t make v the subject

I want v on its own. “Times t” is in the way.

I divide both sides by t: s/t = v.

I flip: v = s/t.

Check with numbers: v = 60 km/h, t = 2 h → s = 120 km. Into the turned formula: v = 120/2 = 60. It fits.

In words: speed is distance divided by time. A formula you know from life (“he rode 120 km in 2 hours, so he went at sixty”), maths only wrote with letters.

Example 2: from o = 2a + 2b make a the subject

Perimeter of a rectangle. I want a.

2b is in the way (it is added): I subtract from both sides → o − 2b = 2a.

The two is in the way (it multiplies a): I divide by two → (o − 2b)/2 = a.

I flip: a = (o − 2b)/2.

Check: a = 5, b = 3 → o = 10 + 6 = 16. Turned formula: a = (16 − 6)/2 = 5. It fits.

Two steps, exactly like the equation 2x + 6 = 16 — only with letters.

Example 3: from V = a · b · c make c the subject

Volume of a cuboid: length times width times height. I know the volume and the base, I want height c.

a · b is in the way (it multiplies c): I divide both sides by a · b →

c = V/(a · b).

Check: a = 2, b = 3, c = 4 → V = 24. Turned formula: c = 24/(2 · 3) = 24/6 = 4. It fits.

Practical use: an aquarium has a base 40 × 30 cm and you want 36 litres in it (36,000 cm³). Water height: 36,000/1200 = 30 cm.

⚠️

Clear step by step and whole sides. From o = 2(a + b) you cannot “cross out the two only on the left” or yank b out of the brackets. Every step is an operation with BOTH sides — just like with equations with numbers.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.

Practise substituting into expressions — a turned formula is useless if you cannot substitute into it reliably.

Practise the questions

On paper

Turn, check with numbers, only then trust. Both steps with every formula.

From the formula o = 4a (perimeter of a square) make a the subject. Check with numbers a = 3.
From the formula s = v · t make t the subject. Then calculate how long a car takes for 150 km at 100 km/h.
From the formula o = 2(a + b) make b the subject and check on a rectangle 4 × 6.
From the formula for the average of two numbers p = (x + y)/2 make y the subject. Check: the average of 3 and 7 is 5, does it fit the other way too?

Now you 💪

I treat letters like numbers: opposite operations, both sides.
I clear from the furthest to the nearest.
I always check a turned formula by substituting simple numbers.

✅

Done when: You have four turned formulae, each checked with numbers, and a run of five correct in practice.

What to take from this lesson

A formula is an equation with letters — it is turned with the same rearrangements.
You want t from s = v · t? Divide both sides by v: t = s/v.
Go from the furthest obstacle to the nearest.
Check a turned formula by substituting simple numbers.
Anyone who can turn formulae remembers three times fewer of them.