Problem 36 - Prealgebra

Question

One year of a dog's life is equivalent to 7 years of human life, as shown in the table. (TABLE CANNOT COPY) What is the age of a dog, if the equivalent human age is \(42 ?\)

Step-by-Step Solution

Answer

The dog's age is 6 years.

Step 1: Understanding the Problem

The problem states that 1 year of a dog's life is equivalent to 7 human years. We need to find out how many dog years correspond to 42 human years.

Step 2: Set Up the Equation

Let the dog's age in years be represented by the variable \( d \). According to the given relationship, \( 1 \text{ dog year} = 7 \text{ human years} \), we can represent the human age as \( 7d \).

Step 3: Solve for Dog's Age

Since we know the equivalent human age is 42, we set up the equation: \( 7d = 42 \). To find \( d \), divide both sides of the equation by 7 to get \( d = \frac{42}{7} \).

Step 4: Calculate

Perform the division: \( d = 6 \). Thus, the dog's age is 6 years.

Key Concepts

Equivalent Relationships

Equivalent relationships are a fundamental concept in understanding ratios and proportions. They show how two different quantities relate to each other in a consistent way. In our exercise, we are looking at how a dog's age correlates with human age. One year of a dog's life is equivalent to seven human years. This means the relationship between dog years and human years is a ratio of 1:7. Whenever one unit changes in the context of dog years, it results in a proportionate change in human years. These kinds of relationships help us make predictions or conversions across different units. Learning about equivalent relationships is essential for solving ratios and proportions in real-life scenarios. This concept is heavily applied in various fields such as finance, science, and everyday problem-solving. It allows us to create simple conversion factors, such as saying 1 dog year equals 7 human years, to easily switch between units.

Division in Mathematics

Division is a mathematical operation that helps us distribute a certain amount evenly over a given number of parts. In the context of the problem, division was used to determine how many dog years correspond to 42 human years. In simple terms, if we know 7 human years equal 1 dog year, and we have a total of 42 human years, we can use division to find out the number of dog years. We set up the equation as \( 7d = 42 \).By dividing both sides of the equation by 7, we simplify \( d = \frac{42}{7} \) to get \( d = 6 \). So, 42 years of human life equate to 6 years of dog life.Here are some quick pointers on division:

• Division helps in evenly distributing values.
• It is often used to find factors or simplify proportions.
• Make sure to divide both sides of an equation when solving for a variable, as shown in the problem.

Variable Representation

In algebra, variables act as symbols that can stand in for unknown numerical values. They allow us to create equations to solve real-world problems.In our exercise, we used the variable \( d \) to represent the dog's age. By establishing that \( 7d \) (7 times the dog's age) equals the human age of 42, we can find the unknown value of \( d \) through calculation.Using variables has several advantages:

• They provide flexibility when solving equations, as they can represent any number or value.
• Variables help simplify complex problem statements into more manageable mathematical expressions.
• They make it easy to update and adapt equations for different values or scenarios.

Understanding and using variable representation is key in algebra and is widely applicable in various contexts, be it science, economics, or technology.