Problem 10 - Fundamentals of Mathematics

Question

Perform the following divisions. $$ \frac{7}{15} \div \frac{14}{15} $$

Step-by-Step Solution

Answer

The result is \( \frac{1}{2} \).

Step 1: Understand Division of Fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the division \( \frac{7}{15} \div \frac{14}{15} \) can be rewritten as \( \frac{7}{15} \times \frac{15}{14} \).

Step 2: Multiply the Fractions

Multiply the fractions \( \frac{7}{15} \times \frac{15}{14} \) by multiplying the numerators together and the denominators together. So, \( 7 \times 15 = 105 \) and \( 15 \times 14 = 210 \), which gives us \( \frac{105}{210} \).

Step 3: Simplify the Fraction

To simplify \( \frac{105}{210} \), find the greatest common divisor (GCD) of 105 and 210. Both numbers are divisible by 105. So, divide both the numerator and the denominator by 105: \( \frac{105}{105} = 1 \) and \( \frac{210}{105} = 2 \). This simplifies to \( \frac{1}{2} \).

Key Concepts

Reciprocal of a Fraction

The concept of finding the reciprocal is a crucial step when dividing fractions. Imagine flipping a fraction upside down: this gives you its reciprocal. In mathematical terms, if you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Here’s why it matters: dividing by a fraction, like \( \frac{7}{15} \div \frac{14}{15} \), is the same as multiplying by its reciprocal. This transforms the division into a more straightforward multiplication. So, \( \frac{7}{15} \div \frac{14}{15} \) becomes \( \frac{7}{15} \times \frac{15}{14} \). By recognizing and using reciprocals, you simplify division tasks and maintain accuracy in calculations. This step makes your work with fractions more efficient and less prone to errors.
When practicing, always double-check that you have flipped the fraction correctly to avoid missteps later in the process.

Multiplying Fractions

In the world of fractions, multiplying is straightforward. Once you have set up the multiplication, like in our transformation of the division problem \( \frac{7}{15} \times \frac{15}{14} \), all it takes is multiplying the numerators together and the denominators together. Here’s how it works: multiply the top numbers (numerators) with each other and the bottom numbers (denominators) with each other.

• For example, \( 7 \times 15 = 105 \)
• And \( 15 \times 14 = 210 \)

Thus, you get \( \frac{105}{210} \). Multiplying fractions does not require making common denominators like addition or subtraction does. It’s just straight multiplication!
Keep multiplicative operations clean and clear, which will simplify the fraction as the final step.

Simplifying Fractions

After multiplying fractions, you often end up with a fraction that can be simplified. Simplification makes the fraction easier to understand and use in future calculations. In our example, the product was \( \frac{105}{210} \). To simplify this fraction, find the greatest common divisor (GCD) of the numerator and the denominator.
Here, both 105 and 210 are divisible by 105:

• \( \frac{105}{105} = 1 \)
• \( \frac{210}{105} = 2 \)

So, \( \frac{105}{210} \) simplifies to \( \frac{1}{2} \). Simplification is essential because it helps in obtaining the most reduced form of a fraction, ensuring clarity and ease when fractions are used alongside other math operations. Always double-check your GCD calculations to make sure you've simplified the fraction as much as possible.