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Understanding Percentages
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Good morning class! Today weโre going to dive into the concept of percentages. Can anyone tell me what a percentage is?
Isnโt a percentage just a way to express a number out of 100?
Exactly! When we say '30%', it means 30 out of 100. Percentages help us understand proportions easily in various real-life scenarios, right? Think about when you're shopping and see sales!
So if something is on sale for 20% off, does that mean I get 20 out of 100, or just that percentage of the total price?
Great question! It means youโll pay 80% of the original price. Remember the primary formula: Final Price = Original Price ร (1 - Percentage Discount).
That sounds useful! Are there specific calculations for increases and decreases?
Yes, we use different calculations for increases and decreases, which we'll discuss shortly. Let's keep that in mind!
Percentage Increase and Decrease
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Now, letโs talk about how to calculate percentage increases and decreases. For example, if a price rises from $100 to $120, what's the percentage increase?
I think itโs $20 increase, right?
Correct! To find the percentage increase, we divide the increase by the original amount and multiply by 100. So, (20 / 100) ร 100 = 20% increase.
What if it decreases instead? Like going from $120 to $100?
Right, weโd find that decrease too. The decrease is also $20, but now for percentage decrease: (20 / 120) ร 100 = approximately 16.67% decrease. It's important to know both!
How do we remember the formula for increase and decrease?
A helpful memory aid is the acronym 'RIM' โ Rise In Money for an increase, and 'DIM' โ Decrease In Money for decrease. Keep practicing with examples!
Profit and Loss Calculations
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Today, weโre going to apply what we've learned about percentages to profit and loss calculations. If an item cost $50 and sold for $75, how do we calculate the profit?
Isn't it $75 - $50? That's $25, right?
Yes! And to find the profit margin as a percentage, we take the profit divided by the cost: (25 / 50) ร 100, giving us a 50% profit margin.
What if we incur a loss? How would that apply?
Good point! If the item costs $50 but sells for $30, that's a loss of $20. So, the percentage loss would be (20 / 50) ร 100 = 40% loss.
That seems very practical! How do businesses use this in real life?
Businesses constantly analyze profits and losses to make decisions. Understanding percentages is key to that evaluation in the market. Let's practice some examples on this!
Reverse Percentages
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Last, let's explore reverse percentages. If a shirt costs $30 after a 20% discount, what was the original price?
Hmm, if it's 80% of the original, how do we calculate that?
Exactly! Since $30 is 80% of the original price, we can express that as: Original Price = Final Price / (1 - Percentage Discount). So, $30 / 0.8 = $37.50.
Got it! So if I know the reduced price and the discount, I can find the original price anytime?
That's right! This skill is especially useful in shopping and budgeting decisions.
Can you give us more practice problems for this?
Absolutely, let's all try a couple now!
Introduction & Overview
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Quick Overview
Standard
The section on percentages introduces essential calculations such as percentage increases, decreases, and profit/loss evaluations. It explores methods to reverse-engineer percentages, allowing students to determine original amounts from known percentages.
Detailed
Percentages
Percentages are a fundamental mathematical concept used to express a number as a fraction of 100, making it easier to relate quantities in various contexts, especially in finance, sales, and data interpretation. In this section, we explore several key topics under percentages:
6.1.1 Percentage Increase and Decrease
Understanding how to calculate percentage increases involves determining how much a quantity has grown relative to its original value. Conversely, percentage decreases focus on reductions in a quantity.
6.1.2 Profit and Loss Calculations
This subsection delves into financial mathematics, teaching students how to calculate profit margins and losses using percentages, fundamental skills in both personal and business finance.
6.1.3 Reverse Percentages (Finding Original Amounts)
Students will also learn about reverse percentages, a valuable skill for calculating original amounts when only the final value and percentage change are known. This is particularly useful in practical scenarios like sales and discounts.
These skills are critical in interpreting information in real-world contexts, aiding in financial decision-making and analytical reasoning.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Percentage: A method to express a number as a fraction of 100.
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Percentage Increase: The increase of a quantity expressed as a percentage of the original quantity.
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Percentage Decrease: The reduction of a quantity expressed as a percentage of the original quantity.
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Profit: Earnings made from sales after costs.
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Loss: A decrease in financial value, when costs exceed revenue.
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Reverse Percentage: Finding the original amount before a percentage change.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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A jacket originally priced at $80 is on sale for 20% off. The discounted price is $80 ร (1 - 0.20) = $64.
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If an item purchased for $40 sells for $60, the profit is $20, which is a 50% profit margin calculated as ($20 / $40) ร 100.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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When you want to calculate change, remember this is not so strange; Increase a bit, or decrease the length, remember the percentages give you strength!
๐ Fascinating Stories
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Once, a clever merchant had a $100 item. In a rush to sell, he offered a 20% discount, calling out how it was now just $80. Customers flocked in, learning to calculate exactly how much they saved!
๐ง Other Memory Gems
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To remember profit and loss: 'Raising income helps, while losses are costly expenses.'
๐ฏ Super Acronyms
Use 'PROFIT' to remember
- P: for Price
- R: for Revenue
- O: for Overhead
- F: for Final Price
- I: for Investment
- and T for Total profits.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Percentage
Definition:
A fraction expressed as a part of 100.
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Term: Percentage Increase
Definition:
The amount by which a quantity increases, expressed as a percentage of the original amount.
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Term: Percentage Decrease
Definition:
The amount by which a quantity decreases, expressed as a percentage of the original amount.
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Term: Profit
Definition:
The financial gain obtained by subtracting costs from revenue.
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Term: Loss
Definition:
The financial deficit incurred when costs exceed revenue.
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Term: Reverse Percentage
Definition:
A calculation used to find the original amount after a percentage change.