Problem 8 - Prealgebra

Question

Find each difference. Write in simplest form. $$-\frac{7}{10}-\frac{2}{15}$$

Step-by-Step Solution

Answer

The difference is \(-\frac{5}{6}\).

Step 1: Find a Common Denominator

To subtract two fractions, they must have a common denominator. The denominators 10 and 15 have a least common multiple of 30. We will convert each fraction to have this common denominator of 30.

Step 2: Convert Fractions to Have a Common Denominator

Convert \(-\frac{7}{10}\) to a fraction with a denominator of 30. Multiply both the numerator and denominator by 3 to get \(-\frac{21}{30}\). Convert \(-\frac{2}{15}\) to a fraction with a denominator of 30. Multiply both the numerator and denominator by 2 to get \(-\frac{4}{30}\).

Step 3: Subtract the Fractions

Subtract the numerators while keeping the common denominator:\[-\frac{21}{30} - \frac{4}{30} = -\frac{21 + 4}{30} = -\frac{25}{30}\]

Step 4: Simplify the Resulting Fraction

The fraction \(-\frac{25}{30}\) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 5.\[-\frac{25}{30} = -\frac{25 \div 5}{30 \div 5} = -\frac{5}{6}\]

Key Concepts

Understanding Common Denominators

Before you can subtract fractions, you need to make sure they have the same denominator. This is because you can only combine fractions when they share a common base, which is the denominator. To find a common denominator, you look for the least common multiple (LCM) of the two denominators involved.

When subtracting fractions like \(-\frac{7}{10}\) and \(-\frac{2}{15}\), the denominators are 10 and 15. The LCM of these numbers is 30. This means both fractions need to be converted to equivalent fractions with the denominator of 30, allowing for easy subtraction. Here’s how you do it:

• Convert \(-\frac{7}{10}\) to a denominator of 30 by multiplying both the numerator and the denominator by 3, getting \(-\frac{21}{30}\).
• Convert \(-\frac{2}{15}\) to a denominator of 30 by multiplying both the numerator and the denominator by 2, resulting in \(-\frac{4}{30}\).

This process ensures that both fractions have the necessary common denominator for subtraction, paving the way for simple arithmetic.

Simplifying Fractions

Once you have subtracted the fractions and found the result, the next step is to simplify the fraction. Simplifying makes a fraction as easy to understand as possible by using the smallest numbers possible. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

After subtracting \(-\frac{21}{30}\) and \(-\frac{4}{30}\), you get \(-\frac{25}{30}\). While this is correct, it can be simplified to make it more straightforward. You find the GCD of 25 and 30, which is 5, and divide both the top and bottom by this number:

• Numerator: \(-25 \div 5 = -5\)
• Denominator: \(30 \div 5 = 6\)

So, \(-\frac{25}{30}\) simplifies to \(-\frac{5}{6}\). In simplified form, the expression is nicer to work with and understand.

Greatest Common Divisor Explained

The greatest common divisor (GCD) is a key part of simplifying fractions. It's the largest number that can divide both the numerator and denominator without leaving a remainder. Finding the GCD helps you reduce fractions to their simplest form.

For the fraction \(-\frac{25}{30}\), the GCD of 25 and 30 is 5. This is because 5 is the largest number that evenly divides both 25 and 30. By dividing both the numerator and the denominator by 5, you effectively shrink the fraction down to its smallest equivalent.

To find a GCD, you can list all the divisors of each number and then choose the greatest one they have in common. Alternatively, you can use the Euclidean algorithm, which is a systematic method for finding the GCD. Understanding this concept is crucial for simplifying fractions as it makes the arithmetic far more manageable.