Ratios, Proportions, and Percentages

TL;DR

Ratios compare quantities, proportions state that two ratios are equal, and percentages are a specific type of ratio out of 100. These concepts help you understand how different parts relate to a whole or to each other. Mastering them allows you to scale recipes, calculate discounts, and understand statistical data.

1. The Mental Model

Think of these as different ways to describe relationships between numbers. Ratios are like comparing apples to oranges, proportions are like saying "this many apples to oranges is the same as that many apples to oranges," and percentages are like saying "this many out of every hundred."

2. The Core Material

Ratios

A ratio compares two or more quantities. You can write ratios in a few ways:
* Using a colon: a : b
* Using the word "to": a to b
* As a fraction: a/b

Ratios can compare parts to parts (e.g., girls to boys) or parts to a whole (e.g., girls to total students). Always simplify ratios to their lowest terms, just like fractions.

Example: If there are 3 red balls and 6 blue balls, the ratio of red to blue is 3 : 6, which simplifies to 1 : 2. The ratio of red balls to total balls is 3 : 9, which simplifies to 1 : 3.

Proportions

A proportion is an equation that states two ratios are equal. If a/b = c/d, then you have a proportion. Proportions are super useful for finding an unknown value when you know three others. The key property is that the cross-products are equal: a * d = b * c.

Example: If you can bake 2 cakes with 4 cups of flour, how many cakes can you bake with 8 cups of flour?
You can set up a proportion: 2 cakes / 4 cups = x cakes / 8 cups
Cross-multiply: 2 * 8 = 4 * x
16 = 4x
x = 4 cakes.

Percentages

A percentage is a ratio where the second number (the denominator) is always 100. The word "percent" literally means "per hundred." You can convert fractions and decimals to percentages and vice-versa.

• Fraction to Percent: Multiply by 100. (e.g., 1/4 * 100% = 25%)
• Decimal to Percent: Multiply by 100. (e.g., 0.75 * 100% = 75%)
• Percent to Decimal: Divide by 100. (e.g., 50% / 100 = 0.50)
• Percent to Fraction: Write over 100 and simplify. (e.g., 20% = 20/100 = 1/5)

The most common percentage problem involves finding a part of a whole: Part = Percentage * Whole.

Example: What is 20% of 50?
Convert 20% to a decimal: 0.20
Part = 0.20 * 50 = 10.

Here's how these concepts relate to each other:

graph TD
    A["Quantity A"] --> R["Ratio (A:B)"]
    B["Quantity B"] --> R
    R --> P["Proportion (A:B = C:D)"]
    C["Quantity C"] --> P
    D["Quantity D"] --> P
    R --> PCT["Percentage (Ratio out of 100)"]
    PCT -- "Is a type of" --> R
    P -- "Solve for unknown" --> U["Unknown Value"]

3. Worked Example

Let's say you're making lemonade. The recipe calls for a ratio of 2 lemons to 3 cups of water. You want to make a big batch using 12 cups of water.

1. Understand the Ratio: The ratio of lemons to water is 2 : 3.
2. Set up a Proportion: You know you'll use 12 cups of water, and you need to find out how many lemons (x).
   2 lemons / 3 cups water = x lemons / 12 cups water
3. Solve the Proportion: Cross-multiply.
   2 * 12 = 3 * x
24 = 3x
x = 24 / 3
x = 8
   So, you need 8 lemons.

Now, let's say you used 8 lemons and 12 cups of water. What percentage of your lemonade volume is lemon (assuming lemons are mostly juice and we can compare directly)?

1. Find the total parts: 8 parts lemon + 12 parts water = 20 total parts.
2. Form a ratio of lemon to total: 8 lemons / 20 total parts.
3. Convert to Percentage:
   (8 / 20) * 100% = 0.40 * 100% = 40%
   So, 40% of your lemonade volume is lemon.

4. Key Takeaways

• Ratios compare quantities using colons, "to," or fractions, and should always be simplified.
• Proportions are equal ratios used to find an unknown value by cross-multiplication.
• Percentages are special ratios expressing a quantity out of 100, useful for parts of a whole.
• You can convert between fractions, decimals, and percentages.
• Part = Percentage * Whole is a common formula for percentage problems.
• Understanding these helps you scale recipes, calculate discounts, and interpret data.
• Always make sure the units in your ratios are consistent (e.g., lemons to lemons, water to water).

Common Mistakes to Avoid:
- Not simplifying ratios to their lowest terms.
- Mixing up the order in a ratio (e.g., a:b is different from b:a).
- Forgetting to convert percentages to decimals (by dividing by 100) before multiplying in calculations.
- Incorrectly setting up proportions, like putting units in the wrong places (always match units across the equals sign).

5. Now Try It

You're mixing paint. The instructions say to use 3 parts blue paint for every 5 parts white paint. If you want to make a total of 80 fluid ounces of mixed paint, how much blue paint and how much white paint do you need?

What to do:
1. Determine the total "parts" in the ratio.
2. Set up a proportion to find the amount of blue paint.
3. Set up another proportion (or subtract) to find the amount of white paint.
4. Check your answer by adding the amounts of blue and white paint to see if they total 80 fluid ounces.

What success looks like: You should have two numbers (one for blue paint, one for white paint) that add up to 80, and their ratio should be 3:5.