Evaluate the expression:
$6 \div 2(1+2)$
The expression to evaluate is \(6 \div 2(1+2)\). To solve this correctly, we follow the standard order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). The expression involves a division operation and an implicit multiplication, where the term \(2(1+2)\) signifies \(2 \times (1+2)\).
First, we must evaluate the operations inside the parentheses. Adding the numbers inside gives us:
\[(1 + 2) = 3\]
Substituting this back into the original expression, we get:
\[6 \div 2 \times 3\]
Now, we have a series of operations involving division and multiplication. According to the order of operations, when division and multiplication appear together, we proceed from left to right.
First, perform the division:
\[6 \div 2 = 3\]
Next, multiply the result by 3:
\[3 \times 3 = 9\]
Therefore, the result of the expression is \(9\).
Understanding the order of operations is essential for ensuring consistency in mathematical communication. In computer programming, for example, compilers follow strict rules of precedence to interpret formulas correctly. If you were writing a program to calculate the cost of items with a discount or tax multiplier, misplacing parentheses or misunderstanding how division and multiplication interact could lead to significant errors in financial reporting. Just as in this problem, clear, step-by-step adherence to these rules prevents ambiguity.