Problem 98 - Prealgebra

Question

Simplify the given expression. \(\frac{6.5(-1.6)-3.35}{-2.75}\)

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Evaluate the Expression in the Numerator

The expression in the numerator is \(6.5(-1.6) - 3.35\). First, calculate \(6.5 \times (-1.6)\), which is \( -10.4\). Then subtract \(3.35\) from \(-10.4\) to obtain \(-10.4 - 3.35 = -13.75\).

Step 2: Rewrite the Full Expression with Calculated Numerator

Now substitute the calculated numerator back into the original expression. The expression becomes \(\frac{-13.75}{-2.75}\).

Step 3: Simplify the Fraction

To simplify \(\frac{-13.75}{-2.75}\), divide \(-13.75\) by \(-2.75\). The division of two negative numbers results in a positive number, so \(\frac{-13.75}{-2.75} = 5\).

Key Concepts

Numerator Evaluation

In mathematics, especially when dealing with fractions, evaluating the numerator is a critical step towards simplification. The numerator is the top part of a fraction that tells you how many parts you have. Let's break down the process of evaluating the numerator in the provided exercise.
First, we have the expression in the numerator: \(6.5(-1.6) - 3.35\). To handle this, we start with the multiplication part: \(6.5 \times (-1.6)\). You multiply these two numbers together, remembering that multiplying a positive number by a negative number gives a negative result. Thus, \(6.5 \times (-1.6) = -10.4\).
Next, we subtract \(3.35\) from \(-10.4\). Subtraction with a negative number is like adding its absolute value, so \(-10.4 - 3.35\) results in \(-13.75\). Make sure to follow the order of operations properly as it can greatly affect the result in complex expressions.

Negative Numbers Division

Dividing negative numbers might seem tricky at first, but it follows simple rules once understood. When dealing with division in the context of simplifying expressions, knowing what happens when you divide negative numbers is crucial.
In the original problem, after evaluating the numerator and finding \(-13.75\), we move on to divide by the denominator \(-2.75\). According to the rules of arithmetic:

• Dividing a negative number by another negative number gives a positive result.
• This is because the two negatives cancel each other out.

So, when dividing \(-13.75\) by \(-2.75\), we get a positive result: \(5\). Remember, whenever you divide negatives, just divide the absolute values of the numbers and ensure your answer is positive.

Prealgebra Operations

Prealgebra operations lay the foundation for more complex mathematical concepts and solving problems like the one we have here involves an understanding of these basic operations.
In our example, we are using several prealgebra operations such as: multiplication, subtraction, and division, each needing its specific attention:

• Multiplication: When you multiply a positive by a negative number, the product is negative. This concept was used when multiplying \(6.5\) by \(-1.6\).
• Subtraction: Subtracting a larger negative value from a smaller one results in moving further into the negative side, which simplifies \(-10.4 - 3.35\) to \(-13.75\).
• Division: As explained, dividing two negative numbers results in a positive quotient. Here, \(-13.75\) divided by \(-2.75\) gave us \(5\).

These operations follow basic arithmetic rules, crucial for understanding more advanced computations, ensuring you're well-equipped for future math challenges.