Problem 5 - Elementary Algebra

Question

Use the grouping symbols to help perform the following operations. $$\frac{1+19}{2+3}$$

Step-by-Step Solution

Answer

Expression: $$ \frac{1+19}{2+3} $$ Answer: The result of the given expression is 4.

Step 1: Perform the Operations within the Grouping Symbols

Add the numbers within the parentheses on both the numerator and the denominator: $$ \frac{1+19}{2+3} = \frac{20}{5} $$

Step 2: Perform the Division

Now, divide the numerator by the denominator: $$ \frac{20}{5} = 4 $$ The result is 4.

Key Concepts

Grouping Symbols

Grouping symbols are crucial in mathematics as they dictate the order in which operations should be performed. They help in clearly organizing complex expressions and indicating which calculations should be prioritized. Typical grouping symbols include:

• Parentheses \(( )\)
• Brackets \([ ]\)
• Braces \(\{ \}\)

In the given problem, the grouping symbols are used in the form of parentheses around the expressions in the numerator and the denominator: \( (1+19) \) and \( (2+3) \). This tells us that we must first perform the addition inside these parentheses before proceeding with any other operation. By following this order, the problem becomes simpler, and we are less likely to make errors.

Fractions

Fractions represent parts of a whole and consist of two main elements: a numerator and a denominator. Understanding how fractions work is essential for simplifying expressions and performing operations such as addition, subtraction, multiplication, and division.A fraction appears in the form \( \frac{a}{b} \) where \(a\) is the numerator, and \(b\) is the denominator. These figures give the fraction's value relative to the whole. If a fraction is set inside an expression, it's important to handle operations involving both the numerator and the denominator correctly, as demonstrated in the problem.Operations on fractions require careful calculation to avoid mistakes. Always continue with numerator/denominator calculations before handling the division or multiplication of the fractions.

Numerator and Denominator

The terms numerator and denominator are widely used in fractions. They help express parts of a single entity or more complex mathematical values. - **Numerator**: This is the top part of the fraction. It shows how many parts we have. For example, in \( \frac{1+19}{2+3} \), the sum \( 1+19 \) equals 20, so 20 becomes the new numerator.- **Denominator**: This is the bottom part of the fraction. It tells us how many equal parts make up a whole. Here, \( 2+3 \) equals 5, which is the denominator of our fraction.Proper assessment of numerator and denominator before solving equations ensures accurate results. By getting the separate identities of these two parts, as done step-by-step in the given exercise, we facilitate correct and precise computations. Afterward, division resolves the fraction, yielding the answer.