Math Problem Statement

Average of 29 numbers is 2586. The average of the first 14 numbers is 2589 and the average of the last 14 numbers is 2584. What is the 15th number?

Solution

To solve this problem, let's break it down step by step.

We know the following:

  • The average of 29 numbers is 2586.
  • The average of the first 14 numbers is 2589.
  • The average of the last 14 numbers is 2584.

We need to find the 15th number.

Step 1: Total Sum of All 29 Numbers

The average of the 29 numbers is 2586, so the total sum of these 29 numbers is: Sum of all 29 numbers=29×2586=75094\text{Sum of all 29 numbers} = 29 \times 2586 = 75094

Step 2: Sum of the First 14 Numbers

The average of the first 14 numbers is 2589, so their total sum is: Sum of the first 14 numbers=14×2589=36246\text{Sum of the first 14 numbers} = 14 \times 2589 = 36246

Step 3: Sum of the Last 14 Numbers

The average of the last 14 numbers is 2584, so their total sum is: Sum of the last 14 numbers=14×2584=36176\text{Sum of the last 14 numbers} = 14 \times 2584 = 36176

Step 4: Find the 15th Number

The sum of the first 14 numbers and the last 14 numbers gives us the sum of 28 numbers. Subtracting this from the total sum will give us the 15th number: 15th number=75094(36246+36176)=7509472422=2672\text{15th number} = 75094 - (36246 + 36176) = 75094 - 72422 = 2672

The 15th number is 2672. Since this number is not one of the given options, the correct answer is: (e)None of these\boxed{(e) \text{None of these}}


Would you like further details on any part of this solution?

Here are 5 related questions:

  1. How would the solution change if the 15th number was part of the first 14 or last 14 numbers?
  2. Can you find the 16th number if the average of numbers from 1 to 16 was given?
  3. How would you calculate the median if all numbers were available?
  4. How can you check if all 29 numbers are equally spaced?
  5. What happens to the total sum if the 15th number is increased by 10?

Tip: When calculating the total of a set of numbers based on the average, remember that multiplying the average by the count of numbers gives you the total sum directly.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Averages
Basic Arithmetic

Formulas

Sum = Average × Number of Terms
15th Number = Total Sum - (Sum of First 14 Numbers + Sum of Last 14 Numbers)

Theorems

-

Suitable Grade Level

Grades 7-10