Which symbol creates a true sentence when placed between these expressions?
$(14-8) \div 2 \quad 14-8 \div 2$
To compare the two expressions, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We are looking to evaluate the expressions \((14 - 8) \div 2\) and \(14 - 8 \div 2\) and determine their relationship.
First, we evaluate the expression \(E_1 = (14 - 8) \div 2\). Following the order of operations, we calculate the expression inside the parentheses first:
\[14 - 8 = 6\]
Then, we perform the division:
\[6 \div 2 = 3\]
So, the value of \(E_1\) is \(3\).
Next, we evaluate the expression \(E_2 = 14 - 8 \div 2\). According to the order of operations, division must be performed before subtraction. Therefore, we first divide:
\[8 \div 2 = 4\]
Then, we subtract this result from 14:
\[14 - 4 = 10\]
So, the value of \(E_2\) is \(10\).
Finally, we compare the two values. We have \(E_1 = 3\) and \(E_2 = 10\). Since \(3 < 10\), we conclude that \((14 - 8) \div 2 < 14 - 8 \div 2\). This corresponds to option (A).
Understanding the order of operations is essential in everyday life, from calculating a shared restaurant bill with a discount to programming automated tasks. For example, if you are splitting a total bill of \(\$14\) minus an \(\$8\) voucher between two people, you calculate \((14-8) \div 2 = \boxed{\$3}\) per person. However, if you incorrectly calculate \(14 - 8 \div 2\), you would wrongly believe each person owes \(\$10\). Correctly applying these rules ensures accuracy in financial transactions and logical problem solving.