Which Of The Following Fractions Lies Between $\frac{1}{3}$ And $\frac{3}{4}$?A. $\frac{1}{2}$ B. $\frac{1}{5}$ C. $\frac{4}{3}$ D. $\frac{3}{2}$

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Introduction

Comparing fractions is an essential skill in mathematics, and it's often used in various real-world applications. In this article, we will explore how to compare fractions and determine which one lies between two given fractions. We will use the fractions 13\frac{1}{3} and 34\frac{3}{4} as examples and compare them with the given options.

Understanding the Problem

The problem asks us to find the fraction that lies between 13\frac{1}{3} and 34\frac{3}{4}. To solve this problem, we need to compare the fractions and determine which one is greater than 13\frac{1}{3} and less than 34\frac{3}{4}.

Comparing Fractions

To compare fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions. In this case, the denominators are 3 and 4, so the LCM is 12.

Converting Fractions to Equivalent Fractions

We can convert the fractions to equivalent fractions with a common denominator of 12.

  • 13=412\frac{1}{3} = \frac{4}{12}
  • 34=912\frac{3}{4} = \frac{9}{12}

Comparing the Equivalent Fractions

Now that we have equivalent fractions with a common denominator, we can compare them.

  • 412\frac{4}{12} is less than 912\frac{9}{12}
  • 412\frac{4}{12} is greater than 112\frac{1}{12} (which is equivalent to 13\frac{1}{3})
  • 912\frac{9}{12} is less than 1212\frac{12}{12} (which is equivalent to 34\frac{3}{4})

Analyzing the Options

Now that we have compared the fractions, let's analyze the options.

  • A. 12=612\frac{1}{2} = \frac{6}{12}, which is greater than 412\frac{4}{12} and less than 912\frac{9}{12}.
  • B. 15=2.412\frac{1}{5} = \frac{2.4}{12}, which is less than 412\frac{4}{12}.
  • C. 43=1612\frac{4}{3} = \frac{16}{12}, which is greater than 912\frac{9}{12}.
  • D. 32=1812\frac{3}{2} = \frac{18}{12}, which is greater than 912\frac{9}{12}.

Conclusion

Based on our analysis, we can conclude that the fraction 12\frac{1}{2} lies between 13\frac{1}{3} and 34\frac{3}{4}.

Why is it Important to Compare Fractions?

Comparing fractions is an essential skill in mathematics, and it's used in various real-world applications. For example, in cooking, you may need to compare the ratio of ingredients to determine the correct amount. In finance, you may need to compare the interest rates of different loans to determine which one is the best option.

Real-World Applications of Comparing Fractions

Comparing fractions has many real-world applications, including:

  • Cooking: Comparing fractions is essential in cooking, where you need to determine the correct ratio of ingredients.
  • Finance: Comparing fractions is used in finance to determine the interest rates of different loans.
  • Science: Comparing fractions is used in science to determine the concentration of solutions.
  • Engineering: Comparing fractions is used in engineering to determine the ratio of materials.

Tips for Comparing Fractions

Here are some tips for comparing fractions:

  • Find a common denominator: To compare fractions, you need to find a common denominator.
  • Convert fractions to equivalent fractions: Convert the fractions to equivalent fractions with a common denominator.
  • Compare the numerators: Compare the numerators of the equivalent fractions.
  • Determine the order: Determine the order of the fractions based on the comparison.

Conclusion

Q: What is the first step in comparing fractions?

A: The first step in comparing fractions is to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions.

Q: How do I find a common denominator?

A: To find a common denominator, you can list the multiples of each denominator and find the smallest multiple that is common to both. Alternatively, you can use the formula for the LCM, which is:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the next step in comparing fractions?

A: After finding a common denominator, the next step is to convert the fractions to equivalent fractions with the common denominator.

Q: How do I convert fractions to equivalent fractions?

A: To convert a fraction to an equivalent fraction with a common denominator, you can multiply the numerator and denominator by the same number. For example, if you want to convert the fraction 1/3 to an equivalent fraction with a denominator of 12, you can multiply the numerator and denominator by 4:

1/3 = (1 × 4) / (3 × 4) = 4/12

Q: How do I compare the fractions?

A: After converting the fractions to equivalent fractions, you can compare the numerators. The fraction with the larger numerator is greater than the fraction with the smaller numerator.

Q: What if the fractions have the same numerator?

A: If the fractions have the same numerator, you can compare the denominators. The fraction with the smaller denominator is greater than the fraction with the larger denominator.

Q: Can I compare fractions with different signs?

A: Yes, you can compare fractions with different signs. To compare fractions with different signs, you can convert them to equivalent fractions with a common denominator and then compare the numerators.

Q: Can I compare fractions with decimals?

A: Yes, you can compare fractions with decimals. To compare fractions with decimals, you can convert the decimals to fractions and then compare them.

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

A: Comparing fractions has many real-world applications, including:

  • Cooking: Comparing fractions is essential in cooking, where you need to determine the correct ratio of ingredients.
  • Finance: Comparing fractions is used in finance to determine the interest rates of different loans.
  • Science: Comparing fractions is used in science to determine the concentration of solutions.
  • Engineering: Comparing fractions is used in engineering to determine the ratio of materials.

Q: How can I practice comparing fractions?

A: You can practice comparing fractions by using online resources, such as fraction comparison games and worksheets. You can also practice comparing fractions by using real-world examples, such as comparing the ratio of ingredients in a recipe.

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: Failing to find a common denominator can lead to incorrect comparisons.
  • Not converting fractions to equivalent fractions: Failing to convert fractions to equivalent fractions can lead to incorrect comparisons.
  • Not comparing the numerators: Failing to compare the numerators can lead to incorrect comparisons.

Conclusion

In conclusion, comparing fractions is an essential skill in mathematics, and it's used in various real-world applications. By following the steps outlined in this article, you can compare fractions and determine which one is greater or less than another fraction. Remember to find a common denominator, convert fractions to equivalent fractions, compare the numerators, and determine the order. With practice, you will become proficient in comparing fractions and be able to apply this skill in various real-world situations.