Problem 68 - 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) $$ \frac{8}{9}-\frac{1}{3} $$

Step-by-Step Solution

Answer

The answer is \(\frac{5}{9}\).

Step 1: Find common denominator

Firstly, identify the lowest common denominator (LCD) of the two fractions. The LCD of 3 and 9 is 9.

Step 2: Convert to similar fractions

Rewrite the fractions \(\frac{8}{9}\) and \(\frac{1}{3}\) as equivalent fractions with denominator 9. The first fraction remains the same and the second fraction becomes \(\frac{3}{9}\) after multiplying the numerator and the denominator of \(\frac{1}{3}\) by 3.

Step 3: Perform subtraction

The two fractions, \(\frac{8}{9}\) and \(\frac{3}{9}\), can now be subtracted because they have the same denominator. Therefore: \(\frac{8}{9}-\frac{3}{9}=\frac{5}{9}\)

Step 4: Simplify the fraction

The fraction \(\frac{5}{9}\) is in its simplest form as the numerator and the denominator have no common factors other than 1. Therefore, the final answer is \(\frac{5}{9}\)

Key Concepts

Understanding Common Denominator

When dealing with fractions, one of the essential concepts you will encounter is the common denominator. This is crucial in fraction addition and subtraction. To subtract fractions, they must have the same denominator. This shared denominator is called the common denominator. Once you have it, you can directly subtract the numerators.

• The lowest common denominator (LCD) is the smallest multiple that both denominators share.
• You can find it by identifying the least common multiple of the two denominators.

In our exercise, the denominators are 9 and 3. The smallest number both can divide into without a remainder is 9, making 9 the common denominator. This step ensures you can combine or subtract fractions accurately.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is a step often performed after subtraction or addition operations to ensure the fraction is as easy to read and understand as possible.Here are some tips:

• Check if the numerator and denominator have common factors other than 1.
• Divide both the numerator and the denominator by their greatest common factor (GCF).

In our specific problem, the result of the subtraction was \(\frac{5}{9}\). Since these numbers have no common divisors other than 1, \(\frac{5}{9}\) is already in its simplest form.
Understanding when and how to simplify fractions not only helps in obtaining a neat answer but also in comparing fractions effectively.

Finding Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. This concept is crucial in operations that involve finding common denominators.To convert fractions into equivalent ones:

• Multiply the numerator and the denominator by the same non-zero number.
• This keeps the value of the fraction the same while offering a different numerical expression.

In our example, to subtract \(\frac{1}{3}\) from \(\frac{8}{9}\), we need both fractions to share the common denominator 9. Thus, we rewrite \(\frac{1}{3}\) as an equivalent fraction \(\frac{3}{9}\) by multiplying both the numerator and the denominator by 3.
It is vital to understand that equivalent fractions allow us to perform arithmetic accurately and compare the size of different fractions conveniently.