The average of three numbers is 16. If two of the numbers are the same and the third number is 18, what is the value of the other two numbers?
We are given three numbers whose average is 16. Let the two identical numbers be represented by the variable \(x\), and the third given number be 18. The average of these three numbers is calculated by summing them up and dividing by the total count, which is 3. The formula for the average is \(\frac{x + x + 18}{3} = 16\). Our goal is to solve for the value of \(x\).
Using the expression derived, we have:
\[\frac{2x + 18}{3} = 16\]
To solve for \(x\), we first eliminate the denominator by multiplying both sides of the equation by 3:
\[2x + 18 = 16 \times 3\]
\[2x + 18 = 48\]
Next, we isolate the term with the variable by subtracting 18 from both sides of the equation:
\[2x = 48 - 18\]
\[2x = 30\]
Finally, divide by 2 to determine the value of \(x\):
\[x = \frac{30}{2} = 15\]
Thus, the two identical numbers are both 15.
In real life, this concept is useful when you need to calculate missing data points based on averages. For example, if you have taken three tests and know that your average score must be 16 to maintain a certain grade, and you know two of your tests resulted in the same score while the third score was 18, you can use this method to find out what those first two scores were. This helps in tracking performance and meeting specific goals in various contexts like budgeting, academic planning, or resource management.