Add. 4 10 + 1 10 = \frac{4}{10} + \frac{1}{10} = 10 4 ​ + 10 1 ​ =

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Understanding the Basics of Fractions

Fractions are a way to represent a part of a whole. They consist of two numbers: a numerator and a denominator. The numerator tells us how many equal parts we have, and the denominator tells us how many parts the whole is divided into. For example, the fraction 12\frac{1}{2} represents one half of a whole.

Adding Fractions with the Same Denominator

When we add fractions with the same denominator, we simply add the numerators and keep the denominator the same. For example, 14+24=34\frac{1}{4} + \frac{2}{4} = \frac{3}{4}.

Adding Fractions with Different Denominators

However, when we add fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. Once we have the common denominator, we can convert each fraction to have that denominator and then add them.

Finding the Common Denominator

To find the common denominator, we need to list the multiples of each denominator and find the smallest multiple that is common to both. For example, if we want to add 14\frac{1}{4} and 16\frac{1}{6}, we need to find the LCM of 4 and 6.

Calculating the LCM

The multiples of 4 are: 4, 8, 12, 16, 20, ... The multiples of 6 are: 6, 12, 18, 24, 30, ...

The smallest multiple that is common to both is 12. Therefore, the common denominator is 12.

Converting the Fractions

Now that we have the common denominator, we can convert each fraction to have that denominator. To convert a fraction, we multiply the numerator and denominator by the same number.

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

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

Adding the Fractions

Now that we have both fractions with the same denominator, we can add them.

312+212=3+212=512\frac{3}{12} + \frac{2}{12} = \frac{3 + 2}{12} = \frac{5}{12}

Conclusion

In conclusion, adding fractions with different denominators requires finding a common denominator, converting each fraction to have that denominator, and then adding them. The common denominator is the least common multiple (LCM) of the two denominators. By following these steps, we can add fractions with different denominators and get the correct answer.

Example Problems

Problem 1

Add 25\frac{2}{5} and 310\frac{3}{10}.

Solution

To add these fractions, we need to find the LCM of 5 and 10, which is 10. We can convert 25\frac{2}{5} to have a denominator of 10 by multiplying the numerator and denominator by 2.

25=2×25×2=410\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}

Now that we have both fractions with the same denominator, we can add them.

410+310=4+310=710\frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10}

Problem 2

Add 16\frac{1}{6} and 14\frac{1}{4}.

Solution

To add these fractions, we need to find the LCM of 6 and 4, which is 12. We can convert 16\frac{1}{6} to have a denominator of 12 by multiplying the numerator and denominator by 2.

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

We can convert 14\frac{1}{4} to have a denominator of 12 by multiplying the numerator and denominator by 3.

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

Now that we have both fractions with the same denominator, we can add them.

212+312=2+312=512\frac{2}{12} + \frac{3}{12} = \frac{2 + 3}{12} = \frac{5}{12}

Tips and Tricks

  • When adding fractions with different denominators, always find the LCM of the two denominators.
  • To convert a fraction to have a different denominator, multiply the numerator and denominator by the same number.
  • When adding fractions, always add the numerators and keep the denominator the same.

Common Denominators

The common denominator is the least common multiple (LCM) of the two denominators. To find the LCM, list the multiples of each denominator and find the smallest multiple that is common to both.

Least Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both. To find the LCM, list the multiples of each number and find the smallest multiple that is common to both.

Multiples of a Number

The multiples of a number are the numbers that can be divided by that number without leaving a remainder. For example, the multiples of 4 are: 4, 8, 12, 16, 20, ...

Finding the LCM

To find the LCM, list the multiples of each number and find the smallest multiple that is common to both.

Example of Finding the LCM

Find the LCM of 4 and 6.

The multiples of 4 are: 4, 8, 12, 16, 20, ... The multiples of 6 are: 6, 12, 18, 24, 30, ...

The smallest multiple that is common to both is 12. Therefore, the LCM of 4 and 6 is 12.

Conclusion

In conclusion, finding the LCM is an important step in adding fractions with different denominators. By listing the multiples of each denominator and finding the smallest multiple that is common to both, we can find the LCM and add the fractions.

Final Answer

The final answer is: 510\boxed{\frac{5}{10}}

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Q: What is the first step in adding fractions with different denominators?

A: The first step in adding fractions with different denominators is to find the least common multiple (LCM) of the two denominators.

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

A: To find the LCM of two numbers, list the multiples of each number and find the smallest multiple that is common to both.

Q: What is the next step after finding the LCM?

A: After finding the LCM, convert each fraction to have the LCM as the denominator.

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

A: To convert a fraction to have a different denominator, multiply the numerator and denominator by the same number.

Q: What is the final step in adding fractions with different denominators?

A: The final step in adding fractions with different denominators is to add the fractions with the same denominator.

Q: Can I add fractions with different denominators without finding the LCM?

A: No, you cannot add fractions with different denominators without finding the LCM. The LCM is necessary to ensure that the fractions have the same denominator.

Q: What if the LCM is not a whole number?

A: If the LCM is not a whole number, you can still add the fractions. However, you may need to simplify the answer by dividing the numerator and denominator by their greatest common divisor (GCD).

Q: Can I add fractions with different denominators using a calculator?

A: Yes, you can add fractions with different denominators using a calculator. However, it's always a good idea to understand the concept of adding fractions with different denominators to ensure that you're getting the correct answer.

Q: What are some common mistakes to avoid when adding fractions with different denominators?

A: Some common mistakes to avoid when adding fractions with different denominators include:

  • Not finding the LCM
  • Not converting the fractions to have the same denominator
  • Adding the fractions without checking if they have the same denominator
  • Not simplifying the answer

Q: How can I practice adding fractions with different denominators?

A: You can practice adding fractions with different denominators by working through examples and exercises. You can also use online resources or math software to help you practice.

Q: What are some real-world applications of adding fractions with different denominators?

A: Adding fractions with different denominators has many real-world applications, including:

  • Cooking: When measuring ingredients, you may need to add fractions with different denominators.
  • Building: When measuring materials, you may need to add fractions with different denominators.
  • Science: When measuring quantities, you may need to add fractions with different denominators.

Q: Can I add fractions with different denominators in a word problem?

A: Yes, you can add fractions with different denominators in a word problem. For example, if you have 1/4 of a pizza and your friend has 1/6 of a pizza, you can add the fractions to find the total amount of pizza.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is 510\boxed{\frac{5}{10}}.

Conclusion

In conclusion, adding fractions with different denominators requires finding the least common multiple (LCM) of the two denominators, converting each fraction to have the LCM as the denominator, and then adding the fractions. By following these steps, you can add fractions with different denominators and get the correct answer.

Tips and Tricks

  • Always find the LCM before adding fractions with different denominators.
  • Convert each fraction to have the LCM as the denominator.
  • Add the fractions with the same denominator.
  • Simplify the answer by dividing the numerator and denominator by their greatest common divisor (GCD).

Common Denominators

The common denominator is the least common multiple (LCM) of the two denominators. To find the LCM, list the multiples of each denominator and find the smallest multiple that is common to both.

Least Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both. To find the LCM, list the multiples of each number and find the smallest multiple that is common to both.

Multiples of a Number

The multiples of a number are the numbers that can be divided by that number without leaving a remainder. For example, the multiples of 4 are: 4, 8, 12, 16, 20, ...

Finding the LCM

To find the LCM, list the multiples of each number and find the smallest multiple that is common to both.

Example of Finding the LCM

Find the LCM of 4 and 6.

The multiples of 4 are: 4, 8, 12, 16, 20, ... The multiples of 6 are: 6, 12, 18, 24, 30, ...

The smallest multiple that is common to both is 12. Therefore, the LCM of 4 and 6 is 12.

Conclusion

In conclusion, finding the LCM is an important step in adding fractions with different denominators. By listing the multiples of each denominator and finding the smallest multiple that is common to both, we can find the LCM and add the fractions.

Final Answer

The final answer is: 510\boxed{\frac{5}{10}}