Overview
This lecture covers solving age-related algebra word problems using present, past, and future scenarios. Each problem is tackled by defining variables, setting up equations, and solving step-by-step.
General Approach to Age Problems
- Define variables for the present ages of each person (e.g., S for Sally, J for John).
- Express future or past ages by adding or subtracting years from present age variables.
- Translate word problem relationships into algebraic equations.
- Use substitution or elimination to solve systems with as many equations as variables.
- Check solutions by plugging values back into original relationships.
Example 1: Sally and John
- Let S = Sally's present age, J = John's present age.
- Equations: S = 3J; S + 8 = 2(J + 8).
- Solving yields J = 8 (John's age), S = 24 (Sally's age).
Example 2: Kim and Timothy
- Let K = Kim's present age, T = Timothy's present age.
- Equations: K = 6 + 2T; K - 2 = 3(T - 2).
- Solution: T = 10 (Timothy), K = 26 (Kim). Kim was 24 two years ago.
Example 3: Leah and Rachel
- Let L = Leah's present age, R = Rachel's present age.
- Equations: L = 3R - 2; L + 3 = 7 + 2(R + 3).
- Solution: R = 12 (Rachel now), L = 34 (Leah now). Rachel is 15 in 3 years.
Example 4: Susan, Becca, Greg
- S = Susan's present age; B = Becca's; G = Greg's.
- Equations: B = 2S; G = S + 9; B - 3 = 3(S - 3) - 9.
- Solution: S = 15, B = 30, G = 24 (Greg’s current age).
Example 5: Lauren, Andrew, Sam
- A = Andrew’s age; S = Sam’s; L = Lauren’s.
- Equations: L = 2A - 3; S + 4 = 2 + 2(A + 4); S - 5 = 3(A - 5).
- Solution: A = 16, S = 38, L = 29 (present); Lauren was 24 five years ago.
Example 6: Larry, Megan, Gabby
- L = Larry's age; M = Megan’s; G = Gabby’s.
- Equations: G = 2L + 1; M + 3 = 2(G + 3) - 27; M - 4 = 3(L - 4) - 1.
- Solution: L = 13, G = 27, M = 30 (present); Megan will be 33 in 3 years.
Key Terms & Definitions
- Present Age — Current age of a person (variable).
- Past/Future Age — Age altered by adding or subtracting years.
- System of Equations — Set of equations with multiple variables to solve together.
- Substitution — Replacing a variable with its equivalent from another equation.
Action Items / Next Steps
- Practice solving additional age word problems.
- Review setting up equations from word descriptions.
- Prepare tables to organize data for each problem (person, present, past, future).