Solving 2 7/15 + 9/20 - 3/10 A Step-by-Step Guide

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Understanding the Problem

We are presented with a mathematical expression involving fractions: 2 7/15 + 9/20 - 3/10. To solve this, we need to perform addition and subtraction operations on these fractions. The key to successfully tackling this problem lies in understanding the fundamental principles of fraction arithmetic. This includes finding a common denominator, converting mixed numbers into improper fractions, and simplifying the final result. Fractions are a fundamental concept in mathematics, representing parts of a whole. They are expressed in the form of a numerator (the number above the fraction bar) and a denominator (the number below the fraction bar). The denominator indicates the total number of equal parts into which a whole is divided, and the numerator indicates how many of those parts are being considered. Mastering operations with fractions is crucial for various mathematical applications, including algebra, geometry, and calculus. In real-world scenarios, fractions are used extensively in cooking, measurement, finance, and many other fields. For instance, recipes often require fractional amounts of ingredients, and understanding fractions is essential for accurately measuring and combining them. In finance, interest rates and investment returns are frequently expressed as fractions or percentages, which are essentially fractions out of 100. Therefore, developing a strong understanding of fraction arithmetic is not only important for academic success but also for practical applications in everyday life. The process of solving this particular fraction problem will involve several steps. First, we need to convert the mixed number (2 7/15) into an improper fraction. This involves multiplying the whole number (2) by the denominator (15) and adding the numerator (7), then placing the result over the original denominator. Next, we need to find a common denominator for all the fractions. This is a crucial step because fractions can only be added or subtracted if they have the same denominator. The common denominator is the least common multiple (LCM) of the denominators (15, 20, and 10). Once we have the common denominator, we need to convert each fraction into an equivalent fraction with this denominator. This involves multiplying both the numerator and the denominator of each fraction by a factor that will make the denominator equal to the common denominator. After converting the fractions, we can perform the addition and subtraction operations. We add or subtract the numerators while keeping the common denominator the same. Finally, we simplify the resulting fraction, if necessary, by dividing both the numerator and the denominator by their greatest common divisor (GCD). If the result is an improper fraction, we can convert it back into a mixed number for easier interpretation.

Converting Mixed Numbers to Improper Fractions

Our initial problem includes a mixed number: 2 7/15. Mixed numbers combine a whole number and a fraction, making them a convenient way to represent quantities greater than one. However, for arithmetic operations like addition and subtraction, it's often easier to work with improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert 2 7/15 into an improper fraction, we follow a simple process. First, we multiply the whole number part (2) by the denominator of the fraction part (15). This gives us 2 * 15 = 30. Next, we add the numerator of the fraction part (7) to the result. So, 30 + 7 = 37. This sum becomes the new numerator of our improper fraction. The denominator remains the same as the original fraction, which is 15. Therefore, the improper fraction equivalent of 2 7/15 is 37/15. This conversion process is crucial because it allows us to combine the whole number and fractional parts into a single fraction, making it easier to perform subsequent arithmetic operations. Without this step, adding and subtracting the fractions would be significantly more complicated. The improper fraction 37/15 represents the same quantity as the mixed number 2 7/15. It indicates that we have 37 parts, each of which is 1/15 of a whole. This may seem less intuitive than the mixed number representation, but it's much more convenient for calculations. Once we have converted all mixed numbers into improper fractions, we can proceed with finding a common denominator and performing the addition and subtraction operations. This systematic approach ensures that we accurately solve the problem and arrive at the correct answer. In summary, converting mixed numbers to improper fractions is a fundamental step in solving fraction problems. It simplifies the process of adding and subtracting fractions and ensures that we can accurately perform the necessary calculations. By following the steps outlined above, we can easily convert any mixed number into its equivalent improper fraction, setting the stage for solving more complex fraction problems. This skill is essential for anyone working with fractions, whether in academic settings or in real-world applications. Understanding and mastering this conversion process will greatly improve your ability to solve a wide range of mathematical problems involving fractions.

Finding the Least Common Denominator (LCD)

To add or subtract fractions, a crucial step is finding the least common denominator (LCD). The LCD is the smallest common multiple of the denominators of the fractions involved. In our problem, the denominators are 15, 20, and 10. To find the LCD, we need to determine the smallest number that is divisible by all three denominators. One way to find the LCD is by listing the multiples of each denominator and identifying the smallest multiple that appears in all the lists. Let's list the multiples of 15, 20, and 10:

  • Multiples of 15: 15, 30, 45, 60, 75, 90, ...
  • Multiples of 20: 20, 40, 60, 80, 100, ...
  • Multiples of 10: 10, 20, 30, 40, 50, 60, 70, ...

By examining these lists, we can see that the smallest multiple common to all three is 60. Therefore, the LCD of 15, 20, and 10 is 60. Another method for finding the LCD is prime factorization. This involves breaking down each denominator into its prime factors. Prime factors are prime numbers that divide a given number without leaving a remainder. Let's find the prime factorization of 15, 20, and 10:

  • 15 = 3 * 5
  • 20 = 2 * 2 * 5 = 2^2 * 5
  • 10 = 2 * 5

To find the LCD using prime factorization, we take the highest power of each prime factor that appears in any of the factorizations. The prime factors involved are 2, 3, and 5. The highest power of 2 is 2^2 (from 20), the highest power of 3 is 3^1 (from 15), and the highest power of 5 is 5^1 (from all three). Therefore, the LCD is 2^2 * 3 * 5 = 4 * 3 * 5 = 60. This confirms our previous result. Once we have the LCD, we can convert each fraction into an equivalent fraction with the LCD as the denominator. This involves multiplying both the numerator and the denominator of each fraction by a factor that will make the denominator equal to the LCD. Finding the LCD is a critical step in adding and subtracting fractions because it ensures that we are working with fractions that have the same-sized parts. Without a common denominator, it's impossible to directly add or subtract the numerators. The LCD provides a common unit for the fractions, allowing us to combine them accurately. In summary, the LCD of 15, 20, and 10 is 60. This means that we will convert each fraction in our problem to have a denominator of 60 before we perform the addition and subtraction operations. This step is essential for accurately solving the problem and arriving at the correct answer. Understanding and mastering the process of finding the LCD is crucial for anyone working with fractions.

Converting Fractions to Equivalent Fractions with the LCD

Now that we have determined the least common denominator (LCD) to be 60, the next step is to convert each fraction in our problem into an equivalent fraction with a denominator of 60. This process involves multiplying both the numerator and the denominator of each fraction by a specific factor that will result in the desired denominator. Let's start with the first fraction, which is 37/15 (the improper fraction equivalent of 2 7/15). To convert 37/15 to an equivalent fraction with a denominator of 60, we need to find the factor by which we must multiply the denominator 15 to get 60. This factor is 60 / 15 = 4. Therefore, we multiply both the numerator and the denominator of 37/15 by 4: (37 * 4) / (15 * 4) = 148 / 60. So, 37/15 is equivalent to 148/60. Next, let's consider the second fraction, which is 9/20. To convert 9/20 to an equivalent fraction with a denominator of 60, we need to find the factor by which we must multiply the denominator 20 to get 60. This factor is 60 / 20 = 3. Therefore, we multiply both the numerator and the denominator of 9/20 by 3: (9 * 3) / (20 * 3) = 27 / 60. So, 9/20 is equivalent to 27/60. Finally, let's convert the third fraction, which is 3/10. To convert 3/10 to an equivalent fraction with a denominator of 60, we need to find the factor by which we must multiply the denominator 10 to get 60. This factor is 60 / 10 = 6. Therefore, we multiply both the numerator and the denominator of 3/10 by 6: (3 * 6) / (10 * 6) = 18 / 60. So, 3/10 is equivalent to 18/60. Now that we have converted all the fractions to equivalent fractions with a common denominator of 60, we can rewrite our original problem as: 148/60 + 27/60 - 18/60. This step is crucial because it allows us to perform the addition and subtraction operations on fractions that have the same-sized parts. Without a common denominator, it would be impossible to directly add or subtract the numerators. By converting the fractions to equivalent fractions with the LCD, we ensure that we are working with comparable quantities. This process maintains the value of each fraction while making them compatible for arithmetic operations. In summary, we have successfully converted the fractions 37/15, 9/20, and 3/10 into the equivalent fractions 148/60, 27/60, and 18/60, respectively. This step is essential for accurately solving the problem and arriving at the correct answer. Understanding and mastering the process of converting fractions to equivalent fractions with the LCD is crucial for anyone working with fractions.

Performing Addition and Subtraction

With all fractions now expressed with the common denominator of 60, we can proceed to perform the addition and subtraction operations. Our problem is now: 148/60 + 27/60 - 18/60. To add and subtract fractions with a common denominator, we simply add or subtract the numerators while keeping the denominator the same. First, let's add the first two fractions: 148/60 + 27/60. To do this, we add the numerators: 148 + 27 = 175. The denominator remains 60. So, 148/60 + 27/60 = 175/60. Next, we subtract the third fraction from the result: 175/60 - 18/60. To do this, we subtract the numerators: 175 - 18 = 157. The denominator remains 60. So, 175/60 - 18/60 = 157/60. Therefore, the result of our calculation is 157/60. This fraction is an improper fraction because the numerator (157) is greater than the denominator (60). While 157/60 is a correct answer, it's often preferable to express the result as a mixed number. This makes it easier to understand the magnitude of the quantity. To convert the improper fraction 157/60 into a mixed number, we divide the numerator (157) by the denominator (60). 157 divided by 60 is 2 with a remainder of 37. The quotient (2) becomes the whole number part of the mixed number, and the remainder (37) becomes the numerator of the fractional part. The denominator remains the same (60). Therefore, 157/60 is equivalent to the mixed number 2 37/60. The mixed number 2 37/60 represents the same quantity as the improper fraction 157/60. It indicates that we have 2 whole units and 37/60 of another unit. This representation is often more intuitive and easier to grasp than the improper fraction. In summary, we have successfully performed the addition and subtraction operations on the fractions, resulting in the improper fraction 157/60, which is equivalent to the mixed number 2 37/60. This step is essential for accurately solving the problem and arriving at the final answer. Understanding and mastering the process of adding and subtracting fractions is crucial for anyone working with fractions.

Simplifying the Result (If Necessary)

After performing the addition and subtraction, we arrived at the result 157/60, which we converted to the mixed number 2 37/60. The final step is to check if the fraction part of the mixed number (37/60) can be simplified. Simplifying a fraction means reducing it to its lowest terms. A fraction is in its lowest terms when the numerator and the denominator have no common factors other than 1. To determine if 37/60 can be simplified, we need to find the greatest common divisor (GCD) of the numerator (37) and the denominator (60). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. One way to find the GCD is by listing the factors of each number and identifying the largest factor that appears in both lists. Let's list the factors of 37 and 60:

  • Factors of 37: 1, 37
  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

By examining these lists, we can see that the only common factor of 37 and 60 is 1. This means that 37 and 60 are relatively prime, and the fraction 37/60 is already in its simplest form. Another method for finding the GCD is the Euclidean algorithm. This involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD. Let's apply the Euclidean algorithm to find the GCD of 37 and 60:

  1. Divide 60 by 37: 60 = 37 * 1 + 23
  2. Divide 37 by 23: 37 = 23 * 1 + 14
  3. Divide 23 by 14: 23 = 14 * 1 + 9
  4. Divide 14 by 9: 14 = 9 * 1 + 5
  5. Divide 9 by 5: 9 = 5 * 1 + 4
  6. Divide 5 by 4: 5 = 4 * 1 + 1
  7. Divide 4 by 1: 4 = 1 * 4 + 0

The last non-zero remainder is 1, which confirms that the GCD of 37 and 60 is 1. Since the GCD of 37 and 60 is 1, the fraction 37/60 cannot be simplified further. Therefore, the mixed number 2 37/60 is the final simplified answer to our problem. In summary, we checked the fraction part of our result (37/60) for simplification and found that it is already in its simplest form. This means that our final answer is 2 37/60. Simplifying fractions is an important step in solving fraction problems because it ensures that we are expressing the result in its most concise and understandable form. A simplified fraction is easier to interpret and compare with other fractions. Understanding and mastering the process of simplifying fractions is crucial for anyone working with fractions.

Final Answer

After following all the steps, we have arrived at the final answer to our problem: 2 7/15 + 9/20 - 3/10. We first converted the mixed number 2 7/15 to the improper fraction 37/15. Then, we found the least common denominator (LCD) of 15, 20, and 10, which is 60. Next, we converted each fraction to an equivalent fraction with a denominator of 60: 37/15 became 148/60, 9/20 became 27/60, and 3/10 became 18/60. We then performed the addition and subtraction operations: 148/60 + 27/60 - 18/60 = 157/60. We converted the improper fraction 157/60 to the mixed number 2 37/60. Finally, we checked if the fraction 37/60 could be simplified, and we found that it is already in its simplest form. Therefore, the final answer to the problem is 2 37/60. This mixed number represents the sum and difference of the original fractions in a clear and concise way. It indicates that the result is 2 whole units and 37/60 of another unit. This comprehensive step-by-step solution demonstrates the importance of understanding the fundamental principles of fraction arithmetic. Each step, from converting mixed numbers to finding the LCD and performing the operations, is crucial for arriving at the correct answer. Mastering these skills is essential for anyone working with fractions, whether in academic settings or in real-world applications. Fractions are a fundamental concept in mathematics, and proficiency in fraction arithmetic is a valuable skill. By following the steps outlined in this solution, you can confidently solve a wide range of fraction problems. The final answer, 2 37/60, represents the solution in its simplest form, providing a clear and understandable representation of the result. This concludes our step-by-step guide to solving the fraction problem. We have successfully navigated each step, from understanding the problem to arriving at the final simplified answer. This process highlights the importance of a systematic approach to problem-solving and the value of mastering fundamental mathematical concepts.