Exploring Mathematical Expressions: Understanding the Value of 10 - 5 × 7 × 6

Mathematics is the language of the universe, and understanding mathematical expressions is crucial for solving a wide range of problems. One common expression that often confuses students is 10 - 5 × 7 × 6. This article will guide you through the process of solving this expression step by step, using the order of operations (PEMDAS or BODMAS).

Understanding the Order of Operations

When solving mathematical expressions, it's essential to follow a specific order. The two most commonly used mnemonics are PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) and BODMAS (Brackets, Orders, Division and Multiplication from left to right, Addition and Subtraction from left to right). Both mnemonics convey the same set of rules but differ slightly in their order of mentioning subtraction and addition.

Solving the Expression: 10 - 5 × 7 × 6

Let's break down the expression 10 - 5 × 7 × 6 and solve it step by step.

Step 1: Identify the order of operations - According to both PEMDAS and BODMAS, we start with multiplication and division from left to right, followed by addition and subtraction from left to right. Step 2: Perform the multiplication

First, we need to perform the multiplication in the expression:

5 × 7 × 6

Let's break it down into two steps:

Multiplication 1: 5 × 7 35 Multiplication 2: 35 × 6 210

So, 5 × 7 × 6 210.

Step 3: Substitute the result into the original expression

Now, substitute 210 back into the original expression:

10 - 210

Step 4: Perform the subtraction

Finally, perform the subtraction:

10 - 210  -200

Therefore, the value of the expression 10 - 5 × 7 × 6 is -200.

Common Misconceptions

Why some might think the answer is 12

Many might mistakenly think the answer is 12, which is the result of evaluating the expression as a whole before applying the operations. This is wrong because, following the order of operations, we must perform the multiplication before the subtraction. This misunderstanding highlights the importance of adhering to the correct order of operations in mathematics.

Exterior example: A misapplied method

For example, if someone were to add 10 and 5 first:

Step 1: Incorrect addition

10   5  15

Step 2: Incorrect multiplication and subtraction

15 × 7 × 6  630
15 - 630  -615

This is incorrect because it disregards the established order of operations. Only after performing the multiplication as per the correct order do we get the right answer, which is -200.

Conclusion

The key to correctly solving mathematical expressions is to follow the order of operations, whether it's PEMDAS or BODMAS. Understanding and applying these rules ensures accurate results and helps in solving more complex mathematical problems. Always remember, multiplication and division take precedence over addition and subtraction.

If you're interested in learning more about mathematics or need help with similar expressions, check out our resources on this website for more tips and practice problems.