Question

For Exercises \(1-74,\) simplify. $$3-10 \div 2$$

Short answer

Question: Simplify the given expression: $$3 - \frac{10}{2}$$ Answer: $$-2$$

Step-by-step solution

Division

In the given expression, perform the division operation first: $$3 - \frac{10}{2}$$

Evaluate division

Now, evaluate the division: $$3 - 5$$

Subtraction

Lastly, perform the subtraction operation: $$3 - 5 = -2$$

Final Answer

So, the simplified expression is: $$-2$$

Key concepts

Division

Division is a fundamental arithmetic operation that involves splitting a number into equal parts. It is often considered the reverse of multiplication. In mathematical expressions, division is typically represented by the symbol \(\div\) or a slash (\(/\)). Understanding division is crucial because it is frequently the first operation to perform in expressions with multiple arithmetic operations.

• Division helps in breaking down problems into simpler parts.
• It is a way to partition a set into equal groups.
• In expressions, it can be represented as a fraction too (e.g., \(\frac{10}{2}\)).

In the example given, we have \(10 \div 2\), which simplifies to \(5\). This means that \(10\) is divided into \(2\) equal parts, each with the value \(5\). Before moving on to other operations like subtraction, take the time to fully evaluate and simplify any division present in the expression.

Subtraction

Subtraction is another basic arithmetic operation which involves taking away one quantity from another. It is represented by the minus sign \(-\). Subtraction helps in finding out the difference between numbers.

• It is the process of deducting one number from another.
• Subtraction can be visualized as moving left on the number line.
• It helps in comparing sizes and quantities.

In our problem, once we had simplified the division to \(5\), the next step was to perform subtraction: \(3 - 5\). This involves taking \(5\) away from \(3\), resulting in \(-2\). Negative results are important in mathematics, as they indicate a value is less than zero.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which calculations should be performed to avoid ambiguity. This is crucial in mathematics when an expression includes more than one type of operation. The standard order of operations can be remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• It helps ensure that expressions yield the same results regardless of who solves them.
• Following these rules prevents errors that arise from calculating in the wrong order.
• Division and multiplication are performed first, followed by addition and subtraction.

In the provided exercise, the order of operations required performing the division \(10 \div 2\) before executing the subtraction \(3 - 5\). Strict adherence to this order ensures the correct solution, which in this case is \(-2\). Always perform operations from left to right for division and multiplication, as well as for addition and subtraction, unless parentheses suggest another arrangement.