Essay Word Count: Expression For Total Words

by TextBrain Team 45 views

Let's break down how to create an expression that calculates the total number of words in a three-page essay, considering the word count variations on each page. This is a classic algebra problem that mixes word problems with expression building. We'll tackle this step by step, making sure everything is crystal clear. So, grab your thinking caps, folks, and let’s dive in!

Understanding the Problem

First, understanding the problem is key. We've got a 3-page essay. The number of words on each page is related, but not the same. We know:

  • Page 1 has M words.
  • Page 2 has 43 fewer words than Page 1.
  • Page 3 has one-third the number of words as Page 1.

Our mission is to find a single expression that represents the total number of words across all three pages. This means we need to represent the word count of each page mathematically and then combine them.

Breaking Down Each Page

Now, let's break down the word count for each page individually:

Page 1

  • This one's straightforward: Page 1 has M words. No calculations needed here!

Page 2

  • Page 2 has 43 fewer words than Page 1. "Fewer than" indicates subtraction. So, Page 2 has M - 43 words. Think of it like this: if Page 1 has 100 words, Page 2 would have 100 - 43 = 57 words.

Page 3

  • Page 3 contains one-third as many words as Page 1. "One-third" means we need to divide the number of words on Page 1 by 3 or multiply by 1/3. Therefore, Page 3 has (1/3) * M words, which can also be written as M/3.

Constructing the Expression

Alright, we've got the word count for each page. Now comes the exciting part: constructing the expression. To find the total number of words, we simply add the word counts of all three pages together.

Total words = Words on Page 1 + Words on Page 2 + Words on Page 3

Substituting the expressions we found earlier:

Total words = M + (M - 43) + (M/3)

This is the expression that represents the total number of words in the essay! We've successfully translated the word problem into an algebraic expression.

Simplifying the Expression (Optional but Recommended)

While the expression we have is correct, we can make it even cleaner by simplifying it. Combining like terms will give us a more compact and easier-to-understand expression.

  1. Combine the M terms: We have M + M + M/3. To add these, we need a common denominator. Think of the first two Ms as 1M and rewrite them with a denominator of 3: (3/3)*M + (3/3)*M + (1/3)*M
  2. Add the fractions: (3/3)*M + (3/3)*M + (1/3)*M = (3+3+1)/3 * M = (7/3)*M
  3. Rewrite the expression: Now our expression looks like this: (7/3)*M - 43

So, the simplified expression representing the total number of words is (7/3)*M - 43. This is a more concise way to represent the same calculation.

Putting It All Together

Let's recap what we've done. We started with a word problem describing the word count on each page of an essay. We then:

  1. Understood the problem: We identified the key information and what we needed to find.
  2. Broke down each page: We expressed the word count for each page using algebraic expressions.
  3. Constructed the expression: We combined the individual page expressions to create an expression for the total word count.
  4. Simplified the expression (optional): We made the expression cleaner and easier to use by combining like terms.

Now we have a powerful tool: an expression that allows us to calculate the total word count of the essay if we know the number of words on the first page (M). Guys, this is how you turn word problems into math problems and conquer them!

Why This Matters

You might be thinking, "Okay, cool, we made an expression. But why is this actually useful?" Well, understanding how to translate real-world scenarios into mathematical expressions is a fundamental skill in algebra and beyond. This skill is super important because:

  • Problem Solving: It helps you break down complex problems into smaller, manageable parts. Instead of being overwhelmed by a big chunk of text, you can focus on individual pieces of information and how they relate.
  • Critical Thinking: It forces you to think critically about the relationships between different quantities. You're not just plugging in numbers; you're understanding the logic behind the calculations.
  • Mathematical Modeling: It's the foundation for mathematical modeling, which is used in everything from engineering to finance to predict outcomes and make decisions.
  • Real-World Applications: Many real-world problems can be solved using algebraic expressions. For example, you could use a similar approach to calculate the cost of a project, the distance traveled on a trip, or the amount of ingredients needed for a recipe.

Let's Look at Some Examples

To really nail this down, let's look at a couple of examples where we plug in a value for M and see how the expression works.

Example 1: M = 150

Suppose Page 1 has 150 words (M = 150). Let's use our simplified expression (7/3)*M - 43 to find the total word count.

  1. Substitute M: (7/3) * 150 - 43
  2. Multiply: (7 * 150) / 3 - 43 = 1050 / 3 - 43
  3. Divide: 350 - 43
  4. Subtract: 307

So, if Page 1 has 150 words, the total word count for the essay is 307 words.

Example 2: M = 90

Now let's try a smaller number. Suppose Page 1 has 90 words (M = 90).

  1. Substitute M: (7/3) * 90 - 43
  2. Multiply: (7 * 90) / 3 - 43 = 630 / 3 - 43
  3. Divide: 210 - 43
  4. Subtract: 167

If Page 1 has 90 words, the total word count for the essay is 167 words. See how easy it is to use the expression once you've created it?

Common Mistakes to Avoid

When working with expressions like this, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Incorrectly Translating Words: The biggest mistake is misinterpreting the words in the problem. For example, confusing "fewer than" with addition instead of subtraction, or not understanding what "one-third" means mathematically. Always double-check your translations.
  • Forgetting Parentheses: When substituting expressions, especially those with subtraction, make sure to use parentheses. For example, Page 2 has M - 43 words. If you're adding this to the expression, you need to write M + (M - 43), not M + M - 43. The parentheses ensure that the subtraction is done correctly.
  • Not Combining Like Terms: While the initial expression is correct, simplifying it makes it much easier to use. Don't skip the step of combining like terms!
  • Arithmetic Errors: Simple arithmetic errors can throw off your entire calculation. Take your time and double-check your work, especially when multiplying and dividing fractions.

Tips for Success

Want to become a pro at solving these types of problems? Here are a few tips for success:

  • Read Carefully: Read the problem slowly and carefully, making sure you understand every word and phrase. Highlight or underline key information.
  • Break It Down: Break the problem down into smaller steps. Don't try to do everything at once. Focus on one piece of information at a time.
  • Write It Out: Write out each step clearly and logically. This makes it easier to follow your work and spot any errors.
  • Check Your Work: After you've solved the problem, take a few minutes to check your work. Does your answer make sense in the context of the problem?
  • Practice, Practice, Practice: The more you practice, the better you'll become. Work through lots of examples and try different types of problems.

Conclusion

Creating expressions from word problems can seem daunting at first, but with a systematic approach and a little practice, you can master it. Remember to break the problem down, translate the words into math, construct the expression, and simplify if possible. By following these steps, you'll be able to tackle any word problem that comes your way. So keep practicing, guys, and you'll be expression-building pros in no time!