Problem 99 - Prealgebra

Question

Simplify the given expression. \(\frac{-14.98-9.6}{17.99-19.99}\)

Step-by-Step Solution

Answer

The simplified expression is 12.29.

Step 1: Simplify the Numerator

The given expression has a numerator of \(-14.98 - 9.6\). We need to perform the subtraction within the numerator first.\(-14.98 - 9.6 = -24.58\).

Step 2: Simplify the Denominator

Now, look at the denominator, which is \(17.99 - 19.99\). Perform the subtraction to simplify.\(17.99 - 19.99 = -2\).

Step 3: Simplify the Fraction

Combine the simplified numerator and denominator into a fraction:\(\frac{-24.58}{-2}\).Since the negatives cancel each other out, divide 24.58 by 2 to simplify this further:\(\frac{-24.58}{-2} = 12.29\).

Key Concepts

Understanding Subtraction

Subtraction is one of the four basic arithmetic operations and involves taking away one number from another. The goal is to find the difference between the two numbers. In the context of fractions and mathematical expressions, subtraction can occur in both the numerator and the denominator. For example, when simplifying a fraction like \(-14.98 - 9.6\), what we're really doing is finding out how much \(14.98\) is larger than \(9.6\) when both values are negative, resulting in a difference of \(-24.58\). The same operation happens in the denominator with \(17.99 - 19.99\), yielding \(-2\). Remember:

• Subtracting a larger number from a smaller number results in a negative number.
• Always align numbers by their decimal points when dealing with decimals.
• Check your work by reversing the subtraction: \(b - a\) should equal the negation of \(a - b\).

By mastering the rules and understanding of subtraction, the process of simplifying fractions becomes much more manageable.

Numerator and Denominator Explained

A fraction consists of two main parts: the numerator and the denominator. The numerator, located above the fraction line, signifies how many parts of the whole are being considered, while the denominator, below the line, indicates the total number of equal parts that the whole is divided into. In the given exercise, we see \(-14.98 - 9.6\) forming our numerator after precisely handling the subtraction, and \(17.99 - 19.99\) is our denominator after the subtraction process, resulting in \(-2\).Understanding how to simplify fractions involves simplifying both the numerator and the denominator. If both parts of a fraction are negative as in our example, their negative signs will cancel out in division, resulting in a positive fraction:

• Numerator: \(-24.58\) (simplified result of subtraction from given numbers).
• Denominator: \(-2\) (simplified difference from given subtraction).

This gives us a simplified fraction, ultimately leading to the result of \(12.29\). Reviewing each component is essential to keep calculations accurate and orderly.

Dealing with Negative Numbers

Negative numbers are numbers less than zero and appear frequently in mathematics. When handling operations involving negative numbers, like subtraction, special attention is needed to ensure proper simplification and results.For example, when we simplified the expression \(\frac{-24.58}{-2}\), both the numerator and the denominator are negative. In division, a negative sign in the numerator and denominator cancels out. Here's why:

• When the signs match—negative over negative, the result is positive.
• This is similar to multiplying two negative numbers, where the negative signs cancel each other out, producing a positive product.

Understanding how negative numbers affect mathematical operations helps students handle complex expressions confidently. In summary, recognizing the behavior of negative numbers can reduce errors and clarify the process of simplification in any mathematical context.