Problem 94 - Prealgebra

Question

Simplify the given expression. \(\frac{1.4-13.25}{-6.84-(-2.1)}\)

Step-by-Step Solution

Answer

The simplified expression is 2.5.

Step 1: Simplify the Numerator

In the numerator, perform the subtraction: \(1.4 - 13.25\). First, align the decimal points, and then subtract. \(1.40 - 13.25 = -11.85\).

Step 2: Simplify the Denominator

In the denominator, handle the double negative: \(-6.84 - (-2.1)\). This is equivalent to \(-6.84 + 2.1\). Align the decimal points and perform the addition. \(-6.84 + 2.10 = -4.74\).

Step 3: Simplify the Fraction

Combine the simplified numerator and denominator to form the new fraction: \(\frac{-11.85}{-4.74}\). Since both the numerator and the denominator are negative, the fraction simplifies to a positive: \(\frac{11.85}{4.74}\).

Step 4: Divide the Numbers

Perform the division \(11.85 \div 4.74\). Calculate this to get approximately 2.5.

Key Concepts

Numerator and Denominator

In mathematics, understanding the terms numerator and denominator is essential when working with fractions. The numerator is the top part of a fraction. It shows how many parts of a whole are considered. Conversely, the denominator is the bottom part. It tells how many equal parts the whole is divided into. Take a fraction like \( \frac{3}{4} \). Here, 3 is the numerator, meaning three parts of the whole are considered. The 4 is the denominator, indicating the whole is divided into four parts. When simplifying expressions, like in this exercise, it's vital first to individually simplify the numerator and denominator. This helps break down the problem into more manageable steps and ensures easier fraction simplification later. Always remember:

• The numerator represents the parts taken.
• The denominator represents the total parts.

Decimal Arithmetic

Decimal arithmetic is crucial for many math problems, especially those requiring precise calculations. Decimals express fractions in a base-10 system, which often makes operations simpler than with fractions. When performing decimal arithmetic, align the decimal points vertically. This alignment is crucial, especially during subtraction and addition. For instance, in this exercise, you need to subtract decimals like \(1.4 - 13.25\). First, adjust 1.4 to 1.40 for proper alignment, making the subtraction straightforward.The same principle applies to addition. When dealing with negative signs, like \(-6.84 - (-2.1)\), remember that subtracting a negative is equivalent to adding its positive equivalent. So it becomes \(-6.84 + 2.10\), once again ensuring decimals are aligned.Key points:

• Always align decimal points for clarity in operations.
• Use equivalent positive values when subtracting negatives.

Fraction Simplification

Simplifying fractions makes them easier to work with or understand. In essence, you're expressing the fraction in its simplest form. It's akin to reducing the fraction to the smallest form where the numerator and denominator have no common factors except 1.For example, in the exercise solution, once the fraction is formed as \( \frac{-11.85}{-4.74} \), both numbers are negative. This means the overall fraction can be represented positively as \( \frac{11.85}{4.74} \). Dividing the numbers gives a simpler expression. Here, as the step-by-step solution shows, results in \(2.5\).The steps to simplify any fraction include:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by their GCD.
• If negatives appear in both, the fraction turns positive.

Simplifying a fraction isn’t about changing its value; it’s about expressing it most compactly.