The sum of the present ages of two cousins is 46 years. Eight years ago, the elder one was twice as old as the younger one. What is the present age of the elder cousin?
28 years
Finding Cousin Ages: Step-by-Step Solution
This problem asks us to find the present age of the elder of two cousins, given the sum of their present ages and a relationship between their ages eight years ago.
Let's define variables for their present ages:
- Let E be the present age of the elder cousin.
- Let Y be the present age of the younger cousin.
Setting up Equations based on the Problem Statement
According to the question, we have two main pieces of information which can be translated into equations:
- The sum of their present ages is 46 years.
\( E + Y = 46 \)
This is our first equation.
- Eight years ago, the elder one was twice as old as the younger one.
First, let's figure out their ages eight years ago:
- Elder cousin's age eight years ago: \( E - 8 \)
- Younger cousin's age eight years ago: \( Y - 8 \)
The relationship given is: The elder's age eight years ago was twice the younger's age eight years ago.
\( E - 8 = 2 \times (Y - 8) \)
This is our second equation. Let's simplify it:
\( E - 8 = 2Y - 16 \)
Solving the System of Linear Equations
We now have a system of two linear equations:
Equation 1: \( E + Y = 46 \)
Equation 2: \( E - 8 = 2Y - 16 \)
We can solve this system using substitution. From Equation 1, we can express Y in terms of E:
\( Y = 46 - E \)
Now substitute this expression for Y into Equation 2:
\( E - 8 = 2(46 - E) - 16 \)
Simplify and solve for E:
\( E - 8 = 92 - 2E - 16 \)
\( E - 8 = 76 - 2E \)
Add 2E to both sides:
\( E + 2E - 8 = 76 \)
\( 3E - 8 = 76 \)
Add 8 to both sides:
\( 3E = 76 + 8 \)
\( 3E = 84 \)
Divide by 3:
\( E = \frac{84}{3} \)
\( E = 28 \)
So, the present age of the elder cousin is 28 years.
Verifying the Solution
Let's check if this age fits the original conditions. If E = 28, then from \( E + Y = 46 \), we get \( 28 + Y = 46 \), so \( Y = 46 - 28 = 18 \). The younger cousin is 18 years old.
Eight years ago, the elder cousin was \( 28 - 8 = 20 \) years old, and the younger cousin was \( 18 - 8 = 10 \) years old.
Is the elder one twice as old as the younger one eight years ago? \( 20 = 2 \times 10 \)? Yes, this is true.
The present age of the elder cousin is indeed 28 years.
| Period | Elder Cousin's Age | Younger Cousin's Age | Relationship |
|---|---|---|---|
| Present | \( E = 28 \) | \( Y = 18 \) | \( E + Y = 28 + 18 = 46 \) (Matches condition) |
| Eight Years Ago | \( E - 8 = 28 - 8 = 20 \) | \( Y - 8 = 18 - 8 = 10 \) | \( E - 8 = 2(Y - 8) \implies 20 = 2(10) \implies 20 = 20 \) (Matches condition) |
Conclusion on the Elder Cousin's Present Age
Based on our calculations, the present age of the elder cousin is 28 years.
Revision Table: Key Concepts in Age Problems
| Concept | Explanation | Example |
|---|---|---|
| Representing Ages | Use variables for unknown ages. Add or subtract years for past or future ages. | If present age is \( A \), age 5 years ago is \( A - 5 \), age in 5 years is \( A + 5 \). |
| Translating Word Problems | Convert sentences into mathematical equations. Look for keywords like "sum", "difference", "ratio", "times", "ago", "in N years". | "A is twice as old as B" becomes \( A = 2B \). "Sum of ages is 30" becomes \( A + B = 30 \). |
| Solving Systems of Equations | Often age problems lead to two or more linear equations. Use substitution or elimination to find variable values. | If \( E+Y=46 \) and \( E-8=2(Y-8) \), solve for E and Y. |
Additional Information: Handling Age-Based Questions
Age problems are a common type in algebra word problems. They typically involve establishing relationships between the ages of individuals at different points in time (past, present, or future). The key steps are consistently:
- Assign variables to the present ages.
- Write expressions for ages at different times mentioned in the problem.
- Formulate equations based on the relationships given between the ages at those times.
- Solve the resulting system of equations.
- Check your answers against the original problem statement to ensure they make sense.
These problems reinforce skills in algebraic translation and solving linear equations. Pay close attention to whether the age relationship applies to the present, past, or future.
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