Introduction to Ratios and Proportions

TL;DR

Ratios compare two quantities, showing how much of one there is relative to another. Proportions state that two ratios are equal, which is useful for scaling things up or down. Understanding these helps you compare things fairly and solve for missing values.

1. The Mental Model

Think of ratios as recipes: if a recipe calls for 2 cups of flour to 1 cup of sugar, that's a ratio. Proportions are like scaling that recipe: if you want to make a double batch, you'd keep the ratio of flour to sugar the same, just increase both amounts proportionally.

2. The Core Material

What's a Ratio?

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A ratio is a way to compare two quantities. It shows you how many times one quantity contains another, or what fraction of one quantity the other quantity is. You can write ratios in a few ways:

• Using a colon: 2:3 (read as "two to three")
• As a fraction: 2/3
• Using the word "to": 2 to 3

No matter how you write it, it means the same thing: for every 2 units of the first quantity, there are 3 units of the second. The order matters! A 2:3 ratio isn't the same as a 3:2 ratio.

Simplifying Ratios

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Just like fractions, you can simplify ratios by dividing both parts by their greatest common factor.

Example:
The ratio of 10 apples to 15 oranges can be written as 10:15.
Both 10 and 15 can be divided by 5.
10 ÷ 5 = 2
15 ÷ 5 = 3
So, the simplified ratio is 2:3.

What's a Proportion?

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A proportion is an equation that says two ratios are equal. If you have two ratios, say a:b and c:d, a proportion states that a:b = c:d, or a/b = c/d. This is super useful because if you know three of the four values, you can always find the fourth.

The key property of proportions is called cross-multiplication. If a/b = c/d, then a * d = b * c.

graph TD
    A["Start with a Ratio (e.g., Ingredients)"] --> B["Simplify the Ratio (if possible)"];
    B --> C["Need to Scale or Compare?"];

    C -- "Yes, Scale" --> D["Set up a Proportion (Equal Ratios)"];
    D --> E["Use Cross-Multiplication"];
    E --> F["Solve for the Unknown"];
    F --> G["Check if Solution Makes Sense"];
    G --> H["End"];

    C -- "No, just Compare" --> H;

Solving Proportions

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Let's say you have the ratio 1:2, and you want to find an equivalent ratio where the first number is 5. You'd set it up as a proportion:

1/2 = 5/x

Using cross-multiplication:
1 * x = 2 * 5
x = 10

So, the equivalent ratio is 5:10. This means if you double the first part of the ratio from 1 to 2, you also double the second part from 5 to 10.

3. Worked Example

You're mixing paint. For a specific shade of green, the ratio of blue paint to yellow paint is 3:5. You need to make a larger batch and you have 12 liters of blue paint. How much yellow paint do you need?

1. Identify the known ratio: Blue : Yellow = 3:5, or 3/5.
2. Identify the known part of the new ratio: You have 12 liters of blue paint.
3. Set up the proportion: Let 'x' be the unknown amount of yellow paint.
   3/5 = 12/x
4. Cross-multiply:
   3 * x = 5 * 12
   3x = 60
5. Solve for x:
   x = 60 / 3
   x = 20

You'll need 20 liters of yellow paint.

4. Key Takeaways

• Ratios are comparisons of two quantities, often written as a:b or a/b.
• The order of quantities in a ratio is important; 2:3 is different from 3:2.
• You can simplify ratios just like fractions by dividing both parts by a common factor.
• A proportion states that two ratios are equal, like a/b = c/d.
• Cross-multiplication (a * d = b * c) is the primary method for solving proportions.
• Proportions help you scale recipes, maps, or any situation where quantities need to maintain a relative balance.
• Always make sure the units are consistent when setting up ratios (e.g., both sides are in liters or both are in dollars).

Common mistakes to avoid:
- Flipping the ratio: If it's "apples to oranges," don't write "oranges to apples."
- Not simplifying ratios when asked, or simplifying incorrectly.
- Incorrectly setting up the proportion, like putting an unknown quantity in the wrong place.
- Forgetting to cross-multiply or making errors in the multiplication/division.

5. Now Try It

You're planning a party, and you want the ratio of savory snacks to sweet snacks to be 4:3. If you've already bought 24 savory snacks, how many sweet snacks do you need to buy to keep the ratio consistent? What does success look like? You should have a single number answer for the amount of sweet snacks, and you should be able to explain how you used a proportion to find it.