Cálculo De Porcentajes: Descuentos Y Aumentos

Hey guys! Today we're diving into the cool world of percentages, specifically how to figure out the rate of change and the final amount when prices go up or down. This stuff is super useful, whether you're trying to snag a deal on a new washing machine or just trying to understand your paycheck. We'll tackle a couple of scenarios to make it crystal clear. So, grab your calculators (or just your brains!), and let's get this percentage party started!

Understanding Percentage Changes

Before we jump into the nitty-gritty examples, let's get a solid grasp on what percentage changes actually mean. When we talk about a percentage change, we're essentially looking at how much something has increased or decreased relative to its original value. It's expressed as a percentage, which is just a fraction out of 100. So, a 10% increase means the value went up by 10 parts out of every 100, and a 15% decrease means it went down by 15 parts out of every 100.

Calculating the Amount of Change: To find out how much the value changed, you multiply the original amount by the percentage change (expressed as a decimal). For example, if something costs $500 and it's discounted by 20%, the discount amount is $500 \times 0.20 = $100$.

Calculating the Final Amount: Once you know the amount of change, you can find the final amount. For a decrease, you subtract the discount amount from the original amount ($500 - $100 = $400$). For an increase, you add the increase amount to the original amount.

A Quicker Way: Using Multipliers: There's a super handy shortcut! Instead of calculating the change and then adding/subtracting, you can use multipliers. If something is discounted by 12%, it means you're paying 100% - 12% = 88% of the original price. So, the multiplier is 0.88. If something increases by 21%, you're paying 100% + 21% = 121% of the original price, making the multiplier 1.21. This method is especially awesome when you have multiple percentage changes in a row, as we'll see in our examples.

Keep these concepts in mind, guys, because they're the bedrock of solving these percentage puzzles. Understanding the relationship between the original value, the percentage change, and the final value is key to mastering this topic.

Scenario A: The Washing Machine Deal

Alright, let's dive into our first scenario, which involves a washing machine. Imagine you've got your eye on a brand-new washing machine that originally costs 300 €. It's on sale, which is always exciting! First, the price is reduced by 12%. Then, as if that wasn't good enough, it's increased by 21%. Your mission, should you choose to accept it, is to find the final price of this washing machine after these two changes.

Step 1: Calculate the first price change (the 12% reduction).

We start with the original price of 300 €.

• Method 1: Calculate the discount amount and subtract.
  • The discount is 12% of 300 €.
  • Discount amount = $300 \times \frac{12}{100} = 300 \times 0.12 = 36 €$.
  • The price after the discount is $300 € - 36 € = 264 €$.
• Method 2: Use the multiplier.
  • If the price is reduced by 12%, you're paying 100% - 12% = 88% of the original price.
  • The multiplier is 0.88.
  • Price after discount = $300 € \times 0.88 = 264 €$.

As you can see, both methods give us the same intermediate price: 264 €. This is the price after the initial discount but before the subsequent increase. Pretty neat, right?

Step 2: Calculate the second price change (the 21% increase).

Now, this new price of 264 € is going to be increased by 21%. Remember, the percentage increase is applied to the current price, not the original 300 €.

• Method 1: Calculate the increase amount and add.
  • The increase is 21% of 264 €.
  • Increase amount = $264 \times \frac{21}{100} = 264 \times 0.21 = 55.44 €$.
  • The final price is $264 € + 55.44 € = 319.44 €$.
• Method 2: Use the multiplier.
  • If the price increases by 21%, you're paying 100% + 21% = 121% of the current price.
  • The multiplier is 1.21.
  • Final price = $264 € \times 1.21 = 319.44 €$.

So, after all the ups and downs, the final price of the washing machine is 319.44 €. It ends up being a bit more expensive than the original price, which is an important takeaway from this sequence of changes!

Overall Percentage Change:

We can also calculate the overall percentage change. The initial price was 300 € and the final price is 319.44 €. The net increase is $319.44 € - 300 € = 19.44 €$.

To find the overall percentage change: $\frac{19.44}{300} \times 100 \approx 6.48\%$. So, the washing machine experienced an overall increase of about 6.48%.

This first example really shows how a discount followed by a larger increase can result in a price higher than the original. It’s all about the order and the amounts of the percentages!

Scenario B: The Shrinking and Growing Value

Now, let's tackle our second scenario, which involves a starting value of 520. This value undergoes a series of changes: first, it decreases by 30%, then it increases by 20%, and finally, it decreases by another 30%. Our goal here is to track this value through all these transformations and find its ultimate final amount.

This scenario is a great way to practice dealing with multiple percentage changes in sequence, and using multipliers will make this a breeze. Remember, each percentage change applies to the result of the previous one.

Step 1: Decrease by 30%.

We start with 520.

• Using the multiplier: A 30% decrease means we are left with 100% - 30% = 70% of the value. The multiplier is 0.70.
• Value after first decrease = $520 \times 0.70 = 364$.

So, after the first 30% drop, our value is 364.

Step 2: Increase by 20%.

Now, we take the current value of 364 and increase it by 20%.

• Using the multiplier: A 20% increase means we now have 100% + 20% = 120% of the current value. The multiplier is 1.20.
• Value after increase = $364 \times 1.20 = 436.8$.

After the increase, the value is 436.8.

Step 3: Decrease by 30% again.

Finally, we take the current value of 436.8 and decrease it by another 30%.

• Using the multiplier: A 30% decrease means we are left with 100% - 30% = 70% of the current value. The multiplier is 0.70.
• Final value = $436.8 \times 0.70 = 305.76$.

And there we have it! The final value after all the fluctuations is 305.76.

Combined Multiplier Approach for Scenario B:

To double-check our work and see the power of multipliers, we can combine all the operations into one calculation.

• The sequence of changes is: decrease by 30% (multiplier 0.70), increase by 20% (multiplier 1.20), decrease by 30% (multiplier 0.70).
• Combined multiplier = $0.70 \times 1.20 \times 0.70 = 0.588$.
• Final value = $520 \times 0.588 = 305.76$.

This confirms our step-by-step calculation. The final value is indeed 305.76.

Overall Percentage Change for Scenario B:

The initial value was 520, and the final value is 305.76. The total decrease is $520 - 305.76 = 214.24$.

To find the overall percentage change: $\frac{-214.24}{520} \times 100 \approx -41.2$.

Interestingly, even though there was an increase in the middle, the two decreases of 30% had a larger impact, resulting in an overall decrease of approximately 41.2%. This highlights that consecutive percentage changes don't simply add up; the order and the base value for each calculation matter immensely.

Key Takeaways and Conclusion

So, what have we learned from these examples, guys? First off, never assume that consecutive percentage changes will simply cancel each other out or add up directly. A 12% decrease followed by a 21% increase does NOT equal a net 9% increase (because the 21% is applied to a smaller number). Similarly, a 30% decrease, then a 20% increase, then another 30% decrease does NOT result in a 40% decrease ($30 + 20 - 30 = 20$, or $30 + 30 - 20 = 40$). You must calculate each step sequentially or use the combined multiplier method.

The multiplier method is your best friend for these types of problems. It simplifies the calculations significantly, especially when you have more than two changes. Remember:

• For a decrease of X%, the multiplier is $(1 - \frac{X}{100})$.
• For an increase of X%, the multiplier is $(1 + \frac{X}{100})$.

By multiplying these factors together, you get one single multiplier that represents the total effect of all the changes. Multiply this combined multiplier by the original amount to get the final amount directly.

Understanding these percentage calculations is a fundamental skill in mathematics and has practical applications everywhere, from shopping and finance to understanding statistics. Keep practicing these kinds of problems, and you'll become a percentage pro in no time! If you ever see a sale or a price change, you'll know exactly how to break it down. Stay curious, keep calculating, and happy percentage hunting!