Question

Evaluate the expression. $$\frac{20+15+22+19}{4}$$

Short answer

The expression evaluates to 19.

Step-by-step solution

Summation

Add the four numbers together: \(20+15+22+19 = 76\).

Division

Divide the sum by 4 to get the average: \(76 / 4 = 19\).

Key concepts

Arithmetic Mean

Understanding the arithmetic mean, commonly referred to as the average, is crucial for interpreting data effectively. When we evaluate the expression \(\frac{20+15+22+19}{4}\), we're in fact calculating the arithmetic mean of four numbers. The arithmetic mean is determined by summing all the numbers in a set and then dividing this total by the count of numbers. In our example, the mean gives us the central value of the dataset, which provides insights into the overall group rather than focusing on individual numbers. The process of finding the arithmetic mean is an essential skill in statistics and everyday life, helping to simplify complex data into a single representative value.

Summation

Summation is the process of adding a sequence of numbers; it is the total accumulated result of all numbers in a list. Considering our given expression, summation is the initial step—adding the given numbers: 20, 15, 22, and 19. By calculating the sum \(20+15+22+19\), you can find the total which is essential before we can proceed to subsequent operations like finding averages. Summation can be done with a small number of terms by simply adding them together, but with larger sequences, techniques like using summation notation, symbolized by the Greek letter sigma (\(\Sigma\)), can provide a shorthand method for the process.

Division

Division is one of the four basic operations in arithmetic, alongside addition, subtraction, and multiplication. It deals with dividing a number, called the dividend, by another, called the divisor, to find a result known as the quotient. In the context of our exercise, after we have summed the numbers (the summation step), we divide the total sum by 4, since we want to find the mean of four numbers. This acts as the divisor. The quotient obtained from this division is the arithmetic mean. Recognizing when and how to apply division is key in problems that involve distributing a quantity evenly or finding an average.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In elementary algebra, they provide a way to generalize arithmetic operations and rules. The expression \(\frac{20+15+22+19}{4}\) is an example of an algebraic expression for finding the arithmetic mean. While this particular expression involves only constants and arithmetic operations, more complex expressions could include variables and a variety of operations. The ability to manipulate and evaluate these expressions is fundamental in algebra, enabling the solution of a wide array of mathematical problems.