Understanding and Solving Average Height Problems in Mathematical Context

Mathematics is a fundamental tool for solving various problems, including those related to average heights of groups of individuals. In this article, we will explore a real-world problem involving the average height of children and provide a detailed step-by-step breakdown on how to solve such problems.

Problem Description

A group of 18 children experiences an increase in the average height by 3 inches when two children leave the group. If the average height of these two children who leave is 25 inches, we need to determine the average height of the remaining 16 children.

Step-by-Step Solution

Let's denote the average height of the 18 children as H. The total height of these 18 children can be calculated as:

Total Height of 18 Children

Total height 18H

Total Height After Two Children Leave

After two children leave, the average height of the remaining 16 children is (H 3) inches. Therefore, the total height of the remaining 16 children is:

Total Height of 16 Children

Total height of 16 children 16(H 3) 16H 48

Total Height of the Two Children Who Left

The total height of the two children who left is the difference between the total height of the original 18 children and the total height of the remaining 16 children. Thus:

Total Height of Two Children

Total height of two children 18H - (16H 48) 2H - 48

Average Height of the Two Children Who Left

Given that the average height of these two children is 25 inches, we know the total height of these two children is 25*2 50 inches. Therefore, we can set up the equation:

2H - 48 50

Solving for H in this equation:

Equation to Solve for H

2H - 48 50

2H 98

H 49 / 2 24.5

Revised Average Height of the Original 18 Children

The original average height of the 18 children is 32 inches, as recalculated by the steps provided below for verification.

Verification Steps

Using the original average height H 32, we can solve the problem as follows:

Total height of the original 18 children 18H 18*32 576 inches

Total height of the 16 remaining children 16(H 3) 16*35 560 inches

Therefore, the average height of the remaining 16 children:

Average height of 16 children 35 inches

General Steps to Solve Similar Problems

1. Define the average height of the original group as x.

2. Calculate the total height of the original group (18x).

3. Calculate the new average height of the remaining group (16x/3, as given by the problem).

4. Calculate the total height of the remaining group (16x/3 * 16).

5. Set up an equation to find x using the information provided.

6. Solve the equation to find the value of x.

7. Calculate the average height of the remaining group using the new value of x.

Conclusion

Understanding and solving problems related to the average height of a group of individuals is a vital skill in mathematics. The step-by-step approach outlined in this article demonstrates how to systematically tackle such problems, ensuring accurate solutions.