Problem 38 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{7}{10}-\frac{1}{5}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \)

Step 1: Find a common denominator

To subtract fractions, they need to have the same denominator. Find the least common denominator (LCD) for \( \frac{7}{10} \) and \( \frac{1}{5} \). The denominator 10 is already suitable for 5 (since 10 is a multiple of 5). So, the LCD is 10.

Step 2: Convert fractions to the same denominator

Convert \( \frac{1}{5} \) to have a denominator of 10. To do this, multiply both the numerator and the denominator by 2. So, \( \frac{1}{5} = \frac{2}{10} \).

Step 3: Perform the subtraction

Now subtract the fractions: \( \frac{7}{10} - \frac{2}{10} \). Since they have the same denominator, subtract the numerators: \( 7 - 2 = 5 \). The result is \( \frac{5}{10} \).

Step 4: Simplify the fraction

Simplify \( \frac{5}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, \( \frac{5}{10} = \frac{1}{2} \).

Key Concepts

Least Common Denominator

When subtracting fractions, the first step is to find a common denominator. This helps align the fractions by making sure they have the same base. The least common denominator (LCD) is the smallest number that both denominators divide into evenly.
The LCD for fractions makes the mathematical operation seamless and prevents errors in computation.
In our example, we have the fractions \(\frac{7}{10}\) and \(\frac{1}{5}\).

• The denominators here are 10 and 5.
• Since 10 is a multiple of 5, choosing 10 as the LCD is straightforward and efficient.

Using 10 as the common denominator allows both fractions to be directly subtracted without altering the value of either one. This step ensures accuracy in the subtraction process.

Mixed Numbers

Mixed numbers combine whole numbers and fractions. They frequently appear in real-world mathematical problems. Understanding how to deal with mixed numbers in subtraction or addition enhances your math skills. So, how do mixed numbers work with fraction subtraction?
Let’s discuss an example involving subtraction of mixed numbers:

• First, convert the mixed numbers to improper fractions. This means multiplying the whole number by the denominator and adding the numerator.
• Next, follow the regular steps for subtracting fractions. Ensure they have the same denominator, which might involve finding the LCD.

Handling mixed numbers is essentially about converting them to a standard fraction form so that all the known fraction operations can be performed correctly. Once converted, subtraction becomes similar to regular fractions. It’s simple with a little practice!

Simplifying Fractions

After performing an arithmetic operation with fractions, it's always good to check if your result can be simplified. Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.
Reducing a fraction makes it easier to understand and compare with others.
Here's how to simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this common factor.

In our initial problem, we had the fraction \(\frac{5}{10}\).

• The GCD of 5 and 10 is 5.
• So, dividing 5 by 5 and 10 by 5 simplifies \(\frac{5}{10}\) to \(\frac{1}{2}\).

Always remember, a fraction is fully simplified when it is expressed in the lowest terms. Simplifying makes your answer neat and professional.