One sum, two answers?

Liri had written down a long chain of calculations: 5 + 3 × 4. And she had got two different answers, depending on how she worked it out.

Liri: “Sometimes I get 32, sometimes 17. Which one is right?”
Erklärus: “Only one – and there are fixed rules for that, which everyone in the world follows.”

✖️ Multiplication and division first

Multiplication (×) and division (÷) are always worked out before addition (+) and subtraction (−) – no matter in which order they are written.

5 + 3 × 4

First 3 × 4 = 12, then 5 + 12 = 17.

Erklärus: “So the right answer was 17. Anyone who works out 5+3 first gets the wrong answer, 32.”
Work out: 4 + 2 × 5 =

🔵 The bracket rule

Whatever is in brackets is always worked out first – even before multiplication and division.

(5 + 3) × 4

First 5 + 3 = 8, then 8 × 4 = 32.

Liri: “So the brackets change what happens first!”
Work out: (6 − 2) × 3 =

🔄 Two important laws of arithmetic

Commutative law (swapping law): the order in multiplication doesn’t matter.

3 × 4 = 4 × 3

Associative law (grouping law): it doesn’t matter which numbers you combine first.

(2 × 3) × 5 = 2 × (3 × 5)

Erklärus: “These laws make mental arithmetic easier: you are allowed to choose the order that is easiest for you.”

🎨 Marking priorities in colour

With long chains of calculations it helps to mark first everything that comes first – then work through it step by step from top to bottom:

(2 + 6) − 2 × 3

1. Brackets: 2+6=8   2. Multiplication/division: 2×3=6   3. Addition/subtraction: 8−6=2

Work out: (2 + 6) − 2 × 3 =
🌌 Rules of arithmetic in the universe

A rocket needs 3 launch stages, each with 2 engines, plus 1 main engine: 1 + 3 × 2 = 7 engines altogether.
Without fixed rules of arithmetic, engineers from different countries could arrive at different – and dangerous – results with complex formulas.

❌ Common mistake

Many people simply work through expressions from left to right, just as they are written: 5 + 3 × 4 as “first 5+3, then ×4” = 32. That is wrong!

Tip: before you calculate, first look for all multiplications and divisions (× and ÷) and brackets – they always come first.

✖️ Exercise 1 – Multiplication before addition
Work out: 10 − 2 × 3
🔵 Exercise 2 – The bracket rule
Work out: (4 + 2) × 3
🔄 Exercise 3 – Commutative law
If 6 × 7 = 42, what is 7 × 6?

Only one right answer

Liri worked out her long chain again – this time with the right rules. Only one answer was left.

Liri: “Now I’m sure: no matter who does the sum, the same answer always comes out.”
Erklärus: “That is exactly what makes these rules so important. Your map home only works if every calculation on it is unambiguous.”
“Rules of arithmetic: brackets first, then multiplication and division (×, ÷) before addition and subtraction (+, −). Commutative law: the order in multiplication doesn’t matter. Associative law: the order of combining doesn’t matter.”
What you learned today
Brackets → × ÷ → + −
5 + 3 × 4 = 17  ·  (5 + 3) × 4 = 32
🌟

Chapter 20 mastered!

Liri now knows the fixed rules that make every calculation unambiguous.

Erklärus’s notebook – page 20:
“Rules of arithmetic: first brackets, then multiplication and division before addition and subtraction. Commutative law: 3×4=4×3. Associative law: (2×3)×5=2×(3×5). These rules apply everywhere – so every calculation has only one right answer.”
← Back to overview 20 – Rules of Arithmetic