Question

The youngest of 4 children has siblings who are 3, 5, and 8 years older than she is. If the average (arithmetic mean) age of the 4 siblings is 21, what is the age of the youngest sibling?

Short answer

The youngest sibling is 17 years old.

Step-by-step solution

- Define Variables

Let the age of the youngest sibling be represented by the variable \( y \). The other siblings' ages can then be written in terms of \( y \): \( y + 3 \), \( y + 5 \), and \( y + 8 \).

- Set Up the Average Age Equation

The sum of the ages of the four siblings is: \[ y + (y + 3) + (y + 5) + (y + 8) \]. Since their average age is 21, set up the equation as: \[ \frac{(y + (y + 3) + (y + 5) + (y + 8))}{4} = 21 \]

- Simplify the Equation

First, combine the terms in the numerator: \[ y + y + 3 + y + 5 + y + 8 = 4y + 16 \]. Rewrite the equation: \[ \frac{4y + 16}{4} = 21 \]

- Solve for y

Multiply both sides of the equation by 4 to eliminate the denominator: \[ 4y + 16 = 84 \]. Next, subtract 16 from both sides: \[ 4y = 68 \]. Finally, divide both sides by 4 to solve for \( y \): \[ y = 17 \]

- Verify the Solution

Check the calculated age of the youngest sibling by finding the total ages: \[ 17, 20, 22, 25 \]. Their sum is \[ 84 \]. The average of these ages is \frac{84}{4} = 21\, which confirms the solution.

Key concepts

Average Age Calculation

Understanding how to calculate the average age involves adding up the ages of all the members in a group and then dividing by the number of individuals.
The formula for the arithmetic mean or average is: \( \text{Average} = \frac{\text{Sum of Ages}}{\text{Number of Individuals}} \).
In the given exercise, the average age of the 4 siblings is given as 21.
By setting up the correct sum with the unknowns, we can solve for the youngest sibling's age efficiently.

Defining Variables in Algebra

Defining variables is crucial in setting up and solving algebraic problems.
When we assign a letter, such as \( y \), to represent an unknown value, it helps us create equations that model real-world scenarios.
In the exercise, we let \( y \) represent the age of the youngest sibling.
We can then express each of the other siblings' ages in terms of \( y \).

• The second sibling is \( y + 3 \)
• The third sibling is \( y + 5 \)
• The fourth sibling is \( y + 8 \)

This systematic approach of defining variables simplifies the problem-solving process significantly.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve only variables raised to the first power.
To solve the exercise, we first set up the equation for the sum of the ages: \( y + (y + 3) + (y + 5) + (y + 8) \).
Combine like terms to get \( 4y + 16 \).
We know the average age is 21, so we set up the equation: \( \frac{4y + 16}{4} = 21 \).
Next, eliminate the denominator by multiplying both sides by 4: \( 4y + 16 = 84 \).
Simplify by subtracting 16 from both sides: \( 4y = 68 \).
Finally, divide both sides by 4 to find \( y \): \( y = 17 \).
Thus, the youngest sibling is 17 years old, and this method shows the power and simplicity of solving linear equations.