Solving the Age Word Problem: Finding Jeremy's Present Age

This problem involves finding the present ages of two individuals, Jeremy and his father, based on given conditions about their ages now and in the future. We can solve this by setting up algebraic equations.

Setting Up the Equations

Let's use variables to represent their present ages:

• Let Jeremy's present age be $J$ years.
• Let his father's present age be $F$ years.

From the first statement, "Jeremy is 26 years younger than his father," we can write the relationship between their present ages:

$$J = F - 26$$

This can be rearranged to express the father's age in terms of Jeremy's age:

$$F = J + 26 \quad (*)$$

Now consider the condition given for the future, "Eight years hence". This means 8 years from now.

• Jeremy's age in 8 years will be $J + 8$ years.
• His father's age in 8 years will be $F + 8$ years.

The second statement says, "his father’s age will be two years less than twice his age" in 8 years. Let's translate this:

• Twice Jeremy's age in 8 years is $2 \times (J + 8)$.
• Two years less than twice Jeremy's age in 8 years is $2(J + 8) - 2$.
• The father's age in 8 years ($F + 8$) will be equal to this value.

So, the second equation is:

$$F + 8 = 2(J + 8) - 2 \quad (**)$$

Solving the System of Equations

We now have a system of two linear equations:

1. $F = J + 26$
2. $F + 8 = 2(J + 8) - 2$

We can substitute the expression for $F$ from equation $(*)$ into equation $(**)$.

Substitute $F = J + 26$ into $F + 8 = 2(J + 8) - 2$:

$$(J + 26) + 8 = 2(J + 8) - 2$$

Now, let's simplify and solve for $J$:

Combine terms on the left side:

$$J + 34 = 2(J + 8) - 2$$

Distribute the 2 on the right side:

$$J + 34 = 2J + 16 - 2$$

Combine constant terms on the right side:

$$J + 34 = 2J + 14$$

Subtract $J$ from both sides:

$$34 = 2J - J + 14$$

$$34 = J + 14$$

Subtract 14 from both sides:

$$34 - 14 = J$$

$$20 = J$$

So, Jeremy's present age ($J$) is 20 years.

Verifying the Answer

Let's check if our answer satisfies the conditions given in the problem.

• Jeremy's present age: 20 years.
• Father's present age: $F = J + 26 = 20 + 26 = 46$ years. (Jeremy is 26 years younger than his father, $46 - 20 = 26$. This checks out).

Now, consider their ages 8 years hence:

• Jeremy's age in 8 years: $J + 8 = 20 + 8 = 28$ years.
• Father's age in 8 years: $F + 8 = 46 + 8 = 54$ years.

The condition for 8 years hence is: "his father’s age will be two years less than twice his age".

• Twice Jeremy's age in 8 years: $2 \times 28 = 56$.
• Two years less than twice Jeremy's age: $56 - 2 = 54$.

The father's age in 8 years is 54, which matches the calculated value. Both conditions are satisfied, confirming that Jeremy's present age is 20 years.

Therefore, Jeremy's present age is 20 years.

Revision Table: Key Information

Additional Information on Age Problems

Age problems are common types of word problems in algebra. They typically involve finding the current ages of individuals based on given information about their ages at different points in time (past, present, or future) and relationships between their ages (difference, ratio, sum, or a combination). The key to solving age problems is to:

1. Carefully read the problem and identify the unknowns (usually present ages).
2. Assign variables to the unknowns.
3. Translate each statement in the problem into a mathematical equation involving the variables. Pay close attention to time references (e.g., "ago," "hence," "in ... years").
4. Form a system of equations based on the statements.
5. Solve the system of equations using methods like substitution or elimination.
6. Check the solution by plugging the values back into the original problem statements to ensure all conditions are met.

These problems help build skills in translating verbal information into algebraic models and solving linear equations.