When you cook rice, you usually use 1 cup of rice for every 2 cups of water. When you look at a map, 1 inch might represent 100 miles. These are everyday examples of Ratio & Proportion.
While addition and subtraction tell us the difference between numbers, Ratio & Proportion tell us the relationship between them. It is the language of scaling, recipes, maps, and financial analysis. Mastering these concepts is key to understanding how different quantities relate to one another in the real world.
1. What is a Ratio?
A Ratio is a comparison of two quantities using division. It tells us how much of one thing there is compared to another.
[Image of ratio comparison diagram of apples and oranges]If a fruit basket contains 3 apples and 2 oranges, the ratio of apples to oranges is "3 to 2".
Ways to Write a Ratio
- With a colon: 3 : 2
- With the word "to": 3 to 2
- As a fraction: 3 / 2
Note: Order matters! The ratio of oranges to apples is 2 : 3, which is different from 3 : 2.
Simplifying Ratios
Like fractions, ratios should be simplified to their lowest terms. If you have a ratio of 10 : 15, you can divide both numbers by 5 to get the simpler ratio of 2 : 3.
2. What is a Proportion?
A Proportion is an equation that says two ratios are equal. It is the mathematical way of saying "this relationship is the same as that relationship."
This is read as "1 is to 2 as 2 is to 4."
The Parts of a Proportion
In the proportion a : b :: c : d:
- Extremes: The outer numbers (a and d).
- Means: The inner numbers (b and c).
The Golden Rule of Proportion: The product of the extremes equals the product of the means.
Example: In 1 : 2 = 2 : 4, we check: 1 * 4 = 4 and 2 * 2 = 4. Since 4 = 4, it is a valid proportion.
3. Direct Proportion
Two quantities are in Direct Proportion if they increase or decrease together at the same rate. If you double one, the other doubles.
[Image of direct proportion graph linear]Example: Cost of Apples
- 1 apple costs $2.
- 2 apples cost $4.
- 5 apples cost $10.
The ratio of Cost to Apples is always constant (2 : 1). We say Cost is directly proportional to Quantity.
4. Inverse Proportion
Two quantities are in Inverse Proportion if one increases while the other decreases. If you double one, the other is cut in half.
[Image of inverse proportion graph curve]Example: Speed and Time
- If you drive at 60 mph, a trip takes 2 hours.
- If you drive at 120 mph (double speed), the trip takes 1 hour (half time).
Here, Speed x Time = Constant Distance. As speed goes up, time goes down.
5. Solving Proportion Problems
One of the most useful skills in math is solving for a missing value in a proportion. This is often called the "Rule of Three."
Problem: If 3 books cost $45, how much do 7 books cost?
Step 1: Set up the proportion.
Ratio 1 (Books : Cost) = Ratio 2 (Books : Cost)
3 / 45 = 7 / x
Step 2: Cross Multiply.
3 * x = 45 * 7
3x = 315
Step 3: Solve for x.
x = 315 / 3
x = 105
So, 7 books cost $105.
6. Real-World Applications
Ratio & Proportion are everywhere.
1. Cooking and Recipes
A recipe serves 4 people but you need to feed 10. You must scale up every ingredient using proportions to keep the taste the same.
2. Maps and Scale Models
Architects use scales like 1 : 100. This means 1 cm on the blueprint equals 100 cm (1 meter) in real life. Without proportions, buildings would collapse and maps would be useless.
3. Finance and Business
Financial analysts use ratios like "Debt-to-Equity" or "Price-to-Earnings" to compare companies of different sizes. A small company can be compared fairly to a giant corporation if you look at their ratios rather than just their raw profits.
7. Conclusion
Ratio & Proportion allow us to make sense of the world by comparing things relative to one another. Whether you are mixing paint to get the perfect color, calculating gas mileage, or resizing a photo on your phone, you are using the fundamental logic of Ratio & Proportion.