Simplify The Following Expression:$\[ \frac{3}{6} + \frac{2}{5} + \frac{2}{3} \\]

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Introduction

Adding fractions can be a challenging task, especially when dealing with different denominators. However, with a clear understanding of the concept and a step-by-step approach, it becomes manageable. In this article, we will simplify the given expression: 36+25+23\frac{3}{6} + \frac{2}{5} + \frac{2}{3}.

Understanding the Concept of Adding Fractions

Before we dive into the solution, let's understand the concept of adding fractions. When we add fractions, we need to have the same denominator. The denominator is the number that appears below the line in a fraction. If the denominators are different, we need to find the least common multiple (LCM) of the denominators to add the fractions.

Finding the Least Common Multiple (LCM)

To find the LCM of the denominators, we need to list the multiples of each denominator and find the smallest multiple that appears in all the lists.

  • The multiples of 6 are: 6, 12, 18, 24, 30, ...
  • The multiples of 5 are: 5, 10, 15, 20, 25, 30, ...
  • The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

The smallest multiple that appears in all the lists is 30. Therefore, the LCM of 6, 5, and 3 is 30.

Simplifying the Expression

Now that we have found the LCM, we can simplify the expression by converting each fraction to have a denominator of 30.

  • 36\frac{3}{6} can be converted to 3Γ—56Γ—5=1530\frac{3 \times 5}{6 \times 5} = \frac{15}{30}
  • 25\frac{2}{5} can be converted to 2Γ—65Γ—6=1230\frac{2 \times 6}{5 \times 6} = \frac{12}{30}
  • 23\frac{2}{3} can be converted to 2Γ—103Γ—10=2030\frac{2 \times 10}{3 \times 10} = \frac{20}{30}

Adding the Fractions

Now that we have the same denominator for all the fractions, we can add them.

1530+1230+2030=4730\frac{15}{30} + \frac{12}{30} + \frac{20}{30} = \frac{47}{30}

Conclusion

In this article, we simplified the given expression: 36+25+23\frac{3}{6} + \frac{2}{5} + \frac{2}{3}. We found the least common multiple (LCM) of the denominators, converted each fraction to have a denominator of 30, and added the fractions. The final answer is 4730\frac{47}{30}.

Frequently Asked Questions

Q: What is the least common multiple (LCM) of 6, 5, and 3?

A: The LCM of 6, 5, and 3 is 30.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find the least common multiple (LCM) of the denominators and convert each fraction to have a denominator of the LCM.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 4730\frac{47}{30}.

Step-by-Step Solution

Step 1: Find the least common multiple (LCM) of the denominators

  • List the multiples of each denominator
  • Find the smallest multiple that appears in all the lists

Step 2: Convert each fraction to have a denominator of the LCM

  • Multiply the numerator and denominator of each fraction by the necessary factor to get a denominator of the LCM

Step 3: Add the fractions

  • Add the numerators of the fractions and keep the denominator the same

Step 4: Simplify the resulting fraction

  • Divide the numerator and denominator by their greatest common divisor (GCD)

Example Use Case

Adding fractions is a common operation in mathematics, and it has many real-world applications. For example, in cooking, you may need to add different ingredients with different measurements. By understanding how to add fractions, you can easily convert between different units of measurement and get the desired result.

Conclusion

In conclusion, adding fractions can be a challenging task, but with a clear understanding of the concept and a step-by-step approach, it becomes manageable. By finding the least common multiple (LCM) of the denominators, converting each fraction to have a denominator of the LCM, and adding the fractions, we can simplify the given expression: 36+25+23\frac{3}{6} + \frac{2}{5} + \frac{2}{3}. The final answer is 4730\frac{47}{30}.

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Q: What is the least common multiple (LCM) of 6, 5, and 3?

A: The LCM of 6, 5, and 3 is 30.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find the least common multiple (LCM) of the denominators and convert each fraction to have a denominator of the LCM.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 4730\frac{47}{30}.

Q: How do I find the least common multiple (LCM) of two or more numbers?

A: To find the LCM of two or more numbers, you can list the multiples of each number and find the smallest multiple that appears in all the lists.

Q: What is the difference between the least common multiple (LCM) and the greatest common divisor (GCD)?

A: The LCM of two or more numbers is the smallest multiple that appears in all the lists of multiples, while the GCD of two or more numbers is the largest number that divides all the numbers without leaving a remainder.

Q: How do I convert a fraction to have a denominator of a different number?

A: To convert a fraction to have a denominator of a different number, you need to multiply the numerator and denominator of the fraction by the necessary factor to get a denominator of the desired number.

Q: What is the rule for adding fractions with different denominators?

A: The rule for adding fractions with different denominators is to find the least common multiple (LCM) of the denominators, convert each fraction to have a denominator of the LCM, and then add the fractions.

Q: Can I add fractions with different signs?

A: Yes, you can add fractions with different signs. When adding fractions with different signs, you need to follow the same rules as adding fractions with the same sign.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find the least common multiple (LCM) of the denominators, convert each fraction to have a denominator of the LCM, and then subtract the fractions.

Q: What is the final answer to the expression 14βˆ’16\frac{1}{4} - \frac{1}{6}?

A: To find the final answer to the expression 14βˆ’16\frac{1}{4} - \frac{1}{6}, you need to find the least common multiple (LCM) of 4 and 6, which is 12. Then, you need to convert each fraction to have a denominator of 12 and subtract the fractions.

14=1Γ—34Γ—3=312\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}

16=1Γ—26Γ—2=212\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}

312βˆ’212=112\frac{3}{12} - \frac{2}{12} = \frac{1}{12}

The final answer to the expression 14βˆ’16\frac{1}{4} - \frac{1}{6} is 112\frac{1}{12}.

Q: How do I add or subtract fractions with like denominators?

A: To add or subtract fractions with like denominators, you can simply add or subtract the numerators and keep the denominator the same.

Q: What is the final answer to the expression 12+12\frac{1}{2} + \frac{1}{2}?

A: To find the final answer to the expression 12+12\frac{1}{2} + \frac{1}{2}, you can simply add the numerators and keep the denominator the same.

12+12=1+12=22=1\frac{1}{2} + \frac{1}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1

The final answer to the expression 12+12\frac{1}{2} + \frac{1}{2} is 1.

Q: How do I add or subtract fractions with unlike denominators?

A: To add or subtract fractions with unlike denominators, you need to find the least common multiple (LCM) of the denominators, convert each fraction to have a denominator of the LCM, and then add or subtract the fractions.

Q: What is the final answer to the expression 13+14\frac{1}{3} + \frac{1}{4}?

A: To find the final answer to the expression 13+14\frac{1}{3} + \frac{1}{4}, you need to find the least common multiple (LCM) of 3 and 4, which is 12. Then, you need to convert each fraction to have a denominator of 12 and add the fractions.

13=1Γ—43Γ—4=412\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}

14=1Γ—34Γ—3=312\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}

412+312=712\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

The final answer to the expression 13+14\frac{1}{3} + \frac{1}{4} is 712\frac{7}{12}.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD).

Q: What is the final answer to the expression 68\frac{6}{8}?

A: To simplify the expression 68\frac{6}{8}, you need to find the greatest common divisor (GCD) of 6 and 8, which is 2. Then, you need to divide the numerator and denominator by 2.

68=6Γ·28Γ·2=34\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}

The final answer to the expression 68\frac{6}{8} is 34\frac{3}{4}.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you need to divide the numerator by the denominator.

Q: What is the decimal equivalent of the fraction 34\frac{3}{4}?

A: To find the decimal equivalent of the fraction 34\frac{3}{4}, you need to divide the numerator by the denominator.

34=0.75\frac{3}{4} = 0.75

The decimal equivalent of the fraction 34\frac{3}{4} is 0.75.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you need to express the decimal as a fraction in its simplest form.

Q: What is the fraction equivalent of the decimal 0.75?

A: To find the fraction equivalent of the decimal 0.75, you need to express the decimal as a fraction in its simplest form.

0.75=340.75 = \frac{3}{4}

The fraction equivalent of the decimal 0.75 is 34\frac{3}{4}.