Evaluating The Expression 1 1/4 - 6 + 2 × 1 1/2: A Step-by-Step Guide

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This article provides a step-by-step guide on how to evaluate the mathematical expression 1 1/4 - 6 + 2 × 1 1/2. We will break down the process, explaining each step in detail, and ensure you understand the order of operations involved. This type of problem is common in basic algebra and arithmetic, and mastering it will enhance your problem-solving skills.

Understanding the Order of Operations

Before we begin, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order dictates the sequence in which we perform mathematical operations to arrive at the correct answer. Ignoring this order can lead to incorrect results, so it's essential to follow it diligently. In our expression, we have a mix of subtraction, addition, and multiplication. According to PEMDAS, we must perform the multiplication first before handling addition and subtraction.

Step 1: Convert Mixed Numbers to Improper Fractions

To effectively work with mixed numbers in mathematical expressions, it's often best to convert them into improper fractions. This simplifies the calculations, especially when dealing with multiplication and division. Let's convert the mixed numbers in our expression:

  • 1 1/4 can be converted to an improper fraction by multiplying the whole number (1) by the denominator (4) and then adding the numerator (1). This gives us (1 * 4) + 1 = 5. We then place this result over the original denominator, resulting in 5/4.
  • Similarly, 1 1/2 can be converted to an improper fraction. Multiplying the whole number (1) by the denominator (2) and adding the numerator (1) gives us (1 * 2) + 1 = 3. Placing this over the original denominator, we get 3/2.

Now, our expression looks like this: 5/4 - 6 + 2 × 3/2.

Step 2: Perform Multiplication

According to the order of operations (PEMDAS), multiplication comes before addition and subtraction. In our expression, we have 2 × 3/2. To multiply a whole number by a fraction, you can think of the whole number as a fraction with a denominator of 1. So, 2 can be written as 2/1. Now, we multiply the numerators (2 * 3) and the denominators (1 * 2):

(2/1) × (3/2) = (2 * 3) / (1 * 2) = 6/2

Simplifying the fraction 6/2, we divide both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:

6/2 = 3

Now, our expression becomes:

5/4 - 6 + 3

Step 3: Perform Addition and Subtraction from Left to Right

Now that we've handled the multiplication, we are left with addition and subtraction. According to PEMDAS, we perform these operations from left to right. So, we first subtract 6 from 5/4.

Step 3a: 5/4 - 6

To subtract a whole number from a fraction, we need to convert the whole number into a fraction with the same denominator as the other fraction. In this case, the denominator is 4. So, we convert 6 into a fraction with a denominator of 4:

6 = 6/1 = (6 * 4) / (1 * 4) = 24/4

Now we can perform the subtraction:

5/4 - 24/4 = (5 - 24) / 4 = -19/4

Step 3b: -19/4 + 3

Next, we add 3 to -19/4. Again, we need to convert the whole number into a fraction with the same denominator, which is 4:

3 = 3/1 = (3 * 4) / (1 * 4) = 12/4

Now we can perform the addition:

-19/4 + 12/4 = (-19 + 12) / 4 = -7/4

Step 4: Convert Improper Fraction to Mixed Number (Optional)

The result we have is -7/4, which is an improper fraction because the numerator is greater than the denominator. To convert it back to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and we keep the original denominator.

Dividing 7 by 4, we get a quotient of 1 and a remainder of 3. So, -7/4 can be written as:

-7/4 = -1 3/4

Therefore, the final answer is -1 3/4.

Alternative Method: Converting All Terms to Fractions

Another approach to solving this problem is to convert all terms into fractions right from the start. This can sometimes make the process more straightforward, especially for those who are more comfortable working with fractions.

Step 1: Convert Mixed Numbers and Whole Numbers to Improper Fractions

  • We already converted 1 1/4 to 5/4 and 1 1/2 to 3/2.
  • We convert the whole number 6 to a fraction with a denominator of 1: 6 = 6/1.

So, our expression becomes:

5/4 - 6/1 + 2 × 3/2

Step 2: Perform Multiplication

As before, we multiply 2 by 3/2:

2 × 3/2 = (2/1) × (3/2) = 6/2 = 3/1

Now the expression is:

5/4 - 6/1 + 3/1

Step 3: Find a Common Denominator

To add and subtract fractions, they must have a common denominator. The least common denominator (LCD) for 4 and 1 is 4. So, we need to convert all fractions to have a denominator of 4:

  • 5/4 already has a denominator of 4.
  • 6/1 = (6 * 4) / (1 * 4) = 24/4
  • 3/1 = (3 * 4) / (1 * 4) = 12/4

Our expression now looks like this:

5/4 - 24/4 + 12/4

Step 4: Perform Addition and Subtraction

Now we can add and subtract the fractions:

(5 - 24 + 12) / 4 = (5 - 12) / 4 = -7/4

Step 5: Convert Improper Fraction to Mixed Number (Optional)

As before, we convert -7/4 to a mixed number:

-7/4 = -1 3/4

This method yields the same result, -1 3/4, providing a confirmation of our initial approach.

Common Mistakes to Avoid

When evaluating mathematical expressions, it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for:

  1. Ignoring the Order of Operations: This is the most common mistake. Always follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to ensure you perform the operations in the correct order.
  2. Incorrectly Converting Mixed Numbers: When converting mixed numbers to improper fractions, double-check your calculations. A small error here can throw off the entire problem.
  3. Forgetting to Find a Common Denominator: When adding or subtracting fractions, you must have a common denominator. Forgetting this step will lead to incorrect results.
  4. Making Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can happen, especially when dealing with negative numbers or fractions. Take your time and double-check your work.
  5. Not Simplifying Fractions: Always simplify your fractions to their lowest terms. This makes the final answer cleaner and easier to understand.

Real-World Applications

Understanding how to evaluate mathematical expressions is not just an academic exercise; it has numerous real-world applications. Here are a few examples:

  • Cooking and Baking: Recipes often involve fractions and require you to adjust quantities. Knowing how to work with fractions and mixed numbers is essential for scaling recipes up or down.
  • Financial Calculations: Calculating interest, taxes, and discounts often involves evaluating expressions with fractions and decimals. For example, calculating a 15% tip on a restaurant bill requires you to multiply the bill amount by 0.15 (which is equivalent to 15/100).
  • Home Improvement: Many home improvement projects involve measurements in feet and inches, which are often expressed as fractions. Whether you're measuring for new flooring or calculating the amount of paint you need, you'll use your knowledge of fractions.
  • Engineering and Construction: Engineers and construction workers use mathematical expressions to calculate dimensions, angles, and material quantities. Accuracy is crucial in these fields, making a solid understanding of order of operations and fraction manipulation essential.
  • Computer Programming: In programming, mathematical expressions are used extensively for calculations, data manipulation, and algorithm design. Knowing how to evaluate expressions correctly is vital for writing efficient and bug-free code.

Practice Problems

To solidify your understanding of evaluating expressions, here are a few practice problems:

  1. 2 1/2 + 3 × 1/4 - 1
  2. 5 - 1 1/3 + 2 × 3/4
  3. 1/2 × (4 + 2 1/2) - 1/4
  4. 3 3/4 - 2 × 1/2 + 1 1/4

Try solving these problems on your own, following the steps we've outlined in this article. Check your answers against the solutions provided below.

Conclusion

Evaluating mathematical expressions accurately requires a clear understanding of the order of operations and proficiency in working with fractions and mixed numbers. By following the steps outlined in this article and practicing regularly, you can master this fundamental skill. Remember to always prioritize multiplication before addition and subtraction, convert mixed numbers to improper fractions for easier calculation, and double-check your work to avoid common mistakes. Whether you're solving problems in the classroom or tackling real-world challenges, a solid grasp of evaluating expressions will serve you well.

By breaking down the problem into manageable steps and understanding the underlying principles, you can confidently approach similar mathematical challenges. Keep practicing, and you'll see your skills improve over time.

Solutions to Practice Problems:

  1. 2 1/2 + 3 × 1/4 - 1 = 2 3/4
  2. 5 - 1 1/3 + 2 × 3/4 = 5 1/6
  3. 1/2 × (4 + 2 1/2) - 1/4 = 3
  4. 3 3/4 - 2 × 1/2 + 1 1/4 = 4