Understanding the Average and Finding Unknown Numbers

The concept of average is a fundamental tool in mathematics used for a wide range of calculations, from everyday problems to more complex statistical analyses. The average (or mean) of two numbers is the sum of the numbers divided by 2. This article will explore how to find an unknown number when the average of two numbers is given.

Tackling the Problem

Consider the following problem: The average of two numbers is 15. If one of the numbers is 17, what is the other number?

Step-by-Step Solution

Identify the known values: The average is 15, and one of the numbers is 17. Formulate the equation: Use the formula for the average of two numbers: [text{Average} frac{text{Number 1} text{Number 2}}{2}] Substitute the known values into the formula: Set up the equation: [15 frac{17 text{Number 2}}{2}] Solve for the unknown number: Multiply both sides of the equation by 2 to eliminate the fraction: 15 × 2 17 text{Number 2} 30 17 text{Number 2} Solve for text{Number 2}: text{Number 2} 30 - 17 text{Number 2} 13

Thus, the other number is 13.

Mental Calculation Techniques

Mental calculation can be a useful skill, especially when solving such problems quickly. Here are alternative methods to find the unknown number:

1. Simplifying the Problem: Understanding that the total of the two numbers should be 30 (since the average is 15 and there are two numbers, the total is 15 × 2), and one number is 17, you can easily subtract:

30 - 17 13

2. Direct Subtraction: Alternatively, use the formula:

[text{Second Number} 2 times 15 - 17 13]

3. Observational Method: The other number must be as much less than 15 as 17 is more than 15 (a difference of 3). Thus, 15 - 3 12.

Summary of Solutions

Three methods to find the unknown number when the average is 15 and one of the numbers is 17:

[text{Number 2} 30 - 17 13] [text{Second Number} 2 times 15 - 17 13] [15 - (18 - 15) 12]

These methods demonstrate the flexibility in approaching problems from different angles. Always try to solve problems in your head first to gain quick insights, then verify your answer for accuracy.