Problem 54 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{1}{4} \div \frac{7}{2}$$

Step-by-Step Solution

Answer

The quotient of \( \frac{1}{4} \div \frac{7}{2} \) is \( \frac{1}{14} \).

Step 1: Understand the Division of Fractions

When dividing fractions, instead of directly dividing them, we multiply by the reciprocal of the divisor fraction. Here, we want to divide \( \frac{1}{4} \) by \( \frac{7}{2} \). So first, we need to find the reciprocal of \( \frac{7}{2} \), which is \( \frac{2}{7} \).

Step 2: Change Division to Multiplication

Convert the division problem into a multiplication problem by multiplying \( \frac{1}{4} \) by the reciprocal of \( \frac{7}{2} \). Thus, the expression \( \frac{1}{4} \div \frac{7}{2} \) becomes \( \frac{1}{4} \times \frac{2}{7} \).

Step 3: Multiply the Fractions

Proceed by multiplying the numerators together and the denominators together for the fractions \( \frac{1}{4} \) and \( \frac{2}{7} \). This results in \( \frac{1 \times 2}{4 \times 7} = \frac{2}{28} \).

Step 4: Simplify the Fraction

To simplify \( \frac{2}{28} \), find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 2 and 28 is 2. Divide both the numerator and the denominator by 2. Thus, \( \frac{2}{28} \) simplifies to \( \frac{1}{14} \).

Key Concepts

Reciprocal

The reciprocal of a number is essentially what you multiply it by to get the value of 1. For a fraction, it involves swapping the positions of the numerator and the denominator. This means for the fraction \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). The reciprocal is crucial in division since we switch division into multiplication using the reciprocal of the divisor.

To put this in perspective using the example \( \frac{7}{2} \) from the exercise:

• Original fraction: \( \frac{7}{2} \)
• Reciprocal: \( \frac{2}{7} \)

When dividing two fractions, like \( \frac{1}{4} \div \frac{7}{2} \), the reciprocal aids in transforming the division into multiplication, simplifying the computation process significantly.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This requires finding a common divisor of both the numerator and the denominator. Once identified, both numbers are divided by this greatest common factor, resulting in a calmer, more elegant fraction.
Here's a simple breakdown with the exercise example:

• Initial fraction: \( \frac{2}{28} \)
• Greatest common divisor (GCD) of \( 2 \) and \( 28 \): \( 2 \)
• Simplifying by dividing both terms by \( 2 \), gives: \( \frac{1}{14} \)

The simplified fraction generally conveys the same proportion with less numerical complexity, and it is more manageable when performing further calculations.

Multiplying Fractions

Multiplying fractions involves a straightforward process where the numerators and denominators are multiplied independently. The product from these calculations provides the new numerator and denominator for the resulting fraction.

Using the example \( \frac{1}{4} \times \frac{2}{7} \):

• Multiply the numerators: \( 1 \times 2 = 2 \)
• Multiply the denominators: \( 4 \times 7 = 28 \)
• Resulting fraction: \( \frac{2}{28} \)

The resulting fraction can then be simplified to find its simplest form. Remember, the key to multiplying fractions efficiently also involves ensuring that simplification is conducted at the end, or even in some cases, midway if common factors are evident prior to multiplication.