We use ratios and proportions when we compare numbers or quantities in math and in everyday life.

A **ratio** is a relationship between two numbers that compares one quantity to the other. Three ways of expressing ratios are using words, colons, or fractions: 2 to 3, 2:3, or 2/3. For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2:3.

A p**roportion**, on the other hand, is an equation stating that two ratios are equivalent. For example, if there are 2 apples for every 3 oranges in one basket, and 4 apples for every 6 oranges in another, the proportion is 2/3 = 4/6, meaning the ratio of apples to oranges is the same in both baskets.

In everyday life, we often use ratios and proportions without even realizing it. When following a recipe, you use ratios to measure ingredients. If you’re doubling a recipe, you’re using proportions to ensure that the increased amounts of ingredients keep the same ratio. When calculating miles per hour for a road trip, you use ratios to express your speed.

### Ratio and Proportion Key Points

- A ratio is a relationship or comparison between two numbers or quantities.
- A proportion is an equation stating that two ratios are equal.
- Ratios are expressions, while proportions are equations.
- Ratios can be simplified just like fractions.
- Direct proportion: as one quantity increases, the other also increases at the same rate.
- Inverse proportion: as one quantity increases, the other decreases.
- Continued proportion: three quantities ‘a’, ‘b’, and ‘c’ are in continued proportion if a:b::b:c.
- In proportions, the product of extremes equals the product of means (a
*d = b*c).

Now, let’s delve deeper into these two important mathematical concepts and explore their properties and applications.

### Ratios

A ratio expresses a relationship or comparison between any quantities. Generally they involve the natural numbers. In the realms of mathematics and science, a ratio finds various uses. For instance, when we talk about speed, it is a ‘rate’ – the ratio of distance covered in the time taken. Ratios are also fundamental in geometry, where they help to compare similar figures and trigonometry.

#### How to Simplify a Ratio

One crucial point is that you can simplify ratios. If you have a ratio of 10:15, it is the same as the simplified ratio 2:3. Here are simple steps for simplifying a ratio:

- Write the ratio a:b in the form of a fraction a/b. The top number of the fraction is its numerator, while the bottom number is the denominator. For example, if the ratio is 18:10, write 18:10.
- Find the greatest common factor of a and b. This is the largest number you can divide them by evenly. For 18 and 10, the greatest common factor is 2.
- Divide the numerator and denominator by the greatest common factor to get the simplified fraction. So, 18/10 becomes 9/5.
- Now, write the fraction is ratio form. 9/5 becomes 9:5.

### Proportions

A proportion, as mentioned earlier, is an equation that equates two ratios. It serves as the foundation for numerous mathematical principles and real-world applications, from scaling models to converting units of measure.

#### Direct Proportion

In a direct proportion, two quantities increase or decrease together at the same rate. If “a” and “b” are two quantities, then the direct proportion is a∝b. If you travel at a constant speed, the distance you cover is directly proportional to the time you travel. This means that if you travel for 2 hours at 60 miles per hour, you cover 120 miles.

#### Inverse Proportion

In an inverse or indirect proportion, as one quantity increases, the other decreases. If “a” and “b” are two quantities, then the inverse proportion is a∝(1/b). For instance, the time taken to complete a task is inversely proportional to the number of people working on it. If 2 people can paint a house in 6 hours, 6 people can paint it in 2 hours, assuming all else stays the same.

#### Continued Proportions

In continued proportions, three quantities are in proportion. If ‘a’, ‘b’, and ‘c’ are in continued proportion, then a:b::b:c. This means that the ratio of ‘a’ to ‘b’ is the same as the ratio of ‘b’ to ‘c’. For example, 2, 6, and 18 are in continued proportion because 2/6 = 6/18.

### Mathematical Properties of Proportions

Proportions have several unique mathematical properties.

The first term of a proportion is the antecedent. The second term is the consequent. For example, in the ratio 4:9, 4 is the antecedent and 9 is the consequent. If you multiply both the antecedent and consequent by the same non-zero number, the ratio remains unaffected.

The ‘extremes’ of a proportion are the first and last terms, while the ‘means’ are the second and third terms. In the proportion a/b = c/d, ‘a’ and ‘d’ are extremes, while ‘b’ and ‘c’ are means. For example, consider the proportion:

3 : 5 :: 4 : 8 or 3/5 = 4/8

Here, 3 and 8 are the extremes, while 5 and 4 are the means.

One key property is that the product of extremes equals the product of means (a*d = b*c). This property, known as the **cross-multiplication rule**, is a fundamental tool for solving proportions.

Here is a quick summary of the proportion properties:

- If a : b = c : d, then a + c : b + d
- If a : b = c : d, then a – c : b – d
- If a : b = c : d, then a – b : b = c – d : d
- If a : b = c : d, then a + b : b = c + d : d
- If a : b = c : d, then a : c = b: d If a : b = c : d, then b : a = d : c
- If a : b = c : d, then a + b : a – b = c + d : c – d

### Additional Information

In higher mathematics, you encounter complex variations and applications of ratios and proportions, including compound ratios, duplicative and triplicate ratios, and the ratios of functions in calculus. The principles of ratios and proportions underpin the concept of scale in geometry, the basis of trigonometric identities, and much more.

### Ratio and Proportion Worked Example Problems

- If 2 books cost $18, how much do 5 books cost?

Here, the ratio of books to cost is 2:18. If we increase the books to 5, we set up a proportion to find the cost: 2/18 = 5/x. Cross-multiplying gives 2x = 90, so x = $45.

- If 5 workers can complete a task in 7 hours, how long will it take 10 workers?

Here, the number of workers is inversely proportional to time. So, 57 = 10x. Solving for x gives x = 3.5 hours.

Understanding ratios and proportions is vital to navigating both academic mathematics and practical everyday situations. Their importance cannot be overstated, as these concepts form the building blocks for many areas of mathematics and real-world problem-solving.

### References

- Ben-Chaim, David; Keret, Yaffa; Ilany, Bat-Sheva (2012).
*Ratio and Proportion: Research and Teaching in Mathematics Teachers. Springer Science & Business Media*. ISBN 9789460917844. - Burrell, Brian (1998).
*Merriam-Webster’s Guide to Everyday Math: A Home and Business Reference*. Merriam-Webster. ISBN 9780877796213. - Smith, D.E. (1925).
*History of Mathematics.*Vol. 2. Ginn and Company. - Van Dooren, Wim; De Bock, Dirk; Evers, Marleen; Verschaffel, Lieven (2009). “Students’ Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions.”
*Journal for Research in Mathematics Education*. 40 (2) 187–211.