Problem 72 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ -\frac{10}{3}\left(\frac{12}{2}\right)(d) \text { when } d=-4 $$

Step-by-Step Solution

Answer

The evaluated expression is 80.

Step 1: Identify and Substitute the Value

First, observe the variable in the expression, which in this case is \(d\). In this problem, \(d\) is given as -4, this value will replace the variable \(d\) in the equation. So, -\(\frac{10}{3}\) * \(\frac{12}{2}\) * (d) becomes -\(\frac{10}{3}\) * \(\frac{12}{2}\) * (-4).

Step 2: Simplify the Expression

Second, following the BODMAS rule, begin by simplifying the fractions. This becomes -\(\frac{10}{3}\) * 6 * (-4). The negative sign at the beginning is part of the expression and it will stay there until the end.

Step 3: Perform the Operations

Now, multiply the terms. This results in positive 80 since multiplying two negative numbers results in a positive number. So, -\(\frac{10}{3}\) * 6 * (-4) = 80.

Key Concepts

Substitution

When working with algebraic expressions, substitution is a crucial concept. In simple terms, substitution involves replacing a variable in an expression with a numeric value that is given or known. This process allows us to evaluate expressions and determine their numerical value. Let's break it down further.

• Identify the variable in the expression. A variable is simply a symbol, often a letter, that stands for an unknown quantity.
• Find the given value for this variable from the problem statement.
• Replace every instance of the variable with its provided value.

This transforms the algebraic expression into a numerical expression, and paves the way for further simplification. For example, consider the expression \(-\frac{10}{3}\left(\frac{12}{2}\right)(d)\), with \(d = -4\). By substituting \(d\) with \(-4\), we write the expression as \(-\frac{10}{3}\left(\frac{12}{2}\right)\left(-4\right)\). The expression is now ready for evaluation using arithmetic rules.

BODMAS Rule

BODMAS is an acronym that stands for Brackets, Order (i.e., powers and roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This rule helps us determine the correct sequence in which operations should be carried out when evaluating mathematical expressions. Let’s go through the specifics.

• Brackets: Solve expressions inside brackets first.
• Order: Evaluate powers or roots next.
• Division and Multiplication: Process these operations as they appear from left to right.
• Addition and Subtraction: Finally, handle these operations from left to right.

In our specific problem, after substituting \(d = -4\), BODMAS directs us to first simplify the expression inside the brackets. So, \(\left(\frac{12}{2}\right)\) needs to be simplified to \(6\) before continuing with the remaining multiplication and accounting for negative signs. Following this rule ensures the expression is evaluated correctly.

Simplifying Fractions

Simplifying fractions is a common mathematical task that makes calculations easier. It involves reducing a fraction to its simplest form where the numerator and denominator have no common factors other than one. Here's how you simplify fractions:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.
• The result is the fraction in its simplest form.

For example, in the expression \(-\frac{10}{3}\left(\frac{12}{2}\right)\), the fraction \(\frac{12}{2}\) can be simplified. Since 12 divided by 2 equals 6, the expression simplifies to an easier form for further calculations: \(-\frac{10}{3}\times 6\times (-4)\). Proper simplification aids in making the multiplication and addition steps straightforward and less error-prone.