Problem 27 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \(\frac{5 a d}{b}\)

Step-by-Step Solution

Answer

The evaluated expression is -4.

Step 1: Substitute the Values

Begin by substituting the given values into the expression \(\frac{5ad}{b}\). The values are \(a = \frac{2}{5}\), \(b = -3\), \(c = 0.5\), and \(d = 6\). Therefore, the expression becomes: \[\frac{5 \times \frac{2}{5} \times 6}{-3}\].

Step 2: Simplify the Numerator

Calculate the product in the numerator. First, multiply \(5\) by \(\frac{2}{5}\), which is \(\frac{10}{5} = 2\). Then multiply this result by \(6\), so the numerator becomes \(2 \times 6 = 12\). The expression now is \(\frac{12}{-3}\).

Step 3: Evaluate the Expression

Divide the numerator \(12\) by the denominator \(-3\). This result is \(-4\). Therefore, the evaluated value of the expression is \(-4\).

Key Concepts

Substitution

Substitution in mathematics simply refers to replacing variables in an expression with given numerical values. For our expression \(\frac{5ad}{b}\), this means replacing \(a\) with \(\frac{2}{5}\), \(b\) with \(-3\), and \(d\) with \(6\). Once these substitutions are made, the expression transforms into \(\frac{5 \times \frac{2}{5} \times 6}{-3}\).

By substituting known values into an expression, we can convert it from a symbolic form to a numeric one. This makes calculation much more straightforward, allowing us to directly solve and find the expression's value. Always ensure to substitute values correctly as this sets the stage for accurate evaluation.

Numerator and Denominator

The terms 'numerator' and 'denominator' are fundamental in fraction evaluation. In the fraction \(\frac{5ad}{b}\), everything above the division line is called the numerator, and everything below is the denominator. In our case, the numerator is \(5ad\) and the denominator is \(b\).

The numerator represents the number of parts we have, while the denominator represents how many parts make up a whole. To simplify fractions or perform operations such as multiplication and division, it's essential to identify both correctly. Proper comprehension of these components aids in seamless mathematical operations.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together to form a new numerator, and multiplying the denominators together to form a new denominator. In the expression \(5 \times \frac{2}{5} \times 6\), multiply as follows:

• First, multiply \(5\) by \(\frac{2}{5}\). Here, \(5\) can be seen as \(\frac{5}{1}\), which gives us \(\frac{5 \times 2}{5 \times 1} = \frac{10}{5} = 2\).
• Next, multiply \(2\) by \(6\) to get \(12\).

Multiplication of fractions can often be simplified by canceling common factors in the numerator and denominator, making the process quicker and easier. Understanding this concept is vital for efficient problem-solving.

Division of Integers

Division of integers is a basic arithmetic operation that involves separating a whole into equal parts. In the expression \(\frac{12}{-3}\), we divide the integer 12 by -3.

Perform the division by splitting 12 into groups of 3 and accounting for the negative sign, which results in \(-4\). Remember, when dividing a positive number by a negative number, or vice versa, the result is always negative. Understanding integer division is crucial to solving algebraic expressions, especially when dealing with negative numbers.