Problem 57 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$ 3(9.20) $$

Step-by-Step Solution

Answer

The simplified expression is \(27.60\)

Step 1: Understand the distributive property

The distributive property is an algebraic property used to simplify expressions, which states that \(a(b + c) = ab + ac\). In this case, the property may not seem apparently necessary due to the simplified nature of the expression.

Step 2: Apply the distributive property

Here, however, this property translates to multiplying the number outside the parenthesis (3) with the number inside the parenthesis (9.20). Thus, the equation becomes \(3 \times 9.20\)

Step 3: Use mental math to carry out the multiplication

Since \(3 \times 9 = 27\) and \(3 \times 0.20 = 0.60\), adding these two values together gets a result of \(27.60\)

Key Concepts

Simplifying Expressions

Simplifying expressions is an important mathematical skill that makes computation easier. It involves reducing an expression to its most basic form. In the case of our expression, "3(9.20)", simplifying through the distributive property allows us to break down the multiplication for easier calculation. The distributive property helps disassemble more complex problems into manageable pieces. For an expression such as \(a(b + c)\), it equates to \(ab + ac\). By applying this idea to our problem, we can directly multiply the separate parts for an easier solution. When simplifying, it's essential to clearly identify all parts of the expression that need to be deduced. This step-by-step process ensures accuracy and understanding throughout the problem-solving process.

Mental Math

Mental math, the art of performing calculations in your mind without the use of paper or calculators, is a valuable skill in daily life. Utilizing mental math when approaching expressions like "3(9.20)" can make solving them faster and more intuitive.In our example, we break down the multiplication into simpler parts. By calculating \(3 \times 9 = 27\) and \(3 \times 0.20 = 0.60\), we utilize mental math to quickly find the solution. This approach emphasizes estimation and the calculation of smaller numbers, enhancing both speed and accuracy.Developing mental math skills involves practice and familiarity with numbers and operations. It can lead to increased confidence when tackling various mathematical problems, making it an essential tool even in more complex calculations.

Multiplication

Multiplication is one of the basic operations of arithmetic, involving the process of adding a number to itself a certain number of times. In our example, "3(9.20)", multiplication is the core operation necessary to find our solution.We apply multiplication here within the framework of the distributive property. Instead of directly multiplying "3(9.20)", we consider the problem in two parts:

• Multiply\(3 \times 9\) resulting in 27.
• Multiply\(3 \times 0.20\) resulting in 0.60.

By adding 27 and 0.60, we arrive at the final result of 27.60.Understanding multiplication and its properties inherently strengthens one's ability to solve various mathematical problems elegantly and efficiently.