Problem 26 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(\frac{9}{20}-\left(-\frac{7}{20}\right)\)

Step-by-Step Solution

Answer

The sum is \( \frac{4}{5} \).

Step 1: Simplify the Expression Inside Parentheses

The expression is given as \( \frac{9}{20} - \left( -\frac{7}{20} \right) \). The subtraction of a negative fraction is the same as adding its positive. Thus, the expression becomes \( \frac{9}{20} + \frac{7}{20} \).

Step 2: Add the Fractions

Now that both fractions have the same denominator, you can simply add the numerators. The calculation is \( 9 + 7 \), which equals \( 16 \). Thus, the result is \( \frac{16}{20} \).

Step 3: Simplify the Fraction

The fraction \( \frac{16}{20} \) can be simplified by finding the greatest common divisor (GCD) of 16 and 20, which is 4. Divide both the numerator and the denominator by 4: \( \frac{16 \div 4}{20 \div 4} = \frac{4}{5} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in math. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. When you simplify a fraction, you make it easier to understand and work with.
To simplify a fraction, you need to:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

For example, consider the fraction \( \frac{16}{20} \). The GCD of 16 and 20 is 4. Dividing both the numerator and the denominator by 4 gives us \( \frac{4}{5} \), which is the fraction in its simplest form. By simplifying, you maintain the value of the fraction while making it easier to interpret and use in calculations.

Adding Fractions

Adding fractions can be straightforward, especially when they share the same denominator. The key is to focus solely on the numerators since the denominators remain constant. In other cases, finding a common denominator becomes necessary.
To add fractions with the same denominator, like \( \frac{9}{20} \) and \( \frac{7}{20} \), simply add the numerators:

• Keep the denominator the same. For our example, it's 20.
• Add the numerators: 9 + 7 = 16.

Thus, \( \frac{9}{20} + \frac{7}{20} = \frac{16}{20} \).
When denominators differ, you'll need to find a common denominator before adding the numerators. This ensures that you are combining equivalent parts.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is a crucial tool in mathematics, especially when simplifying fractions or finding common denominators.
The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCD:

• List the factors of each number.
• Identify the largest common factor.

For example, to find the GCD of 16 and 20:

• The factors of 16 are 1, 2, 4, 8, 16.
• The factors of 20 are 1, 2, 4, 5, 10, 20.
• The common factors are 1, 2, and 4, with 4 being the greatest.

Thus, the GCD of 16 and 20 is 4, and using it to simplify \( \frac{16}{20} \) results in \( \frac{4}{5} \). Recognizing and using the GCD efficiently makes arithmetic operations involving fractions much simpler.