Problem 12 - Algebra 1

Question

Use the distributive property and mental math to simplify the expression. $$ \begin{aligned} 9(1.95) &=9(?-2) \\ &=? \end{aligned}$$

Step-by-Step Solution

Answer

The simplified expression is 17.55.

Step 1: Break Down the Problem

This problem requires use of the distributive property, which states that for all real numbers a, b, and c, \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\). Here, represent 1.95 as a difference from 2, so instead of 1.95 use 2-0.05.

Step 2: Apply the Distributive Property

Apply the distributive property to the modified expression. Instead of calculating \(9 \times 1.95\), calculate \(9 \times (2 - 0.05)\). This becomes \(9 \times 2 - 9 \times 0.05\)

Step 3: Perform Arithmetic

Arithmetic operation can be performed without pen and paper or a calculator. Calculate \(9 \times 2\) to get 18, and \(9 \times 0.05\) to get 0.45.

Step 4: Subtract to Find Final Answer

Subtract the second result from the first to find the final answer. Hence, \(18 - 0.45 = 17.55\).

Key Concepts

Mental Math

Mental math is a powerful tool that allows you to solve math problems in your head without writing anything down or using a calculator. It's all about breaking down complicated problems into smaller, more manageable parts. In the original exercise, we see mental math shine when calculating 9 times 1.95. Instead of multiplying directly, you break it down into simpler elements using 2 and 0.05, as these are easier numbers to handle mentally. By doing so, you turn a complex multiplication task into two simpler ones: \(9 \times 2\) and \(9 \times 0.05\). The brain can handle these smaller parts swiftly, making the process quick and efficient. Practicing mental math not only boosts your computation speed but also enhances your overall mathematical thinking ability.

Arithmetic Operations

Arithmetic operations are the cornerstones of mathematics and involve four basic functions: addition, subtraction, multiplication, and division. In our exercise, we primarily use multiplication and subtraction.

• First, you multiply 9 by 2, giving you 18.
• Then, multiply 9 by 0.05, producing 0.45.
• Finally, the subtraction operation comes in, as you subtract 0.45 from 18.

These operations are fundamental and become tools for simplifying more complex problems. By applying the distributive property, a topic discussed in depth in the context of this exercise, you can effectively rearrange calculations to make these operations less cumbersome.

Simplifying Expressions

Simplifying expressions is about making calculations easier and reducing the complexity of expressions. This involves using mathematical rules and properties to transform the expression into a simpler, yet equivalent, form. For our example, we employed the distributive property, which allows you to break down the expression \(9 \times 1.95\) into smaller parts like \(9 \times (2 - 0.05)\). The process of simplification not only aids in performing arithmetic operations more easily but also helps in understanding the underlying structure of expressions better. By expressing 1.95 as \(2 - 0.05\), and consequently using the distributive approach to rearrange the terms, the computation becomes straightforward. Using tools like the distributive property enhances your ability to simplify, and makes complex problems more approachable, reinforcing both your confidence and competence in mathematics.