Problem 47
Question
PREREQUISITE SKILL Evaluate each expression. $$ \frac{4 \cdot 3}{2 \cdot 1} $$
Step-by-Step Solution
Verified Answer
The value of the expression is 6.
1Step 1: Evaluate the Numerator
To solve this expression, we first calculate the product in the numerator: \(4 \cdot 3 = 12\). So, the expression becomes \( \frac{12}{2 \cdot 1} \).
2Step 2: Evaluate the Denominator
Next, calculate the product in the denominator: \(2 \cdot 1 = 2\). Thus, the expression now simplifies to \( \frac{12}{2} \).
3Step 3: Divide the Numerator by the Denominator
Finally, divide the simplified numerator by the simplified denominator: \( \frac{12}{2} = 6 \).
Key Concepts
Understanding Numerator and DenominatorSimplification TechniquesBasics of Arithmetic Operations
Understanding Numerator and Denominator
In any fraction, it's crucial to identify the numerator and the denominator. The numerator is the top part of the fraction, and it represents how many parts of the whole we're considering. The denominator, on the other hand, is the bottom part of the fraction and shows the total number of equal parts the whole is divided into.
The original expression given was \( \frac{4 \cdot 3}{2 \cdot 1} \). Here, \(4 \cdot 3\) is the numerator, indicating a multiplication of numbers in the top part. Below the line, we have \(2 \cdot 1\) as the denominator. By clearly understanding these positions, you can smoothly navigate through various fractions you encounter.
In fractions:
The original expression given was \( \frac{4 \cdot 3}{2 \cdot 1} \). Here, \(4 \cdot 3\) is the numerator, indicating a multiplication of numbers in the top part. Below the line, we have \(2 \cdot 1\) as the denominator. By clearly understanding these positions, you can smoothly navigate through various fractions you encounter.
In fractions:
- The numerator helps identify what portion is being considered.
- The denominator indicates how many units make up a whole.
Simplification Techniques
Simplification is the process of making an algebraic expression more compact and manageable while keeping its value intact. When we simplify, our goal is to transform it into a form that's easier to work with.
For the expression \( \frac{12}{2} \) that resulted from calculating the products in both the numerator and the denominator, simplification means reducing \( \frac{12}{2} \) to its simplest form. Here, you divide both the numerator and the denominator by their greatest common divisor, which is 2:
Keeping the expression simple doesn't only make it easier to interpret but also easier to use, especially when dealing with more complex operations later on.
For the expression \( \frac{12}{2} \) that resulted from calculating the products in both the numerator and the denominator, simplification means reducing \( \frac{12}{2} \) to its simplest form. Here, you divide both the numerator and the denominator by their greatest common divisor, which is 2:
- Divide the numerator: \(12 \div 2 = 6\)
- Divide the denominator: \(2 \div 2 = 1\) (we typically don't write the denominator as 1)
Keeping the expression simple doesn't only make it easier to interpret but also easier to use, especially when dealing with more complex operations later on.
Basics of Arithmetic Operations
Arithmetic operations form the backbone of algebraic simplifications and evaluations. They include the basic functions of addition, subtraction, multiplication, and division. In the context of our expression, both multiplication and division are used.
Initially, multiplication is needed to evaluate both the numerator \(4 \cdot 3 = 12\) and the denominator \(2 \cdot 1 = 2\). Multiplying ensures we combine values correctly depending on their respective factors.
Finally, division takes the stage as the crucial arithmetic operation to simplify the fraction. Here, division of the evaluated numerator by the denominator \( \frac{12}{2}\) results in a whole number 6, displaying the core necessity of mastering these basic operations.
To summarize:
Initially, multiplication is needed to evaluate both the numerator \(4 \cdot 3 = 12\) and the denominator \(2 \cdot 1 = 2\). Multiplying ensures we combine values correctly depending on their respective factors.
Finally, division takes the stage as the crucial arithmetic operation to simplify the fraction. Here, division of the evaluated numerator by the denominator \( \frac{12}{2}\) results in a whole number 6, displaying the core necessity of mastering these basic operations.
To summarize:
- Multiplication combines values based on distributive properties.
- Division breaks down the fraction into its simplest form.
Other exercises in this chapter
Problem 46
Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=91, d=-4, a_{n}=15 $$
View solution Problem 47
Find the first five terms of each sequence. $$ a_{1}=3, a_{n+1}=2 a_{n}-1 $$
View solution Problem 47
The sum of an infinite geometric series is \(125,\) and the value of \(r\) is 0.4. Find the first three terms of the series.
View solution Problem 47
Find the sum of each geometric series. \(\frac{1}{9}-\frac{1}{3}+1-\cdots\) to 6 terms
View solution