Solving the Average Age Problem: Finding K's Age

This problem involves calculating the ages of friends based on changes in their average age as the group composition changes. We are given information about the average age of different combinations of friends: K, L, M; K, L, M, N; and L, M, N, R. We are also told that the age of R is 2 years more than the age of N. Our goal is to find the age of K.

Setting up Equations from the Given Information

Let the ages of the friends be K, L, M, N, and R respectively.

1. Initially, the average age of K, L, and M is 24 years. \[ \frac{K + L + M}{3} = 24 \] Multiplying both sides by 3, we get the total age of K, L, and M: \[ K + L + M = 24 \times 3 \] \[ K + L + M = 72 \quad \text{(Equation 1)} \]
2. When friend N joins the group, the average age of K, L, M, and N becomes 23 years. \[ \frac{K + L + M + N}{4} = 23 \] Multiplying both sides by 4, we get the total age of K, L, M, and N: \[ K + L + M + N = 23 \times 4 \] \[ K + L + M + N = 92 \quad \text{(Equation 2)} \]
3. Friend R, whose age is 2 years more than N's age, replaces K. The new group is L, M, N, and R, and their average age is 22.5 years. \[ \frac{L + M + N + R}{4} = 22.5 \] Multiplying both sides by 4, we get the total age of L, M, N, and R: \[ L + M + N + R = 22.5 \times 4 \] \[ L + M + N + R = 90 \quad \text{(Equation 3)} \]
4. We are given the relationship between the ages of R and N: \[ R = N + 2 \quad \text{(Equation 4)} \]

Solving the Equations Step-by-Step

Now we will use these equations to find the age of K.

1. Use Equation 1 and Equation 2 to find the age of N. Substitute the value of \(K + L + M\) from Equation 1 into Equation 2: \[ (K + L + M) + N = 92 \] \[ 72 + N = 92 \] Subtract 72 from both sides to find N: \[ N = 92 - 72 \] \[ N = 20 \text{ years} \] So, the age of friend N is 20 years.
2. Use Equation 4 and the age of N to find the age of R. \[ R = N + 2 \] \[ R = 20 + 2 \] \[ R = 22 \text{ years} \] So, the age of friend R is 22 years.
3. Use the ages of N and R and Equation 3 to find the sum of the ages of L and M. \[ L + M + N + R = 90 \] Substitute the values of N and R: \[ L + M + 20 + 22 = 90 \] \[ L + M + 42 = 90 \] Subtract 42 from both sides to find \(L + M\): \[ L + M = 90 - 42 \] \[ L + M = 48 \text{ years} \] So, the sum of the ages of L and M is 48 years.
4. Use the sum of the ages of L and M and Equation 1 to find the age of K. \[ K + L + M = 72 \] Substitute the value of \(L + M\): \[ K + 48 = 72 \] Subtract 48 from both sides to find K: \[ K = 72 - 48 \] \[ K = 24 \text{ years} \] Thus, the age of friend K is 24 years.

Based on our calculations, the age of K is 24 years.

Revision Table: Key Information and Results

Additional Information: Understanding Averages and Algebraic Problems

This problem is a classic example of an age-based word problem that requires setting up and solving linear equations. Understanding how averages work is key.

• Average: The average (or mean) of a set of numbers is calculated by summing all the numbers in the set and then dividing by the count of numbers in the set. Mathematically, \( \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \).
• Total Sum: From the definition of average, we can also find the total sum of values if we know the average and the count: \( \text{Sum of values} = \text{Average} \times \text{Number of values} \). This principle was used repeatedly in this problem to convert average ages into total ages.
• Algebraic Substitution: We used substitution to solve the system of equations. This involves expressing one variable or a group of variables (like \(K + L + M\)) in terms of its value and substituting it into another equation to simplify it and solve for other variables.
• Word Problems: Converting the information given in a word problem into mathematical equations is the crucial first step in solving such problems. Each piece of information or condition usually translates into one or more equations.

Practicing different types of average and age problems helps in mastering the skill of translating verbal descriptions into mathematical models and solving them systematically.