Simplify The Expression:$\[ 8 \frac{5}{6} - 2 \frac{1}{2} \\]

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Introduction

When dealing with fractions, it's essential to understand how to simplify expressions that involve mixed numbers. A mixed number is a combination of a whole number and a fraction. In this article, we will focus on simplifying the expression 856−2128 \frac{5}{6} - 2 \frac{1}{2}, which involves subtracting two mixed numbers.

Understanding Mixed Numbers

Before we dive into simplifying the expression, let's take a closer look at mixed numbers. A mixed number is a combination of a whole number and a fraction. It's denoted by a whole number followed by a fraction. For example, 3123 \frac{1}{2} is a mixed number that represents 33 whole units and 12\frac{1}{2} of a unit.

Converting Mixed Numbers to Improper Fractions

To simplify the expression, we need to convert the mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. Then, we write the result as a fraction with the denominator.

For example, let's convert 3123 \frac{1}{2} to an improper fraction:

312=3×2+1=73 \frac{1}{2} = 3 \times 2 + 1 = 7

So, 312=723 \frac{1}{2} = \frac{7}{2}.

Converting the Given Mixed Numbers to Improper Fractions

Now, let's convert the given mixed numbers to improper fractions:

856=8×6+5=538 \frac{5}{6} = 8 \times 6 + 5 = 53

So, 856=5368 \frac{5}{6} = \frac{53}{6}.

212=2×2+1=52 \frac{1}{2} = 2 \times 2 + 1 = 5

So, 212=522 \frac{1}{2} = \frac{5}{2}.

Subtracting the Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can subtract them:

536−52\frac{53}{6} - \frac{5}{2}

To subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 6 and 2 is 6. So, we can rewrite 52\frac{5}{2} as 156\frac{15}{6}:

536−156\frac{53}{6} - \frac{15}{6}

Now, we can subtract the numerators:

53−156=386\frac{53 - 15}{6} = \frac{38}{6}

Simplifying the Result

The result 386\frac{38}{6} can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:

386=193\frac{38}{6} = \frac{19}{3}

Conclusion

In this article, we simplified the expression 856−2128 \frac{5}{6} - 2 \frac{1}{2} by converting the mixed numbers to improper fractions and then subtracting them. We found that the result is 193\frac{19}{3}.

Frequently Asked Questions

  • What is a mixed number?
  • How do I convert a mixed number to an improper fraction?
  • How do I subtract fractions?

Step-by-Step Solution

  1. Convert the mixed numbers to improper fractions.
  2. Find the least common multiple (LCM) of the denominators.
  3. Rewrite the fractions with the LCM as the denominator.
  4. Subtract the numerators.
  5. Simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Final Answer

The final answer is 193\boxed{\frac{19}{3}}.

Introduction

In our previous article, we simplified the expression 856−2128 \frac{5}{6} - 2 \frac{1}{2} by converting the mixed numbers to improper fractions and then subtracting them. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. It's denoted by a whole number followed by a fraction. For example, 3123 \frac{1}{2} is a mixed number that represents 33 whole units and 12\frac{1}{2} of a unit.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. Then, we write the result as a fraction with the denominator.

For example, let's convert 3123 \frac{1}{2} to an improper fraction:

312=3×2+1=73 \frac{1}{2} = 3 \times 2 + 1 = 7

So, 312=723 \frac{1}{2} = \frac{7}{2}.

Q: How do I subtract fractions?

A: To subtract fractions, we need to have the same denominator. The least common multiple (LCM) of the denominators is the smallest number that both denominators can divide into evenly. We can rewrite the fractions with the LCM as the denominator and then subtract the numerators.

For example, let's subtract 56\frac{5}{6} and 36\frac{3}{6}:

56−36=5−36=26\frac{5}{6} - \frac{3}{6} = \frac{5 - 3}{6} = \frac{2}{6}

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two numbers is the smallest number that both numbers can divide into evenly. For example, the LCM of 6 and 2 is 6, because 6 can be divided by both 6 and 2.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, we can list the multiples of each number and find the smallest number that appears in both lists.

For example, let's find the LCM of 6 and 2:

Multiples of 6: 6, 12, 18, 24, 30, ... Multiples of 2: 2, 4, 6, 8, 10, ...

The smallest number that appears in both lists is 6, so the LCM of 6 and 2 is 6.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that both numbers can divide into evenly. For example, the GCD of 6 and 2 is 2, because 2 can be divided by both 6 and 2.

Q: How do I simplify a fraction?

A: To simplify a fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD).

For example, let's simplify 1218\frac{12}{18}:

GCD of 12 and 18 is 6

1218=12÷618÷6=23\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Conclusion

In this article, we answered some frequently asked questions related to simplifying the expression 856−2128 \frac{5}{6} - 2 \frac{1}{2}. We covered topics such as mixed numbers, improper fractions, subtracting fractions, least common multiple (LCM), greatest common divisor (GCD), and simplifying fractions.

Final Answer

The final answer is 193\boxed{\frac{19}{3}}.