Simplifying Mixed Number Subtraction 7 1/6 - 3 4/9

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In the realm of mathematics, encountering mixed numbers is a common occurrence, and the ability to perform arithmetic operations on them is a fundamental skill. This article delves into the process of subtracting mixed numbers, specifically focusing on the problem of simplifying 7 1/6 - 3 4/9. We will embark on a step-by-step journey, breaking down the process into manageable steps, ensuring a comprehensive understanding of the underlying concepts.

Understanding Mixed Numbers

Before we delve into the subtraction process, let's first establish a clear understanding of mixed numbers. Mixed numbers, as the name suggests, are a combination of a whole number and a fraction. In the given problem, 7 1/6 and 3 4/9 are both mixed numbers. The whole number part represents the number of whole units, while the fractional part represents a portion of a whole unit. For instance, 7 1/6 signifies seven whole units and one-sixth of another unit.

The fraction part of a mixed number is composed of two key components: the numerator and the denominator. The numerator indicates the number of parts we have, while the denominator represents the total number of equal parts that make up a whole. In the fraction 1/6, the numerator is 1, and the denominator is 6, indicating that we have one part out of six equal parts. Similarly, in the fraction 4/9, the numerator is 4, and the denominator is 9, representing four parts out of nine equal parts.

Converting Mixed Numbers to Improper Fractions

To effectively subtract mixed numbers, a crucial initial step is to convert them into improper fractions. Improper fractions are fractions where the numerator is greater than or equal to the denominator. This conversion is necessary because it allows us to perform subtraction operations with greater ease. To convert a mixed number to an improper fraction, we follow a simple procedure:

  1. Multiply the whole number by the denominator of the fraction.
  2. Add the numerator of the fraction to the product obtained in step 1.
  3. Write the sum obtained in step 2 as the new numerator, keeping the original denominator.

Let's apply this procedure to our mixed numbers:

  • Converting 7 1/6 to an improper fraction:
    • Multiply the whole number (7) by the denominator (6): 7 * 6 = 42
    • Add the numerator (1) to the product: 42 + 1 = 43
    • Write the sum (43) as the new numerator, keeping the original denominator (6): 43/6
    • Therefore, 7 1/6 is equivalent to the improper fraction 43/6.
  • Converting 3 4/9 to an improper fraction:
    • Multiply the whole number (3) by the denominator (9): 3 * 9 = 27
    • Add the numerator (4) to the product: 27 + 4 = 31
    • Write the sum (31) as the new numerator, keeping the original denominator (9): 31/9
    • Therefore, 3 4/9 is equivalent to the improper fraction 31/9.

Now that we have successfully converted both mixed numbers into improper fractions, we can proceed with the subtraction operation.

Finding a Common Denominator

Before we can subtract fractions, they must have a common denominator. A common denominator is a shared denominator that allows us to combine or subtract fractions. To find a common denominator for 43/6 and 31/9, we need to determine the least common multiple (LCM) of their denominators, which are 6 and 9.

The LCM of two numbers is the smallest number that is divisible by both of them. There are several methods to find the LCM, but one common approach is to list the multiples of each number and identify the smallest multiple they share.

  • Multiples of 6: 6, 12, 18, 24, 30, 36, ...
  • Multiples of 9: 9, 18, 27, 36, 45, ...

From the lists above, we can see that the smallest multiple shared by both 6 and 9 is 18. Therefore, the LCM of 6 and 9 is 18, and this will be our common denominator.

Now that we have the common denominator, we need to convert each fraction to an equivalent fraction with the denominator of 18. To do this, we multiply both the numerator and denominator of each fraction by a factor that will result in the denominator becoming 18.

  • Converting 43/6 to an equivalent fraction with a denominator of 18:
    • To get a denominator of 18 from 6, we need to multiply by 3 (6 * 3 = 18).
    • Multiply both the numerator and denominator of 43/6 by 3: (43 * 3) / (6 * 3) = 129/18
    • Therefore, 43/6 is equivalent to 129/18.
  • Converting 31/9 to an equivalent fraction with a denominator of 18:
    • To get a denominator of 18 from 9, we need to multiply by 2 (9 * 2 = 18).
    • Multiply both the numerator and denominator of 31/9 by 2: (31 * 2) / (9 * 2) = 62/18
    • Therefore, 31/9 is equivalent to 62/18.

Now we have both fractions with a common denominator: 129/18 and 62/18. We can now proceed with the subtraction.

Subtracting the Fractions

With the fractions now sharing a common denominator, we can subtract them by simply subtracting their numerators while keeping the denominator the same. Therefore, subtracting 62/18 from 129/18, we get:

129/18 - 62/18 = (129 - 62) / 18 = 67/18

Converting the Improper Fraction to a Mixed Number

Our result, 67/18, is an improper fraction. While it is a valid answer, it is often preferred to express the answer as a mixed number. To convert an improper fraction to a mixed number, we divide the numerator by the denominator.

  • Divide 67 by 18: 67 ÷ 18 = 3 with a remainder of 13.

The quotient (3) becomes the whole number part of the mixed number, and the remainder (13) becomes the numerator of the fractional part. The denominator remains the same (18).

Therefore, 67/18 is equivalent to the mixed number 3 13/18.

Simplifying the Fraction (If Possible)

The final step in solving the problem is to check if the fractional part of the mixed number can be simplified. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be further reduced.

In our case, the fractional part is 13/18. To determine if it can be simplified, we need to find the greatest common factor (GCF) of 13 and 18. The GCF is the largest number that divides both numbers without leaving a remainder.

  • Factors of 13: 1, 13
  • Factors of 18: 1, 2, 3, 6, 9, 18

From the lists above, we can see that the only common factor of 13 and 18 is 1. Therefore, the fraction 13/18 is already in its simplest form and cannot be further reduced.

Final Answer

Therefore, the solution to the problem 7 1/6 - 3 4/9, in its simplest form, is 3 13/18.

In conclusion, subtracting mixed numbers involves a series of steps, including converting mixed numbers to improper fractions, finding a common denominator, subtracting the fractions, converting the improper fraction back to a mixed number, and simplifying the fraction if possible. By following these steps carefully, you can confidently solve subtraction problems involving mixed numbers.