Problem 50 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. The sum of two numbers is 80 . If the larger is divided by the smaller, the quotient is 7 , and the remainder is 8 . Find the numbers.

Step-by-Step Solution

Answer

The numbers are 9 and 71.

Step 1: Understand the Information

Let's break down the problem. We have two numbers, whose sum is 80. When the larger number is divided by the smaller, the quotient is 7 and the remainder is 8.

Step 2: Translate into Variables

Let's denote the smaller number by \( x \). The larger number, according to the problem, would then be \( 80 - x \) because their sum is 80.

Step 3: Set Up the Division Equation

Using the division information, \( \text{larger number} = \text{smaller number} \times \text{quotient} + \text{remainder} \), we write: \[ (80 - x) = 7x + 8. \]

Step 4: Rearrange and Solve for x

Our equation is \((80 - x) = 7x + 8\). Rearrange it to find \( x \):\( 80 - x = 7x + 8 \) becomes \( 80 - 8 = 7x + x \) or \( 72 = 8x \), hence, \( x = \frac{72}{8} = 9. \)

Step 5: Find the Larger Number

Since \( x = 9 \), the smaller number is 9. Then, the larger number is: \( 80 - 9 = 71. \)

Key Concepts

Division in Algebra

Division in algebra is a fundamental concept where one number, known as the dividend, is divided by another number, called the divisor, to produce a quotient and sometimes a remainder. In algebraic terms, division is frequently used to isolate a variable to solve equations. For example, in our exercise, when dealing with two numbers, the division is used to express the relation between these numbers through a formula involving division.

The key formula in this context is given by:

• Dividend = Divisor × Quotient + Remainder

This formula helps in writing equations based on the problem's conditions, such as when the larger number divided by the smaller one gives a quotient of 7 and a remainder of 8. Understanding this division process is crucial to breaking down and solving algebraic equations.

Setting Up Equations

Setting up equations is the process of translating a word problem into a mathematical statement using variables and mathematical operations. This process involves identifying the relevant quantities and expressing the relationships between them as equations. In the problem at hand, we first identify the smaller number as a variable, which we call \( x \).

By understanding the conditions provided — that the two numbers sum up to 80 and their division property — we can establish expressions for each condition:

• The first condition: The sum of the numbers is 80, giving us the equation \( x + (80 - x) = 80 \).
• The second condition: Division relation is \( (80 - x) = 7x + 8 \).

These steps show how important it is to clearly translate verbal representations into algebraic expressions. This translation is foundational in approaching any algebraic equation systematically.

Solving Equations Step-by-Step

Solving equations step-by-step is about following a structured process to find the value of unknown variables. Once we have our equation set up, as in the exercise, we need to solve for \( x \) using algebraic techniques. This involves simplifying the equation and isolating the variable.

Here's how you solve an equation step-by-step:

• Start with the equation: \( 80 - x = 7x + 8 \).
• Rearrange terms to get all terms involving \( x \) on one side: \( 80 - 8 = 7x + x \).
• Simplify: \( 72 = 8x \).
• Isolate \( x \) by dividing both sides by 8: \( x = \frac{72}{8} \).
• Solve the division: \( x = 9 \).

Once \( x \) is found, use it to determine other values by substituting back into the relevant equations. Following these steps ensures accuracy and helps in understanding how to solve any similar problems you might encounter.