Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Write an algebraic expression for the verbal expression. The product of two natural numbers whose sum is 25.

Step-by-Step Solution

Answer

The algebraic expression for the verbal phrase is \( x * (25 - x) \)

Step 1: Identify the natural numbers

Let's denote the first natural number as \( x \) and the second one as \( y \). We know that these two numbers add up to 25, so we can write this as an equation: \( x + y = 25 \)

Step 2: Express one number in terms of another

From the the equation, \( x + y = 25 \), we can express \( y \) in terms of \( x \) (or vice versa). Let's find \( y \): \( y = 25 - x \)

Step 3: Write the product of the numbers

According to the problem, we're interested in the product of \( x \) and \( y \). We'd substitute our result from step 2 into this expression, which gives us the product as follows: \( x * y = x * (25 - x) \)

Key Concepts

Natural Numbers

Natural numbers are the basic building blocks of numbers in mathematics. They are the simplest kind of numbers that start from 1 and go all the way up indefinitely, one after the other, like 1, 2, 3, 4, and so on. Notably, zero is not considered a natural number. Natural numbers are useful for counting tangible objects, such as apples, people, or any item that can be counted in whole. Natural numbers have several unique characteristics:

• They are always positive. You will never find a negative natural number.
• Each number is one unit away from its preceding and following numbers.
• They can be used in various arithmetic operations like addition and multiplication easily.

The concept of natural numbers is vital in algebra, especially when you create equations to solve problems based on real-world scenarios, such as finding two numbers whose properties we know.

Product of Numbers

The product of numbers is the result you get when multiplying two or more numbers together. It's like combining two sets of items and seeing how many items you have in total. For example, if you have 3 bags of apples and each bag contains 4 apples, then the product of 3 and 4 gives you 12 apples in total.To understand how to find the product in algebraic terms:

• Identify the numbers or variables you need to multiply.
• Use the asterisk symbol (*) or simply place the variables next to each other to denote multiplication, as seen in the expression, for example, \( x \times y \).

In our specific exercise, finding the product of two natural numbers involves multiplying two variables whose sum is something specific, such as 25. So, if \( x \) represents one number, and \( y \) the other, the product is expressed algebraically by multiplying them with each other, like this: \( x \times (25 - x) \). This expression represents the product of those two numbers when their sum is 25.

Sum of Numbers

The sum of numbers is the total you get when adding two or more numbers together. It encapsulates the idea of combining quantities to find out the total. For instance, if you have 2 oranges and add 3 more, the sum is 5 oranges.In algebra, the sum of numbers is an important operation that helps form equations or expressions:

• Simply add the variables or numbers together using the plus sign (+).
• The result will be a larger number if you're adding positive numbers, such as natural numbers.

Returning to our problem, we set up the sum of two numbers as an equation: \( x + y = 25 \), where \( x \) and \( y \) are the two natural numbers we're trying to identify. This lets us express one number in terms of the other, making it easier to perform more complex calculations like finding their product.