Problem 58 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$ 7(5.98) $$

Step-by-Step Solution

Answer

The simplified expression is 41.86

Step 1: Use Distributive Property

This step involves using the distributive property, i.e., multiplying 7 to each of the numbers inside the parenthesis. Here, since 5.98 cannot be easily multiplied with 7, we can split it as 5 and 0.98. Hence, first multiply 7 with each of these two numbers separately and then add the results. Mathematically, this can be represented as: \(7 * (5 + 0.98)\) which simplifies to \(7*5 + 7*0.98\).

Step 2: Perform the Multiplication

In this step, perform the multiplications separately. \[7*5 = 35\] and \[7*0.98 = 6.86\]

Step 3: Simplify the Expression

In this final step, combine or add the results of the two separate multiplications. We get the simplified expression as: \[35 + 6.86 = 41.86\]

Key Concepts

Simplifying Expressions

Let's dive into simplifying expressions using the distributive property. This mathematical tool lets you break down complex expressions into more manageable parts. Here, for the expression \(7(5.98)\), we can make it easier by splitting 5.98 into 5 and 0.98. Instead of trying to multiply 7 by the cumbersome number 5.98 directly, you simplify the process by using:

• \(7 \times (5 + 0.98)\)
• which turns into \(7 \times 5 + 7 \times 0.98\)

This approach not only simplifies calculations but also helps in mental math, resulting in a simpler answer that can be easily verified.

Mental Math

Using mental math to solve expressions can greatly increase your efficiency and confidence in math. By breaking down numbers into simpler parts, like converting 5.98 into 5 and 0.98, you make calculations swift and less error-prone. Think of mental math as a "shortcut" that doesn't compromise accuracy.

• First, calculate \(7 \times 5 = 35\)
• Next, compute \(7 \times 0.98 = 6.86\)
• Finally, add \(35 + 6.86\) to get \(41.86\)

Breaking problems into smaller, more digestible parts taps into your mental abilities, helping you to perform calculations almost instinctively.

Multiplication

Multiplication is a core skill that becomes much simpler with practice and techniques like the distributive property. For this problem, multiplying 7 by both 5 and 0.98 individually, you'll perform two smaller, easier-to-handle multiplication tasks instead of a larger one.
This method reflects why multiplication tables are essential—they provide a foundational understanding of number interactions.

• First, tackle \(7 \times 5 = 35\), leveraging basic multiplication skills.
• Then, approach \(7 \times 0.98 = 6.86\), applying precision for decimal numbers.
• Combine these results to get a complete answer, \(35 + 6.86 = 41.86\).

With practice, handling more complicated numbers becomes intuitive, especially once you grasp the smaller components of each multiplication task.