Problem 12 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{7}{10}-\frac{3}{5}}{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{1}{5} \).

Step 1: Simplify the Numerator

We begin by simplifying the expression in the numerator \( \frac{7}{10} - \frac{3}{5} \). To do this, we need a common denominator. The common denominator between 10 and 5 is 10. \( \frac{3}{5} \) can be rewritten as \( \frac{6}{10} \). Now, the expression becomes \( \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \).

Step 2: Set Up the Division

Now, we have the complex fraction \( \frac{\frac{1}{10}}{\frac{1}{2}} \). This represents the division of the fraction in the numerator by the fraction in the denominator.

Step 3: Simplify by Multiplying by the Reciprocal

To simplify \( \frac{\frac{1}{10}}{\frac{1}{2}} \), multiply the numerator by the reciprocal of the denominator. The reciprocal of \( \frac{1}{2} \) is \( 2 \). Thus, the expression becomes \( \frac{1}{10} \times 2 = \frac{2}{10} \).

Step 4: Final Simplification

Simplify the resulting fraction \( \frac{2}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives \( \frac{1}{5} \).

Key Concepts

Common Denominator

Understanding how to find a common denominator is an essential skill when working with fractions, especially when adding or subtracting them. A common denominator is a shared multiple of the denominators of two or more fractions. It's much like finding a common unit to allow comparisons or operations between fractions.

To find a common denominator, identify the smallest number that is a multiple of both denominators. For instance, if you have the fractions \( \frac{7}{10} \) and \( \frac{3}{5} \), you look for a number that both 10 and 5 can divide into without leaving a remainder. In this example, 10 is such a number, making it the common denominator.

• Rewrite each fraction so that they have this common denominator.
• In our example, \( \frac{3}{5} \) is rewritten as \( \frac{6}{10} \).
• Then, you can subtract as \( \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \).

This step is crucial, especially when simplifying complex fractions, as it allows easy manipulation of the fraction values.

Reciprocal

The concept of a reciprocal is important when dividing fractions. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \). It's like flipping the fraction upside down.

When you want to divide by a fraction, you multiply by its reciprocal. This is because multiplying by a reciprocal is equivalent to performing the division operation, but it's often more straightforward computationally.

• For the fraction \( \frac{1}{2} \), the reciprocal is \( 2 \).
• Instead of dividing by \( \frac{1}{2} \), you multiply by its reciprocal. This changes the operation to a multiplication by \( 2 \).
• For example, \( \frac{1}{10} \) divided by \( \frac{1}{2} \) becomes \( \frac{1}{10} \times 2 \), simplifying your calculation.

Remember that finding the reciprocal is key to transforming a complex division problem into a simple multiplication problem, making it easier to solve.

Fraction Division

Dividing fractions might seem tricky at first, but it's made simple by using the reciprocal. Fraction division is where you take one fraction and divide it by another.

This starts with taking the fraction in the denominator, finding its reciprocal, and then performing multiplication instead. This operation turns what could be a complicated fraction into a more straightforward multiplication problem.

• Take the complex fraction \( \frac{\frac{1}{10}}{\frac{1}{2}} \).
• By finding the reciprocal of \( \frac{1}{2} \), which is \( 2 \), you multiply: \( \frac{1}{10} \times 2 \).
• This yields \( \frac{2}{10} \), a simpler fraction you can easily reduce.

This method unravels what at first glance seems complex and makes solving fraction-related problems straightforward.

Greatest Common Divisor

The greatest common divisor (GCD) is a helpful concept when simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.

Simplifying a fraction involves reducing it to its simplest form by dividing the numerator and the denominator by their GCD.

• In \( \frac{2}{10} \), the GCD of 2 and 10 is 2.
• Divide both the numerator and the denominator by this GCD: \( \frac{2}{10} \divide 2 = \frac{1}{5} \).
• This gives the fraction in its simplest form, \( \frac{1}{5} \).

Using the GCD is a clear and efficient way to make sure your fractions are as simple as possible, leading to clean and precise results.