Simplifying Ratios 5 Cm 5 M 0.02 Km A Step By Step Solution

Ratios are a fundamental concept in mathematics, serving as a cornerstone for various applications across different fields. In essence, a ratio is a comparison of two or more quantities, providing insights into their relative sizes. This comparison is typically expressed in a simplified form, allowing for easy comprehension and analysis. Understanding ratios is crucial not only for academic pursuits but also for practical problem-solving in everyday life.

The importance of ratios extends beyond the classroom. In everyday scenarios, we often encounter ratios without even realizing it. For example, when cooking, a recipe might call for a specific ratio of ingredients, such as flour to sugar. Similarly, in finance, ratios are used to assess the financial health of a company, comparing its assets to its liabilities. In fields like engineering and architecture, ratios play a vital role in scaling designs and ensuring structural integrity.

Ratios are expressed in several ways, but the most common method is using a colon (:) to separate the quantities being compared. For instance, a ratio of 2:3 indicates that for every two units of the first quantity, there are three units of the second quantity. This representation allows for a clear and concise comparison. Another way to express ratios is as fractions, where the ratio 2:3 can be written as 2/3. This fractional representation is particularly useful when performing calculations or comparing ratios. Additionally, ratios can be expressed as decimals or percentages, providing alternative perspectives on the relationship between quantities.

To simplify ratios, the goal is to reduce the quantities to their smallest whole number equivalents while maintaining the proportional relationship. This is achieved by dividing each quantity in the ratio by their greatest common divisor (GCD). For example, the ratio 10:15 can be simplified by dividing both numbers by 5, resulting in the simplified ratio 2:3. Simplification makes ratios easier to understand and compare, which is essential in many mathematical and real-world applications. Understanding how to manipulate and simplify ratios is a crucial skill in mathematics, enabling effective problem-solving and decision-making in various contexts.

Problem Statement Deconstructing the 5 cm 5 m 0.02 km Ratio

In this particular mathematical challenge, we are presented with the ratio 5 cm: 5 m: 0.02 km. This ratio involves three quantities, each expressed in different units of measurement. To effectively compare these quantities and simplify the ratio, it is imperative that we first convert them to a common unit. This initial step is crucial because it ensures that we are comparing like with like, thereby avoiding any misinterpretations or inaccuracies. Without a common unit, the numerical values would not accurately reflect the true proportional relationship between the quantities.

The first quantity in the ratio is 5 centimeters (cm). Centimeters are a unit of length in the metric system, commonly used for measuring relatively small distances. The second quantity is 5 meters (m). Meters are also a unit of length in the metric system, but they are larger than centimeters. One meter is equivalent to 100 centimeters. This means that 5 meters is equal to 500 centimeters. This conversion is essential because it allows us to express the second quantity in the same unit as the first quantity, facilitating a direct comparison. The third quantity is 0.02 kilometers (km). Kilometers are a larger unit of length in the metric system, with one kilometer being equivalent to 1000 meters or 100,000 centimeters. Therefore, 0.02 kilometers is equal to 0.02 multiplied by 100,000 centimeters, which equals 2000 centimeters. This conversion ensures that all three quantities are expressed in the same unit, centimeters, allowing for accurate comparison and simplification of the ratio.

By converting all quantities to a common unit, we set the stage for a meaningful comparison and simplification of the ratio. This step is a fundamental aspect of working with ratios, particularly when dealing with different units of measurement. The subsequent steps will involve simplifying the ratio by finding the greatest common divisor and expressing the quantities in their simplest form. This process will ultimately lead us to the correct answer and enhance our understanding of ratio simplification.

Unit Conversion cm to m to km

To accurately compare and simplify the ratio 5 cm: 5 m: 0.02 km, the crucial first step involves converting all the quantities to a common unit. The metric system, being a decimal system, provides a straightforward way to perform these conversions. Understanding the relationships between centimeters, meters, and kilometers is essential for this process.

Let's begin with the conversion of meters to centimeters. There are 100 centimeters in a meter. Therefore, to convert 5 meters to centimeters, we multiply 5 by 100. This calculation yields 5 * 100 = 500 centimeters. So, 5 meters is equivalent to 500 centimeters. This conversion allows us to express the second quantity in the same unit as the first quantity, which is already in centimeters. The importance of this conversion lies in the fact that it enables us to compare the quantities directly, without the confusion of different units. Without this conversion, the ratio would not accurately reflect the proportional relationship between the quantities.

Next, we need to convert kilometers to centimeters. There are 1000 meters in a kilometer, and each meter contains 100 centimeters. Therefore, there are 1000 * 100 = 100,000 centimeters in a kilometer. To convert 0.02 kilometers to centimeters, we multiply 0.02 by 100,000. This calculation results in 0.02 * 100,000 = 2000 centimeters. Thus, 0.02 kilometers is equivalent to 2000 centimeters. This conversion is vital for the same reasons as the previous one: it ensures that all quantities are expressed in the same unit, allowing for a fair and accurate comparison. Converting kilometers to centimeters might seem like a large conversion, but it is necessary to align the third quantity with the units of the other two quantities.

Now that we have converted all the quantities to centimeters, the ratio becomes 5 cm: 500 cm: 2000 cm. Expressing the ratio in a common unit is a fundamental step in simplifying it. This conversion process not only facilitates the simplification but also ensures that the final ratio accurately represents the proportional relationships between the original quantities. With all the quantities now in centimeters, we can proceed to the next step, which involves simplifying the ratio to its lowest terms.

Ratio Simplification Finding the Greatest Common Divisor (GCD)

Now that we have expressed the ratio 5 cm: 5 m: 0.02 km in a common unit as 5 cm: 500 cm: 2000 cm, the next crucial step is to simplify this ratio. Simplification involves reducing the numbers in the ratio to their smallest whole number equivalents while maintaining the same proportional relationship. To achieve this, we need to find the greatest common divisor (GCD) of the numbers 5, 500, and 2000.

The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest number that divides evenly into each of the numbers in the ratio. Finding the GCD is essential for simplifying ratios because dividing each number by the GCD ensures that the resulting numbers are the smallest possible whole numbers that maintain the original ratio. There are several methods to find the GCD, including listing factors, prime factorization, and the Euclidean algorithm. In this case, listing factors is a straightforward approach due to the relatively small numbers involved.

Let's identify the factors of each number. The factors of 5 are 1 and 5. The factors of 500 include 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, and 500. The factors of 2000 include 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000, and 2000. By examining these factors, we can identify the largest number that is common to all three lists. In this case, the greatest common divisor of 5, 500, and 2000 is 5. The GCD is the key to simplifying the ratio effectively. Once we have found the GCD, we can proceed to divide each number in the ratio by this value. This step will reduce the numbers to their smallest possible whole number equivalents, thereby simplifying the ratio while preserving the original proportional relationship. The process of finding the GCD is a fundamental aspect of simplifying ratios and is crucial for various mathematical applications.

Dividing by the GCD Simplifying the Ratio to its Simplest Form

With the greatest common divisor (GCD) identified as 5 for the ratio 5 cm: 500 cm: 2000 cm, the next step is to divide each quantity in the ratio by this GCD. This division will reduce the numbers in the ratio to their smallest whole number equivalents, providing the simplified form of the ratio. Dividing each quantity by the GCD is a fundamental process in simplifying ratios, ensuring that the proportional relationship between the quantities is maintained while the numbers are as small as possible.

To begin, we divide the first quantity, 5 cm, by the GCD, which is 5. The calculation is 5 ÷ 5 = 1. This means that the first term in the simplified ratio will be 1. The division process is straightforward but critical, as it reduces the original number to its simplest form while maintaining its proportion relative to the other quantities in the ratio.

Next, we divide the second quantity, 500 cm, by the GCD, 5. The calculation is 500 ÷ 5 = 100. Thus, the second term in the simplified ratio is 100. This step is essential for ensuring that the ratio accurately reflects the original proportions in the simplest possible terms. The result of this division shows how the second quantity compares to the first quantity in the simplified ratio.

Finally, we divide the third quantity, 2000 cm, by the GCD, 5. The calculation is 2000 ÷ 5 = 400. Therefore, the third term in the simplified ratio is 400. This division completes the process of simplifying the ratio, ensuring that all quantities are expressed in their smallest whole number forms while preserving their original proportional relationship. After performing these divisions, the simplified ratio is 1: 100: 400. This ratio represents the same proportional relationship as the original ratio but in a more concise and understandable form. The process of dividing by the GCD is a key step in simplifying ratios and is crucial for solving various mathematical problems involving ratios and proportions.

Solution and Answer Rationalizing the Final Ratio

After performing the necessary conversions and simplifications, we have arrived at the simplified ratio 1: 100: 400. This ratio represents the original ratio of 5 cm: 5 m: 0.02 km in its simplest form. Each term in the simplified ratio is the smallest whole number that maintains the proportional relationship between the original quantities.

The process began with the ratio 5 cm: 5 m: 0.02 km. To compare these quantities effectively, we converted them to a common unit, which was centimeters. This conversion resulted in the ratio 5 cm: 500 cm: 2000 cm. The next step involved finding the greatest common divisor (GCD) of the numbers 5, 500, and 2000, which was determined to be 5. Dividing each term in the ratio by the GCD simplified the ratio to 1: 100: 400. This final ratio is the answer to the problem, providing a clear and concise representation of the proportional relationship between the original quantities.

Now, let’s consider the given options to identify the one that matches our simplified ratio. The options provided were:

A. 1: 10: 400 B. 1: 100: 400 C. 1: 300: 2000 D. 2: 10: 500

By comparing our simplified ratio of 1: 100: 400 with the given options, it is evident that option B, 1: 100: 400, is the correct answer. This option matches the simplified ratio we derived through our calculations, confirming that our solution is accurate. Option B accurately represents the proportional relationship between the original quantities in the simplest possible terms. Therefore, the final answer to the question “Which of the following is equal to the ratio 5 cm: 5 m: 0.02 km?” is B. 1: 100: 400.

Conclusion Mastering Ratios A Key to Mathematical Proficiency

In conclusion, simplifying the ratio 5 cm: 5 m: 0.02 km to its simplest form, 1: 100: 400, has been a comprehensive exercise in understanding and applying fundamental mathematical concepts. This process underscores the importance of unit conversion, finding the greatest common divisor (GCD), and dividing by the GCD to achieve simplification. Each step in this process is crucial for ensuring accuracy and clarity in mathematical problem-solving.

The initial step, converting all quantities to a common unit, is essential for comparing and simplifying ratios effectively. Without this step, the different units of measurement would obscure the true proportional relationship between the quantities. In this case, converting meters and kilometers to centimeters allowed us to express all quantities in the same unit, facilitating a direct comparison. This highlights the significance of unit conversion in various mathematical and scientific applications, where quantities are often expressed in different units.

Finding the greatest common divisor (GCD) is another critical step in simplifying ratios. The GCD is the largest number that divides evenly into each term of the ratio, and dividing by the GCD reduces the terms to their smallest whole number equivalents while maintaining the proportional relationship. In this problem, the GCD of 5, 500, and 2000 was 5. Dividing each term by 5 resulted in the simplified ratio 1: 100: 400. Understanding how to find and use the GCD is a valuable skill in simplifying ratios and fractions.

The final simplified ratio, 1: 100: 400, provides a clear and concise representation of the proportional relationship between the original quantities. This ratio is easier to understand and work with compared to the original ratio with different units. It demonstrates the power of simplification in making complex mathematical relationships more accessible. Mastering the process of simplifying ratios is a valuable skill that extends beyond mathematics, finding applications in various fields such as science, engineering, and everyday problem-solving.