Problem 64 - Prealgebra

Question

Add or subtract the following fractions, as indicated. $$\frac{7}{10}-\frac{3}{5}$$

Step-by-Step Solution

Answer

The result of the subtraction is \( \frac{1}{10} \).

Step 1: Identify a Common Denominator

For the fractions \( \frac{7}{10} \) and \( \frac{3}{5} \), the common denominator needs to be found for subtraction. The denominators 10 and 5 share a least common multiple of 10.

Step 2: Adjust Fractions to Common Denominator

Since \( \frac{3}{5} \) does not have the common denominator, convert it by multiplying both the numerator and the denominator by 2. Thus, \( \frac{3}{5} \) becomes \( \frac{6}{10} \).

Step 3: Subtract the Fractions

With a common denominator of 10, subtract the numerators: \( 7 - 6 = 1 \). Place this over the common denominator: \( \frac{1}{10} \).

Step 4: Simplify the Result

The result \( \frac{1}{10} \) is already in its simplest form as there are no common factors between the numerator and the denominator. Thus, the simplified subtracted fraction is \( \frac{1}{10} \).

Key Concepts

Understanding a Common Denominator

When performing fraction addition or subtraction, having a common denominator is crucial. A common denominator is the same bottom number in two or more fractions that allows them to be easily added or subtracted. Think of it like a shared stage where both fractions can perform their mathematical dance. In our example, with fractions \( \frac{7}{10} \) and \( \frac{3}{5} \), the denominators are 10 and 5. To determine the common denominator, you'll need to find a denominator that both fractions can "agree" on. This involves making sure that both fractions have the same number on the bottom, allowing for straightforward addition or subtraction. Passing mathematical notes between the fractions becomes simple once they share the same "language" - the common denominator.

Discovering the Least Common Multiple

The Least Common Multiple (LCM) is a key player in finding a common denominator. It’s the smallest multiple that two or more numbers share. Let's dive into how we find this for our example. For the denominators \(10\) and \(5\), you identify multiples of each number:

• Multiples of 10: 10, 20, 30, 40, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...

Here, \(10\) is the first and smallest multiple both numbers share, making it the least common multiple. With the LCM identified, you can easily rewrite the fractions if necessary so that the denominators match. This technique makes the addition or subtraction of fractions smooth and error-free.

The Art of Simplifying Fractions

Simplifying fractions is like streamlining a path to attain its purest form. It involves reducing the fraction to its simplest terms where the numerator and the denominator have no other common factor besides 1. For instance, consider the final step of our problem. After subtracting the fractions, we arrive at \( \frac{1}{10} \). To check if this is simplified, examine the highest number that can evenly divide both 1 (numerator) and 10 (denominator):

• The factors of 1: 1
• The factors of 10: 1, 2, 5, 10

There is no greater common divisor than 1, which means \( \frac{1}{10} \) is already as simple as it gets. Streamlining fractions makes them neater and often easier to use in further calculations! Keeping fractions in their simplest form is not only mathematically efficient but also aesthetically satisfying.