If the arithmetic mean of five numbers is 12 and four of the numbers are 10, 14, 8, and 16, find the fifth number.
["Finding the Missing Fifth Number Using the Arithmetic Mean", "Understanding how to calculate the arithmetic mean is essential in math and everyday problem-solving, especially when dealing with averages. One common scenario involves finding an unknown number when the average of a set of values is known. In this article, we’ll explore a practical example: if the arithmetic mean of five numbers is 12, and four of those numbers are 10, 14, 8, and 16, we’ll determine the missing fifth number.", "---", "### What is Arithmetic Mean?", "The arithmetic mean (or average) is calculated by adding all the numbers in a set and dividing by the count of numbers. If the average of five numbers is 12, then the sum of all five numbers divided by 5 equals 12:", "[\n\ ext{Mean} = \frac{\ ext{Sum of all numbers}}{5} = 12\n]", "To find the total sum:", "[\n\ ext{Sum} = 12 \ imes 5 = 60\n]", "So, the five numbers together must add up to 60.", "---", "### Using the Known Numbers", "We are given four of the five numbers: 10, 14, 8, and 16. Let’s add these together:", "[\n10 + 14 + 8 + 16 = 48\n]", "---", "### Calculating the Missing Number", "Let the unknown fifth number be ( x ). Since the total sum must be 60:", "[\n48 + x = 60\n]", "Solving for ( x ):", "[\nx = 60 - 48 = 12\n]", "---", "### Verification", "To verify, add all five numbers: 10, 14, 8, 16, and 12.", "[\n10 + 14 + 8 + 16 + 12 = 60\n]", "The average is:", "[\n\frac{60}{5} = 12\n]", "which confirms our solution.", "---", "### Conclusion", "Finding the fifth number in a set with a known arithmetic mean is straightforward once you understand how averages work. By computing the total sum from the mean and subtracting the known values, you efficiently solve for the missing value. This method is widely applicable in statistics, education, finance, and real-life data analysis.", "Remember:\nArithmetic mean = (Sum of numbers) ÷ (Count of numbers)\nSo, to find an unknown without altering the average, just use:\nMissing number = Total sum (from mean and count) – Sum of known values", "---", "Keywords: arithmetic mean, average calculation, missing number problem, five numbers average, math problem solving, data analysis, statistics.\nMeta Description: Learn how to find the missing number in a set with an arithmetic mean of 12. Using known values 10, 14, 8, and 16, discover that the fifth number is 12. Step-by-step solution explained."]
