Evaluating Fractions 3/20 - 11/8 Simplest Form Guide

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Evaluate the expression $\frac{3}{20}-\frac{11}{8}$ and write your answer as a fraction or mixed number in simplest form.

When dealing with fraction subtraction, it's crucial to grasp the underlying principles to arrive at the correct solution. In this guide, we'll meticulously break down the process of evaluating the expression 320−118\frac{3}{20} - \frac{11}{8}, ensuring a clear understanding of each step involved. Our main focus will be on obtaining the answer as a fraction or mixed number in its simplest form.

Understanding the Basics of Fraction Subtraction

To effectively subtract fractions, a fundamental requirement is that they share a common denominator. This common denominator serves as the foundation for combining the numerators, which represent the number of parts we are considering. If the fractions lack a common denominator, we must first identify the least common multiple (LCM) of the denominators. The LCM becomes our new common denominator, allowing us to rewrite the fractions with equivalent values.

Let's delve into the concept of equivalent fractions. Equivalent fractions are fractions that, despite having different numerators and denominators, represent the same value. To create equivalent fractions, we multiply both the numerator and the denominator of a fraction by the same non-zero number. This maintains the fraction's value while adjusting its representation.

Finding the Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest positive integer that is divisible by both denominators. Several methods can be employed to determine the LCM, including listing multiples, prime factorization, and using the greatest common divisor (GCD). We'll illustrate the listing multiples method here, as it's straightforward and intuitive.

Consider the denominators 20 and 8 in our expression. To find the LCM, we list the multiples of each number:

Multiples of 20: 20, 40, 60, 80, 100... Multiples of 8: 8, 16, 24, 32, 40, 48...

By examining these lists, we identify the smallest multiple common to both, which is 40. Therefore, the LCM of 20 and 8 is 40.

Rewriting Fractions with a Common Denominator

Now that we've established the LCM as 40, we need to rewrite both fractions in the expression 320−118\frac{3}{20} - \frac{11}{8} with this common denominator. To achieve this, we'll multiply the numerator and denominator of each fraction by a suitable factor.

For the first fraction, 320\frac{3}{20}, we ask ourselves: "What number multiplied by 20 gives us 40?" The answer is 2. So, we multiply both the numerator and denominator of 320\frac{3}{20} by 2:

320∗22=640\frac{3}{20} * \frac{2}{2} = \frac{6}{40}

Similarly, for the second fraction, 118\frac{11}{8}, we ask: "What number multiplied by 8 gives us 40?" The answer is 5. We multiply both the numerator and denominator of 118\frac{11}{8} by 5:

118∗55=5540\frac{11}{8} * \frac{5}{5} = \frac{55}{40}

Now, our expression becomes 640−5540\frac{6}{40} - \frac{55}{40}, which has a common denominator, allowing us to proceed with the subtraction.

Performing the Subtraction

With the fractions now sharing a common denominator of 40, we can subtract the numerators directly. We have 640−5540\frac{6}{40} - \frac{55}{40}. Subtracting the numerators, we get 6 - 55 = -49. Therefore, the result of the subtraction is −4940\frac{-49}{40}.

This fraction represents a negative value, indicating that the second fraction ( rac{11}{8}) is larger than the first fraction ( rac{3}{20}). The result, −4940\frac{-49}{40}, is an improper fraction because the absolute value of the numerator (49) is greater than the denominator (40).

Converting Improper Fractions to Mixed Numbers

While −4940\frac{-49}{40} is a valid answer, it's often preferred to express improper fractions as mixed numbers. A mixed number consists of a whole number part and a proper fraction part. To convert an improper fraction to a mixed number, we perform division.

We divide the absolute value of the numerator (49) by the denominator (40): 49 ÷ 40 = 1 with a remainder of 9. The quotient, 1, becomes the whole number part of the mixed number. The remainder, 9, becomes the numerator of the fractional part, and the denominator remains 40. Since our original fraction was negative, the mixed number will also be negative.

Therefore, −4940\frac{-49}{40} is equivalent to the mixed number -1940\frac{9}{40}.

Simplifying Fractions

The final step in evaluating the expression is to ensure the result is in its simplest form. A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. In other words, the fraction is reduced to its lowest terms.

To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest positive integer that divides both numbers without leaving a remainder. Once we've identified the GCD, we divide both the numerator and denominator by it.

In our mixed number, -1940\frac{9}{40}, we focus on the fractional part, 940\frac{9}{40}. We need to determine the GCD of 9 and 40. The factors of 9 are 1, 3, and 9. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The only common factor between 9 and 40 is 1.

Since the GCD of 9 and 40 is 1, the fraction 940\frac{9}{40} is already in its simplest form. Therefore, the mixed number -1940\frac{9}{40} is also in its simplest form.

Final Answer: Evaluating 3/20 - 11/8

In conclusion, after meticulously following the steps of finding a common denominator, subtracting the fractions, converting the improper fraction to a mixed number, and simplifying the result, we have successfully evaluated the expression 320−118\frac{3}{20} - \frac{11}{8}. The answer, expressed as a mixed number in its simplest form, is -1940\frac{9}{40}. This detailed explanation provides a clear understanding of the process involved in subtracting fractions and arriving at the correct solution.

Further Practice

To solidify your understanding of fraction subtraction, practice with various examples. Try different combinations of fractions, both positive and negative, and with varying denominators. This will enhance your skills and build confidence in tackling such problems.

Consider exploring online resources and textbooks for additional practice problems and explanations. Many websites offer interactive exercises and step-by-step solutions to help you master fraction arithmetic.

Common Mistakes to Avoid

When working with fractions, it's essential to be aware of common mistakes that can lead to incorrect answers. One frequent error is attempting to subtract fractions without first establishing a common denominator. Remember, fractions must have the same denominator before their numerators can be subtracted.

Another mistake is neglecting to simplify the final answer. Always ensure that the fraction is reduced to its lowest terms by dividing the numerator and denominator by their greatest common divisor.

By avoiding these common pitfalls and practicing consistently, you can develop a strong foundation in fraction arithmetic and confidently tackle more complex mathematical problems.

Applications of Fraction Subtraction

Fraction subtraction isn't just a theoretical concept confined to textbooks; it has numerous practical applications in everyday life. From cooking and baking to measuring materials for construction projects, fractions are an integral part of many tasks.

For instance, imagine you're following a recipe that calls for 34\frac{3}{4} cup of flour, but you only have 12\frac{1}{2} cup on hand. To determine how much more flour you need, you would subtract 12\frac{1}{2} from 34\frac{3}{4}.

Similarly, in construction, precise measurements are crucial. If you need to cut a piece of wood to a specific length and have already cut off a portion, you would use fraction subtraction to calculate the remaining length.

Understanding fraction subtraction empowers you to solve real-world problems effectively and efficiently. It's a fundamental skill that proves valuable in various aspects of life.

By mastering the concepts and techniques outlined in this guide, you'll be well-equipped to handle fraction subtraction problems with confidence and accuracy. Remember, practice is key to success, so continue to challenge yourself with new examples and explore the diverse applications of fractions in the world around you.