Problem 41 - Prealgebra

Question

Translate each sentence into an equation. Five more than three times a number is \(20 .\)

Step-by-Step Solution

Answer

The equation is \( 3x + 5 = 20 \).

Step 1: Understand the Problem

The problem states: "Five more than three times a number is 20." Here, we need to express it as a mathematical equation. We are asked to translate this sentence into an equation.

Step 2: Define the Variable

Let's denote the unknown number as \( x \). This is the number we will use in our equation to represent 'a number'.

Step 3: Translate 'Three Times a Number'

The phrase 'three times a number' translates to the expression \( 3x \). Here, \( 3 \) is multiplied by the variable \( x \).

Step 4: Translate 'Five More Than'

The term 'five more than' translates to the operation of adding 5 to something. So we will add 5 to our previous expression \( 3x \).

Step 5: Combine and Form the Equation

Combining these translations from the earlier steps, the phrase 'Five more than three times a number' becomes the expression \( 3x + 5 \).

Step 6: Translate 'Is 20'

The word 'is' translates mathematically to an equals sign \( = \). Therefore, 'is 20' becomes \( = 20 \).

Step 7: Write the Complete Equation

Combine all parts to form the final equation: \( 3x + 5 = 20 \). This equation represents the sentence given in the problem.

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and arithmetic operations. In the context of our problem, we are trying to convert a sentence into a mathematical expression. The sentence is "Five more than three times a number is 20." We do this by using mathematical symbols to represent words. For example, "three times a number" becomes the expression \(3x\), where \(x\) is the variable representing the unknown number. When we add 5 to this expression, we get \(3x + 5\). Here, mathematical expressions allow us to succinctly and effectively communicate mathematical ideas that can represent real-world situations in a common language of math.

Defining Variables

Defining variables is a crucial part of transforming a problem into a mathematical equation. A variable is a symbol, often a letter, that stands in for an unknown value in an equation or expression. In our exercise, the unknown number is represented by the variable \(x\). By defining \(x\) as the unknown quantity, we can develop relationships using mathematics without knowing the exact value of \(x\). This step is essential because it allows us to use algebra to solve for \(x\) eventually, answering the question posed by the problem. Variables are placeholders that make abstract thinking and problem-solving in math much more manageable.

Forming Equations

Forming equations involves creating a balanced mathematical statement using expressions, variables, and numbers. An equation indicates that two expressions are equal. In our exercise, we need to create an equation from the sentence "Five more than three times a number is 20." Here’s how we can break this down:

1. **'Three times a number'** translates to \(3x\).
2. **'Five more than'** indicates we add 5 to this expression, forming \(3x + 5\).
3. **'Is'** means we need an equal sign \(=\).
4. **'20'** is the number on the other side of the equation.

When combined, these steps give us the complete equation: \(3x + 5 = 20\). This equation represents the balance of the problem—everything on one side equals everything on the other.

Prealgebra Concepts

Prealgebra is a foundational level of mathematics that helps set the stage for more advanced algebraic concepts. It involves basic arithmetic operations, number sense, and understanding variables and equations, just like in our exercise. In the example, prealgebra helps us learn how to express everyday words using math language—like translating the sentence "Five more than three times a number is 20" into an algebraic equation.

Understanding prealgebra concepts allows students to see the relationships between numbers and operations more clearly and prepares them for solving more complicated algebraic equations. It's a critical skill for developing logical thinking and problem-solving strategies that students will use in more awkward and challenging mathematical contexts later on. By practicing such problems, students gain confidence in speaking 'math' and begin to appreciate the power and utility of mathematics in everyday life.