Problem 24 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{7}{9}}{\frac{5}{9}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{7}{5} \).

Step 1: Understand the Problem

The given problem is a complex fraction, which is a fraction where the numerator or the denominator (or both) is also a fraction. Here, we have \( \frac{\frac{7}{9}}{\frac{5}{9}} \). Our task is to simplify this expression.

Step 2: Recognize the Division of Fractions

A complex fraction like \( \frac{\frac{7}{9}}{\frac{5}{9}} \) is essentially a division problem where the numerator (\( \frac{7}{9} \)) is divided by the denominator (\( \frac{5}{9} \)). The rule for dividing fractions is to multiply by the reciprocal of the divisor.

Step 3: Multiply by the Reciprocal

To simplify \( \frac{\frac{7}{9}}{\frac{5}{9}} \), we multiply the numerator by the reciprocal of the denominator. The reciprocal of \( \frac{5}{9} \) is \( \frac{9}{5} \). So, \( \frac{7}{9} \div \frac{5}{9} = \frac{7}{9} \times \frac{9}{5} \).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \( \frac{7 \times 9}{9 \times 5} = \frac{63}{45} \).

Step 5: Simplify the Resulting Fraction

Now, simplify \( \frac{63}{45} \) by finding the greatest common divisor (GCD) of 63 and 45, which is 9. Divide both the numerator and the denominator by 9: \( \frac{63 \div 9}{45 \div 9} = \frac{7}{5} \).

Step 6: Final Answer

The simplified form of the complex fraction \( \frac{\frac{7}{9}}{\frac{5}{9}} \) is \( \frac{7}{5} \).

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we mean reducing the fraction to its smallest possible form while keeping its value unchanged. This is typically done by finding the greatest common divisor (GCD) of the numerator and the denominator. We then divide both parts by this common divisor. Let’s consider the fraction \( \frac{63}{45} \).

• Identify the greatest common divisor of 63 and 45. In this case, it’s 9.
• Divide both the numerator and the denominator by 9. This gives us \( \frac{63 \div 9}{45 \div 9} = \frac{7}{5} \).

After simplifying, the fraction is \( \frac{7}{5} \), which cannot be reduced further. Simplifying fractions makes calculations easier and results clearer.
Always remember, simplifying does not change the proportion the fraction represents but makes it easier to work with.

Dividing Fractions

Dividing fractions might seem tricky at first, but it becomes straightforward once you use the correct method: multiply by the reciprocal. To divide fractions, follow these steps:

• Take the reciprocal of the divisor (the fraction you are dividing by).
• Multiply the dividend (the fraction you start with) by this reciprocal.

Let's use our example \( \frac{\frac{7}{9}}{\frac{5}{9}} \). Here:

• The dividend is \( \frac{7}{9} \).
• The divisor is \( \frac{5}{9} \). Its reciprocal is \( \frac{9}{5} \).

Multiply the two fractions: \( \frac{7}{9} \times \frac{9}{5} = \frac{63}{45} \). Now, you have a simple multiplication problem instead of a division. Easy, right? Dividing fractions this way makes the operation not only easier but also error-free with practice.

Reciprocal of a Fraction

The concept of a reciprocal is fundamental in simplifying complex fractions. The reciprocal of a fraction essentially flips the fraction upside down. This means that the numerator becomes the denominator and the denominator becomes the numerator. If you have a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

• For the fraction \( \frac{5}{9} \), the reciprocal is \( \frac{9}{5} \).

Why is this important? Because multiplying a fraction by its reciprocal results in 1. This property is useful when dividing fractions. For example, when you encounter a complex fraction such as \( \frac{\frac{7}{9}}{\frac{5}{9}} \), you use the reciprocal of \( \frac{5}{9} \), which is \( \frac{9}{5} \), and multiply it by \( \frac{7}{9} \) to simplify the expression. This understanding simplifies fraction division significantly and helps solve complex fractions easily.