Problem 52
Question
Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{2 n+3}{4}+\frac{4 n-1}{7}$$
Step-by-Step Solution
Verified Answer
The answer is \(\frac{30n + 17}{28}\).
1Step 1: Identify the Denominators
The given fractions are \( \frac{2n+3}{4} \) and \( \frac{4n-1}{7} \). The denominators are 4 and 7, respectively. To add these fractions, we must find a common denominator.
2Step 2: Find the Least Common Denominator (LCD)
The least common denominator of 4 and 7 is 28, as 28 is the smallest number that both 4 and 7 divide into evenly.
3Step 3: Rewrite Fractions with the LCD
Rewrite each fraction with the common denominator of 28. Multiply the numerator and denominator of the first fraction by 7: \[\frac{2n+3}{4} = \frac{7(2n+3)}{28} = \frac{14n+21}{28}\]Next, multiply the numerator and denominator of the second fraction by 4:\[\frac{4n-1}{7} = \frac{4(4n-1)}{28} = \frac{16n-4}{28}\]
4Step 4: Add the Fractions
Since the fractions now have the same denominator, you can add them by adding the numerators together:\[\frac{14n+21}{28} + \frac{16n-4}{28} = \frac{(14n+21) + (16n-4)}{28} = \frac{30n + 17}{28}\]
5Step 5: Simplify the Result
Check to see if the resulting fraction \(\frac{30n + 17}{28}\) can be simplified. Since there are no common factors between 30, 17, and 28, this is the simplest form of the expression.
Key Concepts
Common DenominatorLeast Common DenominatorSimplifying Fractions
Common Denominator
When adding or subtracting algebraic fractions, finding a common denominator is crucial. The common denominator allows us to combine fractions that have different denominators by rewriting them over a shared base.
Let's think of the common denominator as a translator between fractions. It provides a common language so they can be easily added or subtracted. In the problem where we have the fractions \( \frac{2n+3}{4} \) and \( \frac{4n-1}{7} \), their denominators are 4 and 7.
To perform operations, we must express both fractions with a denominator that both 4 and 7 can divide into evenly. This creates a unified base for calculating the fractions together.
Let's think of the common denominator as a translator between fractions. It provides a common language so they can be easily added or subtracted. In the problem where we have the fractions \( \frac{2n+3}{4} \) and \( \frac{4n-1}{7} \), their denominators are 4 and 7.
To perform operations, we must express both fractions with a denominator that both 4 and 7 can divide into evenly. This creates a unified base for calculating the fractions together.
Least Common Denominator
The least common denominator (LCD) is the smallest number that all denominators considered can divide evenly.
For example, let's revisit 4 and 7: we determined that the smallest multiple they both share is 28.
To find it, you'd begin by listing the multiples or using the prime factorization and determining the highest power of each factor needed.
The LCD is more than a common denominator; it’s the most efficient choice when simplifying the problem further. It's like taking the shortest route on a map, saving time and space in calculations.
For example, let's revisit 4 and 7: we determined that the smallest multiple they both share is 28.
To find it, you'd begin by listing the multiples or using the prime factorization and determining the highest power of each factor needed.
The LCD is more than a common denominator; it’s the most efficient choice when simplifying the problem further. It's like taking the shortest route on a map, saving time and space in calculations.
Simplifying Fractions
After obtaining fractions with a common denominator, the final step often involves simplifying the result.
Simplifying means expressing the fraction in its simplest form, where the numerator and the denominator share no common factors other than 1.
Once the fractions are combined—like in our example, resulting in \( \frac{30n + 17}{28} \)—the next step is to ensure the expression is as simplified as possible.
If the numerator and the denominator can no longer be divisible by the same number, we have reached the simplest form.
This ensures the final answer is neat and clear, making it easier to understand and recognize.
Simplifying means expressing the fraction in its simplest form, where the numerator and the denominator share no common factors other than 1.
Once the fractions are combined—like in our example, resulting in \( \frac{30n + 17}{28} \)—the next step is to ensure the expression is as simplified as possible.
If the numerator and the denominator can no longer be divisible by the same number, we have reached the simplest form.
This ensures the final answer is neat and clear, making it easier to understand and recognize.
Other exercises in this chapter
Problem 51
For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{1}{y}-\frac{4}{x^{2}}}{\frac{7}{x}-\frac{3}{y}} $$
View solution Problem 51
Kent drives his Mazda 270 miles in the same time that Dave drives his Nissan 250 miles. If Kent averages 4 miles per hour faster than Dave, find their rates.
View solution Problem 52
Simplify each algebraic fraction. $$\frac{y^{2}+20 y+96}{y^{2}+23 y+120}$$
View solution Problem 52
For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{4}{a b}+\frac{2}{b}}{\frac{8}{a}+\frac{1}{b}} $$
View solution