Problem 44 - Algebra 1

Question

Simplify. \frac{2}{7} \div \frac{14}{24}

Step-by-Step Solution

Answer

The simplified form of the fraction is \(\frac{24}{7}\).

Step 1: Converting the Division to Multiplication

Fraction division is actually multiplying by a reciprocal. So to simplify this expression, begin by changing the operation from division to multiplication by flipping the divisor fraction. This gives: \(\frac{2}{7} \times \frac{24}{14}\).

Step 2: Simplifying the Fraction

Once the operation is converted to multiplication, we can simplify by cancelling out common factors between the numerators and the denominators. In this case, 2 and 14 can be simplified to 1 and 7 respectively, while 7 and 24 do not share any common factor. So we end up with: \(1 \times \frac{24}{7}\).

Step 3: Final Simplification

Now, simply multiply the numerators together and the denominators together to get your final answer. This is: \(\frac{24}{7}\)

Key Concepts

Reciprocal

The concept of a reciprocal is foundational in mathematics and is especially useful when dealing with multiplication and division of fractions. A reciprocal is simply another fraction that, when multiplied by the original, yields 1. In technical terms, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

This operation fundamentally involves flipping the numerator and denominator of the original fraction. For example, the reciprocal of \(\frac{14}{24}\) is \(\frac{24}{14}\). This becomes important when dividing fractions, because division by a fraction is equivalent to multiplication by its reciprocal.

By replacing the division operation with multiplication by a reciprocal, the process of simplifying the exercise becomes straightforward and filters down to basic multiplication of fractions. This serves as a vital skill in mathematics, making complex problems simpler to solve.

Multiplication of Fractions

Multiplying fractions involves a relatively simple process. When multiplying two fractions, you multiply the numerators together and the denominators together. For instance, if you have \(\frac{a}{b} \times \frac{c}{d}\), it results in \(\frac{a \cdot c}{b \cdot d}\).

This principle is clearly demonstrated in our exercise. After finding the reciprocal of \(\frac{14}{24}\), the problem transforms into \(\frac{2}{7} \times \frac{24}{14}\).

Here's a quick summary of how to multiply fractions:

• Multiply the numerators, placing the product in the numerator of the result.
• Multiply the denominators, placing the product in the denominator of the result.

Once you have your result, you should always check if further simplification is possible to get the simplest form of the fraction, which leads us to common factors.

Common Factors

Understanding common factors is crucial for simplifying fractions efficiently. A common factor is a number that divides exactly into two or more numbers. This means that both the numerator and the denominator have the same factor that can simplify the expression.

During our exercise simplification, after setting the multiplication with the reciprocal, we have \(\frac{2}{7} \times \frac{24}{14}\). The numbers 2 and 14 have a common factor of 2. This means we can divide both by 2 to simplify: \(\frac{2}{14} \rightarrow \frac{1}{7}\).

When simplifying, it is helpful to list the factors of the numerator and the denominator:

• Factors of 2: 1, 2
• Factors of 14: 1, 2, 7, 14

Common factors simplify expressions, making calculations more manageable and the solutions more readable, leading us to the simplest form of the fraction.