Ralph Is 3 Times As Old As Sara. In 6 Years, Ralph Will Be Only Twice As Old As Sara Will Be Then. Find Ralph's Age Now.Which Of The Following Equations Could Be Used To Solve The Problem?A. 3 X + 2 X 3x + 2x 3 X + 2 X B. X + 6 = 2 ( 3 X + 6 X + 6 = 2(3x + 6 X + 6 = 2 ( 3 X + 6 ]C. $3x +
Introduction
Algebra is a powerful tool for solving problems involving unknowns and variables. In this article, we will explore how to use algebra to solve a common type of problem: age problems. We will use a specific example to illustrate the steps involved in solving an age problem using algebra.
The Problem
Ralph is 3 times as old as Sara. In 6 years, Ralph will be only twice as old as Sara will be then. Find Ralph's age now.
Step 1: Define the Variables
Let's define two variables to represent the ages of Ralph and Sara:
- Let R be Ralph's current age.
- Let S be Sara's current age.
Step 2: Write the First Equation
Since Ralph is 3 times as old as Sara, we can write an equation based on this information:
R = 3S
This equation states that Ralph's age is equal to 3 times Sara's age.
Step 3: Write the Second Equation
In 6 years, Ralph will be R + 6 years old, and Sara will be S + 6 years old. According to the problem, in 6 years, Ralph will be only twice as old as Sara will be then. We can write an equation based on this information:
R + 6 = 2(S + 6)
This equation states that Ralph's age in 6 years is equal to twice Sara's age in 6 years.
Step 4: Simplify the Second Equation
We can simplify the second equation by distributing the 2 to the terms inside the parentheses:
R + 6 = 2S + 12
Step 5: Substitute the First Equation into the Second Equation
We can substitute the first equation (R = 3S) into the second equation (R + 6 = 2S + 12) to get:
3S + 6 = 2S + 12
Step 6: Solve for S
We can solve for S by subtracting 2S from both sides of the equation:
S + 6 = 12
Subtracting 6 from both sides gives us:
S = 6
Step 7: Find Ralph's Age
Now that we know Sara's age (S = 6), we can find Ralph's age by substituting S into the first equation (R = 3S):
R = 3(6) R = 18
Therefore, Ralph's age is 18.
Which of the Following Equations Could Be Used to Solve the Problem?
A. B. C.
The correct answer is B. .
This equation is equivalent to the second equation we wrote in Step 3:
R + 6 = 2(S + 6)
By substituting R = 3S into this equation, we can solve for S and then find Ralph's age.
Conclusion
Q: What is an age problem?
A: An age problem is a type of problem that involves finding the age of one or more people based on certain conditions or relationships between their ages.
Q: How do I know which variable to use for each person's age?
A: When solving an age problem, it's a good idea to use a variable to represent each person's age. For example, if we're solving a problem involving two people, we might use "x" to represent one person's age and "y" to represent the other person's age.
Q: What if the problem involves more than two people?
A: If the problem involves more than two people, we can use multiple variables to represent each person's age. For example, if we're solving a problem involving three people, we might use "x", "y", and "z" to represent each person's age.
Q: How do I write the first equation in an age problem?
A: The first equation in an age problem is usually based on the given information about the people's ages. For example, if the problem states that one person is 3 times as old as another person, we can write an equation like this:
x = 3y
Q: How do I write the second equation in an age problem?
A: The second equation in an age problem is usually based on the future ages of the people. For example, if the problem states that one person will be twice as old as another person in 6 years, we can write an equation like this:
x + 6 = 2(y + 6)
Q: How do I simplify the second equation?
A: To simplify the second equation, we can distribute the 2 to the terms inside the parentheses:
x + 6 = 2y + 12
Q: How do I substitute the first equation into the second equation?
A: To substitute the first equation into the second equation, we can replace the variable in the second equation with the expression from the first equation. For example, if the first equation is x = 3y and the second equation is x + 6 = 2y + 12, we can substitute x = 3y into the second equation like this:
3y + 6 = 2y + 12
Q: How do I solve for the variable?
A: To solve for the variable, we can use algebraic techniques such as addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.
Q: What if I get stuck or can't solve the problem?
A: If you get stuck or can't solve the problem, try the following:
- Read the problem carefully and make sure you understand what's being asked.
- Check your work and make sure you haven't made any mistakes.
- Try using a different approach or method to solve the problem.
- Ask for help from a teacher, tutor, or classmate.
Q: Can I use algebra to solve other types of problems?
A: Yes, algebra can be used to solve a wide range of problems, including:
- Linear equations and inequalities
- Quadratic equations and inequalities
- Systems of equations and inequalities
- Graphing and functions
Conclusion
In this article, we answered some frequently asked questions about solving age problems with algebra. We covered topics such as variables, equations, and algebraic techniques. We also provided tips and advice for solving age problems and other types of problems using algebra. By following these tips and practicing your algebra skills, you can become proficient in solving age problems and other types of problems.