Problem 61 - Prealgebra

Question

Find each product. $$48 \times 5$$

Step-by-Step Solution

Answer

The product of 48 and 5 is 240.

Step 1: Break Down the Multiplication

We will break down 48 as 50 - 2 to make the calculation easier. So, the expression becomes \[ (50 - 2) \times 5 \]

Step 2: Distribute the 5

Distribute the 5 across the terms inside the parentheses: \[ 50 \times 5 - 2 \times 5 \]

Step 3: Calculate Each Product Separately

Calculate each product: \[ 50 \times 5 = 250 \] \[ 2 \times 5 = 10 \]

Step 4: Subtract to Find the Final Product

Subtract the second product from the first product: \[ 250 - 10 = 240 \]

Key Concepts

Distributive Property

The distributive property is a core component of multiplication strategies that allows you to break down a complex multiplication problem into simpler parts. Essentially, it lets you spread a number over addition or subtraction inside of parentheses. This method makes calculations more manageable and is particularly helpful in mental math. For example, if we have \((a + b) \times c\), the distributive property tells us this is equivalent to \(a \times c + b \times c\). This can also work with subtraction, as we used in our problem: \((50 - 2) \times 5\). By distributing the 5, the calculation becomes \(50 \times 5 - 2 \times 5\).
This technique reveals the versatility and power of the distributive property, making it a highly useful tool not just in mathematics but in any scenario that requires efficient mental calculation.

Mental Math

Mental math is a powerful skill that allows you to perform arithmetic operations quickly without the need for a calculator or paper. By utilizing the distributive property, you can simplify movements like multiplication or addition to perform them in your head. By breaking down \(48 \times 5\) into \((50 - 2) \times 5\), you're simplifying the task into smaller, more digestible multiplication problems: \(50 \times 5\) and \(2 \times 5\).
This strategy turns a potentially complicated calculation into one that can be solved more straightforwardly, relying only on basic multiplication and subtraction.

• Practice visualizing the numbers in your head.
• Break numbers down into more manageable parts, like tens and units.
• Systematically solve smaller problems quickly, leading to an overall solution faster.

Arithmetic Operations

Arithmetic operations, including multiplication, addition, and subtraction, form the backbone of mathematics. Each operation follows a specific set of rules and properties, like the commutative and associative properties, which also support the distributive property. In our original exercise, the focus was on two operations: multiplication and subtraction.
By breaking down 48 into simpler terms (50 - 2) and applying it to \( \times 5 \), you perform two multiplication operations: \(50 \times 5\) and \(2 \times 5\). These results (250 and 10) are then combined through subtraction to find the final product: \(250 - 10 = 240\).
Understanding each of these operations and how they interlink is vital. It provides a solid foundation for solving more complex mathematical problems and improves mental calculation speed. Practice regularly will embed these principles as a part of your natural problem-solving toolkit.