Solve It: 20% More Than 20% Less Problem Explained

Hey guys! Ever stumbled upon a math problem that felt like deciphering a secret code? Today, we're going to crack one of those codes together. Our mission, should we choose to accept it (and we totally do!), is to find a number that, after some percentage play, lands us squarely at 144. Sounds intriguing, right? Let's dive in!

The Puzzle: Decoding the Percentage Problem

The heart of our challenge lies in understanding how percentages work their magic. The problem throws a curveball by involving two percentage changes – an increase and a decrease – before revealing the final number. This is where many get tripped up, but fear not! We're going to break it down step by step, making it as clear as crystal.

Keywords are important here guys, make sure we understand them. When we see “20% more than,” we know we're adding 20% of the original number to itself. Think of it as the number growing a little. On the flip side, “20% less than” means we're subtracting 20% from the number – it's shrinking slightly. The key is that these percentages are calculated based on different values. The first 20% decrease is from the original number, while the 20% increase is from the new number after the decrease.

To truly grasp this, let's imagine we're dealing with 100 as our original number (just for illustration purposes). If we decrease 100 by 20%, we get 80. Now, if we increase 80 by 20%, we don't end up back at 100! We end up at 96 (20% of 80 is 16, and 80 + 16 = 96). See how the order and the base value matter? This is the essence of the puzzle we're solving.

So, our mission is to reverse-engineer this process. We know the final number is 144, and we need to figure out the original number that went through this percentage wringer. To do this, we'll need to translate the words into math – the language of problem-solving!

Translating Words into Math: The Equation Unveiled

Alright, let's arm ourselves with the most powerful tool in our problem-solving arsenal: the equation. Equations are like secret decoders, turning word problems into solvable mathematical expressions. In our case, we need to represent the series of percentage changes using mathematical symbols.

First, let's call our mystery number “x”. This is a common trick in algebra – using a letter to stand in for the unknown. Now, let's tackle the first part of the problem: “20% less than a number.” To decrease x by 20%, we're essentially finding 80% of x (because 100% - 20% = 80%). Mathematically, we write this as 0.80x. Remember, percentages are just fractions in disguise (20% is the same as 20/100, which simplifies to 0.20), and multiplying by 0.80 gives us 80% of the number.

Next up is the “20% more than” part. This means we're increasing the result of the previous step (0.80x) by 20%. So, we're finding 120% of 0.80x (because 100% + 20% = 120%). We write this as 1.20 * (0.80x). We use parentheses to show that we're applying the 20% increase to the result of the 20% decrease.

The final piece of the puzzle is knowing that this whole process equals 144. We can now write our complete equation: 1. 20 * (0.80x) = 144. This equation is the key to unlocking the mystery number. It's a concise mathematical representation of the word problem, and now we can use our algebraic skills to solve for x.

Solving for 'x': Unmasking the Original Number

With our equation in hand, we're ready to roll up our sleeves and solve for “x.” This is where the magic of algebra comes into play, guys. Our goal is to isolate “x” on one side of the equation, revealing its true value.

Let's start by simplifying the left side of the equation. We have 1.20 * (0.80x). Multiplication is associative, meaning we can multiply the numbers in any order. So, let's multiply 1.20 and 0.80 first. Using a calculator (or good old-fashioned multiplication), we find that 1.20 * 0.80 = 0.96. Our equation now looks like this: 0. 96x = 144.

We're getting closer! Now, we need to get “x” all by itself. It's currently being multiplied by 0.96. To undo this multiplication, we'll perform the opposite operation: division. We'll divide both sides of the equation by 0.96. Remember, whatever we do to one side of an equation, we must do to the other to keep things balanced. This is a fundamental principle of algebra.

Dividing both sides by 0.96, we get: x = 144 / 0.96. Now, it's calculator time again! 144 divided by 0.96 is 150. Eureka! We've found our mystery number. x = 150. This means the original number that satisfies the conditions of the problem is 150.

But wait, we're not done yet! A good problem-solver always checks their work. Let's make sure 150 actually fits the problem's description.

Double-Checking Our Solution: The Proof is in the Numbers

We've found a potential solution, but we're not going to take it on faith alone, guys. Let's put our answer to the test and make sure it holds up under scrutiny. This is a crucial step in problem-solving – verifying your results to avoid any silly mistakes.

Our claim is that 150, when decreased by 20% and then increased by 20%, will equal 144. Let's put it to the test! First, we decrease 150 by 20%. 20% of 150 is (0.20 * 150) = 30. Subtracting this from 150, we get 150 - 30 = 120. So far, so good.

Next, we increase this new number (120) by 20%. 20% of 120 is (0.20 * 120) = 24. Adding this to 120, we get 120 + 24 = 144. Boom! It works! Our calculations check out, and 150 is indeed the correct answer.

This process of checking our work is so important, guys. It's like having a built-in safety net. It catches any errors we might have made along the way, whether it's a simple arithmetic mistake or a misunderstanding of the problem. Plus, it gives us that satisfying feeling of confidence, knowing that we've not only solved the problem but also verified our solution.

The Takeaway: Mastering the Percentage Puzzle

We've conquered a tricky percentage problem, and in doing so, we've reinforced some valuable problem-solving skills. Let's recap the key steps we took, so you can apply them to future challenges:

1. Understand the Problem: We started by carefully reading the problem and identifying the core challenge – finding a number that satisfies a specific set of percentage changes. We paid close attention to the order of operations and the different base values for the percentages.
2. Translate Words into Math: We transformed the word problem into a mathematical equation. This involved representing the unknown number with a variable (“x”) and expressing the percentage changes using decimal multiplication. The equation 1. 20 * (0.80x) = 144 was our key to unlocking the solution.
3. Solve the Equation: We used algebraic techniques to isolate “x” and find its value. This involved simplifying the equation, performing inverse operations (division to undo multiplication), and keeping the equation balanced.
4. Double-Check the Solution: We verified our answer by plugging it back into the original problem and ensuring that it satisfied the given conditions. This crucial step helped us catch any potential errors and build confidence in our result.

By following these steps, you'll be well-equipped to tackle all sorts of percentage problems and other mathematical challenges that come your way. Remember, guys, math isn't about memorizing formulas – it's about understanding the underlying concepts and applying them creatively. Keep practicing, keep exploring, and keep those problem-solving gears turning! You got this!