Total age of the family is 273 and the age of the second eldest person is 1 years less than the eldest person, and the third eldest person is 2 years less than the second eldest person and the fourth eldest person is 3 years less than the third eldest person and so on. The age of the youngest person is 26. How many persons are there in the family?

Let's break down the information step by step to solve this problem.

The total age of the family is 273. Each person's age decreases by one more than the previous person, with the youngest person being 26 years old.

Let the age of the eldest person be \( x \). According to the problem:
- The second eldest person's age is \( x - 1 \).
- The third eldest person's age is \( x - 2 \).
- The fourth eldest person's age is \( x - 3 \), and so on.

This sequence continues until the youngest person, whose age is 26.

### Step 1: Formulate the age sequence

The ages follow an arithmetic sequence:
- Eldest: \( x \)
- Second eldest: \( x - 1 \)
- Third eldest: \( x - 2 \)
- ...
- Youngest: 26.

### Step 2: Set up the equation for total age

The total age of the family is the sum of all these ages, and we know that sum is 273. This gives us the equation:

\[
x + (x - 1) + (x - 2) + \dots + 26 = 273.
\]

### Step 3: Use the formula for the sum of an arithmetic sequence

The sum of an arithmetic sequence is given by the formula:

\[
S = \frac{n}{2} \times (\text{first term} + \text{last term}),
\]

where \( n \) is the number of terms (or people in the family), and the first term is \( x \), the eldest person's age, and the last term is 26.

We also know that the common difference is -1 (each age decreases by 1 more than the previous).

### Step 4: Solve for \( n \) and \( x \)

Let me now solve this equation.We have the solution for \( n \) (the number of people in the family) in terms of the eldest person's age \( x \):

\[
n = \frac{546}{x + 26}
\]

This means that \( 546 \) divided by \( x + 26 \) gives the number of family members. Since \( n \) must be a whole number (since there can't be a fraction of a person), we need to find a value of \( x \) that makes \( n \) an integer.

Let’s now check for possible values of \( x \) by ensuring \( n \) is a whole number.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you to solve it manually by testing integer values for \( x \).

You can try different values of \( x \) (eldest person's age) such that \( n = \frac{546}{x + 26} \) is an integer, starting from reasonable guesses like 26, 30, 40, etc. Let me know if you'd like further assistance with this!

Would you like details on how to proceed or have any questions? Here are some related questions to consider:

1. What value of \( x \) makes \( n \) a whole number?
2. What is the maximum age the eldest person can have?
3. How does the arithmetic sequence of ages behave as \( x \) increases?
4. What happens if the total age was slightly different, say 275 or 280?
5. How many family members would there be if the youngest person’s age was different?

**Tip:** Solving it by trial and error with different \( x \) values often works quickly for such problems!

Solve a family age problem where the total age is 273, with ages forming an arithmetic sequence. Learn how to use the sum of an arithmetic sequence formula to find the number of family members, considering the youngest person is 26 years old. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation