Final answer:
Madeleine points out that sometimes following the order of operations will give you the same result as solving from left to right, often when operations have the same precedence or do not affect each other. Two examples illustrate this principle, showing that the rules of mathematics are universally valid and should always be consistently applied to ensure accuracy.
Step-by-step explanation:
Madeleine is referring to situations where the normal rules of the order of operations (typically remembered by the acronym PEMDAS for Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) can sometimes provide the same result as simply performing calculations from left to right, without considering these rules. This usually happens when operations are of the same precedence or when they do not affect each other.
For example:
- Expression 1: 6 + 3 × 2. Using order of operations, you multiply first: 3 × 2 = 6, and then add: 6 + 6 = 12. Going from left to right without following the order of operations would incorrectly suggest: 6 + 3 = 9, then 9 × 2 = 18, which is incorrect. So, in this case, the same precedence must be maintained.
- Expression 2: 4 × 5 × 2. Both operations are multiplication, which has the same precedence. You can multiply from left to right: 4 × 5 = 20, then 20 × 2 = 40. Or you can multiply in any order because multiplication is commutative: 5 × 2 = 10, then 4 × 10 = 40. These provide the same result: 40.
Several key points reinforce this concept:
- Multiplication or division applies to every term enclosed in parentheses.
- When all operations have the same precedence, you can compute from left to right.
Understanding that the rules of mathematics are universally valid, regardless of context, helps us appreciate the importance of consistently applying these rules to achieve correct results.