Problem 100 - Elementary Algebra

Question

For exercises 97-114, evaluate. $$ \frac{4}{5}+\frac{1}{10} \div \frac{1}{5}-\frac{1}{2} $$

Step-by-Step Solution

Answer

\( \frac{4}{5} \)

Step 1: Follow the order of operations (PEMDAS)

Start by following the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Identify the operations involved in the expression: \[ \frac{4}{5} + \frac{1}{10} \text{ (divided by) } \frac{1}{5} - \frac{1}{2} \]Division takes priority over addition and subtraction.

Step 2: Perform the division

Calculate the division first: \[ \frac{1}{10} \text{ divided by } \frac{1}{5} = \frac{1}{10} \times \frac{5}{1} = \frac{1 \times 5}{10 \times 1} = \frac{5}{10} = \frac{1}{2} \]Now substitute this result back into the original expression: \[ \frac{4}{5} + \frac{1}{2} - \frac{1}{2} \]

Step 3: Simplify the expression

Next, simplify the expression step by step. Notice that \( +\frac{1}{2} \) and \( -\frac{1}{2} \) cancel each other out: \[ \frac{4}{5} + \frac{1}{2} - \frac{1}{2} = \frac{4}{5} \]

Key Concepts

Order of Operations (PEMDAS)

When evaluating algebraic expressions, it's essential to follow the order of operations. The acronym PEMDAS helps us remember the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
This ensures we perform calculations in the correct order and get accurate results.
For example, in the given expression \[\frac{4}{5} + \frac{1}{10} \div \frac{1}{5} - \frac{1}{2}\], we start with the division according to PEMDAS.

Fractions in Algebra

Working with fractions in algebra involves understanding how to manipulate these numbers correctly. Fractions represent parts of a whole number and are written as \(\frac{numerator}{denominator}\).
In algebra, we often need to perform operations such as addition, subtraction, multiplication, and division with fractions.
For example, in the expression given, we see multiple fractions that we need to work with correctly.

Simplification of Expressions

Simplifying expressions means making them as concise as possible. This typically involves combining like terms, reducing fractions, and performing arithmetic operations.
In our expression \[\frac{4}{5} + \frac{1}{10} \div \frac{1}{5} - \frac{1}{2}\], we simplified it to\[\frac{4}{5} + \frac{1}{2} - \frac{1}{2}\].
Noticing that \(\frac{1}{2}\) and \(-\frac{1}{2}\) cancel each other out, we arrived at the final simplified result: \[\frac{4}{5}\].

Division of Fractions

To divide fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is simply flipping its numerator and denominator.
For instance, when dividing \(\frac{1}{10}\ \div\ \frac{1}{5}\), we rewrite it as \(\frac{1}{10} \times \frac{5}{1}\) and then simplify: \(\frac{5}{10} = \frac{1}{2}\).
This step is crucial and is performed before any addition or subtraction in the expression.

Addition and Subtraction of Fractions

To add or subtract fractions, they need to have a common denominator. This means the bottom numbers (denominators) must be the same.
In the expression, after performing the division, we had: \[\frac{4}{5} + \frac{1}{2} - \frac{1}{2} \].
The fractions \(\frac{1}{2}\) and \(\frac{1}{2}\) had the same denominator and thus canceled each other out, simplifying the calculation to \(\frac{4}{5}\).
Remember, finding a common denominator is essential for these operations.