Lesson 5

Percentages

Percentages - Mathematics

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Why This Matters

Imagine you're sharing a pizza with friends. How do you make sure everyone gets a fair slice? Or maybe you're at a store, and they announce a big '50% off!' sale. How much money do you actually save? That's where percentages come in! They help us understand parts of a whole in a super easy, standardized way. Percentages are everywhere in real life: from calculating discounts and tips to understanding statistics in the news, like '70% of people prefer chocolate ice cream.' On the SAT, percentages are a big deal because they test your ability to think about numbers in different ways and solve practical problems. Mastering percentages means you'll be able to tackle word problems with confidence, understand financial math, and even impress your friends with your quick mental math skills when splitting a bill. It's a fundamental math tool that you'll use far beyond the SAT!

Key Words to Know

01
Percentage — A way to express a number as a fraction of 100, meaning 'per hundred'.
02
Decimal — A number that includes a decimal point, used to represent fractions or percentages in calculations.
03
Fraction — A way to represent a part of a whole, written as one number over another (e.g., 1/2).
04
Original Amount — The starting value or base number before any change occurs.
05
Amount of Change — The difference between the new value and the original value.
06
Percentage Increase — How much a quantity has grown, expressed as a percentage of the original amount.
07
Percentage Decrease — How much a quantity has shrunk, expressed as a percentage of the original amount.
08
Part — A portion or section of a whole.
09
Whole — The entire amount or total quantity.

What Is This? (The Simple Version)

Think of a percentage like a special fraction where the bottom number (the denominator) is always 100. It's a way to show a part of a whole, but instead of saying 'half a pizza,' you say '50% of the pizza.' The word 'percent' literally means 'per hundred' or 'out of every hundred.'

Imagine you have 100 little squares in a grid. If you color in 25 of them, you've colored in 25% (read as 'twenty-five percent') of the squares. It's a universal language for comparing parts of different things. For example, if your friend ate 1/4 of a cake and you ate 25% of a cake, you both ate the same amount!

Key idea: Percentages are just fractions with a secret denominator of 100. So, 75% is the same as 75/100, which can be simplified to 3/4. And 100% means the whole thing, like eating the entire pizza!

Real-World Example

Let's say you want to buy a cool new video game that costs $40. The store announces a special sale: 20% off! How much money do you save, and what's the new price?

  1. Understand the discount: '20% off' means you save 20 out of every 100 dollars. Since the game is $40, we need to find 20% of $40.
  2. Convert percentage to a decimal: To do math with percentages, it's usually easiest to change them into a decimal (a number with a point, like 0.25). To do this, just divide the percentage by 100. So, 20% becomes 20 ÷ 100 = 0.20.
  3. Calculate the savings: Multiply the original price by the decimal. $40 * 0.20 = $8. This means you save $8!
  4. Find the new price: Subtract the savings from the original price. $40 - $8 = $32. So, the new price for your game is $32. Awesome!*

How It Works (Step by Step)

Let's learn how to find a percentage of a number, like finding 15% of 60.

  1. Convert the percentage to a decimal: Divide the percentage by 100. (Example: 15% becomes 0.15).
  2. Multiply the decimal by the number: This gives you the 'part' you're looking for. (Example: 0.15 * 60).
  3. Calculate the result: 0.15 * 60 = 9. So, 15% of 60 is 9.

What if you want to find the whole when you only know a part and its percentage? Like, '9 is 15% of what number?'

  1. Convert the percentage to a decimal: Again, divide the percentage by 100. (Example: 15% becomes 0.15).
  2. Divide the 'part' by the decimal: This will give you the 'whole' number. (Example: 9 ÷ 0.15).
  3. Calculate the result: 9 ÷ 0.15 = 60. So, 9 is 15% of 60. It's like working backward!

Percentage Increase and Decrease

Sometimes you need to know how much something has grown or shrunk in percentage terms. This is super common for prices or populations.

Percentage Increase: Imagine your allowance went from $10 to $12. How much did it increase?

  1. Find the amount of change: Subtract the old amount from the new amount. ($12 - $10 = $2).
  2. Divide the change by the original amount: ($2 ÷ $10 = 0.2).
  3. Multiply by 100 to get the percentage: (0.2 * 100 = 20%). Your allowance increased by 20%!*

Percentage Decrease: What if your game score went from 200 points down to 150 points?

  1. Find the amount of change: Subtract the new amount from the old amount. (200 - 150 = 50).
  2. Divide the change by the original amount: (50 ÷ 200 = 0.25).
  3. Multiply by 100 to get the percentage: (0.25 * 100 = 25%). Your score decreased by 25%.*

Common Mistakes (And How to Avoid Them)

Here are some traps students often fall into with percentages:

  1. Mixing up 'of' and 'is': ❌ Wrong: Thinking '50% of 20 is what number?' means 50 / 20. ✅ Right: Remember 'of' usually means multiply, and 'is' means equals. So, 0.50 * 20 = x.

  2. Forgetting to convert to decimal or fraction: ❌ Wrong: Trying to multiply 25 by 50 when finding '25% of 50'. ✅ Right: Always convert the percentage first! 25% becomes 0.25 (decimal) or 1/4 (fraction). Then multiply: 0.25 * 50 or (1/4) * 50.

  3. Confusing original vs. new amount in increase/decrease: ❌ Wrong: When calculating percentage change, dividing the change by the new amount instead of the original amount. ✅ Right: Always divide by the original amount (the starting number) when calculating percentage increase or decrease. Think of it as the 'base' for your comparison.

  4. Adding/subtracting percentages directly: ❌ Wrong: If a price goes up by 10% and then down by 10%, thinking it's back to the original price (10% - 10% = 0%). ✅ Right: You must calculate each percentage change based on the current value. If $100 goes up 10% ($10), it's $110. If it then goes down 10% of $110 ($11), it becomes $99. It's not back to $100!*

Exam Tips

  • 1.Convert percentages to decimals (divide by 100) or fractions (put over 100 and simplify) before doing calculations; it's usually easier.
  • 2.For percentage change problems, always divide by the **original amount** (the starting number) to find the correct percentage.
  • 3.Don't add or subtract percentages directly; calculate each percentage change based on the *current* value, not the original.
  • 4.When a question asks for a 'percentage of a number,' think 'multiply' (e.g., 20% of 50 means 0.20 * 50).
  • 5.Practice common conversions: 25% = 1/4, 50% = 1/2, 75% = 3/4, 10% = 1/10, 20% = 1/5. Knowing these saves time!