$\frac{4b-9}{2b}+\frac{5b-1}{6b}$

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wolfs4lfe15 wolfs4lfe15

01-11-2026
Mathematics

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$\frac{4b-9}{2b}+\frac{5b-1}{6b}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T19:05:32-05:00

Identify the least common denominator as $6b$.
Adjust the first fraction to match the common denominator: $\frac{4b-9}{2b} = \frac{12b-27}{6b}$.
Combine the numerators while keeping the common denominator: $\frac{12b-27+5b-1}{6b}$.
Simplify the resulting expression to $\boxed{\frac{17b-28}{6b}}$.

Explanation

Analysis of the Expression

To simplify the expression $\frac{4b - 9}{2b} + \frac{5b - 1}{6b}$, we first analyze the terms. We are adding two rational expressions with different denominators: $2b$ and $6b$. For the expression to be defined, the variable $b$ must not be equal to zero, as this would result in a division by zero in both denominators. Our goal is to combine these into a single simplified fraction.

Finding a Common Denominator

To add fractions, they must have a common denominator. We look at the denominators $2b$ and $6b$. The least common denominator (LCD) is the smallest expression that is divisible by both $2b$ and $6b$. Since $6b = 3 \times 2b$, the LCD is $6b$. We keep the second fraction as is and adjust the first fraction by multiplying both the numerator and the denominator by $3$:
\[ \frac{4b - 9}{2b} = \frac{(4b - 9) \times 3}{2b \times 3} = \frac{12b - 27}{6b} \]

Combining and Simplifying the Numerator

Now that both fractions have the common denominator $6b$, we can add them by summing their numerators and keeping the denominator:
\[ \frac{12b - 27}{6b} + \frac{5b - 1}{6b} = \frac{(12b - 27) + (5b - 1)}{6b} \]
Distributing the addition in the numerator, we get:
\[ \frac{12b + 5b - 27 - 1}{6b} \]
Combining the like terms ($12b + 5b = 17b$ and $-27 - 1 = -28$), the expression becomes:
\[ \frac{17b - 28}{6b} \]

Final Simplification

Finally, we check if the expression $\frac{17b - 28}{6b}$ can be simplified further by looking for common factors between the numerator and the denominator. The numerator is $17b - 28$ and the denominator is $6b$. Since $17$ is a prime number and does not divide $28$, there are no common factors to cancel. Thus, the fraction is already in its simplest form. The final result is $\boxed{\frac{17b - 28}{6b}}$.

Examples

Imagine you are managing an inventory system where the total cost of two different items needs to be combined. If the cost of the first item is represented by $\frac{4b - 9}{2b}$ and the second by $\frac{5b - 1}{6b}$, simplifying these expressions allows you to find a single, unified cost formula. This is a common practice in accounting and engineering to streamline calculations when dealing with multiple variables or interdependent costs.