Adding Fractions How To Solve 1/9 + 7/12

Adding fractions might seem daunting at first, but with a clear understanding of the underlying principles, it becomes a straightforward process. In this comprehensive guide, we will delve into the intricacies of adding fractions, focusing specifically on the example of $\frac{1}{9} + \frac{7}{12}$. Our goal is to provide you with a step-by-step approach that not only solves this particular problem but also equips you with the knowledge to tackle any fraction addition challenge.

Understanding Fractions

Before we dive into the addition process, it's essential to grasp the fundamental concept of fractions. A fraction represents a part of a whole, expressed as a ratio between two numbers: the numerator and the denominator. The numerator (the top number) indicates how many parts we have, while the denominator (the bottom number) indicates the total number of equal parts that make up the whole. For example, in the fraction $\frac{1}{9}$, 1 is the numerator, and 9 is the denominator, signifying that we have one part out of a total of nine equal parts.

Why Common Denominators Matter

The cornerstone of adding fractions lies in the concept of common denominators. To add fractions accurately, they must share the same denominator. This is because we can only add or subtract quantities that are measured in the same units. Think of it like trying to add apples and oranges directly – it doesn't quite work. You need a common unit, like "fruit," to combine them meaningfully. Similarly, fractions need a common denominator to be added together.

Finding the Least Common Multiple (LCM)

The most efficient way to find a common denominator is to identify the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. For our example, $\frac{1}{9} + \frac{7}{12}$, we need to find the LCM of 9 and 12. Let's explore two methods for doing this:

1. Listing Multiples

One way to find the LCM is by listing the multiples of each denominator until we find a common one:

• Multiples of 9: 9, 18, 27, 36, 45, 54, ...
• Multiples of 12: 12, 24, 36, 48, 60, ...

The smallest multiple that appears in both lists is 36. Therefore, the LCM of 9 and 12 is 36.

2. Prime Factorization

A more systematic approach involves prime factorization. We break down each denominator into its prime factors:

• 9 = 3 x 3 = $3^2$
• 12 = 2 x 2 x 3 = $2^2$ x 3

To find the LCM, we take the highest power of each prime factor that appears in either factorization:

• $2^2$ (from 12)
• $3^2$ (from 9)

LCM = $2^2$ x $3^2$ = 4 x 9 = 36

Both methods confirm that the LCM of 9 and 12 is 36.

Converting to Equivalent Fractions

Now that we've found the LCM, we need to convert both fractions to equivalent fractions with a denominator of 36. An equivalent fraction is a fraction that represents the same value but has a different numerator and denominator. To create an equivalent fraction, we multiply both the numerator and the denominator by the same non-zero number. This doesn't change the value of the fraction because we are essentially multiplying by 1.

Converting $\frac{1}{9}$

To convert $\frac{1}{9}$ to an equivalent fraction with a denominator of 36, we need to determine what number to multiply 9 by to get 36. Since 9 x 4 = 36, we multiply both the numerator and the denominator of $\frac{1}{9}$ by 4:

$\frac{1}{9}$ x $\frac{4}{4}$ = $\frac{1 x 4}{9 x 4}$ = $\frac{4}{36}$

Converting $\frac{7}{12}$

Similarly, to convert $\frac{7}{12}$ to an equivalent fraction with a denominator of 36, we need to find the number that multiplies by 12 to give 36. Since 12 x 3 = 36, we multiply both the numerator and the denominator of $\frac{7}{12}$ by 3:

$\frac{7}{12}$ x $\frac{3}{3}$ = $\frac{7 x 3}{12 x 3}$ = $\frac{21}{36}$

Adding the Equivalent Fractions

Now that we have equivalent fractions with a common denominator, we can add them. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same:

$\frac{4}{36}$ + $\frac{21}{36}$ = $\frac{4 + 21}{36}$ = $\frac{25}{36}$

Reducing to Lowest Terms

The final step is to reduce the resulting fraction to its lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To reduce a fraction, we find the Greatest Common Divisor (GCD) of the numerator and denominator and divide both by it.

Finding the Greatest Common Divisor (GCD)

The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In our case, we need to find the GCD of 25 and 36. Let's use the listing factors method:

• Factors of 25: 1, 5, 25
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The only common factor of 25 and 36 is 1. Therefore, the GCD is 1.

Reducing the Fraction

Since the GCD of 25 and 36 is 1, the fraction $\frac{25}{36}$ is already in its simplest form. This means that it cannot be reduced further.

The Final Answer

Therefore, the sum of $\frac{1}{9}$ and $\frac{7}{12}$, expressed as a fraction reduced to its lowest terms, is $\frac{25}{36}$.

Key Takeaways

Let's recap the key steps involved in adding fractions:

1. Find the Least Common Multiple (LCM) of the denominators.
2. Convert each fraction to an equivalent fraction with the LCM as the denominator.
3. Add the numerators of the equivalent fractions, keeping the denominator the same.
4. Reduce the resulting fraction to its lowest terms by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).

Practice Problems

To solidify your understanding, try solving these practice problems:

1. $\frac{1}{4} + \frac{2}{5}$
2. $\frac{3}{8} + \frac{1}{6}$
3. $\frac{5}{12} + \frac{3}{16}$

Conclusion

Adding fractions is a fundamental skill in mathematics. By mastering the concepts of common denominators, equivalent fractions, and reducing to lowest terms, you can confidently tackle any fraction addition problem. Remember to practice regularly to reinforce your understanding and build your proficiency. This comprehensive guide has provided you with the tools and knowledge to successfully add fractions, and specifically, to solve $\frac{1}{9} + \frac{7}{12}$ with ease. Keep practicing, and you'll become a fraction addition expert in no time!