The sum of Joyce and Vangie's ages is 36. Twelve years ago, Vangie was twice as old as Joyce. Find their present ages. solve for x 

Let's solve for the present ages of Joyce and Vangie using algebra.

### Step 1: Set up the equations
Let:
- \( x \) be Joyce's current age.
- \( y \) be Vangie's current age.

We are given the following information:
1. The sum of Joyce and Vangie's ages is 36:
   \[
   x + y = 36 \quad \text{(Equation 1)}
   \]
2. Twelve years ago, Vangie was twice as old as Joyce:
   \[
   y - 12 = 2(x - 12) \quad \text{(Equation 2)}
   \]

### Step 2: Solve Equation 2
Expand and simplify Equation 2:
\[
y - 12 = 2x - 24
\]
Add 12 to both sides to isolate \( y \):
\[
y = 2x - 12 \quad \text{(Equation 3)}
\]

### Step 3: Substitute Equation 3 into Equation 1
Substitute \( y = 2x - 12 \) from Equation 3 into Equation 1:
\[
x + (2x - 12) = 36
\]
Simplify:
\[
3x - 12 = 36
\]
Add 12 to both sides:
\[
3x = 48
\]
Divide both sides by 3:
\[
x = 16
\]
So, Joyce's current age is 16.

### Step 4: Find Vangie's age
Substitute \( x = 16 \) back into Equation 1:
\[
16 + y = 36
\]
Subtract 16 from both sides:
\[
y = 20
\]
So, Vangie's current age is 20.

### Final Answer
- Joyce is 16 years old.
- Vangie is 20 years old.

Would you like more details on any part of the solution? Here are some related questions to explore:

1. How would the equation change if the difference in their ages was given instead?
2. What if Joyce was older than Vangie? How would that affect the solution?
3. How can we verify the solution using the original conditions?
4. What other scenarios could change the structure of the equations?
5. How would this problem be solved graphically?

**Tip:** When solving word problems, always define variables clearly and set up equations that represent the given conditions.

The sum of Joyce and Vangie's ages is 36. Twelve years ago, Vangie was twice as old as Joyce. Find their present ages.

Discover how to solve a classic age-related word problem involving Joyce and Vangie using algebraic equations. Learn step-by-step how to set up and solve systems of equations to find their current ages. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation