Solving For N: 85%+1 5/6 X 16/9 = N

by Tim Redaksi 36 views
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Alright guys, let's break down this math problem step by step! The equation we're tackling is 85%+1 5/6 x 16/9 = n. Our goal? To find the value of n. This involves understanding percentages, mixed numbers, and fractions, and then applying the correct order of operations.

First things first: Let's convert that percentage into a decimal. 85% is the same as 85 out of 100, so we can write it as 0.85. This makes it easier to work with in the equation. Next up, we've got a mixed number: 1 5/6. To make things simpler, let's convert this into an improper fraction. We do this by multiplying the whole number (1) by the denominator (6), and then adding the numerator (5). This gives us (1 * 6) + 5 = 11. So, 1 5/6 is the same as 11/6. Now our equation looks like this: 0.85 + 11/6 x 16/9 = n.

Now comes the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our equation, we have addition and multiplication. According to PEMDAS, we need to do the multiplication before the addition. So, we'll multiply 11/6 by 16/9. To multiply fractions, we simply multiply the numerators together and the denominators together. That gives us (11 * 16) / (6 * 9) = 176/54. We can simplify this fraction by finding the greatest common divisor (GCD) of 176 and 54. Both numbers are divisible by 2, so we can simplify the fraction to 88/27. So, the equation is now: 0.85 + 88/27 = n. We are getting closer to the solution. To add 0.85 and 88/27, we need to convert 88/27 to a decimal. 88 divided by 27 is approximately 3.259. Now our equation is: 0.85 + 3.259 = n. Adding these two numbers together, we get 4.109. Therefore, n = 4.109.

Diving Deeper: Understanding the Math Concepts

To really nail this down, let's explore the core math concepts involved. Understanding these concepts will make similar problems a breeze in the future.

Percentages to Decimals

Percentages are a way of expressing a number as a fraction of 100. The word "percent" actually means "per hundred." So, when we say 85%, we mean 85 out of every 100. To convert a percentage to a decimal, we simply divide it by 100. For example:

  • 50% = 50/100 = 0.5
  • 25% = 25/100 = 0.25
  • 120% = 120/100 = 1.2

In our problem, 85% becomes 0.85. This conversion is crucial because decimals are much easier to work with in calculations than percentages. Trying to directly multiply or add a percentage can get messy, so converting to a decimal is always the way to go.

Mixed Numbers to Improper Fractions

A mixed number is a combination of a whole number and a proper fraction (where the numerator is less than the denominator). In our equation, we had the mixed number 1 5/6. Converting mixed numbers to improper fractions makes multiplication and division much easier. The process involves multiplying the whole number by the denominator of the fraction, adding the numerator, and then placing the result over the original denominator. So, for 1 5/6:

  • Multiply the whole number (1) by the denominator (6): 1 * 6 = 6
  • Add the numerator (5): 6 + 5 = 11
  • Place the result (11) over the original denominator (6): 11/6

Therefore, 1 5/6 is equal to 11/6. This conversion allows us to perform fraction operations smoothly.

Order of Operations (PEMDAS/BODMAS)

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. This ensures that everyone arrives at the same answer for a given equation. The most common acronyms to remember the order are PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). The key is that Multiplication and Division are done from left to right, as are Addition and Subtraction. In our equation, we had addition and multiplication. According to PEMDAS, we perform multiplication before addition.

Understanding the order of operations is critical for solving any mathematical equation correctly. Ignoring this order can lead to drastically wrong answers. For example, if we had added 0.85 and 11/6 before multiplying by 16/9, we would have ended up with a completely different result.

Step-by-Step Solution Breakdown

Let's recap the entire solution step by step to make sure we've got it all down:

  1. Convert the percentage to a decimal: 85% = 0.85
  2. Convert the mixed number to an improper fraction: 1 5/6 = 11/6
  3. Rewrite the equation: 0.85 + 11/6 x 16/9 = n
  4. Perform the multiplication: (11/6) x (16/9) = 176/54
  5. Simplify the fraction: 176/54 = 88/27
  6. Convert the fraction to a decimal: 88/27 ≈ 3.259
  7. Rewrite the equation: 0.85 + 3.259 = n
  8. Perform the addition: 0.85 + 3.259 = 4.109
  9. Therefore, n = 4.109

By following these steps carefully, we can confidently solve the equation and find the value of n.

Common Mistakes and How to Avoid Them

Even with a good understanding of the concepts, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Forgetting the order of operations: This is the most common mistake. Always remember PEMDAS/BODMAS. If you are not using the correct order, you will get the incorrect answer. Always do multiplication and division before addition and subtraction.
  • Incorrectly converting mixed numbers: Make sure you multiply the whole number by the denominator before adding the numerator. A lot of people can rush through the steps and make a mistake.
  • Calculator errors: When using a calculator, be careful to enter the numbers and operations correctly. Double-check your entries to avoid simple mistakes.
  • Rounding errors: If you round numbers during intermediate steps, it can affect the final answer. Try to keep as many decimal places as possible until the very end.

By being aware of these common mistakes, you can minimize your chances of making them.

Practice Problems

To really solidify your understanding, try solving these similar problems:

  1. 60% + 2 1/4 x 8/5 = n
  2. 125% - 1 2/3 x 9/4 = n
  3. 75% + 3 1/2 x 10/7 = n

Work through these problems step by step, paying attention to the order of operations and the conversions. Check your answers with a calculator to make sure you're on the right track.

Conclusion

So, there you have it! We've successfully solved for n in the equation 85%+1 5/6 x 16/9 = n. The key takeaways are understanding percentages, mixed numbers, fractions, and the order of operations. By mastering these concepts and avoiding common mistakes, you'll be well-equipped to tackle similar math problems with confidence. Keep practicing, and you'll become a math whiz in no time!