Problem 48 - Prealgebra

Question

Give an example of an expression involving multiplication and subtraction in which you would subtract first.

Step-by-Step Solution

Answer

The expression \((10 - 3) \times 2\) requires that subtraction be performed first, resulting in 14.

Step 1: Review Order of Operations

In mathematics, the order of operations is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. The order is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This can be remembered by the acronym PEMDAS.

Step 2: Understanding When to Subtract First

According to PEMDAS, operations should be solved in the order of multiplication before subtraction. However, when subtraction comes before multiplication in the expression and there are no parentheses affecting these operations, the subtraction operation which comes first will be dealt with before you perform the multiplication.

Step 3: Create an Expression

Let's create an expression to fit this scenario: \(10 - 3 \times 2\). Here, we must multiply first according to the order of operations, but let's focus on creating an expression where we want to prioritize subtraction.

Step 4: Use Parentheses to Prioritize Subtraction

To subtract first in the expression, we introduce parentheses: \((10 - 3) \times 2\). Parentheses here change the usual order and force the subtraction to be evaluated before the multiplication.

Step 5: Evaluate the Subtraction

Evaluate the expression inside the parentheses: \(10 - 3 = 7\). This result will then be used in the next operation.

Step 6: Perform Multiplication

After evaluating the subtraction, perform the multiplication: \(7 \times 2 = 14\). This yields the final result of the expression.

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in mathematics. It involves the process of adding a number to itself a certain number of times. For example, multiplying 4 by 3 can be understood as adding 4 together three times:

• 4 + 4 + 4 = 12

Multiplication is often represented by the symbol \( \times \) or sometimes simply as placing numbers next to each other in parentheses, like \( (4)(3) \). It's crucial to understand that multiplication is not only about repeated addition; it plays a vital role when dealing with fractions, decimals, and algebraic expressions as well.
When working with equations that include multiple operations, remember multiplication typically takes precedence over addition and subtraction unless influenced by parentheses.
In the example provided, \( (10 - 3) \times 2 \), the multiplication is carried out after the parentheses operations are completed, affecting the usual order.

Subtraction

Subtraction is the process of deducting one number from another, essentially finding the difference between two values. When you subtract, you are calculating how much less one quantity is compared to another.

• For instance, in the expression \( 10 - 3 \), 3 is taken away from 10, resulting in 7.

Subtraction is represented by the minus sign \(-\). In complex equations, where subtraction and other operations occur, the placement of subtraction relative to other operations, like multiplication, can affect the calculation's outcome. This is because operations need to follow the specific sequence outlined by the order of operations. Without parentheses indicating a different priority, subtraction usually comes after operations like multiplication and division.
To subtract before other operations, like in \((10 - 3) \times 2\), parentheses are used to alter the order, ensuring subtraction is prioritized before multiplication.

PEMDAS

PEMDAS is an acronym that helps remember the sequence of operations in mathematical equations. It stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order helps resolve mathematical expressions accurately. For example, when evaluating an expression, you handle operations inside Parentheses first, then work with any Exponents.

• Multiplication and Division are handled next, proceeding from left to right. These operations have equal precedence, unlike what the acronym might misleadingly suggest, so they should be addressed in the order they appear.
• Finally, Addition and Subtraction are the last operations to tackle, also moving from left to right.

In the given situation of \((10 - 3) \times 2\), despite multiplication generally coming before subtraction when following PEMDAS without modification, parentheses allow subtraction to be evaluated first. This demonstrates the flexibility in adjusting operation sequences by using parentheses to control the process.