Why it happens: In 5 × 4 + 3 there are two terms — the product 5 × 4 and the lone 3 — so only the 4 gets multiplied by 5. In 5 × (4 + 3) the bracket makes 4 + 3 one single quantity, so both the 4 and the 3 get multiplied by 5. The difference is 5 × 3 – 3 = 12.
Tip: Picture it: 5 rows of 4 dots plus 3 loose dots (23 dots) is clearly not the same as 5 rows of 7 dots (35 dots).
Why it happens: Two rules are at work. Inside the bracket, 4 + 3 = 3 + 4 because addition is commutative. Outside, 5 × 7 = 7 × 5 because multiplication is also commutative. So the order of neither the addition nor the multiplication matters.
Check it yourself: 7 rows of 5 chairs and 5 rows of 7 chairs both seat 35 people — the hall is just turned sideways.