Understanding Proportions Which Ratio Shows Equivalence

In the realm of mathematics, proportions play a crucial role in establishing relationships between quantities. A proportion is essentially a statement asserting the equality of two ratios. Understanding proportions is fundamental not only in academic settings but also in various real-world applications, ranging from scaling recipes in cooking to calculating distances on maps. This article delves into the concept of proportions, focusing on how to identify equivalent ratios and solve related problems. We'll dissect the given options to determine which one accurately represents a true proportion, providing a comprehensive explanation to enhance your understanding.

Defining Ratios and Proportions

Before we dive into the specifics of the problem at hand, let's solidify our understanding of ratios and proportions. A ratio is a comparison of two quantities, often expressed as a fraction. For instance, if you have 4 apples and 5 oranges, the ratio of apples to oranges is 4:5 or 4/5. A proportion, on the other hand, states that two ratios are equal. It's a way of saying that the relationship between two sets of quantities is the same. A proportion is typically written in the form a/b = c/d or a:b = c:d, where a, b, c, and d are numbers.

The key to understanding proportions lies in recognizing that equivalent ratios represent the same proportional relationship. This means that the fractions representing the ratios can be simplified to the same value, or that one ratio can be obtained from the other by multiplying or dividing both terms by the same non-zero number. For example, the ratios 1:2 and 2:4 are equivalent because 2/4 can be simplified to 1/2. Similarly, multiplying both terms of 1:2 by 3 gives us 3:6, which is also an equivalent ratio. Recognizing these equivalent relationships is crucial for solving proportion problems.

In real-world scenarios, proportions are used extensively. Consider scaling a recipe. If a recipe for 4 servings calls for 2 cups of flour, you can use a proportion to determine how much flour is needed for 8 servings. The ratio of servings to flour is 4:2, which simplifies to 2:1. To find the amount of flour needed for 8 servings, you can set up the proportion 4/2 = 8/x, where x is the unknown amount of flour. Solving for x gives you 4 cups of flour. This simple example highlights the practical utility of proportions in everyday life. Understanding how to set up and solve proportions is a valuable skill that extends beyond the classroom and into various aspects of decision-making and problem-solving.

Evaluating the Given Proportions

Now, let's apply our understanding of ratios and proportions to the given options. We need to determine which of the following proportions correctly demonstrates the equivalence of two ratios:

A) 2:3 = 27:18 B) 13:39 = 6:2 C) 14:35 = 7:2 D) 4:5 = 32:40

To evaluate these proportions, we will examine each option individually. The fundamental principle we'll use is that two ratios are equivalent if they simplify to the same fraction or if one ratio can be obtained from the other by multiplying or dividing both terms by the same number. We'll start by expressing each ratio as a fraction and then simplifying it to its lowest terms. This will allow us to directly compare the ratios and determine if they are equivalent. If the simplified fractions are equal, then the proportion is correct. If not, the proportion is incorrect.

For option A, we have the proportion 2:3 = 27:18. This can be written as the fractions 2/3 and 27/18. The fraction 2/3 is already in its simplest form. To simplify 27/18, we can divide both the numerator and the denominator by their greatest common divisor, which is 9. Dividing 27 by 9 gives us 3, and dividing 18 by 9 gives us 2. So, 27/18 simplifies to 3/2. Comparing the simplified fractions, we have 2/3 and 3/2. These fractions are not equal, indicating that the proportion in option A is incorrect.

Next, we consider option B, which presents the proportion 13:39 = 6:2. As fractions, this is 13/39 and 6/2. To simplify 13/39, we notice that both 13 and 39 are divisible by 13. Dividing 13 by 13 gives us 1, and dividing 39 by 13 gives us 3. Thus, 13/39 simplifies to 1/3. For 6/2, both numbers are divisible by 2. Dividing 6 by 2 gives us 3, and dividing 2 by 2 gives us 1. So, 6/2 simplifies to 3/1, or simply 3. Comparing the simplified fractions, we have 1/3 and 3. These are clearly not equal, meaning the proportion in option B is also incorrect.

Moving on to option C, the proportion is 14:35 = 7:2. The fractional representation is 14/35 and 7/2. To simplify 14/35, we look for a common divisor. Both 14 and 35 are divisible by 7. Dividing 14 by 7 gives us 2, and dividing 35 by 7 gives us 5. Therefore, 14/35 simplifies to 2/5. The fraction 7/2 is already in its simplest form. Comparing the simplified fractions, we have 2/5 and 7/2. These fractions are not equal, so option C does not represent a correct proportion.

Finally, we evaluate option D, which states the proportion 4:5 = 32:40. As fractions, this is 4/5 and 32/40. The fraction 4/5 is in its simplest form. To simplify 32/40, we can divide both the numerator and denominator by their greatest common divisor, which is 8. Dividing 32 by 8 gives us 4, and dividing 40 by 8 gives us 5. Therefore, 32/40 simplifies to 4/5. Comparing the simplified fractions, we have 4/5 and 4/5. Since these fractions are equal, the proportion in option D is correct. This means that the ratios 4:5 and 32:40 represent the same proportional relationship. In summary, only option D accurately demonstrates the equivalence of two ratios.

Detailed Analysis of the Correct Proportion (Option D)

Option D, which states the proportion 4:5 = 32:40, is the correct answer. To fully understand why this proportion is valid, let's delve deeper into the relationship between the two ratios. The ratio 4:5 can be interpreted as for every 4 units of one quantity, there are 5 units of another quantity. The ratio 32:40 expresses a similar relationship but with larger numbers. The key to determining the equivalence of these ratios lies in recognizing that one ratio can be obtained from the other by multiplying both terms by the same factor.

In this case, we can see that the ratio 32:40 is a multiple of the ratio 4:5. To get from 4 to 32, we multiply by 8 (4 * 8 = 32). Similarly, to get from 5 to 40, we also multiply by 8 (5 * 8 = 40). Since both terms of the ratio 4:5 are multiplied by the same factor (8), the resulting ratio 32:40 is equivalent. This illustrates a fundamental property of proportions: if you multiply or divide both terms of a ratio by the same non-zero number, the resulting ratio is equivalent to the original.

Another way to verify the equivalence is by simplifying both ratios to their lowest terms. The ratio 4:5 is already in its simplest form, as 4 and 5 have no common factors other than 1. The ratio 32:40 can be simplified by dividing both terms by their greatest common divisor, which is 8. Dividing 32 by 8 gives us 4, and dividing 40 by 8 gives us 5. Thus, the simplified form of 32:40 is 4:5, which is identical to the first ratio. This confirms that the two ratios are indeed equivalent, and the proportion 4:5 = 32:40 is correct.

The equivalence of these ratios can also be visualized graphically. Imagine dividing a pie into 5 equal slices and taking 4 of them. This represents the ratio 4:5. Now, imagine dividing the same pie into 40 equal slices and taking 32 of them. The proportion of the pie taken in both cases is the same. This visual representation helps to solidify the understanding that equivalent ratios represent the same proportional relationship, even though the numbers involved may be different.

In practical terms, this proportion could represent various scenarios. For example, if a recipe requires 4 cups of flour for every 5 cups of water, then scaling up the recipe would require maintaining this proportion. If you needed to make a larger batch and used 32 cups of flour, you would need 40 cups of water to maintain the correct consistency. This demonstrates the real-world applicability of understanding and working with proportions. The ability to recognize and manipulate equivalent ratios is essential for problem-solving in a wide range of fields, from cooking and baking to engineering and finance.

Why the Other Options Are Incorrect

To gain a complete understanding of proportions, it's crucial not only to identify correct proportions but also to recognize why incorrect proportions are flawed. Let's revisit the incorrect options (A, B, and C) and dissect their inaccuracies:

Option A: 2:3 = 27:18

This proportion is incorrect because the ratios 2:3 and 27:18 do not represent the same relationship. As we established earlier, equivalent ratios must simplify to the same fraction or be obtainable from each other by multiplying or dividing both terms by the same factor. In this case, the ratio 2:3 is already in its simplest form. The ratio 27:18 can be simplified by dividing both terms by their greatest common divisor, which is 9. Dividing 27 by 9 gives us 3, and dividing 18 by 9 gives us 2. So, 27:18 simplifies to 3:2.

Comparing the simplified ratios, we have 2:3 and 3:2. These are not the same. The ratio 3:2 is actually the inverse of 2:3. To further illustrate why this proportion is incorrect, consider if we were to try to multiply the terms of 2:3 by a single factor to obtain 27:18. To get from 2 to 27, we would need to multiply by 13.5 (2 * 13.5 = 27). However, if we multiply 3 by 13.5, we get 40.5, not 18. This confirms that there is no single factor that can be used to transform 2:3 into 27:18, and therefore, the proportion is incorrect.

Option B: 13:39 = 6:2

The flaw in this proportion lies in the fact that the ratios 13:39 and 6:2 do not share the same proportional relationship. To determine this, we can simplify both ratios. The ratio 13:39 can be simplified by dividing both terms by their greatest common divisor, which is 13. Dividing 13 by 13 gives us 1, and dividing 39 by 13 gives us 3. Thus, 13:39 simplifies to 1:3.

The ratio 6:2 can be simplified by dividing both terms by their greatest common divisor, which is 2. Dividing 6 by 2 gives us 3, and dividing 2 by 2 gives us 1. So, 6:2 simplifies to 3:1. Comparing the simplified ratios, we have 1:3 and 3:1. These are not equivalent. In fact, they are reciprocals of each other. This means that the relationship expressed by 1:3 is the opposite of the relationship expressed by 3:1. Therefore, the proportion 13:39 = 6:2 is incorrect.

Option C: 14:35 = 7:2

This proportion is also incorrect because the ratios 14:35 and 7:2 do not represent the same proportional relationship. Let's simplify both ratios to demonstrate this. The ratio 14:35 can be simplified by dividing both terms by their greatest common divisor, which is 7. Dividing 14 by 7 gives us 2, and dividing 35 by 7 gives us 5. So, 14:35 simplifies to 2:5.

The ratio 7:2 is already in its simplest form, as 7 and 2 have no common factors other than 1. Comparing the simplified ratios, we have 2:5 and 7:2. These are clearly not equivalent. There is no single factor that can be used to multiply both terms of 2:5 to obtain 7:2. This confirms that the proportion 14:35 = 7:2 is incorrect. Understanding why these proportions are incorrect reinforces the concept that a proportion is a statement of equality between two equivalent ratios, and this equivalence must be verifiable through simplification or by finding a common multiplier.

Conclusion

In conclusion, the proportion that correctly shows the equivalence of two ratios is D) 4:5 = 32:40. This proportion holds true because the ratio 32:40 can be obtained by multiplying both terms of the ratio 4:5 by the same factor (8), or by simplifying 32:40 to 4:5. The other options (A, B, and C) were incorrect because their ratios did not simplify to the same form and could not be obtained from each other through multiplication by a common factor. Understanding the concept of proportions and equivalent ratios is fundamental in mathematics and has numerous practical applications in everyday life. By mastering this concept, you can confidently solve problems involving scaling, comparisons, and proportional relationships. This knowledge equips you with a valuable tool for both academic pursuits and real-world problem-solving scenarios, enhancing your analytical and decision-making skills.

By dissecting the given options and thoroughly explaining the underlying principles of proportions, this article aims to provide a comprehensive understanding of how to identify equivalent ratios. The ability to correctly assess proportions is a crucial skill in mathematics and beyond, enabling individuals to make accurate comparisons and informed decisions in various contexts.