Problem 79 - Elementary Algebra

Question

\(\frac{1}{8} \div \frac{1}{2}-\frac{3}{4}\)

Step-by-Step Solution

Answer

-\(\frac{1}{2}\)

Step 1: Understand the Problem

The given expression is \(\frac{1}{8} \div \frac{1}{2}-\frac{3}{4}\). It involves division and subtraction of fractions.

Step 2: Perform the Division

To divide fractions, multiply \(\frac{1}{8}\) by the reciprocal of \(\frac{1}{2}\). The reciprocal of \(\frac{1}{2}\) is \(2\), so the expression becomes \(\frac{1}{8} \times 2\).

Step 3: Simplify the Division Result

Multiply the fractions: \(\frac{1 \times 2}{8} = \frac{2}{8}\). Then simplify \(\frac{2}{8}\) to \(\frac{1}{4}\).

Step 4: Perform the Subtraction

Subtract \(\frac{3}{4}\) from \(\frac{1}{4}\). Align the fractions: \(\frac{1}{4} - \frac{3}{4}\).

Step 5: Simplify the Subtraction Result

Perform the subtraction: \(\frac{1}{4} - \frac{3}{4} = -\frac{2}{4}\). Simplify \(-\frac{2}{4}\) to \(-\frac{1}{2}\).

Key Concepts

Fraction Division

When dividing fractions, you will use a specific method that involves the 'reciprocal.' To divide one fraction by another, you actually multiply the first fraction by the reciprocal of the second fraction.

For example, to solve \(\frac{1}{8} \div \frac{1}{2}\), you need to turn the division into multiplication by flipping the second fraction (finding its reciprocal).

So the calculation changes to \(\frac{1}{8} \times 2\). This makes the problem simpler, as multiplying fractions follows an easier rule: multiply the numerators together and the denominators together. Here, it becomes \(\frac{1 \times 2}{8} = \frac{2}{8}\).

Make sure to simplify your result. \(\frac{2}{8}\) can be simplified to \(\frac{1}{4}\). This method works for all fraction division problems!

The Reciprocal

The reciprocal of a fraction is simply the fraction flipped upside down. If you have \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

This concept is critical when dividing fractions. For instance, when dividing \(\frac{1}{2}\) by \(\frac{3}{4}\), you'll need the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\).

In our exercise, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1} = 2\). This step transforms the division into a multiplication problem, making it easier to solve. Always remember:

• Find the reciprocal of the second fraction.
• Change the division sign to multiplication.
• Solve by multiplying the fractions as usual.

Fraction Subtraction

Fraction subtraction requires a common denominator to make the process easier. To subtract fractions, follow these steps:

• Find a common denominator.
• Convert each fraction to have this common denominator.
• Subtract the numerators and keep the denominator the same.

Let's apply this to the problem \(\frac{1}{4} - \frac{3}{4}\). Both fractions already have the same denominator, which is 4.

So, we just need to subtract the numerators: \(\frac{1 - 3}{4} = -\frac{2}{4}\).

Finally, simplify \(-\frac{2}{4}\) to \(-\frac{1}{2}\).

Remember that handling negative results follows the same principles. This makes fraction subtraction straightforward once you ensure both fractions share a common denominator!