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Chapter 7: Problem 89

Perform the operations. Simplify, if possible. $$ \frac{4 b}{3}-\frac{5 b}{12} $$

Short Answer

Expert verified
\( \frac{11b}{12} \)

Step by step solution

01

Identify the Least Common Denominator (LCD)

Begin by identifying the least common denominator for the fractions in the expression. The denominators are 3 and 12. The least common denominator is 12.
02

Convert Fractions to Equivalent Fractions with LCD

Convert each fraction so they have the same denominator of 12. \( \frac{4b}{3} \) becomes \( \frac{16b}{12} \) because \( 4b \times 4 = 16b \), when multiplying by 4 to match the denominator 12.
03

Subtract the Fractions

Subtract the fractions: \( \frac{16b}{12} - \frac{5b}{12} = \frac{(16b - 5b)}{12} \). Combine the numerators to get \( \frac{11b}{12} \).
04

Simplify the Expression

The fraction \( \frac{11b}{12} \) is already in its simplest form, as 11 and 12 have no common factors other than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Common Denominator
When working with algebraic fractions, identifying the least common denominator (LCD) is crucial when it comes to performing operations like addition or subtraction. The LCD is the smallest number that both denominators can divide into without leaving a remainder.
  • In our problem, the denominators are 3 and 12.
  • To find the LCD, we compare multiples of each denominator: multiples of 3 are 3, 6, 9, 12, 15... and multiples of 12 are 12, 24, 36...
  • The smallest common multiple is 12, which makes it the least common denominator.
Identifying the LCD allows us to convert both fractions in our expression to equivalent fractions with a common denominator, crucial for the next steps in operations like subtraction.
Fraction Subtraction
Once the fractions have a common denominator, you can easily subtract them. This process is similar to subtracting integers or other like terms:
  • Keep the denominator the same for both fractions. In our example, it's 12.
  • Subtract the numerators: In the fraction \( \frac{16b}{12} - \frac{5b}{12} \), only the numerators are affected as 16b and 5b.
  • This subtraction gives us: \( 16b - 5b = 11b \).
The result will be a new fraction with the same denominator, \( \frac{11b}{12} \), simplifying the problem significantly before addressing further simplification.
Simplification of Expressions
After performing subtraction, simplifying the expression is the final step to reach the simplest form of your answer. Simplification involves checking for any common factors between the numerator and denominator.
  • For the fraction \( \frac{11b}{12} \), consider the numerator and denominator. The numerator has a factor of 11 and the denominator 12.
  • Determine if there are any common factors: here, 11 and 12 are coprime (they have no common factors apart from 1).
  • Since no simplification is possible, \( \frac{11b}{12} \) is already in its simplest form.
Remember, always check to see if you can simplify further to ensure your answer is as concise as possible. In some cases, expressions might simplify further by canceling common factors.

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