Problem Statement:
Elvin is 2 more than three times older than his daughter, Ezra. In 16 years, he will be twice as old as his daughter. What are their present ages?
---
Solution:
#### Step 1: Define Variables
Let:
- \( E \) = Elvin's current age.
- \( Z \) = Ezra's current age.
#### Step 2: Translate the Problem into Equations
1.
First Condition: Elvin is 2 more than three times older than Ezra.
\[
E = 3Z + 2
\]
2.
Second Condition: In 16 years, Elvin will be twice as old as Ezra.
- In 16 years, Elvin's age will be \( E + 16 \).
- In 16 years, Ezra's age will be \( Z + 16 \).
- According to the problem:
\[
E + 16 = 2(Z + 16)
\]
#### Step 3: Solve the System of Equations
We now have two equations:
1. \( E = 3Z + 2 \)
2. \( E + 16 = 2(Z + 16) \)
##### Substitute \( E \) from the first equation into the second equation:
From the first equation:
\[
E = 3Z + 2
\]
Substitute \( E \) in the second equation:
\[
(3Z + 2) + 16 = 2(Z + 16)
\]
##### Simplify the equation:
\[
3Z + 2 + 16 = 2(Z + 16)
\]
\[
3Z + 18 = 2Z + 32
\]
##### Isolate \( Z \):
\[
3Z - 2Z = 32 - 18
\]
\[
Z = 14
\]
##### Find \( E \) using \( E = 3Z + 2 \):
\[
E = 3(14) + 2
\]
\[
E = 42 + 2
\]
\[
E = 44
\]
#### Step 4: Verify the Solution
1.
Check the first condition: Elvin is 2 more than three times Ezra's age.
\[
E = 3Z + 2
\]
\[
44 = 3(14) + 2
\]
\[
44 = 42 + 2
\]
\[
44 = 44 \quad \text{(True)}
\]
2.
Check the second condition: In 16 years, Elvin will be twice as old as Ezra.
- In 16 years, Elvin's age will be:
\[
E + 16 = 44 + 16 = 60
\]
- In 16 years, Ezra's age will be:
\[
Z + 16 = 14 + 16 = 30
\]
- Check if Elvin's age is twice Ezra's age:
\[
60 = 2 \times 30
\]
\[
60 = 60 \quad \text{(True)}
\]
Both conditions are satisfied, so the solution is correct.
---
Final Answer:
\[
\boxed{44 \text{ and } 14}
\]
Parent Tip: Review the logic above to help your child master the concept of age problems worksheet.
