Problem 35 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{1}{2}(2.3+2.5)$$

Step-by-Step Solution

Answer

The simplified expression is 2.4.

Step 1: Evaluate the Expression Inside the Parentheses

Add the numbers inside the parentheses: \(2.3 + 2.5\). Calculate \(2.3 + 2.5 = 4.8\). Now the expression reads \(\frac{1}{2} \times 4.8\).

Step 2: Multiply by the Fraction

Multiply \(\frac{1}{2}\) by \(4.8\). This is equivalent to dividing \(4.8\) by 2. \(4.8 \div 2 = 2.4\).

Key Concepts

Decimals

Decimals are a way of representing numbers that are not whole. They use a decimal point to separate the whole number part from the fractional part. This format is incredibly useful for calculations because it allows for precise representation of values like parts of a whole.
When working with decimals, always start by identifying the places to the right of the decimal. For example, in the number 4.8, "4" is the whole number part and "8" is in the tenths place. The number represents 4 whole units and 8 tenths of another unit. Understanding the place value helps make arithmetic with decimals simpler.
When adding decimals, line up the decimal points. This alignment ensures that each digit is in the correct place value, whether tenths, hundredths, or thousandths. It simplifies any carry-over you might need when the sum exceeds 10 in a given place.

Fractions

Fractions are another method of expressing numbers that are less than whole. They consist of a numerator and a denominator, separated by a slash. The numerator specifies how many parts you have, while the denominator tells how many equal parts make up a whole.
In the expression given, you see the fraction \(\frac{1}{2}\). This fraction represents one part of something divided into two equal parts. When multiplying fractions by numbers such as decimals, think of it as multiplying by the numerator and dividing by the denominator.
For example, multiplying \(\frac{1}{2} (4.8)\) effectively becomes dividing 4.8 by 2. Break down fractions into simple steps makes calculations more manageable, ensuring you stay on track to reach simplified results.

Order of Operations

The order of operations is a rule that tells us how to simplify and calculate expressions with multiple operations like addition, multiplication, and parentheses. The typical sequence is remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
In our example, we first solve the operation within the parentheses \((2.3 + 2.5)\). According to the order of operations, anything inside parentheses gets calculated first. So we add these numbers to get 4.8.
Next, we multiply by \(\frac{1}{2}\), which involves dividing 4.8 by 2 to get 2.4. Following the order of operations is crucial to obtaining the correct outcome when simplifying or evaluating any mathematical expression.