Which Expression Is Equivalent To \[$-1 \frac{4}{5} + 5 \frac{1}{3}\$\]?A. \[$5 \frac{1}{3} - \left(-1 \frac{4}{5}\right)\$\]B. \[$1_{\frac{4}{5}}^{\frac{4}{2}} - 5 \frac{1}{3}\$\]C. \[$-1 \frac{4}{5} - 5

by ADMIN 205 views

Which Expression is Equivalent to −145+513{-1 \frac{4}{5} + 5 \frac{1}{3}}?

Understanding the Problem

When dealing with mixed numbers, it's essential to understand how to add and subtract them. A mixed number is a combination of a whole number and a fraction. In this problem, we have two mixed numbers: −145{-1 \frac{4}{5}} and 513{5 \frac{1}{3}}. We need to find an equivalent expression for their sum.

Adding and Subtracting Mixed Numbers

To add or subtract mixed numbers, we need to follow a specific procedure. First, 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.

Converting Mixed Numbers to Improper Fractions

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

For example, let's convert −145{-1 \frac{4}{5}} to an improper fraction:

−145=(−1)×5+45=−5+45=−15{-1 \frac{4}{5} = \frac{(-1) \times 5 + 4}{5} = \frac{-5 + 4}{5} = \frac{-1}{5}}

Similarly, let's convert 513{5 \frac{1}{3}} to an improper fraction:

513=5×3+13=15+13=163{5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}}

Finding the Equivalent Expression

Now that we have converted the mixed numbers to improper fractions, we can find the equivalent expression for their sum.

−145+513=−15+163{-1 \frac{4}{5} + 5 \frac{1}{3} = \frac{-1}{5} + \frac{16}{3}}

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 5 and 3 is 15.

−15+163=−1×35×3+16×53×5=−315+8015{\frac{-1}{5} + \frac{16}{3} = \frac{-1 \times 3}{5 \times 3} + \frac{16 \times 5}{3 \times 5} = \frac{-3}{15} + \frac{80}{15}}

Now, we can add the fractions:

−315+8015=−3+8015=7715{\frac{-3}{15} + \frac{80}{15} = \frac{-3 + 80}{15} = \frac{77}{15}}

Evaluating the Options

Now that we have found the equivalent expression for the sum of the two mixed numbers, we can evaluate the options.

A. 513−(−145){5 \frac{1}{3} - \left(-1 \frac{4}{5}\right)}

To evaluate this option, we need to convert the mixed numbers to improper fractions and subtract them:

513−(−145)=163−−15{5 \frac{1}{3} - \left(-1 \frac{4}{5}\right) = \frac{16}{3} - \frac{-1}{5}}

To subtract these fractions, we need to find a common denominator. The LCM of 3 and 5 is 15.

163−−15=16×53×5−−1×35×3=8015−−315{\frac{16}{3} - \frac{-1}{5} = \frac{16 \times 5}{3 \times 5} - \frac{-1 \times 3}{5 \times 3} = \frac{80}{15} - \frac{-3}{15}}

Now, we can subtract the fractions:

8015−−315=80+315=8315{\frac{80}{15} - \frac{-3}{15} = \frac{80 + 3}{15} = \frac{83}{15}}

This option is not equivalent to the original expression.

B. 14542−513{1_{\frac{4}{5}}^{\frac{4}{2}} - 5 \frac{1}{3}}

This option is not a valid expression, as it contains an invalid mixed number.

C. −145−513{-1 \frac{4}{5} - 5 \frac{1}{3}}

To evaluate this option, we need to convert the mixed numbers to improper fractions and subtract them:

−145−513=−15−163{-1 \frac{4}{5} - 5 \frac{1}{3} = \frac{-1}{5} - \frac{16}{3}}

To subtract these fractions, we need to find a common denominator. The LCM of 5 and 3 is 15.

−15−163=−1×35×3−16×53×5=−315−8015{\frac{-1}{5} - \frac{16}{3} = \frac{-1 \times 3}{5 \times 3} - \frac{16 \times 5}{3 \times 5} = \frac{-3}{15} - \frac{80}{15}}

Now, we can subtract the fractions:

−315−8015=−3−8015=−8315{\frac{-3}{15} - \frac{80}{15} = \frac{-3 - 80}{15} = \frac{-83}{15}}

This option is not equivalent to the original expression.

Conclusion

In this problem, we were asked to find an equivalent expression for the sum of two mixed numbers: −145{-1 \frac{4}{5}} and 513{5 \frac{1}{3}}. We converted the mixed numbers to improper fractions and found the equivalent expression for their sum. We then evaluated the options and found that none of them were equivalent to the original expression.

Key Takeaways

  • To add or subtract mixed numbers, we need to convert them to improper fractions.
  • To convert a mixed number to an improper fraction, we need to multiply the whole number by the denominator and add the numerator.
  • To find a common denominator for fractions, we need to find the least common multiple (LCM) of the denominators.
  • To add or subtract fractions, we need to find a common denominator and then add or subtract the numerators.

Final Answer

The final answer is not among the options provided. The equivalent expression for the sum of the two mixed numbers is 7715{\frac{77}{15}}.
Q&A: Mixed Numbers and Improper Fractions

Understanding Mixed Numbers and Improper Fractions

In the previous article, we discussed how to add and subtract mixed numbers and improper fractions. We also learned how to convert mixed numbers to improper fractions and vice versa. In this article, we will answer some frequently asked questions about mixed numbers and improper fractions.

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. For example, 312{3 \frac{1}{2}} is a mixed number.

Q: What is an improper fraction?

A: An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 52{\frac{5}{2}} is an improper fraction.

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

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

For example, let's convert 312{3 \frac{1}{2}} to an improper fraction:

312=3×2+12=6+12=72{3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}}

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

A: To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the result as a whole number and a fraction.

For example, let's convert 72{\frac{7}{2}} to a mixed number:

72=312{\frac{7}{2} = 3 \frac{1}{2}}

Q: What is the difference between a mixed number and an improper fraction?

A: The main difference between a mixed number and an improper fraction is the way they are written. A mixed number is written as a combination of a whole number and a fraction, while an improper fraction is written as a fraction with a numerator greater than or equal to the denominator.

Q: When do I use mixed numbers and when do I use improper fractions?

A: You use mixed numbers when you need to represent a quantity that is a combination of a whole number and a fraction. You use improper fractions when you need to perform arithmetic operations with fractions.

Q: Can I add and subtract mixed numbers and improper fractions?

A: Yes, you can add and subtract mixed numbers and improper fractions. However, you need to follow the rules for adding and subtracting fractions, which include finding a common denominator and then adding or subtracting the numerators.

Q: How do I find a common denominator for mixed numbers and improper fractions?

A: To find a common denominator for mixed numbers and improper fractions, you need to find the least common multiple (LCM) of the denominators.

For example, let's find the common denominator for 312{3 \frac{1}{2}} and 54{\frac{5}{4}}:

The LCM of 2 and 4 is 4. Therefore, the common denominator is 4.

Q: Can I multiply and divide mixed numbers and improper fractions?

A: Yes, you can multiply and divide mixed numbers and improper fractions. However, you need to follow the rules for multiplying and dividing fractions, which include multiplying the numerators and denominators separately and then simplifying the result.

Conclusion

In this article, we answered some frequently asked questions about mixed numbers and improper fractions. We learned how to convert mixed numbers to improper fractions and vice versa, how to add and subtract mixed numbers and improper fractions, and how to find a common denominator. We also learned when to use mixed numbers and when to use improper fractions.

Key Takeaways

  • A mixed number is a combination of a whole number and a fraction.
  • 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, you need to multiply the whole number by the denominator and add the numerator.
  • To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the result as a whole number and a fraction.
  • You use mixed numbers when you need to represent a quantity that is a combination of a whole number and a fraction.
  • You use improper fractions when you need to perform arithmetic operations with fractions.

Final Answer

The final answer is that mixed numbers and improper fractions are both used to represent quantities that are a combination of a whole number and a fraction. However, mixed numbers are used when you need to represent a quantity that is a combination of a whole number and a fraction, while improper fractions are used when you need to perform arithmetic operations with fractions.