Problem 29 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{5}{8}-\frac{1}{4}}{\frac{1}{8}+\frac{1}{2}}$$

Step-by-Step Solution

Answer

\( \frac{3}{5} \)

Step 1: Simplify the Numerator

First, simplify the numerator of the complex fraction, \( \frac{5}{8} - \frac{1}{4} \). Find a common denominator, which is 8 in this case. Convert \( \frac{1}{4} \) to \( \frac{2}{8} \). Now subtract: \( \frac{5}{8} - \frac{2}{8} = \frac{3}{8} \).

Step 2: Simplify the Denominator

Next, simplify the denominator of the complex fraction, \( \frac{1}{8} + \frac{1}{2} \). Find a common denominator, which is 8. Convert \( \frac{1}{2} \) to \( \frac{4}{8} \). Now add: \( \frac{1}{8} + \frac{4}{8} = \frac{5}{8} \).

Step 3: Divide the Simplified Fractions

Now divide the simplified numerator by the simplified denominator: \( \frac{3}{8} \div \frac{5}{8} \). This is equivalent to multiplying \( \frac{3}{8} \) by the reciprocal of \( \frac{5}{8} \), which is \( \frac{8}{5} \).

Step 4: Perform the Multiplication

Carry out the multiplication: \( \frac{3}{8} \times \frac{8}{5} \). Multiply the numerators together and the denominators together: \( \frac{3 \times 8}{8 \times 5} = \frac{24}{40} \).

Step 5: Simplify the Resulting Fraction

Finally, simplify \( \frac{24}{40} \). The greatest common divisor of 24 and 40 is 8. Divide both the numerator and the denominator by 8: \( \frac{24 \div 8}{40 \div 8} = \frac{3}{5} \).

Key Concepts

Understanding the Common Denominator

When dealing with fractions, finding a common denominator is crucial for addition or subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. It allows you to rewrite fractions so they have the same denominator, making addition or subtraction possible.
For example, to simplify the fraction \( \frac{1}{4} \) and \( \frac{5}{8} \) together, we need to find a common denominator. Here, that number is 8.

• Convert \( \frac{1}{4} \) to an equivalent fraction with a denominator of 8 by multiplying the numerator and the denominator by 2. This gives us \( \frac{2}{8} \).
• Once both fractions share common denominators, you can easily perform addition or subtraction: \( \frac{5}{8} - \frac{2}{8} = \frac{3}{8} \).

Always remember: finding and converting to a common denominator is a key step in fraction operations.

What is a Reciprocal?

Reciprocals might sound fancy, but they are actually quite simple. The reciprocal of a fraction is just flipping its numerator and denominator. It's like turning the fraction upside down.
For instance, the reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \).

• Reciprocals are very useful when dividing fractions. Instead of dividing, you multiply by the reciprocal.
• For example, dividing \( \frac{3}{8} \) by \( \frac{5}{8} \) is the same as multiplying \( \frac{3}{8} \) by \( \frac{8}{5} \).

Using reciprocals makes complex calculations simpler and more intuitive once you get the hang of it.

Mastering Fraction Simplification

Simplifying fractions reduces them to their simplest form. This means the fraction should represent the same quantity but with the smallest possible numerator and denominator.
After finding the simplified numerator and denominator of a complex fraction, you often get a result that can be simplified even more.

For instance:

• After multiplying \( \frac{3}{8} \times \frac{8}{5} \), you get \( \frac{24}{40} \).
• To simplify, identify a number that divides both the numerator and denominator—this is often the greatest common divisor (GCD).
• Here the GCD of 24 and 40 is 8. Dividing both by 8 simplifies the fraction to \( \frac{3}{5} \).

Always check if there's a larger divisor than 1 that can make any fraction simpler.

Greatest Common Divisor in Simplification

The greatest common divisor (GCD) is the largest number that evenly divides two numbers. It is key when simplifying fractions, as it allows you to reduce fractions to their simplest form efficiently.
To find the GCD, list the divisors of both numbers and pick the largest common one.

• For instance, with 24 and 40: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, and for 40 they are 1, 2, 4, 5, 8, 10, 20, 40.
• The largest common divisor here is 8.
• Therefore, divide both the numerator (24) and the denominator (40) by 8 to get \( \frac{3}{5} \).

Using the GCD for simplification ensures you arrive at the simplest expression quickly and effectively.