Welcome to the World of Ratios, Rates, and Percentages!

Hello there! Today, we are going to dive into one of the most useful parts of math: Ratio, Rate, and Percentage. These aren't just numbers on a page; they are tools we use every day. Whether you are mixing the perfect drink, checking the speed of a car, or looking for a discount at your favorite shop, you are using these concepts!

Don't worry if these seem a bit confusing at first. We will break them down into small, easy steps. By the end of these notes, you'll be a pro at comparing numbers!

1. Ratios: Comparing Things of the Same Kind

A ratio is a way to compare two or more quantities that have the same units. For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2 to 3.

How to Write a Ratio

We usually use a colon (:) to show a ratio.
- Example: \( 2 : 3 \)
- We read this as "two to three."

Simplifying Ratios

Just like fractions, we like to keep ratios in their simplest form. This means we divide both numbers by the same largest number possible (the Highest Common Factor).

Example: Simplify the ratio \( 10 : 15 \).
1. Both 10 and 15 can be divided by 5.
2. \( 10 \div 5 = 2 \)
3. \( 15 \div 5 = 3 \)
4. The simplest form is \( 2 : 3 \).

Important Tip: No Units!

Because we are comparing the same kinds of things, ratios do not have units. If you are comparing 5cm to 10cm, the ratio is simply \( 5 : 10 \), which simplifies to \( 1 : 2 \).

Common Mistake: Always make sure the units are the same before writing the ratio. If you have 1 meter and 20 centimeters, change the 1 meter to 100 centimeters first! The ratio is \( 100 : 20 \), which simplifies to \( 5 : 11 \).

Key Takeaway:

Ratios compare "parts to parts." Always simplify them and make sure your units match before you start!

2. Rates: Comparing Different Kinds of Things

A rate is different from a ratio because it compares two quantities with different units. Think about how fast a car goes or how much a kilogram of grapes costs.

Common Examples of Rates

- Speed: Distance traveled over time (e.g., \( 60 \) kilometers per hour or \( 60 \) km/h).
- Price: Cost per item (e.g., \( \$20 \) per kg).

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Calculating the Unit Rate

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A "unit rate" tells us how much for one single unit. To find it, just divide the first number by the second number.

\nExample: If 4 towels cost \( \$100 \), what is the rate per towel?
1. \( \$100 \div 4 = \$25 \)
2. The rate is \( \$25 \) / towel.

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Did you know? We use rates to find the "Best Buy" at the supermarket. If a big bottle of juice is cheaper per ml than a small bottle, you are saving money!

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Key Takeaway:
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Rates must include units (like km/h or $/kg). They tell us how two different things change together.

3. Percentages: Parts of 100

The word percentage comes from "per cent," which literally means "per 100." Imagine a big square cut into 100 tiny pieces. A percentage tells you how many of those pieces you have.

Converting Numbers to Percentages

To turn any fraction or decimal into a percentage, just multiply by \( 100\% \).

- Decimal to %: \( 0.75 \times 100\% = 75\% \)
- Fraction to %: \( \frac{1}{4} \times 100\% = 25\% \)

Finding the Percentage of a Number

If you want to find \( 20\% \) of \( 80 \), you can follow these steps:
1. Change the percentage to a fraction: \( \frac{20}{100} \)
2. Multiply by the number: \( \frac{20}{100} \times 80 \)
3. Calculate: \( 0.2 \times 80 = 16 \).

Percentage Change (Increase or Decrease)

This is very common in exams! Use this simple formula:

\( \text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% \)

- If the answer is positive, it's an increase.
- If the answer is negative, it's a decrease.

Profit, Loss, and Discount

These are just special names for percentage changes:
- Profit: Making extra money (Selling Price > Cost Price).
- Loss: Losing money (Selling Price < Cost Price).
- Discount: A reduction in the original price.

Memory Trick: Always divide by the Original (Starting) Price, never the new one!

Key Takeaway:

Percentages are just fractions with a denominator of 100. They make it easy to compare different sizes!

Summary & Quick Review

Quick Review Box:
- Ratio: Same units, no units in answer (e.g., \( 1:2 \)).
- Rate: Different units, units must be in answer (e.g., \( 5 \) m/s).
- Percentage: Means "out of 100."
- Change: Always \( \frac{\text{change}}{\text{original}} \times 100\% \).

Don't worry if you find word problems tricky at first. The secret is to read slowly and identify if you are comparing the same things (Ratio) or different things (Rate). You've got this!