Problem 119 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}-\frac{1}{3}\right) \div \frac{5}{8}$$

Step-by-Step Solution

Answer

The result of the operation \(\left(\frac{1}{2}-\frac{1}{3}\right) \div \frac{5}{8}\) is \(\frac{4}{15}\).

Step 1: Subtraction within Parentheses

The first operation to be done is the one inside the parentheses. Subtract \(\frac{1}{2}\)by \(\frac{1}{3}\). You first find a common denominator, which is 6, then convert each fraction to have that denominator and perform the subtraction: \(\frac{3}{6} - \(\frac{2}{6} = \(\frac{1}{6}\). Parameter inside parentheses changes to \(\frac{1}{6}\).

Step 2: Division Operation

The second operation to perform is the division, \(\frac{1}{6} \div \frac{5}{8}\). To divide by a fraction, you multiply by its reciprocal. The reciprocal of \(\frac{5}{8}\)is \(\frac{8}{5}\), so it becomes \(\frac{1}{6} * \(\frac{8}{5}\).

Step 3: Multiplication Operation

To perform multiplication, simply multiply the numerators and the denominators. \(\frac{1}{6} * \(\frac{8}{5} = \(\frac{8}{30}\). The fraction \(\frac{8}{30}\) can be simplified by dividing both the numerator and denominator by 2, getting our solution \(\frac{4}{15}\).

Key Concepts

Order of Operations

When tackling mathematical expressions, especially those involving fractions, it's imperative to follow the universal principle known as the Order of Operations. This protocol ensures accuracy by setting a standard for which operations to carry out first. To remember the order, you can use the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).

Let's apply this to a fraction problem. You're presented with \(\left(\frac{1}{2}-\frac{1}{3}\right) \div \frac{5}{8}\). The first step is simplifying the expression within the parentheses because that's our 'P' in PEMDAS. With the result from the parentheses, we then move to Division, as it comes before multiplication in the expression. These systematic steps are crucial to ensure that you end up with the correct final answer.

Subtracting Fractions

Subtracting fractions seems tricky, but it gets easier with practice. The key here is to find a common denominator for the fractions involved. A common denominator allows you to combine the fractions without changing their values.

For instance, if you need to subtract \(\frac{1}{2}\) from \(\frac{1}{3}\), you'll need to make the denominators the same. The smallest number both 2 and 3 can divide into without a remainder is 6. So, we convert each fraction: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\). Now, you can subtract them easily: just subtract the numerators to get \(\frac{1}{6}\).

Exercise Improvement Advice:

Explicitly illustrate each step in finding the common denominator and converting the fractions before performing the subtraction to ensure full understanding.

Dividing Fractions

When it comes to dividing fractions, you'll want to remember the mantra 'invert and multiply.' Dividing by a fraction is the same as multiplying by its reciprocal. What's a reciprocal, you ask? It's simply the original fraction flipped upside down — the numerator becomes the denominator and the denominator becomes the numerator.

Let's see this in action: to divide \(\frac{1}{6}\) by \(\frac{5}{8}\), we multiply \(\frac{1}{6}\) by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\). This move transforms the division into a multiplication problem, which is often much easier to solve. Understanding this concept is a game-changer for efficiently working through fraction divisions.

Multiplying Fractions

Multiplying fractions is straightforward: multiply the numerators together and the denominators together. Taking our previous example, we multiply \(\frac{1}{6}\) by \(\frac{8}{5}\). This gives you \(\frac{1\times8}{6\times5} = \frac{8}{30}\). However, never forget the last step: simplifying the fraction. Just like you wouldn’t leave your hair uncombed in the morning, don't leave your fraction looking messy!

In our example, both numbers can be divided by 2, giving us a simplified answer of \(\frac{4}{15}\). Always check if your fraction can be simplified. It's not just about elegance; simplified fractions are easier to understand and use in subsequent calculations.

Keep It Neat:

Simplify fractions whenever possible to obtain the 'neatest' version of the answer.