Problem 95 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{\frac{10}{3}}{\frac{2}{3}} $$

Step-by-Step Solution

Answer

5

Step 1: Understand the expression

The given expression is a complex fraction. It is written as follows: ewline ewline \(\frac{\frac{10}{3}}{\frac{2}{3}} \). ewline ewline This means the numerator and the denominator are both fractions.

Step 2: Simplify the complex fraction

To simplify the complex fraction, divide the numerator by the denominator. Dividing by a fraction is the same as multiplying by its reciprocal:ewline ewline \(\frac{10}{3} \) divided by \(\frac{2}{3} \) is the same as ewline ewline \(\frac{10}{3} \times \frac{3}{2} \).

Step 3: Multiply the fractions

Multiply the numerators and the denominators: ewline ewline \(\frac{10}{3} \times \frac{3}{2} = \frac{10 \times 3}{ 3 \times 2 } \).

Step 4: Simplify the result

Simplify the fraction by canceling out common factors in the numerator and the denominator: ewline ewline \(\frac{10 \times 3}{ 3 \times 2 } = \frac{10}{2} = 5 \).

Key Concepts

Fraction Division

Fraction division might seem tricky at first, but it's actually straightforward once you understand the basic rules.
When you are dividing by a fraction, it is the same as multiplying by the reciprocal of that fraction.
In the provided exercise, we start with the complex fraction \(\frac{\frac{10}{3}}{\frac{2}{3}} \). To simplify this, we need to perform a fraction division.

• Identify the numerator: \(\frac{10}{3} \)
• Identify the denominator: \(\frac{2}{3} \)
• Remember, dividing by a fraction is the same as multiplying by its reciprocal.

Once you understand these steps, you'll see that fraction division isn't so challenging after all.

Reciprocals

Understanding reciprocals is a key part of mastering fraction division.
The reciprocal of a fraction is simply flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{2}{3} \) is \(\frac{3}{2} \).
Let's apply this to our problem.
Instead of dividing \(\frac{10}{3} \) by \(\frac{2}{3} \), we multiply \(\frac{10}{3} \) by the reciprocal of \(\frac{2}{3} \), which is \(\frac{3}{2} \).

• Original fraction: \(\frac{2}{3} \)
• Reciprocal: \(\frac{3}{2} \)
• Simplified multiplication: \(\frac{10}{3} \times \frac{3}{2} \)

This transformation simplifies the process of division significantly.
Next, we multiply the numerators together and the denominators together.

Simplifying Fractions

Simplifying fractions is an essential skill in many areas of math.
After multiplying the fractions together, you may find that your result can be simplified.
In our problem, after calculating \(\frac{10}{3} \times \frac{3}{2} \), we get \(\frac{30}{6} \).
Simplifying this fraction involves finding the Greatest Common Divisor (GCD) of the numerator and the denominator.

• Numerator: 30
• Denominator: 6
• GCD of 30 and 6: 6

Divide both the numerator and the denominator by their GCD:
\(\frac{30}{6} = 5 \).
Now, our fraction is simplified, and the final answer is 5.
This process of simplifying helps in making calculations easier and results cleaner.