Problem 48 - Algebra A Combined Function

Question

Simplify. See Sections 1.5 and \(1.6 .\) \(\frac{12-3}{10-9}\)

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Simplify the Numerator

The numerator of the fraction is \(12 - 3\). Compute the subtraction: \(12 - 3 = 9\).

Step 2: Simplify the Denominator

The denominator of the fraction is \(10 - 9\). Compute the subtraction: \(10 - 9 = 1\).

Step 3: Perform Division

With the numerator being \(9\) and the denominator being \(1\), divide to simplify the fraction: \(\frac{9}{1} = 9\).

Key Concepts

Numerator

In a fraction, the numerator plays an essential role as it sits above the line, representing the number of parts we are interested in. When simplifying a fraction like \( \frac{12-3}{10-9} \), the first step involves focusing on this top figure. Here, the numerator is expressed as \( 12 - 3 \). To simplify, we simply perform the arithmetic operation involved—subtraction. We calculate \( 12 - 3 = 9 \).
Understanding how to determine and simplify the numerator is crucial. Remember:

• The numerator is the top part of the fraction.
• It tells you how many parts you have or are considering.
• It requires performing the designated arithmetic operation to simplify it.

Denominator

The denominator is equally important in a fraction. It appears below the line and tells us into how many parts the whole is divided. In this example, we have the fraction \( \frac{12-3}{10-9} \), with \( 10 - 9 \) as the denominator. Simplifying this involves solving the subtraction, which gives \( 10 - 9 = 1 \).
Key things to remember about the denominator:

• It is the bottom number, indicating the division of the whole.
• A smaller denominator means larger pieces, and a larger denominator means smaller pieces in terms of division.
• Without simplifying the denominator properly, the fraction's value may not be accurately represented.

Arithmetic Operations

Simplifying fractions often involves basic arithmetic operations: addition, subtraction, multiplication, and division. In our example, the operations simplify both the numerator and the denominator.

• A fraction is essentially a division operation \( \frac{a}{b} \), where \( a \) is divided by \( b \).
• In the fraction \( \frac{12-3}{10-9} \), we subtract to simplify both the numerator and the denominator separately.
• Once both parts are simplified, the division is straightforward: \( \frac{9}{1} = 9 \).

Arithmetic operations allow for the reduction and simplification of fractions, which is vital for easier calculation and clearer understanding of the quantity represented. Being comfortable with these operations will make manipulating fractions far simpler.