Add The Fractions: 5 7 + 1 4 \frac{5}{7} + \frac{1}{4} 7 5 ​ + 4 1 ​ Simplify To The Lowest Terms. Enter The Numerator Of The Simplified Fraction: [?]

Introduction

In mathematics, fractions are a way to represent a part of a whole. When we add fractions, we need to find a common denominator to combine them. In this article, we will learn how to add fractions and simplify them to their lowest terms.

What are Fractions?

A fraction is a way to represent a part of a whole. It consists of two numbers: a numerator and a denominator. The numerator is the number on top, and the denominator is the number on the bottom. For example, the fraction $\frac{1}{2}$ represents one half of a whole.

Adding Fractions

To add fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions. For example, to add $\frac{1}{2}$ and $\frac{1}{3}$, we need to find the LCM of 2 and 3, which is 6.

Step 1: Find the Common Denominator

To find the common denominator, we need to list the multiples of each denominator. For example, the multiples of 2 are 2, 4, 6, 8, 10, ... and the multiples of 3 are 3, 6, 9, 12, 15, ....

Step 2: Write Each Fraction with the Common Denominator

Once we have found the common denominator, we need to write each fraction with that denominator. For example, to add $\frac{1}{2}$ and $\frac{1}{3}$, we need to write each fraction with a denominator of 6.

$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Step 3: Add the Numerators

Now that we have written each fraction with the common denominator, we can add the numerators. For example, to add $\frac{3}{6}$ and $\frac{2}{6}$, we need to add the numerators.

$\frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}$

Simplifying Fractions

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. For example, to simplify $\frac{5}{6}$, we need to find the GCD of 5 and 6.

Step 1: Find the GCD

To find the GCD, we need to list the factors of each number. For example, the factors of 5 are 1 and 5, and the factors of 6 are 1, 2, 3, and 6.

Step 2: Divide the Numerator and Denominator by the GCD

Once we have found the GCD, we need to divide the numerator and the denominator by the GCD. For example, to simplify $\frac{5}{6}$, we need to divide the numerator and the denominator by 1.

$\frac{5}{6} = \frac{5 \div 1}{6 \div 1} = \frac{5}{6}$

Conclusion

In this article, we learned how to add fractions and simplify them to their lowest terms. We found the common denominator, wrote each fraction with the common denominator, added the numerators, and simplified the fraction by finding the GCD and dividing the numerator and the denominator by the GCD.

Example Problem

Add the fractions $\frac{5}{7}$ and $\frac{1}{4}$ and simplify to the lowest terms.

Solution

To add the fractions, we need to find the common denominator. The common denominator is the LCM of 7 and 4, which is 28.

$\frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28}$

$\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}$

Now that we have written each fraction with the common denominator, we can add the numerators.

$\frac{20}{28} + \frac{7}{28} = \frac{20 + 7}{28} = \frac{27}{28}$

To simplify the fraction, we need to find the GCD of 27 and 28. The GCD is 1.

$\frac{27}{28} = \frac{27 \div 1}{28 \div 1} = \frac{27}{28}$

Therefore, the simplified fraction is $\frac{27}{28}$.

Final Answer

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

A: When adding fractions, we need to find a common denominator, whereas when adding whole numbers, we can simply add the numbers together.

Q: How do I find the common denominator of two fractions?

A: To find the common denominator, you need to list the multiples of each denominator and find the least common multiple (LCM).

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

A: The LCM is the smallest number that is a multiple of both numbers.

Q: How do I write each fraction with the common denominator?

A: To write each fraction with the common denominator, you need to multiply the numerator and denominator of each fraction by the necessary factor to get the common denominator.

Q: Can I simplify a fraction before adding it to another fraction?

A: Yes, you can simplify a fraction before adding it to another fraction. However, make sure to simplify the fraction to its lowest terms.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

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

A: The GCD is the largest number that divides both numbers without leaving a remainder.

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 subtract the absolute values of the fractions.

Q: How do I add fractions with different signs?

A: To add fractions with different signs, you need to subtract the absolute values of the fractions and then apply the sign of the larger fraction.

Q: Can I add fractions with zero?

A: Yes, you can add fractions with zero. When adding a fraction with zero, the result is the other fraction.

Q: How do I add fractions with zero?

A: To add a fraction with zero, you simply ignore the zero and add the other fraction.

Q: Can I add fractions with negative numbers?

A: Yes, you can add fractions with negative numbers. When adding fractions with negative numbers, you need to follow the rules of adding negative numbers.

Q: How do I add fractions with negative numbers?

A: To add fractions with negative numbers, you need to subtract the absolute values of the fractions and then apply the sign of the larger fraction.

Q: Can I add mixed numbers?

A: Yes, you can add mixed numbers. When adding mixed numbers, you need to convert the mixed numbers to improper fractions and then add them.

Q: How do I add mixed numbers?

A: To add mixed numbers, you need to convert the mixed numbers to improper fractions by multiplying the whole number by the denominator and adding the numerator, and then add the fractions.

Q: Can I subtract fractions?

A: Yes, you can subtract fractions. When subtracting fractions, you need to find a common denominator and then subtract the numerators.

Q: How do I subtract fractions?

A: To subtract fractions, you need to find a common denominator and then subtract the numerators.

Q: Can I multiply fractions?

A: Yes, you can multiply fractions. When multiplying fractions, you need to multiply the numerators and denominators separately.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and denominators separately.

Q: Can I divide fractions?

A: Yes, you can divide fractions. When dividing fractions, you need to invert the second fraction and then multiply.

Q: How do I divide fractions?

A: To divide fractions, you need to invert the second fraction and then multiply.

Q: Can I simplify a fraction after multiplying or dividing it?

A: Yes, you can simplify a fraction after multiplying or dividing it. However, make sure to simplify the fraction to its lowest terms.

Q: How do I simplify a fraction after multiplying or dividing it?

A: To simplify a fraction after multiplying or dividing it, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.