Mastering Percentages: Your Guide to GCE O-Level Success

Welcome to the world of Percentages! Whether you are shopping for a sale, looking at your exam scores, or checking the interest on a bank account, percentages are everywhere. Don't worry if you've found this topic confusing before. We are going to break it down step-by-step until you're an expert!

In this chapter, we will learn how to compare amounts, handle values greater than 100%, and even "travel back in time" to find original prices using reverse percentages.

1. What exactly is a Percentage?

The word "percent" comes from the Latin "per centum", which literally means "out of 100". Think of a percentage as a fraction where the denominator is always 100.

Analogy: Imagine a huge bar of chocolate with 100 tiny squares. If you eat 20 squares, you have eaten 20% of the bar. If you eat the whole thing, you've eaten 100%!

Quick Review:

\( 1\% = \frac{1}{100} = 0.01 \)
\( 50\% = \frac{50}{100} = 0.5 \)
\( 100\% = \frac{100}{100} = 1 \)

2. Expressing One Quantity as a Percentage of Another

To find out what percentage one number is of another, we follow a simple two-step recipe:

Step 1: Write the two numbers as a fraction: \( \frac{\text{Part}}{\text{Whole}} \)
Step 2: Multiply by \( 100\% \)

Example: If you scored 15 out of 20 in a math quiz, what is your percentage?
\( \frac{15}{20} \times 100\% = 75\% \)

Common Mistake: Always make sure both quantities are in the same units before you start! (e.g., convert both to grams or both to kilograms).

Key Takeaway: Always put the "Part" on top and the "Total/Whole" on the bottom.

3. Comparing Two Quantities by Percentage

Sometimes we want to know how much bigger or smaller one thing is compared to another using percentages.

Example: Box A weighs 5kg and Box B weighs 8kg. What percentage of Box B's weight is Box A's weight?
Here, Box B is the "Whole" we are comparing against.
\( \frac{\text{Weight of A}}{\text{Weight of B}} \times 100\% = \frac{5}{8} \times 100\% = 62.5\% \)

4. Percentages Greater than 100%

Can a percentage be more than 100? Yes! This just means the "Part" is actually larger than the original "Whole."

Analogy: If you had \$10 last year and you have \$30 this year, you have 300% of what you had last year. You've tripled your money!

Example: A company produced 200 toys in January and 500 toys in February. Express February's production as a percentage of January's.
\( \frac{500}{200} \times 100\% = 250\% \)

5. Increasing and Decreasing by a Percentage

This is very common in "Sale" or "Price Hike" questions. The most important thing to remember is that the original value is always 100%.

Method: The Multiplier Way

To find the new value directly:

For an Increase: Add the percentage to 100%
For a Decrease: Subtract the percentage from 100%

Example (Increase): A \$80 pair of shoes increases in price by 15%.
New Percentage = \( 100\% + 15\% = 115\% \)
New Price = \( 115\% \times 80 = 1.15 \times 80 = \$92 \)

Example (Decrease): A laptop costing \$1200 is on a 20% discount.
New Percentage = \( 100\% - 20\% = 80\% \)
New Price = \( 80\% \times 1200 = 0.8 \times 1200 = \$960 \)

Did you know? A "100% increase" means the value has doubled! A "200% increase" means the value has tripled.

6. Reverse Percentages (Finding the Original)

This is the part that many students find tricky, but here is a secret: Never calculate the percentage of the NEW price to get back to the old price. It doesn't work that way!

The Goal: Find the original value (the 100% value) after an increase or decrease has already happened.

Step-by-Step Strategy:
1. Identify what percentage the given value represents.
2. Find 1% by dividing.
3. Find 100% by multiplying by 100.

Example: A phone is sold for \$630 after a 10% discount. What was the original price?
Step 1: The discount was 10%, so \$630 must be \( 100\% - 10\% = 90\% \) of the original price.
Step 2: \( 90\% = \$630 \)
Step 3: \( 1\% = \frac{630}{90} = \$7 \)
Step 4: \( 100\% = 7 \times 100 = \$700 \)
The original price was \$700.

Memory Aid: In reverse percentages, you are trying to find the 100% mark. Always ask yourself: "What percent is the number I'm looking at right now?"

7. Common Pitfalls to Avoid

1. Adding percentages of different totals: You cannot simply add 10% of a small box to 10% of a big box and say it's 20% of the total. Percentages are relative to their specific "whole."

2. The "Backwards" Error: If a price goes up by 20% and then down by 20%, you do not end up back at the original price. Try it! (Original \$100 \(\rightarrow\) Up 20% = \$120 \(\rightarrow\) Down 20% of \$120 = \$96. You lost \$4!)

3. Forgetting the 100%: In reverse percentages, students often calculate 10% of the sale price and add it back. Don't do this! Always link the sale price to its percentage (like 90% or 115%).

Final Checklist for Exams

- Expressing as %: \( \frac{\text{Is}}{\text{Of}} \times 100\% \)
- Finding % of a value: \( \frac{\text{Percentage}}{100} \times \text{Value} \)
- Percentage Change: \( \frac{\text{Change}}{\text{Original Value}} \times 100\% \)
- Reverse %: Match the given value to its percentage, find 1%, then find 100%.

Don't worry if this seems tricky at first! The more you practice "finding the 100%", the easier it becomes. You've got this!