Ratio Of 2 Hours To 30 Minutes In Lowest Term A Step-by-Step Guide

In the realm of mathematics, ratios play a crucial role in comparing quantities and understanding their proportional relationships. A ratio is essentially a way to express how much of one thing there is compared to another. It's a fundamental concept that finds applications in various fields, from everyday cooking and measurements to complex scientific calculations. When dealing with ratios, it's often necessary to simplify them to their lowest terms, which means expressing the ratio in its simplest form where the numbers have no common factors other than 1. This makes the comparison clearer and easier to understand. Let's dive into the process of finding the lowest term ratio, specifically focusing on the example of comparing two hours to 30 minutes.

When we talk about ratios, we're essentially comparing two or more quantities. These quantities can be anything – distances, weights, amounts of ingredients, or even time, as in our case. The ratio is expressed using a colon (:) between the quantities being compared. For instance, if we have 2 apples and 3 oranges, the ratio of apples to oranges is 2:3. This simply means that for every 2 apples, there are 3 oranges. The order of the quantities in the ratio is crucial, as changing the order changes the comparison being made. So, the ratio of oranges to apples would be 3:2, which is a different comparison altogether. Ratios help us understand the relative sizes of different quantities and are a powerful tool for making comparisons and solving problems in various contexts. In the context of our problem, we're looking to compare time durations – two hours and 30 minutes – and express this comparison as a ratio in its simplest form.

The first step in finding the lowest term ratio is to ensure that the quantities being compared are expressed in the same units. This is crucial because you can't directly compare quantities that are measured in different units. It's like trying to add apples and oranges – you need a common unit to make the comparison meaningful. In our case, we have two hours and 30 minutes. Since hours and minutes are different units of time, we need to convert one of them to the other. The most straightforward approach is to convert hours to minutes because we know that 1 hour is equal to 60 minutes. So, two hours would be 2 multiplied by 60, which equals 120 minutes. Now, we have both quantities in the same unit: 120 minutes and 30 minutes. This conversion is essential for accurate comparison and simplification of the ratio. Once the units are consistent, we can proceed to express the comparison as a ratio and then simplify it to its lowest terms.

Now that we have both quantities in the same unit (minutes), we can express the ratio. We have 120 minutes and 30 minutes. The ratio is written as 120:30. This means we are comparing 120 minutes to 30 minutes. However, this ratio is not in its simplest form. To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers and divide both numbers by it. The GCD is the largest number that divides both quantities without leaving a remainder. In our case, we need to find the GCD of 120 and 30. One way to find the GCD is to list the factors of both numbers and identify the largest one they have in common. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 120 include 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Looking at the lists, we can see that the largest factor common to both 30 and 120 is 30. Therefore, the GCD of 120 and 30 is 30. This GCD will be the key to simplifying our ratio to its lowest terms. Simplifying ratios helps us understand the proportional relationship in its most basic form.

With the GCD identified as 30, we can now simplify the ratio 120:30 to its lowest terms. To do this, we divide both sides of the ratio by the GCD. So, we divide 120 by 30, which gives us 4, and we divide 30 by 30, which gives us 1. Therefore, the simplified ratio is 4:1. This means that for every 4 minutes in the first quantity (two hours), there is 1 minute in the second quantity (30 minutes). The ratio 4:1 is the simplest form of the comparison between two hours and 30 minutes. It's much easier to understand and interpret than the original ratio of 120:30. Simplifying ratios to their lowest terms makes it easier to see the proportional relationship between the quantities being compared. In this case, we can clearly see that two hours is four times longer than 30 minutes. This simplified ratio provides a clear and concise understanding of the relationship between the two time durations.

Final Answer: The Lowest Term Ratio of Two Hours to 30 Minutes

In summary, the ratio in the lowest term of two hours to 30 minutes is 4:1. We arrived at this answer by first converting both time durations to the same unit (minutes), expressing the comparison as a ratio (120:30), finding the greatest common divisor (GCD) of the two quantities (30), and then dividing both sides of the ratio by the GCD to simplify it to its lowest terms. The simplified ratio of 4:1 clearly illustrates the proportional relationship between the two time durations, showing that two hours is four times longer than 30 minutes. This process of simplifying ratios is a fundamental skill in mathematics and has wide-ranging applications in various fields where comparisons and proportions are important. The ability to express ratios in their simplest form allows for clearer communication and understanding of quantitative relationships.

Therefore, the correct answer is B. 4:1.