Simplifying Ratios A Step-by-Step Guide

In mathematics, a ratio is a comparison of two quantities of the same kind. It indicates how many times one quantity contains another. A proportion, on the other hand, is an equality between two ratios. Understanding ratios and proportions is fundamental in various fields, including mathematics, science, economics, and everyday life. This article delves into expressing ratios in their simplest form, providing clear explanations and examples to solidify your understanding. We will explore several practical scenarios and demonstrate how to reduce ratios to their simplest form, making them easier to interpret and use in calculations.

Understanding Ratios

Before we dive into expressing ratios in their simplest form, let's first understand what a ratio is. A ratio compares two quantities of the same unit. It can be expressed in several ways, such as using a colon (:), a fraction, or the word "to." For example, if there are 5 apples and 3 oranges in a basket, the ratio of apples to oranges can be written as 5:3, 5/3, or 5 to 3. This means that for every 5 apples, there are 3 oranges. The order of the quantities in a ratio is crucial. The ratio of oranges to apples would be 3:5, which is different from 5:3.

Ratios are used to compare quantities, show proportions, and solve problems involving scaling and division. They are particularly useful when dealing with mixtures, rates, and proportions. In mathematics, ratios are often simplified to their lowest terms, making them easier to work with. Simplifying a ratio involves dividing both quantities by their greatest common divisor (GCD). This process ensures that the ratio is expressed in the smallest possible whole numbers while maintaining the same proportional relationship. Understanding the basic principles of ratios is essential for solving various mathematical problems and real-world applications. Whether it's calculating mixtures, comparing quantities, or determining proportions, a solid grasp of ratios provides a valuable tool for analysis and decision-making. The ability to simplify ratios further enhances their utility, allowing for clearer comparisons and easier calculations. Let's explore how to express different types of ratios in their simplest forms, providing a foundational understanding for more complex mathematical concepts.

Expressing Ratios in Simplest Form

To express a ratio in its simplest form, you need to reduce it to its lowest terms. This involves finding the greatest common divisor (GCD) of the two numbers and dividing both numbers by the GCD. The simplest form of a ratio is when the two numbers have no common factors other than 1. Let’s illustrate this process with the examples provided.

(a) 5.6 to 28 cm

In this scenario, we aim to express the ratio 5.6 to 28 in its simplest form. The initial ratio is 5.6 cm to 28 cm. The first step in simplifying this ratio is to eliminate the decimal. We can do this by multiplying both sides of the ratio by 10. This gives us a new ratio of 56 to 280. Now, we need to find the greatest common divisor (GCD) of 56 and 280. The GCD is the largest number that divides both 56 and 280 without leaving a remainder. One way to find the GCD is to list the factors of each number and identify the largest factor they have in common. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The factors of 280 are 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, and 280. The largest factor that both 56 and 280 share is 56. Therefore, the GCD of 56 and 280 is 56. To simplify the ratio, we divide both numbers by the GCD, which is 56. So, 56 divided by 56 is 1, and 280 divided by 56 is 5. Thus, the simplified ratio is 1 to 5. This means that for every 1 unit of the first quantity, there are 5 units of the second quantity. Expressing the ratio in its simplest form makes it easier to understand the proportional relationship between the two quantities. It also simplifies any further calculations involving the ratio.

(b) Dozen to a Score

Here, we are tasked with expressing the ratio of a dozen to a score in its simplest form. We need to remember what these terms mean in numerical values. A dozen is equal to 12 items, and a score is equal to 20 items. Therefore, the ratio we are dealing with is 12 to 20. To simplify this ratio, we need to find the greatest common divisor (GCD) of 12 and 20. The GCD is the largest number that can divide both 12 and 20 without leaving a remainder. To find the GCD, we can list the factors of both numbers. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 20 are 1, 2, 4, 5, 10, and 20. By comparing the lists, we can see that the largest common factor is 4. So, the GCD of 12 and 20 is 4. Now, we divide both numbers in the ratio by the GCD. Dividing 12 by 4 gives us 3, and dividing 20 by 4 gives us 5. Thus, the simplified ratio is 3 to 5. This ratio means that for every 3 units of the first quantity (a dozen), there are 5 units of the second quantity (a score). Expressing the ratio in this simplified form makes it easier to grasp the proportional relationship between the two quantities. It also aids in making comparisons and performing further calculations. Simplifying ratios like this is a fundamental skill in mathematics, used in various real-world applications, such as cooking, construction, and financial analysis. Understanding and simplifying ratios helps in making informed decisions and solving problems efficiently.

(c) Score to a Gross

In this example, we are required to express the ratio of a score to a gross in its simplest form. First, we need to know the numerical values of a score and a gross. As we established earlier, a score is equal to 20 items. A gross, on the other hand, is equal to 144 items. So, the ratio we are dealing with is 20 to 144. To simplify this ratio, we must find the greatest common divisor (GCD) of 20 and 144. The GCD is the largest number that can divide both 20 and 144 without leaving a remainder. To find the GCD, we can list the factors of both numbers. The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. By comparing the lists, we can see that the largest common factor is 4. Therefore, the GCD of 20 and 144 is 4. Now, we divide both numbers in the ratio by the GCD. Dividing 20 by 4 gives us 5, and dividing 144 by 4 gives us 36. Thus, the simplified ratio is 5 to 36. This simplified ratio means that for every 5 units of the first quantity (a score), there are 36 units of the second quantity (a gross). Expressing the ratio in this form makes it much easier to understand the relationship between the quantities. It helps in quickly grasping how many scores are equivalent to a certain number of gross, or vice versa. The ability to simplify ratios is an important skill in many areas of mathematics and real-world applications, allowing for clearer comparisons and more efficient problem-solving.

(d) 6 hours to a day

For this part, we need to express the ratio of 6 hours to a day in its simplest form. The first step is to ensure that both quantities are in the same units. We know that there are 24 hours in a day. Therefore, the ratio is 6 hours to 24 hours. Now we can write this ratio as 6 to 24. To simplify this ratio, we need to find the greatest common divisor (GCD) of 6 and 24. The GCD is the largest number that divides both 6 and 24 without leaving a remainder. The factors of 6 are 1, 2, 3, and 6. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The largest factor that both 6 and 24 share is 6. Therefore, the GCD of 6 and 24 is 6. Next, we divide both numbers in the ratio by the GCD. Dividing 6 by 6 gives us 1, and dividing 24 by 6 gives us 4. Thus, the simplified ratio is 1 to 4. This means that for every 1 unit of time in the first quantity (6 hours), there are 4 units of time in the second quantity (24 hours or 1 day). In other words, 6 hours is one-fourth of a day. Simplifying ratios in this manner is crucial for making quick comparisons and understanding proportions. It allows for easier visualization of the relationship between the quantities. In various applications, such as scheduling, time management, and planning, the ability to simplify ratios involving time is invaluable. This skill enables efficient calculations and clear interpretations of time-related proportions.

(e) 20 litres to 0.75 litres

In this instance, we aim to express the ratio of 20 litres to 0.75 litres in its simplest form. The ratio is initially presented as 20 to 0.75. The first step in simplifying this ratio is to eliminate the decimal from 0.75. To do this, we can multiply both numbers in the ratio by 100. This gives us a new ratio of 2000 to 75. Now, we need to find the greatest common divisor (GCD) of 2000 and 75. The GCD is the largest number that divides both 2000 and 75 without leaving a remainder. The prime factorization method can be used to find the GCD. First, we find the prime factors of each number. The prime factors of 2000 are 2 × 2 × 2 × 2 × 5 × 5 × 5, and the prime factors of 75 are 3 × 5 × 5. The common prime factors are 5 × 5, which equals 25. Therefore, the GCD of 2000 and 75 is 25. Next, we divide both numbers in the ratio by the GCD. Dividing 2000 by 25 gives us 80, and dividing 75 by 25 gives us 3. Thus, the simplified ratio is 80 to 3. This means that for every 80 units of the first quantity (20 litres), there are 3 units of the second quantity (0.75 litres). Simplifying ratios involving decimals often requires this initial step of multiplying by a power of 10 to eliminate the decimal, followed by finding the GCD and dividing. This process is crucial in various scientific, engineering, and practical applications where precise comparisons of quantities are needed. Expressing the ratio in its simplest form allows for a clearer understanding of the proportional relationship between the two volumes.

(f) 1728 : 2400

Finally, we address the ratio 1728 : 2400 and seek to express it in its simplest form. The ratio is given as 1728 to 2400. To simplify this ratio, we need to find the greatest common divisor (GCD) of 1728 and 2400. The GCD is the largest number that can divide both 1728 and 2400 without leaving a remainder. The prime factorization method can be employed to find the GCD efficiently. First, we find the prime factors of each number. The prime factors of 1728 are 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, and the prime factors of 2400 are 2 × 2 × 2 × 2 × 3 × 5 × 5. By comparing the prime factorizations, we identify the common prime factors and their lowest powers: 2 × 2 × 2 × 2 × 3, which equals 144. Thus, the GCD of 1728 and 2400 is 144. Now, we divide both numbers in the ratio by the GCD. Dividing 1728 by 144 gives us 12, and dividing 2400 by 144 gives us 16. So, the ratio is now 12 to 16. However, we can simplify this ratio further. The GCD of 12 and 16 is 4. Dividing both numbers by 4, we get 3 to 4. Thus, the simplest form of the ratio 1728 to 2400 is 3 to 4. This simplified ratio indicates that for every 3 units of the first quantity, there are 4 units of the second quantity. This process of finding the GCD and simplifying ratios is fundamental in various mathematical contexts, including proportional reasoning, scaling, and comparative analysis. Expressing ratios in their simplest form allows for easier understanding and application in practical scenarios.

Conclusion

Expressing ratios in their simplest form is a fundamental skill in mathematics. By finding the greatest common divisor (GCD) and dividing both quantities by it, we can simplify ratios to their lowest terms. This process makes it easier to compare and understand the relationship between different quantities. The examples discussed above provide a clear understanding of how to simplify various types of ratios, from those involving decimals to those involving different units of measurement. Mastering this skill is crucial for solving a wide range of mathematical problems and real-world applications.