Problem 26 - Basic College Mathematics

Question

Jason's brothers are \(174 \mathrm{~cm}, 180 \mathrm{~cm},\) \(179 \mathrm{~cm},\) and \(172 \mathrm{~cm}\) tall. The average male is \(176.5 \mathrm{~cm}\) tall. How tall is Jason if he and his brothers have an average height of \(176.5 \mathrm{~cm}\) ?

Step-by-Step Solution

Answer

Jason is 177.5 cm tall.

Step 1 - Understand the problem

We are given the heights of Jason's four brothers and need to find Jason's height such that the average height of all five brothers is 176.5 cm.

Step 2 - Set up the equation for the average height

The average height formula for five people is given by \[ \text{Average height} = \frac{H_1 + H_2 + H_3 + H_4 + H_j}{5} \]where \(H_1, H_2, H_3, H_4 \) are the heights of Jason's brothers and \(H_j\) is Jason's height.

Step 3 - Substitute the known values

Substitute the given heights of Jason's brothers and the average height into the formula: \[ 176.5 = \frac{174 + 180 + 179 + 172 + H_j}{5} \]

Step 4 - Solve for Jason's height

Multiply both sides of the equation by 5 to isolate the sum of the heights on the right side: \[ 176.5 \times 5 = 174 + 180 + 179 + 172 + H_j \]This simplifies to: \[ 882.5 = 705 + H_j \]Subtract 705 from both sides to solve for \(H_j\): \[ H_j = 882.5 - 705 \]\[ H_j = 177.5 \]

Key Concepts

Basic Algebra

Basic algebra involves using mathematical operations and symbols to solve problems. In this exercise, we use algebra to find Jason's height so that all five brothers have the same average height of 176.5 cm.

First, we set up an equation. We know the average height formula, and we plug in what we know. The unknown value is represented with a variable. In our problem, Jason's height is our unknown, represented by the variable H_j. Once the variable is in place, we can use algebraic operations to solve for it.

Operations like addition, subtraction, multiplication, and division help isolate the variable. These steps allow us to first gather all the known quantities on one side of the equation and perform the necessary operations to find the unknown value. In this case, we multiply both sides by 5 to simplify the equation and then isolate H_j by subtracting the sum of Jason's brothers' heights from both sides. This results in Jason's height being 177.5 cm.

Average Formula

To calculate the average height, we use the average formula. The average (or mean) is calculated by adding up all the values and then dividing by the number of values.

In our problem, the heights of the five brothers are involved. The formula for average height is: \[ \text{Average height} = \frac{H_1 + H_2 + H_3 + H_4 + H_j}{5} \]

We know four of the heights and the average. By substituting these values into our formula, we can solve for the unknown height. The step-by-step process involves basic arithmetic operations that align with what we do in our regular calculations.

This method is broadly used in many situations, such as finding the average score, temperature, or other measurements, making it crucial for students to grasp it well.

Height Measurement

Height measurement is an important concept in various fields like biology, medicine, and sports. In this exercise, we measured the heights in centimeters.

Understanding how to properly measure height and convert units if necessary is essential. Consistent units are required for correct calculations. For instance, all heights in this problem were measured in centimeters. This prevents any confusion that might arise from mixing units like inches and centimeters.

Accurate measurement can affect the outcome significantly, so it's critical to use precise measurements for any average calculations. The formula and the algebraic steps we used illustrate how measurements fit into mathematical operations and their real-world applications.

Learning to handle measurements and apply them correctly ensures accuracy in data analysis, fostering better decision-making in various contexts.