Problem 49 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. The sum of the present ages of Angie and her mother is 64 years. In eight years Angie will be three-fifths as old as her mother at that time. Find the present ages of Angie and her mother.

Step-by-Step Solution

Answer

Angie is 22 years old, and her mother is 42 years old.

Step 1: Define Variables

Let Angie's present age be \( A \) and her mother's present age be \( M \). These will be our two unknowns.

Step 2: Set Up Equations

We have two pieces of information:1. The sum of Angie’s and her mother’s present ages is 64, so \( A + M = 64 \).2. In eight years, Angie will be three-fifths the age of her mother, giving us the equation \( A + 8 = \frac{3}{5}(M + 8) \).

Step 3: Solve the First Equation for One Variable

From the equation \( A + M = 64 \), solve for \( M \) to get \( M = 64 - A \).

Step 4: Substitute into the Second Equation

Replace \( M \) in the second equation with \( 64 - A \), yielding:\[A + 8 = \frac{3}{5}((64 - A) + 8)\] This simplifies to:\[ A + 8 = \frac{3}{5}(72 - A) \]

Step 5: Clear the Fraction

Multiply every term by 5 to get rid of the fraction:\[ 5(A + 8) = 3(72 - A) \]which simplifies to:\[ 5A + 40 = 216 - 3A \]

Step 6: Solve for A

Rearrange the equation:\[ 5A + 3A = 216 - 40 \]\[ 8A = 176 \]Divide by 8:\[ A = 22 \]

Step 7: Solve for M

Substitute \( A = 22 \) back into the equation \( M = 64 - A \):\[ M = 64 - 22 = 42 \]

Step 8: Verify Solution

Check the solution: - Present ages: Angie is 22, and her mother is 42. - In 8 years, Angie will be 30, and her mother will be 50. - Check: 30 is indeed three-fifths of 50. The calculated ages satisfy both conditions.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical sentences that involve variables, numbers, and operations like addition, subtraction, multiplication, and division. They are used to express relationships between different quantities. In age-related word problems, these equations help us represent the relationships between ages and solve for unknowns.

• In the given problem, we formulated two equations based on the information: one using the sum of present ages, and the other using a future age comparison.
• The first equation is straightforward: \( A + M = 64 \), where \( A \) and \( M \) are the present ages of Angie and her mother.
• The second equation involves a fraction to represent a future situation: \( A + 8 = \frac{3}{5}(M + 8) \).

Understanding how to set up these equations is crucial to solving any age-related word problem effectively.

Variable Substitution

Variable substitution is a technique that involves solving one equation for one variable and then substituting that expression into another equation. This simplifies the problem, allowing us to solve for the other variable first.

• In our case, from the first equation \( A + M = 64 \), we expressed \( M \) in terms of \( A \): \( M = 64 - A \).
• This expression was then substituted into the second equation, effectively reducing the number of variables, which allows us to solve for \( A \) more easily.

This technique is a powerful tool in algebra for solving systems of equations, helping simplify the computations needed.

Equation Solving

The art of equation solving involves a series of strategic steps to isolate and find the value of unknown variables. It starts with simplifying the equations as much as possible by eliminating terms or fractions.

• After substitution, our equation becomes \( A + 8 = \frac{3}{5}(72 - A) \).
• To eliminate the fraction, multiply every term by 5, which changes the equation to \( 5(A + 8) = 3(72 - A) \).
• This gives us a clearer view and simplifies the equation to \( 5A + 40 = 216 - 3A \).
• Combine like terms and solve for \( A \) as follows: \( 8A = 176 \), then divide both sides by 8 to find \( A = 22 \).

Solving equations is a methodical process, where each step follows logically from the previous one to ensure accuracy.

Fraction Operations

Working with fractions in equations can seem challenging at first, but understanding the basics makes them manageable. Fractions in algebraic equations often denote a portion or ratio between different quantities.

• In our second equation, the fraction \( \frac{3}{5} \) represents the portion of the mother’s future age that corresponds to Angie’s future age in eight years.
• The goal is to eliminate the fraction by multiplying all terms by the denominator, which simplifies calculations. In this problem, multiplying through by 5 converts \( \frac{3}{5}(72 - A) \) to a simpler whole number operation: \( 3(72 - A) \).

Mastering fraction operations allows for smoother problem solving, especially in more complex algebraic scenarios involving ratios or proportions.

Verification Method

Verification is an essential step to ensure that the solution derived from the equations is correct. After finding potential solutions, substituting them back into the original equations tests their validity.

• With Angie’s age calculated as 22 and her mother’s age as 42, we first check if they add up to 64: \( 22 + 42 = 64 \).
• Then verify the future age condition: In eight years, Angie will be 30 and her mother 50. Check if Angie’s age is indeed \( \frac{3}{5} \) of her mother’s future age: \( \frac{3}{5} \times 50 = 30 \).

This cross-verification ensures the solution not only solves the equations but also fits the real-world scenario described. It acts as a final checkpoint for both computation and understanding.