Which Of The Following Inequalities Is True?A. 3 4 \textless 5 7 \frac{3}{4}\ \textless \ \frac{5}{7} 4 3 ​ \textless 7 5 ​ B. 2 3 \textgreater 5 6 \frac{2}{3}\ \textgreater \ \frac{5}{6} 3 2 ​ \textgreater 6 5 ​ C. 5 8 \textgreater 6 10 \frac{5}{8}\ \textgreater \ \frac{6}{10} 8 5 ​ \textgreater 10 6 ​ D. $\frac{4}{5}\ \textless \

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Understanding the Basics of Fractions

When it comes to comparing fractions, it's essential to understand the concept of equivalent ratios. Two fractions are equivalent if they have the same value, even if their numerators and denominators are different. For example, the fractions 1/2 and 2/4 are equivalent because they both represent the same ratio of 1 part to 2 parts.

Comparing Fractions with Different Denominators

To compare 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 compare the fractions by comparing their numerators.

Solving Inequality A: 34 \textless 57\frac{3}{4}\ \textless \ \frac{5}{7}

To solve this inequality, we need to find a common denominator for 4 and 7. The LCM of 4 and 7 is 28. We can rewrite the fractions with the common denominator:

34=3×74×7=2128\frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28}

57=5×47×4=2028\frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28}

Now that we have the fractions with the same denominator, we can compare them. Since 21 is greater than 20, we can conclude that 34 \textless 57\frac{3}{4}\ \textless \ \frac{5}{7} is true.

Solving Inequality B: 23 \textgreater 56\frac{2}{3}\ \textgreater \ \frac{5}{6}

To solve this inequality, we need to find a common denominator for 3 and 6. The LCM of 3 and 6 is 6. We can rewrite the fractions with the common denominator:

23=2×23×2=46\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}

56=5×16×1=56\frac{5}{6} = \frac{5 \times 1}{6 \times 1} = \frac{5}{6}

Now that we have the fractions with the same denominator, we can compare them. Since 4 is less than 5, we can conclude that 23 \textless 56\frac{2}{3}\ \textless \ \frac{5}{6} is false.

Solving Inequality C: 58 \textgreater 610\frac{5}{8}\ \textgreater \ \frac{6}{10}

To solve this inequality, we need to find a common denominator for 8 and 10. The LCM of 8 and 10 is 40. We can rewrite the fractions with the common denominator:

58=5×58×5=2540\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40}

610=6×410×4=2440\frac{6}{10} = \frac{6 \times 4}{10 \times 4} = \frac{24}{40}

Now that we have the fractions with the same denominator, we can compare them. Since 25 is greater than 24, we can conclude that 58 \textgreater 610\frac{5}{8}\ \textgreater \ \frac{6}{10} is true.

Solving Inequality D: 45 \textless 34\frac{4}{5}\ \textless \ \frac{3}{4}

To solve this inequality, we need to find a common denominator for 5 and 4. The LCM of 5 and 4 is 20. We can rewrite the fractions with the common denominator:

45=4×45×4=1620\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}

34=3×54×5=1520\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}

Now that we have the fractions with the same denominator, we can compare them. Since 16 is greater than 15, we can conclude that 45 \textgreater 34\frac{4}{5}\ \textgreater \ \frac{3}{4} is false.

Conclusion

In conclusion, the correct answer is A. 34 \textless 57\frac{3}{4}\ \textless \ \frac{5}{7} is true. The other options are false. When comparing fractions, it's essential to find a common denominator and compare the numerators. This will help you determine which fraction is greater or lesser than the other.

Tips and Tricks

  • When comparing fractions, always find a common denominator.
  • Compare the numerators once you have the common denominator.
  • Use equivalent ratios to simplify fractions.
  • Practice, practice, practice! The more you practice, the better you'll become at comparing fractions.

Real-World Applications

Comparing fractions has many real-world applications. For example, in cooking, you may need to compare the ratio of ingredients to determine the correct amount. In science, you may need to compare the ratio of molecules to determine the correct concentration. In finance, you may need to compare the ratio of investments to determine the correct portfolio.

Common Mistakes

  • Not finding a common denominator.
  • Not comparing the numerators.
  • Not using equivalent ratios.
  • Not practicing enough.

Conclusion

In conclusion, comparing fractions is a crucial skill that has many real-world applications. By following the steps outlined in this article, you'll be able to compare fractions with ease. Remember to find a common denominator, compare the numerators, and use equivalent ratios. With practice, you'll become a pro at comparing fractions in no time!

Q: What is the best way to compare fractions?

A: The best way to compare fractions is to find a common denominator and compare the numerators. This will help you determine which fraction is greater or lesser than the other.

Q: How do I find a common denominator?

A: To find a common denominator, you need to find the least common multiple (LCM) of the two denominators. You can use a calculator or a chart to find the LCM.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

Q: How do I compare fractions with different denominators?

A: To compare fractions with different denominators, you need to find a common denominator and compare the numerators. This will help you determine which fraction is greater or lesser than the other.

Q: What is the difference between equivalent ratios and equivalent fractions?

A: Equivalent ratios and equivalent fractions are the same thing. They are fractions that have the same value, even if their numerators and denominators are different.

Q: How do I simplify fractions?

A: To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

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

A: The greatest common divisor (GCD) is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I convert fractions to decimals?

A: To convert fractions to decimals, you need to divide the numerator by the denominator.

Q: How do I convert decimals to fractions?

A: To convert decimals to fractions, you need to find the decimal as a fraction. For example, 0.5 can be written as 1/2.

Q: What are some real-world applications of comparing fractions?

A: Comparing fractions has many real-world applications, such as cooking, science, and finance. In cooking, you may need to compare the ratio of ingredients to determine the correct amount. In science, you may need to compare the ratio of molecules to determine the correct concentration. In finance, you may need to compare the ratio of investments to determine the correct portfolio.

Q: What are some common mistakes to avoid when comparing fractions?

A: Some common mistakes to avoid when comparing fractions include not finding a common denominator, not comparing the numerators, not using equivalent ratios, and not practicing enough.

Q: How can I practice comparing fractions?

A: You can practice comparing fractions by using online resources, such as worksheets and games. You can also practice by comparing fractions in real-world situations, such as cooking or science.

Q: What are some tips for comparing fractions?

A: Some tips for comparing fractions include finding a common denominator, comparing the numerators, using equivalent ratios, and practicing enough. Additionally, you can use visual aids, such as charts and graphs, to help you compare fractions.

Q: How can I use technology to compare fractions?

A: You can use technology, such as calculators and computer programs, to compare fractions. You can also use online resources, such as worksheets and games, to practice comparing fractions.

Q: What are some advanced topics in comparing fractions?

A: Some advanced topics in comparing fractions include comparing fractions with negative numbers, comparing fractions with decimals, and comparing fractions with percents.

Q: How can I apply comparing fractions to real-world situations?

A: You can apply comparing fractions to real-world situations by using the concepts and skills you have learned. For example, you can compare the ratio of ingredients in a recipe to determine the correct amount. You can also compare the ratio of molecules in a chemical reaction to determine the correct concentration.

Q: What are some common misconceptions about comparing fractions?

A: Some common misconceptions about comparing fractions include thinking that comparing fractions is only for math problems, thinking that comparing fractions is only for simple fractions, and thinking that comparing fractions is only for positive numbers.

Q: How can I overcome common misconceptions about comparing fractions?

A: You can overcome common misconceptions about comparing fractions by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some resources for learning more about comparing fractions?

A: Some resources for learning more about comparing fractions include online tutorials, worksheets, and games. You can also use textbooks and other educational materials to learn more about comparing fractions.

Q: How can I use comparing fractions to solve real-world problems?

A: You can use comparing fractions to solve real-world problems by applying the concepts and skills you have learned. For example, you can compare the ratio of ingredients in a recipe to determine the correct amount. You can also compare the ratio of molecules in a chemical reaction to determine the correct concentration.

Q: What are some career paths that involve comparing fractions?

A: Some career paths that involve comparing fractions include cooking, science, finance, and engineering. In these careers, you may need to compare fractions to determine the correct amount of ingredients, to compare the ratio of molecules, or to compare the ratio of investments.

Q: How can I use comparing fractions to improve my critical thinking skills?

A: You can use comparing fractions to improve your critical thinking skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common applications of comparing fractions in everyday life?

A: Some common applications of comparing fractions in everyday life include cooking, science, finance, and engineering. In these applications, you may need to compare fractions to determine the correct amount of ingredients, to compare the ratio of molecules, or to compare the ratio of investments.

Q: How can I use comparing fractions to improve my problem-solving skills?

A: You can use comparing fractions to improve your problem-solving skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common mistakes to avoid when using comparing fractions in real-world applications?

A: Some common mistakes to avoid when using comparing fractions in real-world applications include not finding a common denominator, not comparing the numerators, not using equivalent ratios, and not practicing enough.

Q: How can I use comparing fractions to improve my communication skills?

A: You can use comparing fractions to improve your communication skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common applications of comparing fractions in business and industry?

A: Some common applications of comparing fractions in business and industry include finance, engineering, and manufacturing. In these applications, you may need to compare fractions to determine the correct amount of materials, to compare the ratio of products, or to compare the ratio of investments.

Q: How can I use comparing fractions to improve my analytical skills?

A: You can use comparing fractions to improve your analytical skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common applications of comparing fractions in education?

A: Some common applications of comparing fractions in education include math, science, and engineering. In these applications, you may need to compare fractions to determine the correct amount of materials, to compare the ratio of products, or to compare the ratio of investments.

Q: How can I use comparing fractions to improve my decision-making skills?

A: You can use comparing fractions to improve your decision-making skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common applications of comparing fractions in healthcare?

A: Some common applications of comparing fractions in healthcare include medicine, nursing, and healthcare management. In these applications, you may need to compare fractions to determine the correct amount of medication, to compare the ratio of patients, or to compare the ratio of resources.

Q: How can I use comparing fractions to improve my time management skills?

A: You can use comparing fractions to improve your time management skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

Q: What are some common applications of comparing fractions in technology?

A: Some common applications of comparing fractions in technology include software development, data analysis, and computer science. In these applications, you may need to compare fractions to determine the correct amount of data, to compare the ratio of algorithms, or to compare the ratio of resources.

Q: How can I use comparing fractions to improve my creativity skills?

A: You can use comparing fractions to improve your creativity skills by practicing comparing fractions in different contexts, such as cooking, science, and finance. You can also use visual aids, such as charts and graphs, to help you compare fractions.

**Q: What are some common applications of comparing fractions