3 10/15 Minus 1 12/15

Solving the Subtraction Problem: 3 10/15 - 1 12/15

This article will guide you through the process of solving the mixed number subtraction problem: 3 10/15 - 1 12/15. We'll break down the process step-by-step, explain the underlying mathematical principles, and address common misconceptions. Understanding how to subtract mixed numbers is a crucial skill in elementary mathematics and forms the foundation for more advanced algebraic concepts. This detailed explanation will ensure you not only understand how to solve this specific problem but also how to approach similar problems with confidence.

Understanding Mixed Numbers

Before diving into the subtraction, let's clarify what mixed numbers are. A mixed number combines a whole number and a fraction. In our problem, 3 10/15 and 1 12/15 are mixed numbers. The whole number represents the number of complete units, while the fraction represents a part of a whole unit.

Step-by-Step Solution: 3 10/15 - 1 12/15

The immediate challenge in this subtraction problem is that the fraction in the second mixed number (12/15) is larger than the fraction in the first mixed number (10/15). We can't directly subtract 12/15 from 10/15. This requires us to borrow from the whole number part.

Step 1: Borrowing from the Whole Number

We need to borrow one whole unit (1) from the 3 in the first mixed number. This "borrowed" unit is then converted into a fraction with the same denominator as the existing fractions (15 in this case). One whole unit is equal to 15/15.

Therefore, 3 10/15 becomes:

3 10/15 = 2 + 1 + 10/15 = 2 + 15/15 + 10/15 = 2 25/15

Step 2: Rewriting the Problem

Now our problem looks like this:

2 25/15 - 1 12/15

Step 3: Subtracting the Fractions

We can now subtract the fractions:

25/15 - 12/15 = 13/15

Step 4: Subtracting the Whole Numbers

Subtract the whole numbers:

2 - 1 = 1

Step 5: Combining the Result

Combine the results from Step 3 and Step 4 to get the final answer:

1 13/15

Simplifying Fractions: A Crucial Step

While 1 13/15 is a correct answer, it's always good practice to check if the resulting fraction can be simplified. In this case, 13 and 15 don't share any common factors other than 1, meaning the fraction is already in its simplest form.

Alternative Method: Converting to Improper Fractions

Another approach to solving this problem is to convert both mixed numbers into improper fractions before subtracting. An improper fraction has a numerator larger than or equal to its denominator.

Step 1: Converting to Improper Fractions

• Convert 3 10/15: (3 * 15) + 10 = 55. So, 3 10/15 becomes 55/15.
• Convert 1 12/15: (1 * 15) + 12 = 27. So, 1 12/15 becomes 27/15.

Step 2: Subtracting the Improper Fractions

Now subtract the improper fractions:

55/15 - 27/15 = 28/15

Step 3: Converting Back to a Mixed Number

The result, 28/15, is an improper fraction. To convert it back to a mixed number, divide the numerator (28) by the denominator (15):

28 ÷ 15 = 1 with a remainder of 13.

Therefore, 28/15 is equal to 1 13/15. This matches our previous answer.

The Importance of Common Denominators

Throughout this problem, we've maintained a common denominator of 15. This is crucial when adding or subtracting fractions. You can only directly add or subtract fractions that have the same denominator. If you encounter fractions with different denominators, you must first find a common denominator before proceeding with the calculation. This involves finding the least common multiple (LCM) of the denominators.

Practical Applications and Real-World Examples

Understanding mixed number subtraction is essential in various real-world scenarios. Imagine you're baking and a recipe calls for 3 10/15 cups of flour, but you've already used 1 12/15 cups. Knowing how to subtract mixed numbers will help you determine exactly how much flour you have left. Similarly, this skill is applicable in carpentry, construction, sewing, and many other fields where precise measurements are crucial.

Addressing Common Mistakes

Many students struggle with mixed number subtraction, often making these common mistakes:

• Not borrowing correctly: Failing to borrow a whole unit from the larger mixed number when the fraction part is smaller.
• Incorrectly converting between mixed numbers and improper fractions: Mistakes in the multiplication and addition steps during the conversion process.
• Ignoring common denominators: Attempting to subtract fractions with different denominators directly.

Frequently Asked Questions (FAQ)

Q: Can I use a calculator to solve this problem?

A: While calculators can certainly perform the calculation, it's highly beneficial to understand the underlying steps involved. This understanding will help you solve similar problems even without a calculator and develop a deeper grasp of mathematical principles.

Q: What if the fractions have different denominators?

A: You'll need to find a common denominator before you can subtract. Find the least common multiple (LCM) of the denominators and convert both fractions to have that common denominator.

Q: Is there only one way to solve this type of problem?

A: No, there are multiple approaches. You can choose the method that you find most comfortable and efficient. Both the borrowing method and the improper fraction method are valid and will lead to the correct answer.

Q: What if the result is a negative number?

A: In cases where the second mixed number is larger than the first, the result will be a negative mixed number. The same principles apply, but you will subtract the smaller number from the larger number and the result will have a negative sign.

Q: How can I practice solving more problems like this?

A: Practice is key! Look for online resources, workbooks, or educational apps that provide ample practice problems with mixed number subtraction. Gradually increase the difficulty of the problems to solidify your understanding.

Conclusion

Subtracting mixed numbers might seem challenging at first, but by understanding the fundamental principles of borrowing, converting to improper fractions, and working with common denominators, you can master this essential skill. Remember to always check your work and consider simplifying the resulting fraction. With consistent practice and attention to detail, you'll gain confidence in solving even more complex mathematical problems. The solution to 3 10/15 - 1 12/15 is 1 13/15, a result achieved through careful application of these fundamental mathematical concepts. This understanding forms a solid base for more advanced mathematical learning.