Problem 53 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of the fraction is \( \frac{24}{49} \).

Step 1: Identify the fractions

The fractions involved in this calculation are \( \frac{2}{7} \) and \( \frac{14}{24} \). The task is to divide the first fraction by the second fraction.

Step 2: Reciprocal of the divisor

Next, find the reciprocal of the second fraction. The reciprocal of a fraction is obtained by reversing the numerator and the denominator. So the reciprocal of \( \frac{14}{24} \) is \( \frac{24}{14} \).

Step 3: Multiply fractions

Now, replace the division of fractions with multiplication by the reciprocal of the second fraction. Therefore, \( \frac{2}{7} \div \frac{14}{24} \) can be written as \( \frac{2}{7} \times \frac{24}{14} \).

Step 4: Simplify the fractions

To simplify this calculation, first simplify the fraction \( \frac{24}{14} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So \( \frac{24}{14} \) simplifies to \( \frac{12}{7} \). Therefore, the multiplication now is \( \frac{2}{7} \times \frac{12}{7} \). Now perform the multiplication which gives \( \frac{24}{49} \).

Key Concepts

Reciprocal of a Fraction

In mathematics, the reciprocal of a fraction serves a vital role, especially when dividing fractions. To find the reciprocal, you simply swap or flip the numerator (the top part of the fraction) with the denominator (the bottom part). For example, if you have the fraction \( \frac{14}{24} \), its reciprocal is \( \frac{24}{14} \).
This process is crucial because division by a fraction is the same as multiplication by its reciprocal.

• Think of it like this: if you want to "undo" a fraction that is dividing your value, you multiply by its reciprocal.
• The principle is much like turning around to go back the same way you came when you're driving.

Reciprocals simplify the calculation process and convert complex division into an easier multiplication task. Being able to identify reciprocals can clear up confusion and streamline your work.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. This means expressing the fraction in its lowest terms by finding the greatest common divisor (GCD) of the numerator and the denominator.
Once you find it, you divide both the numerator and the denominator by this number.

• For instance, the fraction \( \frac{24}{14} \) can be simplified. The GCD of 24 and 14 is 2.
• By dividing the numerator and denominator by 2, the fraction simplifies to \( \frac{12}{7} \).

Simplification helps in two main ways: it makes numbers easier to work with and enhances the accuracy of calculations, as you ensure no factors are unnecessarily complicating the fraction. Ultimately, it's about making life simpler, not just the math.

Multiplying Fractions

Multiplying fractions is simpler than it seems. When multiplying, the process is straightforward: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. For example, when multiplying \( \frac{2}{7} \) and \( \frac{12}{7} \):

• Multiply the numerators: \( 2 \times 12 = 24 \).
• Multiply the denominators: \( 7 \times 7 = 49 \).
• The result is \( \frac{24}{49} \).

Once you have multiplied, it’s always good practice to check if the resulting fraction can be simplified further. However, in this case, \( \frac{24}{49} \) is already in its simplest form. Multiplication of fractions doesn't usually require common denominators, which is why it often feels easier than addition or subtraction of fractions. This straightforward process is why multiplying fractions is often seen as one of the "friendlier" tasks in fraction arithmetic.