Problem 14 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{3}{8}}{\frac{4}{15}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{45}{32} \).

Step 1: Understanding the Problem

We have a complex fraction \( \frac{\frac{3}{8}}{\frac{4}{15}} \). This fraction involves a fraction in both its numerator and its denominator. Our goal is to simplify this expression to a simpler fraction.

Step 2: Rewrite as Division

A complex fraction \( \frac{\frac{3}{8}}{\frac{4}{15}} \) can be rewritten as a division of two fractions: \( \frac{3}{8} \div \frac{4}{15} \). This is because dividing by a fraction is the same as multiplying by the reciprocal of that fraction.

Step 3: Find the Reciprocal

The reciprocal of \( \frac{4}{15} \) is \( \frac{15}{4} \). So, instead of dividing by \( \frac{4}{15} \), we will multiply by \( \frac{15}{4} \). The expression becomes \( \frac{3}{8} \times \frac{15}{4} \).

Step 4: Multiply the Fractions

To multiply the fractions \( \frac{3}{8} \) and \( \frac{15}{4} \), multiply the numerators and the denominators separately. The product of the numerators is \( 3 \times 15 = 45 \), and the product of the denominators is \( 8 \times 4 = 32 \). Thus, the expression is \( \frac{45}{32} \).

Step 5: Simplification

Check if \( \frac{45}{32} \) can be simplified. Since 45 and 32 have no common factors other than 1, \( \frac{45}{32} \) is already in its simplest form. Therefore, the simplified form of the complex fraction is \( \frac{45}{32} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible by reducing them to their lowest terms. When you have a fraction, your aim is to find the greatest common divisor (GCD) of the numerator and the denominator.
Doing so allows you to divide both by this common number to get the simplest form.

For example, if you have a fraction \( \frac{6}{9} \), you can simplify it. Both 6 and 9 are divisible by 3, the GCD. So, you divide them both by 3, which gives \( \frac{2}{3} \) as the simpler fraction.

• Always look for the largest number that divides both parts of the fraction.
• Simplification ensures fractions are easier to understand and work with.
• Nothing changes about the value; it's just a more straightforward version of the original.

If factors of a fraction aren’t easily noticeable, consider using prime factorization. It helps in pinpointing any common factors!

Reciprocal

A reciprocal flips any given fraction upside down, essentially exchanging its numerator and denominator.
When fractions are involved in division, understanding reciprocals is particularly useful.

Take a fraction like \( \frac{7}{3} \). Its reciprocal is \( \frac{3}{7} \).

• To find a reciprocal, simply switch the numerator with the denominator.
• Reciprocals are crucial for changing division problems into multiplication ones.
• They help simplify expressions and solve equations.

This operation doesn’t alter the value of numbers but creates an expression easier to work with, especially in operations like dividing fractions.

Division of Fractions

Dividing fractions might seem tricky, but it's straightforward when approached correctly. Instead of dividing directly, you multiply by the reciprocal.
This approach simplifies complex operations dramatically.

Suppose you have the division \( \frac{3}{8} \div \frac{4}{15} \). By multiplying \( \frac{3}{8} \) by the reciprocal of \( \frac{4}{15} \) (which is \( \frac{15}{4} \)), the division transforms into a more manageable multiplication.

• Always convert division into multiplication by using reciprocals.
• This method prevents common pitfalls and makes calculations easier.
• Remember, dividing fractions doesn’t follow direct arithmetic logic.

Taking this approach ensures you correctly simplify the entire expression, replacing division challenges with multiplication rules.

Multiplication of Fractions

Multiplying fractions is a straightforward process where you work with both numerators and denominators separately.
You multiply the numerators together and the denominators together to get the result.

For example, if you are multiplying \( \frac{3}{8} \times \frac{15}{4} \), you will calculate: - Numerators: \( 3 \times 15 = 45 \)- Denominators: \( 8 \times 4 = 32 \)This results in the fraction \( \frac{45}{32} \).

• Multiplication offers a way to combine fractions quickly.
• Unlike addition or subtraction of fractions, you don’t need a common denominator to multiply.
• Always consider simplifying the final result if possible.

This technique helps streamline calculations, particularly when used with reciprocals to transform division into a multiplication strategy.