Understanding Ratios and Proportions

A ratio compares two quantities of the same unit using division. You can write ratios in different ways: as a fraction $8/28$, with a colon (8:28), or with the word "to" (8 to 28). The first number is called the antecedent, while the second is the consequent.

Ratios come in two main forms. A part-to-part ratio compares two separate groups, like "For every 3 boys, there are 5 girls" (3:5). A part-to-whole ratio compares one group to the total, such as "Out of 16 total students, 6 are boys" (6:16).

Simplifying ratios works just like simplifying fractions - divide both numbers by their greatest common factor (GCF). For example, 8:28 simplifies to 2:7 when both are divided by 4.

💡 When solving real-world problems, remember that ratios maintain the same relationship even when the actual quantities change. This is why we can simplify them!

A proportion is an equation showing that two ratios are equal. When setting up percent proportions, use the formula part/whole = percent/100. To solve proportions, cross multiply and solve for the unknown value. For example, to find 10% of 30, set up x/30 = 10/100, cross multiply to get 30×10 = 100x, and solve to find x = 3.

Relationships between quantities can be direct proportions (both increase or decrease together) or inverse proportions (when one increases, the other decreases, and vice versa).