Problem 70 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction or as a mixed number in simplest form. (Skills Review p.764) $$ \frac{3}{5}-\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution to the fraction subtraction \(\frac{3}{5}-\frac{1}{2}\) is \(\frac{1}{10}\).

Step 1: Finding the Least Common Denominator (LCD)

To subtract fractions, they need to have the same denominator. In this case, the denominators are 5 and 2. The smallest number that both these numbers divide into evenly is 10. So, 10 is the least common denominator.

Step 2: Changing the fractions to have the same denominator

Change both fractions to equivalent fractions with denominator 10. This is achieved by multiplying the numerator and denominator of the first fraction \(\frac{3}{5}\) by 2, getting \(\frac{6}{10}\). The second fraction \(\frac{1}{2}\) should be multiplied by 5 in both the numerator and denominator, resulting in \(\frac{5}{10}\).

Step 3: Subtracting the fractions

After aligning the denominators, subtract the two fractions \(\frac{6}{10}-\frac{5}{10}\). The subtraction of the fractions results in \(\frac{1}{10}\).

Step 4: Simplifying the final fraction

The resulting fraction \(\frac{1}{10}\) is already in the simplest form since numerator and denominator are coprimes (their greatest common divisor is 1).

Key Concepts

Least Common Denominator

When subtracting fractions, they must share a common denominator so they can be easily worked with. The least common denominator (LCD) is the smallest number that both original denominators can divide into without leaving a remainder. For fractions like \(\frac{3}{5}\) and \(\frac{1}{2}\), the denominators are 5 and 2. We look for a number that both 5 and 2 can go into evenly. In this case, that number is 10. So, the LCD for these fractions is 10.

Finding the LCD ensures that the fractions are on the same "number ground." This can simplify the subtraction process significantly. By aligning the denominators, you only need to subtract the numerators when you perform the subtraction.

To find the least common denominator:

• List multiples of each denominator.
• Find the smallest multiple that appears in both lists.
• Use this as your common denominator to adjust the fractions accordingly.

Simplifying Fractions

Once the fractions have been subtracted and you have an answer like \(\frac{1}{10}\), it's important to check if this fraction is in its simplest form. A fraction is in simplest form if the greatest common divisor (GCD) of the numerator and the denominator is 1.

In our example, \(\frac{1}{10}\) can't be simplified further because 1 and 10 share no other common factors besides 1. Simplifying fractions involves the following steps:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• If the GCD is 1, the fraction is already in its simplest form.

This process ensures that the fraction remains in its most efficient form possible, making calculations simpler and providing a clearer understanding of the quantity represented by the fraction.
Remember, working with simple terms makes the fraction easier to comprehend and use, especially in further calculations.

Equivalent Fractions

Creating equivalent fractions is necessary when aligning denominators. Equivalent fractions are different fractions that represent the same value.

Consider our fractions \(\frac{3}{5}\) and \(\frac{1}{2}\). To make them equivalent with a common denominator of 10, we needed to multiply the numerator and denominator of \(\frac{3}{5}\) by 2, resulting in \(\frac{6}{10}\), and multiply both parts of \(\frac{1}{2}\) by 5 to get \(\frac{5}{10}\).

This process does not change the value of the fraction, only how it is expressed. To form equivalent fractions:

• Multiply or divide the numerator and the denominator by the same non-zero number.
• Ensure the fraction's value remains unchanged.
• Confirm that the new denominator matches the LCD.

Equivalent fractions are powerful tools for simplifying operations like addition and subtraction since the denominators are aligned, simplifying the arithmetic involved.