The ages of 7 family members are 2, 5, 12, 18, 38, 40 and 60 years respectively. After 5 years a new member aged x years is added. If the mean age of the family now goes up by 1.5 years, then what is the value of x?

Question

Prepp Team · Accepted Answer

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:

Each of the original 7 family members becomes 5 years older.
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

Description	Calculation	Result
Initial Number of Members		7
Initial Sum of Ages	\(2+5+12+18+38+40+60\)	175 years
Initial Mean Age	\(175 / 7\)	25 years
Time Elapsed		5 years
Increase in Sum for Original Members	\(7 \times 5\)	35 years
Sum of Original Ages After 5 Years	\(175 + 35\)	210 years
Age of New Member		\(x\)
Total Number of Members Now	\(7 + 1\)	8
Total Sum of Ages Now	\(210 + x\)	\(210 + x\) years
Increase in Mean Age		1.5 years
New Mean Age	\(25 + 1.5\)	26.5 years
Equation	\(\frac{210 + x}{8} = 26.5\)
Value of x	\(26.5 \times 8 - 210\)	2

Revision Table: Key Concepts

Concept	Definition/Formula	Application in Problem
Mean (Average)	Sum of all values divided by the number of values. \(\text{Mean} = \frac{\sum x}{n}\)	Used to find initial and final mean ages.
Sum of Ages	Total age when adding up all individual ages.	Calculated initially and after 5 years with the new member.
Algebraic Equation	A statement that two mathematical expressions are equal, often containing a variable.	Used to represent the relationship between the new mean, total sum, and the unknown age \(x\).

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.

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