Solving the Family Mean Age Problem

This problem involves calculating the age of a new family member based on how the average age of the family changes over time.

Understanding the Initial Situation

We are given the ages of 7 family members:

• 2 years
• 5 years
• 12 years
• 18 years
• 38 years
• 40 years
• 60 years

The number of family members initially is 7.

Calculating the Initial Sum of Ages

The sum of their current ages is:

\begin{equation*} \text{Sum} = 2 + 5 + 12 + 18 + 38 + 40 + 60 \end{equation*}

\begin{equation*} \text{Sum} = 175 \text{ years} \end{equation*}

Calculating the Initial Mean Age

The mean age is the sum of ages divided by the number of members:

\begin{equation*} \text{Initial Mean Age} = \frac{\text{Sum of ages}}{\text{Number of members}} \end{equation*}

\begin{equation*} \text{Initial Mean Age} = \frac{175}{7} = 25 \text{ years} \end{equation*}

Situation After 5 Years

After 5 years, two things happen:

1. Each of the original 7 family members becomes 5 years older.
2. A new member aged 'x' years is added to the family.

Ages of Original Members After 5 Years

Each of the 7 members ages by 5 years. The total increase in age for these 7 members is \(7 \times 5 = 35\) years.

The sum of the ages of these original 7 members after 5 years is:

\begin{equation*} \text{Sum of original ages after 5 years} = \text{Initial Sum} + (\text{Number of original members} \times 5) \end{equation*}

\begin{equation*} \text{Sum of original ages after 5 years} = 175 + (7 \times 5) = 175 + 35 = 210 \text{ years} \end{equation*}

Adding the New Member

A new member aged \(x\) years is added. The total number of family members is now \(7 + 1 = 8\).

The total sum of ages of all 8 family members after 5 years is:

\begin{equation*} \text{Total Sum of Ages Now} = (\text{Sum of original ages after 5 years}) + (\text{Age of new member}) \end{equation*}

\begin{equation*} \text{Total Sum of Ages Now} = 210 + x \end{equation*}

Calculating the New Mean Age

The problem states that the mean age of the family now goes up by 1.5 years.

\begin{equation*} \text{New Mean Age} = \text{Initial Mean Age} + \text{Increase in Mean Age} \end{equation*}

\begin{equation*} \text{New Mean Age} = 25 + 1.5 = 26.5 \text{ years} \end{equation*}

Setting up the Equation

The new mean age is also calculated by dividing the new total sum of ages by the new number of members:

\begin{equation*} \text{New Mean Age} = \frac{\text{Total Sum of Ages Now}}{\text{Total Number of Members Now}} \end{equation*}

Substituting the values we found:

\begin{equation*} 26.5 = \frac{210 + x}{8} \end{equation*}

Solving for x

Now, we solve the equation for \(x\) to find the age of the new member:

Multiply both sides by 8:

\begin{equation*} 26.5 \times 8 = 210 + x \end{equation*}

Calculate the left side:

\begin{equation*} 212 = 210 + x \end{equation*}

Subtract 210 from both sides:

\begin{equation*} x = 212 - 210 \end{equation*}

\begin{equation*} x = 2 \end{equation*}

So, the age of the new member is 2 years.

Summary of Calculations

Revision Table: Key Concepts

Additional Information: Mean Age Problems

Mean age problems are common in statistics and mathematics. They often involve calculating the average age of a group and seeing how that average changes when new members are added, members leave, or time passes.

Key things to remember when solving such problems:

• Understand how the number of members changes.
• Understand how the sum of ages changes (due to time passing for existing members and the age of new/leaving members).
• Use the formula for the mean correctly: Mean = Sum / Number of members.
• Set up an equation based on the given information to solve for the unknown variable.

In this specific problem, it was crucial to account for the fact that the original members also aged by 5 years over the 5-year period, in addition to the new member being added.