Contents
Ratios
A ratio is a mathematical concept that expresses the quantitative relationship between two or more quantities. It serves as a comparison tool, enabling us to understand the relative size or magnitude of different values. Ratios can be represented in various formats, such as fractions, decimals, or using the colon symbol (:). The order of the terms in a ratio holds significance, as it can alter the meaning conveyed. Ratios are extensively used in everyday life, ranging from financial analysis and sports statistics to cooking recipes and construction plans. [1]
Proportions
Proportion is a mathematical concept that signifies the equality of two ratios. It emphasizes the relationship between parts and the whole, ensuring a balanced distribution. Proportions are represented using the “=” sign, indicating that two ratios are equivalent. They are commonly encountered in various fields, including geometry, physics, and statistics, enabling us to solve problems involving scaling, similarity, and percentages. [1]
Ratios & Proportions
Understanding the concepts of ratio and proportion is essential in various fields such as mathematics, finance, and science. While these terms are often used interchangeably, they have distinct meanings and applications. In this article, we will delve into the key differences between ratio and proportion, highlighting their definitions, advantages, disadvantages, and similarities. [1]
| Ratio | Proportion |
|---|---|
| Defines the quantitative relationship between two or more quantities | Indicates the equality of two ratios |
| Can be expressed as a fraction, using the colon symbol (:), or in decimal form | Usually represented using the “=” sign |
| Focuses on the comparison of quantities | Emphasizes the relationship between parts and the whole |
| A ratio can exist independently without a comparison | Requires a comparison between two ratios |
| Does not guarantee an equal distribution | Ensures a balanced distribution |
| Example: The ratio of boys to girls is 3:2 | Example: In a group of 50 students, the ratio of boys to girls is 3:2 |
| Used in various real-life scenarios, such as financial analysis and cooking recipes | Widely employed in solving proportion word problems and scaling measurements |
| Can have an infinite number of possible values | Generally, a proportion has only one solution |
| Ratios can be simplified or expanded by multiplying or dividing both sides by the same number | Proportions can be solved using cross-multiplication or the unitary method |
| The order of the terms in a ratio matters (e.g., 2:1 is different from 1:2) | The order of the terms in a proportion does not matter (e.g., 2:1 = 1:2) |
Ratio and proportion are distinct mathematical concepts with unique definitions, characteristics, advantages, and disadvantages. Understanding their differences is crucial for accurate problem-solving, data analysis, and decision-making. While ratio focuses on the comparison of quantities, proportion emphasizes the equality of two ratios. Both concepts play vital roles in various fields and provide valuable tools for mathematical reasoning and analysis. By comprehending the dissimilarities and similarities between ratio and proportion, we can harness their power effectively in practical applications and academic pursuits. [1]
Examples
There are total 80 students in class, out of which 30 are boys and rest of the students are girls. Now find out the following:
(i) Ratio of boys to girls and girls to boys
(ii) Proportion of boys and girls in the class
Solutions
(i) Ratio of boys to girls = Boys : Girls = 30 : 50 or 3 : 5
Ratio of girls to boys = Girls: Boys = 50 : 30 or 5 : 3
Thus, For every three boys there are five girls or for every five girls, there are three boys.
(ii) Proportion of boys = 30/80 or 3/8
Proportion of girls = 50/80 or 5/8
Thus, 3 in every 8 students is a boy and 5 in every 8 students is a girl.
Rates
In math, a rate is a ratio that compares two different quantities which have different units. For example, if we say John types 50 words in a minute, then his rate of typing is 50 words per minute. The word “per” gives a clue that we are dealing with a rate. The word “per” can be further replaced by the symbol “/” in problems. Let us consider an example of a car that is traveling at a speed of 150 miles in 3 hours. This can be expressed as 150 miles divided by 3 hours which is equal to 150 miles/3 hours or 50 miles/hour. Here the word ‘per’ means for every hour. 50 miles/ hour is the average speed at which the car travels. 50 miles/hour is an example of a unit rate. [2]
| Rate | Ratio |
|---|---|
| Different units are taken into comparison. | Same units are taken in to comparison. |
| Rate of two quantities are expressed using the word “per” or the symbol “/”. | Ratio of two numbers are expressed using the word “to” or the symbol “:”. |
| Example: 100 miles per hour, or 100 miles/hour. | Example : 3 apples to 2 cups of milk. This can be expressed as 3:2, which is the number of apples to the measure of milk. |
Example 1
Sam makes 3 pancakes every 6 minutes, that is a rate of:
- 3 pancakes per 6 minutes
- 0.5 pancakes per minute
- 30 pancakes per hour
- an hourly rate of 30
- etc.
Example 2
200 sausages were eaten by 50 people. that is:
- 200 sausages per 50 people
- 100 sausages per 25 people
- 4 sausages per person
Example 3
Alex invested $100 for a month and made $3
- the interest rate is 3/100 = 3% per month
When we compare to a single unit quantity we call it a Unit Rate.
References
[1] “Difference Between Ratio and Proportion: Know the Differences.” 2023. testbook. Accessed October 29. https://testbook.com/key-differences/difference-between-ratio-and-proportion.
[2] “Rate Definition – Definition of Rate, Unit Rate, Ratio Definition, Ratio and Rate Difference, Rate Calculation and Solved Examples.” 2023. CUEMATH. Accessed October 29. https://www.cuemath.com/commercial-math/rate-definition/.
Additional Reading
Government of Canada, Statistics Canada. 2022. “Statistics 101: Proportions, Ratios and Rates.” Government of Canada, Statistics Canada. February 25. https://www.statcan.gc.ca/en/wtc/data-literacy/catalogue/892000062021003.
“Ratio and Proportion – Basics, Definitions, Examples.” 2023. CUEMATH. Accessed October 29. https://www.cuemath.com/commercial-math/ratio-and-proportion/.
⭐ “Ratio and Proportion – Definition, Formulas and Examples.” 2023. BYJUS. BYJU’S. October 3. https://byjus.com/maths/ratios-and-proportion/.
“Ratios and Proportions | Lesson (Article).” 2023. Khan Academy. Khan Academy. Accessed October 29. https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp–praxis-math–article–ratios-and-proportions–lesson.
“Ratios and Proportions | Worked Example (Video).” 2023. Khan Academy. Khan Academy. Accessed October 29. https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/v/gtp–praxis-math–video–ratios-and-proportions.
“Rates | Lesson (Article) | Lessons.” 2023. Khan Academy. Khan Academy. Accessed October 29. https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp–praxis-math–article–rates–lesson.
“Rates | Worked Example (Video) | Lessons.” 2023. Khan Academy. Khan Academy. Accessed October 29. https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/v/gtp–praxis-math–video–rates.
S, Surbhi. 2017. “Difference Between Ratio and Proportion (with Comparison Chart).” Key Differences. September 9. https://keydifferences.com/difference-between-ratio-and-proportion.html.
“Unit Rate – Definition, Examples and Problems.” 2020. BYJUS. BYJU’S. July 28. https://byjus.com/maths/unit-rate/.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the Ratios and Proportion webpage on the BJYU’s website.
Mathematical Mysteries 