Simplifying The Ratio 1.2 To 33 1/3% A Step-by-Step Guide

Understanding Ratios and Percentages

In mathematics, ratios are used to compare two or more quantities. A ratio indicates how many times one quantity contains another. They are fundamental in various mathematical concepts and real-world applications, from scaling recipes to analyzing financial data. Expressing ratios in their simplest form allows for easier comparison and understanding. The ratio 1.2 : 33 rac{1}{3}\% might seem complex at first glance, but it can be simplified through a series of steps. This process involves converting the percentage to a decimal or fraction, ensuring both parts of the ratio are in the same format, and then reducing the ratio to its simplest form by dividing both sides by their greatest common factor. Understanding these initial concepts is crucial for tackling the problem efficiently and accurately. The importance of simplifying ratios lies in the clarity and ease of comparison they provide. A simplified ratio makes it easier to grasp the relative proportions of the quantities being compared, which is essential in decision-making processes and problem-solving scenarios. For instance, in business, simplifying ratios can help in analyzing profit margins or debt-to-equity ratios, providing a clear picture of the company's financial health. Similarly, in cooking, simplified ratios of ingredients allow for easy scaling of recipes while maintaining the correct proportions. Therefore, mastering the techniques of simplifying ratios is not just a mathematical skill but also a practical tool applicable in various aspects of life. By understanding the fundamental principles of ratios and percentages, and by applying the appropriate conversion and simplification techniques, we can effectively tackle complex ratios and make informed decisions based on clear and concise information.

Step 1: Convert the Percentage to a Fraction

The first step in simplifying the ratio 1.2 : 33 rac{1}{3}\% is to convert the percentage into a fraction. Percentages are essentially fractions with a denominator of 100, making this conversion a crucial step in making the ratio more manageable. The percentage 33 rac{1}{3}\% is a mixed number percentage, which can be a bit tricky to work with directly. To convert it to a fraction, we first need to convert the mixed number percentage into an improper fraction. 33 rac{1}{3} can be written as $\frac{(33 \times 3) + 1}{3} = \frac{100}{3}$. This gives us $\frac{100}{3}\%$. Now, we recall that percent means "out of one hundred," so we divide the fraction by 100 to convert the percentage to a fraction. This gives us $\frac{100}{3} \div 100 = \frac{100}{3} \times \frac{1}{100}$. When we multiply these fractions, the 100 in the numerator and the 100 in the denominator cancel each other out, leaving us with $\frac{1}{3}$. Thus, 33 rac{1}{3}\% is equivalent to the fraction $\frac{1}{3}$. This conversion is a fundamental step in simplifying the ratio because it allows us to express both parts of the ratio in a common format, either as decimals or fractions. By converting the percentage to a fraction, we eliminate the percentage sign and deal with a more straightforward numerical value. This makes the subsequent steps of simplification much easier to perform. Understanding this conversion process is not just about following a set of rules; it’s about grasping the underlying relationship between percentages and fractions. This understanding enables us to apply this knowledge in various contexts, such as calculating discounts, understanding financial statements, or even adjusting recipes. By mastering this skill, we lay a solid foundation for more advanced mathematical concepts and problem-solving techniques.

Step 2: Convert the Decimal to a Fraction

Having converted the percentage portion of the ratio to a fraction, the next step is to address the decimal component. In the ratio 1.2 : 33 rac{1}{3}\%, 1.2 is a decimal number, and to simplify the ratio effectively, we need to express it as a fraction as well. Converting decimals to fractions involves understanding the place value system. The decimal 1.2 has a 1 in the ones place and a 2 in the tenths place. This means that 1.2 can be read as "one and two tenths." To convert this to a fraction, we can write it as a mixed number: $1 \frac{2}{10}$. However, to make it easier to work with in a ratio, it’s best to convert it to an improper fraction. To do this, we multiply the whole number (1) by the denominator (10) and add the numerator (2), then place the result over the original denominator: $\frac{(1 \times 10) + 2}{10} = \frac{12}{10}$. Now, we have 1.2 expressed as the fraction $\frac{12}{10}$. This fraction can be further simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 12 and 10 is 2. Dividing both the numerator and the denominator by 2, we get $\frac{12 \div 2}{10 \div 2} = \frac{6}{5}$. So, the simplified fractional representation of 1.2 is $\frac{6}{5}$. This step is crucial because it ensures that both parts of the ratio are expressed in the same format – fractions. This consistency is essential for performing accurate comparisons and simplifications. By converting the decimal to a fraction, we eliminate the decimal point, making it easier to find a common denominator and perform the necessary arithmetic operations to simplify the ratio. Understanding the conversion of decimals to fractions is a fundamental skill in mathematics. It allows us to work seamlessly between different representations of numbers, which is vital in various mathematical contexts and real-world applications. Whether we are calculating proportions, solving equations, or analyzing data, the ability to convert between decimals and fractions is an invaluable tool.

Step 3: Express the Ratio with Fractions

Now that we have converted both the percentage and the decimal to fractions, we can express the original ratio 1.2 : 33 rac{1}{3}\% using fractions. We found that $1.2$ is equivalent to the fraction $\frac{6}{5}$, and 33 rac{1}{3}\% is equivalent to the fraction $\frac{1}{3}$. Therefore, the ratio can be rewritten as $\frac{6}{5} : \frac{1}{3}$. Expressing the ratio with fractions is a critical step because it sets the stage for simplifying the ratio to its simplest form. Working with fractions allows us to use the rules of fraction manipulation to our advantage. To simplify a ratio of fractions, we need to eliminate the denominators. This is typically done by finding the least common multiple (LCM) of the denominators and multiplying both parts of the ratio by this LCM. In this case, the denominators are 5 and 3. The LCM of 5 and 3 is 15. Multiplying both parts of the ratio by 15 will clear the fractions and give us a ratio of whole numbers. The process of expressing the ratio with fractions also highlights the importance of understanding equivalent fractions. When we convert decimals and percentages to fractions, we are essentially finding equivalent representations of the same value. This understanding is fundamental in various mathematical contexts, including algebra, calculus, and beyond. By expressing the ratio with fractions, we make it easier to see the relationship between the two quantities being compared. Fractions provide a clear representation of proportions, which is crucial for understanding and interpreting ratios. This step also reinforces the idea that ratios are essentially comparisons of quantities, and by expressing them as fractions, we can apply the rules of fraction arithmetic to simplify these comparisons. Understanding how to express ratios with fractions is not just a mathematical technique; it’s a way of thinking about proportions and relationships. This skill is valuable in various fields, from finance to engineering, where understanding and manipulating ratios is essential for problem-solving and decision-making.

Step 4: Eliminate the Denominators

With the ratio expressed as $\frac{6}{5} : \frac{1}{3}$, the next crucial step is to eliminate the denominators. This is essential for simplifying the ratio and expressing it in its simplest form using whole numbers. As mentioned earlier, the key to eliminating denominators in a ratio of fractions is to find the least common multiple (LCM) of the denominators. In our case, the denominators are 5 and 3. The LCM of 5 and 3 is 15. To eliminate the denominators, we multiply both parts of the ratio by the LCM. This means we multiply both $\frac{6}{5}$ and $\frac{1}{3}$ by 15. Multiplying $\frac{6}{5}$ by 15 gives us: $\frac{6}{5} \times 15 = \frac{6 \times 15}{5} = \frac{90}{5} = 18$. Multiplying $\frac{1}{3}$ by 15 gives us: $\frac{1}{3} \times 15 = \frac{1 \times 15}{3} = \frac{15}{3} = 5$. So, after multiplying both parts of the ratio by the LCM, we get a new ratio of 18 : 5. This new ratio is equivalent to the original ratio but is now expressed using whole numbers, making it much easier to understand and work with. Eliminating the denominators is a fundamental technique in simplifying ratios and proportions. It allows us to compare quantities directly without the complication of fractions. This step is particularly important in practical applications where ratios are used to make decisions or solve problems. For example, in cooking, if a recipe calls for ingredients in a fractional ratio, eliminating the denominators can help you easily scale the recipe up or down. The process of finding the LCM and multiplying both parts of the ratio is also a great way to reinforce the concept of equivalent fractions. When we multiply both the numerator and the denominator of a fraction by the same number, we are creating an equivalent fraction. In this case, we are essentially multiplying each fraction by a form of 1 (15/15), which doesn't change the value of the ratio but allows us to express it in a more convenient form. This skill is valuable in various mathematical contexts, including algebra, calculus, and beyond.

Step 5: Simplify the Ratio

After eliminating the denominators, we have the ratio expressed as 18 : 5. The final step in simplifying the ratio 1.2 : 33 rac{1}{3}\% is to ensure that the ratio is in its simplest form. This means that the two numbers in the ratio, 18 and 5, should have no common factors other than 1. In other words, we need to check if the ratio can be further reduced by dividing both numbers by their greatest common divisor (GCD). To determine if 18 and 5 have any common factors, we can list the factors of each number. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 5 are 1 and 5. Comparing the lists of factors, we can see that the only common factor between 18 and 5 is 1. This means that 18 and 5 are relatively prime, and the ratio 18 : 5 is already in its simplest form. Therefore, the simplified form of the ratio 1.2 : 33 rac{1}{3}\% is 18 : 5. This simplified ratio provides a clear and concise comparison of the two quantities. It tells us that for every 18 units of the first quantity, there are 5 units of the second quantity. Simplifying ratios to their simplest form is crucial for making meaningful comparisons and solving problems involving proportions. A simplified ratio makes it easier to understand the relationship between the quantities being compared and to perform calculations based on this relationship. For example, if we know that two quantities are in the ratio 18 : 5, we can use this information to determine the amount of one quantity given the amount of the other. Understanding how to simplify ratios is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and finance. It is a skill that allows us to make sense of numerical relationships and to solve problems involving proportions efficiently and accurately. By ensuring that a ratio is in its simplest form, we can communicate the relationship between quantities in the clearest and most concise way possible.

Conclusion

In conclusion, simplifying the ratio 1.2 : 33 rac{1}{3}\% involves a series of essential steps that highlight fundamental mathematical concepts. Starting with the conversion of the percentage to a fraction, we transform 33 rac{1}{3}\% into $\frac{1}{3}$. This step underscores the relationship between percentages and fractions and sets the stage for expressing both parts of the ratio in a common format. Next, converting the decimal 1.2 to a fraction gives us $\frac{6}{5}$, further emphasizing the versatility of fractions in representing numerical values. By expressing the ratio with fractions, we rewrite the original ratio as $\frac{6}{5} : \frac{1}{3}$. This representation allows us to apply fraction manipulation techniques to simplify the ratio effectively. Eliminating the denominators by finding the least common multiple (LCM) of 5 and 3, which is 15, we multiply both parts of the ratio by 15. This step transforms the ratio into 18 : 5, removing the complexity of fractions and making the comparison more straightforward. Finally, simplifying the ratio by checking for common factors confirms that 18 and 5 have no common factors other than 1. This ensures that the ratio is in its simplest form, 18 : 5, providing a clear and concise comparison of the two quantities. The process of simplifying ratios not only enhances our understanding of proportions but also reinforces the importance of mastering fundamental mathematical skills such as fraction manipulation, decimal-to-fraction conversion, and finding the LCM. These skills are crucial in various mathematical contexts and have practical applications in real-world scenarios, from scaling recipes to analyzing financial data. By following these steps, we can confidently simplify complex ratios and gain a deeper appreciation for the elegance and efficiency of mathematical problem-solving. The ability to simplify ratios is a valuable tool in various fields, enabling us to make informed decisions and solve problems with clarity and precision.