Problem 5 - Elementary Algebra

Question

Translate the following into algebraic equations. Five times the sum of 6 and some number is 20 .

Step-by-Step Solution

Answer

The algebraic equation is \( 5(6 + x) = 20 \).

Step 1: Understand the Problem Statement

The problem statement is 'Five times the sum of 6 and some number is 20.' We need to translate this statement into an algebraic equation.

Step 2: Define the Unknown

Identify the unknown element in the problem. Here, 'some number' is the unknown, which we can call \( x \).

Step 3: Interpret 'the sum of 6 and some number'

The phrase 'the sum of 6 and some number' translates to the expression \( 6 + x \). This represents adding 6 to the unknown number \( x \).

Step 4: Interpret 'Five times the sum of 6 and some number'

The phrase 'five times the sum of 6 and some number' means that you multiply the entire sum (\( 6 + x \)) by 5. Thus, it translates to \( 5(6 + x) \).

Step 5: Set Up the Equation

The problem states that 'five times the sum of 6 and some number is 20.' This means \( 5(6 + x) = 20 \). Hence, the required algebraic equation is \( 5(6 + x) = 20 \).

Key Concepts

Translating Verbal Expressions

Turning words into algebraic equations is like translating a language. When we convert a verbal expression into a math equation, we break down a sentence into components that mathematical rules can handle.

• "Five times" suggests a multiplication operation.
• "The sum of 6 and some number" indicates an addition operation.
• "Is 20" represents the equality symbol, \( = \).

When you read the phrase "five times the sum of 6 and some number is 20," it essentially guides you to create an equation. Start by identifying the operation words. "Sum" means addition—here it refers to 6 added to an unknown number. "Five times" indicates that you will be multiplying the entire previous sum by 5. Finally, "is 20" means the result equals 20. These steps help form a coherent mathematical statement and are crucial in problem-solving.Converting verbal statements to algebraic ones requires attention to language specifics and arithmetic operations. Recognizing keywords in a sentence can help you determine what actions to take mathematically.

Unknown Variables

When dealing with algebra, one common obstacle is understanding and working with unknown variables. A variable in math is a symbol, often a letter, that stands for a number that we either don’t know yet or want to find out.
The problem gives us “some number” as the unknown, which we can conveniently represent with a variable, most commonly \( x \). This unknown \( x \) represents a number that paired with 6, in a sum, is then multiplied and equals 20.

• Using a symbol like \( x \) simplifies writing and managing equations.
• It's important to remain consistent with symbols to avoid confusion.

By calling this unknown number \( x \), we simplify further operations. It makes it easier to perform calculations needed to solve for this unknown later. Understanding that a variable is something standing in for a real number—and can be manipulated like any other number—is fundamental to solving equations.

Mathematical Expressions

Creating a mathematical expression from a problem statement involves systematically transforming words into a sequence of numbers, operators, and variables. This logical sequence forms the building block of solving algebraic problems.

• "The sum of 6 and some number" becomes \( 6 + x \).
• "Five times the sum" translates to \( 5(6 + x) \), where you multiply the entire sum of 6 and the variable.

Expressions are crafted by translating phrases like "sum" or "times" into addition or multiplication. Once each part of the verbal statement is converted into symbols and numbers, it becomes an equation ready to solve. The purpose of arranging these elements is to express the precise relationship described in words. In our problem, the equation \( 5(6 + x) \) represents first adding 6 and the unknown \( x \), then multiplying the whole sum by 5.A well-crafted mathematical expression captures the problem’s essence in numerical form, setting the stage for further operations that lead to finding the value of the variable.