Problem 71 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{1}{3} \div \frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \( \frac{4}{3} \).

Step 1: Understand Division of Fractions

To divide by a fraction, you need to multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

Step 2: Find the Reciprocal

The fraction given in the denominator is \( \frac{1}{4} \). The reciprocal of \( \frac{1}{4} \) is \( 4 \) since \( \frac{a}{b} \) becomes \( \frac{b}{a} \).

Step 3: Perform the Multiplication

Multiply the reciprocal obtained in Step 2 by the numerator fraction. So, you have \( \frac{1}{3} \times 4 \). Simplify to get \( \frac{4}{3} \).

Step 4: Simplify the Result (If Needed)

Since \( \frac{4}{3} \) is already in its simplest form, there are no further simplifications to be made. It's an improper fraction.

Key Concepts

Reciprocal of a Fraction

When you hear the term "reciprocal," it simply means "flipping" a fraction upside down. This means that the numerator becomes the denominator, and the denominator becomes the numerator. For example, if you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Swapping these parts is necessary whenever you are dividing fractions.Understanding the reciprocal is crucial in fractions because it easily turns division problems into multiplication ones.

• For instance, if you need to divide \( \frac{1}{3} \) by \( \frac{1}{4} \), you don't divide, but instead multiply by the reciprocal of \( \frac{1}{4} \), which is \( 4 \).

Knowing how to find a reciprocal is especially useful in mathematics, as it is a foundational skill that simplifies dealing with fractions.

Multiplication of Fractions

When multiplying fractions, the process is quite straightforward. You multiply the numerators together and the denominators together. This means simplifying tasks because there's no need to find a common denominator.For example, if you are given to multiply \( \frac{1}{3} \times 4 \) as part of your operation, do:

• Multiply the numerators: \( 1 \times 4 = 4 \).
• Multiply the denominators: \( 3 \times 1 = 3 \).

Hence, the result of this multiplication is \( \frac{4}{3} \).Multiplying by the reciprocal makes division of fractions much simpler, turning it into a problem we can solve easily. So, instead of dealing with a complex fraction division, convert it to something more manageable: multiplication.

Simplifying Fractions

Once you've multiplied your fractions, you might need to simplify them. Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1.Consider an example of a fraction \( \frac{4}{6} \):

• Divide both the numerator and the denominator by 2, which is their greatest common factor.
• The simplified fraction becomes \( \frac{2}{3} \).

In some cases, like with the fraction \( \frac{4}{3} \), it is already in its simplest form and cannot be simplified further.In other words, if you can't simplify a result due to the lack of a common factor, you're done. Understanding and being able to simplify a fraction is vital as it ensures your answers are as clear and concise as possible.