Add The Following Together:$\frac{1}{x-3} + \frac{4}{x-3}$

Understanding the Problem

When adding fractions, it's essential to have the same denominator. In this case, we're given two fractions with the same denominator, $\frac{1}{x-3}$ and $\frac{4}{x-3}$. Our goal is to add these fractions together and simplify the resulting expression.

The Importance of Having the Same Denominator

Having the same denominator is crucial when adding fractions. If the denominators are different, we need to find the least common multiple (LCM) of the denominators and then convert each fraction to have the LCM as the new denominator. However, in this case, we're lucky to have the same denominator, making the addition process much simpler.

Adding the Numerators

Since the denominators are the same, we can add the numerators directly. The numerator of the first fraction is 1, and the numerator of the second fraction is 4. To add these numerators, we simply add 1 and 4, which gives us a total of 5.

Writing the Resulting Fraction

Now that we have the sum of the numerators, we can write the resulting fraction. The denominator remains the same, which is $x-3$. The numerator is the sum of the original numerators, which is 5. Therefore, the resulting fraction is $\frac{5}{x-3}$.

Simplifying the Expression (If Necessary)

In this case, the fraction $\frac{5}{x-3}$ is already in its simplest form. However, if the numerator or denominator had any common factors, we would need to simplify the expression further. For example, if the numerator and denominator had a common factor of 5, we could simplify the expression by dividing both the numerator and denominator by 5.

Real-World Applications

Adding fractions with the same denominator is a fundamental concept in mathematics that has numerous real-world applications. For instance, in finance, we often need to add fractions to calculate interest rates or investment returns. In science, we may need to add fractions to calculate probabilities or concentrations. In engineering, we may need to add fractions to calculate stress or strain on a material.

Conclusion

Adding fractions with the same denominator is a straightforward process that involves adding the numerators and keeping the same denominator. By following these simple steps, we can add fractions and simplify the resulting expression. Whether you're a student, a professional, or simply someone who enjoys mathematics, understanding how to add fractions with the same denominator is an essential skill that will serve you well in a variety of contexts.

Example Problems

Problem 1

Add the fractions $\frac{2}{x+2}$ and $\frac{3}{x+2}$.

Solution

To add these fractions, we simply add the numerators, which gives us a total of 5. The denominator remains the same, which is $x+2$. Therefore, the resulting fraction is $\frac{5}{x+2}$.

Problem 2

Add the fractions $\frac{1}{x-1}$ and $\frac{2}{x-1}$.

Solution

To add these fractions, we simply add the numerators, which gives us a total of 3. The denominator remains the same, which is $x-1$. Therefore, the resulting fraction is $\frac{3}{x-1}$.

Tips and Tricks

• When adding fractions, make sure to have the same denominator.
• If the denominators are different, find the least common multiple (LCM) of the denominators and convert each fraction to have the LCM as the new denominator.
• When adding numerators, simply add the numbers together.
• When writing the resulting fraction, keep the same denominator and use the sum of the numerators as the new numerator.

Common Mistakes to Avoid

• Not having the same denominator when adding fractions.
• Not finding the least common multiple (LCM) of the denominators when the denominators are different.
• Not simplifying the expression further if the numerator or denominator has any common factors.
• Not following the correct order of operations when adding fractions.

Conclusion

Adding fractions with the same denominator is a fundamental concept in mathematics that has numerous real-world applications. By following the simple steps outlined in this article, we can add fractions and simplify the resulting expression. Whether you're a student, a professional, or simply someone who enjoys mathematics, understanding how to add fractions with the same denominator is an essential skill that will serve you well in a variety of contexts.

Q: What is the first step in adding fractions with the same denominator?

A: The first step in adding fractions with the same denominator is to add the numerators. This means that you simply add the numbers together.

Q: What if the denominators are different? How do I add fractions in that case?

A: If the denominators are different, you need to find the least common multiple (LCM) of the denominators and convert each fraction to have the LCM as the new denominator. Once you have the same denominator, you can add the fractions by adding the numerators.

Q: Can I simplify the expression after adding the fractions?

A: Yes, you can simplify the expression after adding the fractions. If the numerator or denominator has any common factors, you can simplify the expression by dividing both the numerator and denominator by the common factor.

Q: What is the difference between adding fractions and adding whole numbers?

A: The main difference between adding fractions and adding whole numbers is that fractions have a denominator, which is the number of equal parts that the whole is divided into. When adding fractions, you need to have the same denominator, whereas when adding whole numbers, you can simply add the numbers together.

Q: Can I add fractions with different signs?

A: Yes, you can add fractions with different signs. When adding fractions with different signs, you need to follow the rules of addition, which state that when you add two numbers with different signs, you need to subtract the smaller number from the larger number.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 and 6.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that is common to both lists. Alternatively, you can use the formula LCM(a, b) = (a × b) / GCD(a, b), where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can list the factors of each number and find the largest number that is common to both lists. Alternatively, you can use the Euclidean algorithm to find the GCD.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is zero. The last non-zero remainder is the GCD.

Q: Can I use a calculator to find the GCD of two numbers?

A: Yes, you can use a calculator to find the GCD of two numbers. Most calculators have a built-in function for finding the GCD, which you can access by pressing the "GCD" button or by using the "MOD" function.

Q: What is the difference between the GCD and the LCM of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder, whereas the least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. The GCD and LCM are related by the formula GCD(a, b) × LCM(a, b) = a × b.

Q: Can I use the GCD and LCM to simplify fractions?

A: Yes, you can use the GCD and LCM to simplify fractions. If the numerator and denominator of a fraction have a common factor, you can simplify the fraction by dividing both the numerator and denominator by the common factor. The GCD and LCM can be used to find the common factor.

Q: What is the difference between simplifying a fraction and reducing a fraction?

A: Simplifying a fraction involves dividing both the numerator and denominator by the greatest common divisor (GCD), whereas reducing a fraction involves dividing both the numerator and denominator by the least common multiple (LCM). Simplifying a fraction makes the fraction easier to read, whereas reducing a fraction makes the fraction more compact.

Q: Can I simplify or reduce a fraction that has a variable in the denominator?

A: Yes, you can simplify or reduce a fraction that has a variable in the denominator. However, you need to be careful when simplifying or reducing fractions with variables, as the variable may cancel out or become a different variable.

Q: What is the difference between adding fractions with the same denominator and adding fractions with different denominators?

A: Adding fractions with the same denominator involves adding the numerators and keeping the same denominator, whereas adding fractions with different denominators involves finding the least common multiple (LCM) of the denominators and converting each fraction to have the LCM as the new denominator.

Q: Can I add fractions with different signs and different denominators?

A: Yes, you can add fractions with different signs and different denominators. However, you need to follow the rules of addition and find the least common multiple (LCM) of the denominators before adding the fractions.

Q: What is the difference between adding fractions and adding decimals?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding decimals involves adding the numbers as if they were whole numbers and then dividing by the place value of the decimal point.

Q: Can I add fractions and decimals together?

A: Yes, you can add fractions and decimals together. However, you need to convert the fractions to decimals or the decimals to fractions before adding them.

Q: What is the difference between adding fractions and adding mixed numbers?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding mixed numbers involves adding the whole number part and the fractional part separately.

Q: Can I add fractions and mixed numbers together?

A: Yes, you can add fractions and mixed numbers together. However, you need to convert the mixed numbers to improper fractions before adding them.

Q: What is the difference between adding fractions and adding integers?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding integers involves adding the numbers as if they were whole numbers.

Q: Can I add fractions and integers together?

A: Yes, you can add fractions and integers together. However, you need to convert the fractions to decimals or the integers to fractions before adding them.

Q: What is the difference between adding fractions and adding algebraic expressions?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding algebraic expressions involves combining like terms and simplifying the expression.

Q: Can I add fractions and algebraic expressions together?

A: Yes, you can add fractions and algebraic expressions together. However, you need to convert the fractions to decimals or the algebraic expressions to fractions before adding them.

Q: What is the difference between adding fractions and adding trigonometric expressions?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding trigonometric expressions involves combining like terms and simplifying the expression.

Q: Can I add fractions and trigonometric expressions together?

A: Yes, you can add fractions and trigonometric expressions together. However, you need to convert the fractions to decimals or the trigonometric expressions to fractions before adding them.

Q: What is the difference between adding fractions and adding exponential expressions?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding exponential expressions involves combining like terms and simplifying the expression.

Q: Can I add fractions and exponential expressions together?

A: Yes, you can add fractions and exponential expressions together. However, you need to convert the fractions to decimals or the exponential expressions to fractions before adding them.

Q: What is the difference between adding fractions and adding logarithmic expressions?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding logarithmic expressions involves combining like terms and simplifying the expression.

Q: Can I add fractions and logarithmic expressions together?

A: Yes, you can add fractions and logarithmic expressions together. However, you need to convert the fractions to decimals or the logarithmic expressions to fractions before adding them.

Q: What is the difference between adding fractions and adding rational expressions?

A: Adding fractions involves adding the numerators and keeping the same denominator, whereas adding rational expressions involves combining like terms and simplifying the expression.

Q: Can I add fractions and rational expressions together?

A: Yes, you can add fractions and rational expressions together. However, you need to convert the fractions to decimals or the rational expressions to fractions before adding them.

Q: What is the difference between adding fractions and adding complex expressions?

A: Adding fractions involves adding the numer