Problem 63 - Prealgebra

Question

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$\frac{2}{3} \div 4$$

Step-by-Step Solution

Answer

\( \frac{2}{3} \div 4 = \frac{1}{6} \).

Step 1: Understand the Problem

We are asked to divide the fraction \( \frac{2}{3} \) by the whole number 4. Dividing by a number is the same as multiplying by its reciprocal.

Step 2: Convert Whole Number to Fraction

The whole number 4 can be expressed as a fraction by putting it over 1. So, 4 becomes \( \frac{4}{1} \).

Step 3: Take the Reciprocal

The reciprocal of \( \frac{4}{1} \) is \( \frac{1}{4} \).

Step 4: Set Up Multiplication Problem

Now change the division problem \( \frac{2}{3} \div 4 \) to a multiplication problem with the reciprocal: \( \frac{2}{3} \times \frac{1}{4} \).

Step 5: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together: \( \frac{2 \times 1}{3 \times 4} = \frac{2}{12} \).

Step 6: Simplify the Fraction

The fraction \( \frac{2}{12} \) can be simplified. Find the greatest common divisor (GCD) of 2 and 12, which is 2, and divide both the numerator and the denominator by 2: \( \frac{2 \div 2}{12 \div 2} = \frac{1}{6} \).

Key Concepts

Basic Operations with Fractions

Working with fractions involves four basic operations: addition, subtraction, multiplication, and division. Understanding these operations is essential in math, especially when dealing with problems that require different calculations.

• **Addition and Subtraction:** To add or subtract fractions, you need a common denominator. This means the denominators (bottom numbers) should be the same.
• **Multiplication:** When multiplying fractions, you multiply the numerators (top numbers) together and the denominators together. This creates a new fraction.
• **Division:** Dividing fractions involves multiplying by the reciprocal. This is a key trick that simplifies many problems where division is involved.

In the example given, dividing the fraction \( \frac{2}{3} \) by 4 is the same as multiplying \( \frac{2}{3} \) by \( \frac{1}{4} \), the reciprocal of 4.

Multiplying Fractions

Multiplying fractions is simpler than it sounds, and this operation doesn’t require a common denominator. Here’s a step-by-step guide:

• **Multiply Numerators:** First, multiply the numerators together. In our example, the numerators are 2 and 1. So, 2 times 1 equals 2.
• **Multiply Denominators:** Next, do the same for the denominators. You have 3 and 4, so 3 times 4 equals 12.
• **Resulting Fraction:** The resulting fraction from multiplying the numerators and denominators is \( \frac{2}{12} \).

Multiplying doesn’t require modifying the denominators beforehand, which makes it straightforward. However, you may need to simplify the result later.

Simplifying Fractions

Simplifying or reducing fractions helps in expressing them in their simplest form. It often makes calculations easier and results clearer.

• **Find the GCD:** To simplify \( \frac{2}{12} \), find the greatest common divisor (GCD) of the numerator and the denominator. In this case, it is 2.
• **Divide Numerator and Denominator:** Divide both the numerator and the denominator by the GCD. For \( \frac{2}{12} \), divide 2 by 2 and 12 by 2 to get \( \frac{1}{6} \).

The result is a simplified fraction, which is always a tidier and often more useful representation. Simplifying is always the last step after operations like multiplication or addition.