Solve For Unknowns: 75% Of What Number Is 150?

When faced with a word problem involving percentages, it's natural to wonder, "What equation can be used to solve the problem?" Specifically, when you encounter a question like "75 percent of what number is 150?", the key is to translate the words into a mathematical expression. This article will guide you through understanding and solving this type of percentage problem, exploring the options provided and explaining the underlying mathematical principles. We'll break down how to set up the equation correctly and why certain approaches work better than others. Get ready to demystify percentage problems and build your confidence in tackling them!

Understanding the Components of a Percentage Problem

Let's start by dissecting the problem: "75 percent of what number is 150?" Each part of this sentence carries specific mathematical meaning. The phrase "75 percent" translates directly to the fraction rac{75}{100} or the decimal 0.75. The word "of" in mathematics almost always signifies multiplication. The phrase "what number" is our unknown quantity, which we can represent with a variable, often 'x'. Finally, "is" translates to the equals sign (=). So, we can rephrase the problem as: $0.75 imes x = 150$ or rac{75}{100} imes x = 150. This fundamental translation is the crucial first step in solving any percentage word problem. Without this accurate conversion, even the most sophisticated algebraic manipulation will lead to an incorrect answer. It's about building a solid foundation of understanding the language of mathematics and how it applies to real-world scenarios. Think of it as learning a new language; once you understand the grammar and vocabulary, you can start constructing meaningful sentences and comprehending complex ideas. In this case, the 'grammar' involves recognizing that 'percent' means 'out of one hundred,' 'of' means 'multiply,' and 'is' means 'equals.' By mastering this translation, you empower yourself to tackle a wide array of percentage-related challenges, from calculating discounts and tips to understanding financial statements and statistical data. The beauty of mathematics lies in its universality, and this principle of translation applies across various contexts, making it a truly invaluable skill to develop. We are essentially building a bridge between the abstract world of numbers and the concrete language we use every day.

Analyzing the Equation Options

Now, let's examine the given options and see which one accurately represents the problem and leads to a solution. We are looking for an equation that, when solved for 'x' (our unknown number), will give us the correct answer. Remember, our goal is to find the number that, when multiplied by 75% (or 0.75), equals 150.

Option A: rac{75 imes 1}{150 imes 1}=rac{75}{150} This option seems to be setting up a ratio, but it doesn't directly translate our problem. If we were to try and solve for 'x' using this, it's unclear how 'x' would be incorporated or what it would represent. The structure doesn't align with the "part over whole equals percent over 100" or the "decimal times unknown equals total" format we established. It appears to be manipulating the numbers 75 and 150 without a clear connection to the unknown 'what number.' This is a common pitfall – getting caught up in numerical manipulation without first establishing the correct relationship. The equation needs to reflect that 75% (or 0.75) is the rate, 'x' is the base (the number we're looking for), and 150 is the amount or the part. Option A doesn't show this relationship. It's essential to always go back to the translated sentence: 0.75 * x = 150. Option A does not isolate 'x' in a way that reflects this original equation.

Option B: rac{150 imes 2}{75 imes 2}=rac{300}{150} Similar to Option A, this option also presents a ratio. While the numbers 75 and 150 are present, the multiplication by 2 on both the numerator and denominator is arbitrary in the context of our problem. It doesn't introduce or isolate the unknown 'what number.' We are trying to find a value for 'x' such that when 75% is applied to it, we get 150. Option B does not set up an equation where 'x' is the variable we need to solve for. It seems to be performing an unrelated calculation. The core issue here is that the equation must contain the unknown 'x' as part of the setup that mirrors the original problem statement. This option, like A, doesn't achieve that. The multiplication by 2 looks like an attempt to scale the numbers, perhaps to find a common denominator or simplify a fraction, but it doesn't directly address the relationship $0.75 imes x = 150$. Therefore, it's not the correct equation to represent our percentage problem.

Option C: rac{200 imes 2}{75 imes 2}=rac{400}{150} This option introduces even more numbers (200 and 2) that are not present in the original problem statement. The structure remains a ratio, and there is still no clear representation of the unknown 'what number.' It appears to be a completely unrelated calculation or a misinterpretation of how to set up a proportion. The numbers chosen here do not logically stem from the given information: "75 percent of what number is 150." When evaluating mathematical options, always return to the problem's core question and the translated equation. Does this option help you find 'x' in the equation $0.75 imes x = 150$? Clearly, it does not. The introduction of '200' is particularly baffling, as it has no grounding in the problem. This highlights the importance of carefully reading and understanding each component of a mathematical problem before attempting to construct or evaluate solutions. This option is a distraction and does not represent a valid approach to solving the stated problem.

Let's consider the correct way to set up the equation to solve for the unknown number.

We know:

• Percent: 75% = 0.75
• "Of" means multiply
• "What number" is our unknown, let's call it $x$
• "Is" means equals
• The result is 150

So, the equation is: $0.75 imes x = 150$

To solve for $x$, we need to isolate it. We do this by dividing both sides of the equation by 0.75:

x = rac{150}{0.75}

Now, let's calculate this value:

$x = 200$

So, 75 percent of 200 is 150.

While none of the provided options directly show the solution x = rac{150}{0.75} or $x=200$, we can infer how the problem might be intended to be solved using proportions, which is a common method for percentage problems.

A proportion compares two ratios. We know that 75 percent is equivalent to the ratio rac{75}{100}. We are looking for a number ($x$) such that 150 is 75 percent of it. This can be set up as:

rac{ ext{part}}{ ext{whole}} = rac{ ext{percent}}{100}

In our case, the 'part' is 150, the 'percent' is 75, and the 'whole' is our unknown number ($x$).

So, the proportion is:

rac{150}{x} = rac{75}{100}

To solve for $x$, we can cross-multiply:

$150 imes 100 = 75 imes x$

$15000 = 75x$

Now, divide both sides by 75:

x = rac{15000}{75}

$x = 200$

This confirms our previous calculation. Let's re-examine the options to see if any of them could be interpreted as a step towards this proportional setup or a related concept, even if not perfectly expressed. It's possible the options are designed to test understanding of ratios and equivalence. However, based on a strict interpretation of setting up the equation to solve for 'x', none of the provided options A, B, or C are correct. They do not isolate the unknown 'x' or represent the initial problem accurately. The correct approach involves setting up either $0.75x = 150$ or the proportion rac{150}{x} = rac{75}{100}.

Conclusion: Finding the Right Equation

Successfully solving percentage problems hinges on accurate translation from words to mathematical equations. For the problem "75 percent of what number is 150?", the core equation to solve is $0.75x = 150$, or when using proportions, rac{150}{x} = rac{75}{100}. Both of these correctly set up the relationship between the known percentage, the unknown number, and the resulting value. After performing the necessary algebraic steps (isolating $x$ by dividing 150 by 0.75, or cross-multiplying and then dividing), we find that the unknown number is 200. It's essential to critically evaluate each provided option, ensuring it logically represents the problem and provides a clear path to finding the unknown. None of the given options (A, B, C) accurately reflect this fundamental setup. They appear to be unrelated calculations or misinterpretations of how to form an equation or proportion. Always remember to break down the word problem into its core mathematical components: identify the percentage, the unknown quantity, and the result, then use the appropriate operations and symbols to form your equation. This systematic approach will ensure you choose the correct method and arrive at the accurate answer.

For more insights into solving percentage problems and understanding mathematical concepts, you can explore resources from reputable educational websites. A great place to start is the Khan Academy, which offers comprehensive lessons and practice exercises on a wide range of mathematical topics, including percentages and algebra. You can also find valuable information and additional problem-solving strategies on the National Council of Teachers of Mathematics (NCTM) website, which is dedicated to promoting excellence in mathematics teaching and learning.