Which Of The Following Is A True Statement?A. $\frac{3}{5}\ \textless \ \frac{4}{7}$B. $\frac{3}{4}\ \textgreater \ \frac{9}{12}$C. $\frac{5}{12}\ \textgreater \ \frac{3}{8}$D. $\frac{6}{7}\ \textless \ \frac{5}{8}$

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Introduction

Comparing fractions is a fundamental concept in mathematics that involves determining the relative size of two or more fractions. In this article, we will explore the process of comparing fractions and evaluate the truth of four given statements. We will use the concept of equivalent ratios and the comparison of numerators and denominators to determine the relative size of the fractions.

Understanding Equivalent Ratios

Equivalent ratios are fractions that have the same value, but may be expressed differently. For example, the fractions 12\frac{1}{2} and 24\frac{2}{4} are equivalent because they represent the same ratio of 1:2. To compare fractions, we can convert them to equivalent ratios by finding the least common multiple (LCM) of the denominators.

Comparing Numerators and Denominators

When comparing fractions, we can compare the numerators and denominators separately. If the numerators are equal, the fraction with the smaller denominator is larger. If the denominators are equal, the fraction with the larger numerator is larger. If the numerators and denominators are not equal, we can compare the fractions by finding the LCM of the denominators and converting both fractions to equivalent ratios.

Evaluating Statement A

Statement A claims that 35 \textless 47\frac{3}{5}\ \textless \ \frac{4}{7}. To evaluate this statement, we can compare the numerators and denominators separately. The numerator of the first fraction is 3, and the numerator of the second fraction is 4. Since 3 is less than 4, the first fraction has a smaller numerator. The denominator of the first fraction is 5, and the denominator of the second fraction is 7. Since 5 is less than 7, the first fraction has a smaller denominator. Therefore, the first fraction is smaller than the second fraction, and statement A is true.

Evaluating Statement B

Statement B claims that 34 \textgreater 912\frac{3}{4}\ \textgreater \ \frac{9}{12}. To evaluate this statement, we can compare the numerators and denominators separately. The numerator of the first fraction is 3, and the numerator of the second fraction is 9. Since 3 is less than 9, the first fraction has a smaller numerator. The denominator of the first fraction is 4, and the denominator of the second fraction is 12. Since 4 is less than 12, the first fraction has a smaller denominator. Therefore, the first fraction is smaller than the second fraction, and statement B is false.

Evaluating Statement C

Statement C claims that 512 \textgreater 38\frac{5}{12}\ \textgreater \ \frac{3}{8}. To evaluate this statement, we can compare the numerators and denominators separately. The numerator of the first fraction is 5, and the numerator of the second fraction is 3. Since 5 is greater than 3, the first fraction has a larger numerator. The denominator of the first fraction is 12, and the denominator of the second fraction is 8. Since 12 is greater than 8, the first fraction has a larger denominator. Therefore, the first fraction is larger than the second fraction, and statement C is true.

Evaluating Statement D

Statement D claims that 67 \textless 58\frac{6}{7}\ \textless \ \frac{5}{8}. To evaluate this statement, we can compare the numerators and denominators separately. The numerator of the first fraction is 6, and the numerator of the second fraction is 5. Since 6 is greater than 5, the first fraction has a larger numerator. The denominator of the first fraction is 7, and the denominator of the second fraction is 8. Since 7 is less than 8, the first fraction has a smaller denominator. Therefore, the first fraction is larger than the second fraction, and statement D is false.

Conclusion

In conclusion, we have evaluated four statements involving the comparison of fractions. We have used the concept of equivalent ratios and the comparison of numerators and denominators to determine the relative size of the fractions. We have found that statement A is true, statement B is false, statement C is true, and statement D is false. By understanding how to compare fractions, we can make informed decisions and solve problems involving ratios and proportions.

Key Takeaways

  • Equivalent ratios are fractions that have the same value, but may be expressed differently.
  • To compare fractions, we can compare the numerators and denominators separately.
  • If the numerators are equal, the fraction with the smaller denominator is larger.
  • If the denominators are equal, the fraction with the larger numerator is larger.
  • If the numerators and denominators are not equal, we can compare the fractions by finding the LCM of the denominators and converting both fractions to equivalent ratios.

Final Thoughts

Comparing fractions is a fundamental concept in mathematics that involves determining the relative size of two or more fractions. By understanding how to compare fractions, we can make informed decisions and solve problems involving ratios and proportions. We hope that this article has provided a clear and concise guide to evaluating true statements involving the comparison of fractions.

Introduction

Comparing fractions is a fundamental concept in mathematics that involves determining the relative size of two or more fractions. In this article, we will answer some frequently asked questions about comparing fractions and provide additional guidance on how to evaluate true statements involving the comparison of fractions.

Q: What is the best way to compare fractions?

A: The best way to compare fractions is to compare the numerators and denominators separately. If the numerators are equal, the fraction with the smaller denominator is larger. If the denominators are equal, the fraction with the larger numerator is larger. If the numerators and denominators are not equal, we can compare the fractions by finding the least common multiple (LCM) of the denominators and converting both fractions to equivalent ratios.

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

A: To find the LCM of two numbers, we can list the multiples of each number and find the smallest multiple that is common to both. For example, to find the LCM of 4 and 6, we can list the multiples of each number as follows:

Multiples of 4: 4, 8, 12, 16, 20, ... Multiples of 6: 6, 12, 18, 24, 30, ...

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

Q: How do I convert a fraction to an equivalent ratio?

A: To convert a fraction to an equivalent ratio, we can multiply the numerator and denominator by the same number. For example, to convert the fraction 12\frac{1}{2} to an equivalent ratio, we can multiply the numerator and denominator by 3 as follows:

12×33=36\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}

The fraction 36\frac{3}{6} is an equivalent ratio of the original fraction 12\frac{1}{2}.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. A decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part. For example, the fraction 12\frac{1}{2} can be expressed as the decimal 0.5.

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

A: To compare a fraction to a decimal, we can convert the fraction to a decimal by dividing the numerator by the denominator. For example, to compare the fraction 12\frac{1}{2} to the decimal 0.5, we can divide the numerator (1) by the denominator (2) as follows:

12=0.5\frac{1}{2} = 0.5

Since the decimal 0.5 is equal to the fraction 12\frac{1}{2}, we can conclude that the two are equal.

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

A: A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, the fraction 12\frac{1}{2} is a proper fraction, while the fraction 32\frac{3}{2} is an improper fraction.

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

A: To convert an improper fraction to a mixed number, we can divide the numerator by the denominator and write the remainder as a fraction. For example, to convert the improper fraction 32\frac{3}{2} to a mixed number, we can divide the numerator (3) by the denominator (2) as follows:

3÷2=1123 \div 2 = 1 \frac{1}{2}

The mixed number 1121 \frac{1}{2} is equivalent to the improper fraction 32\frac{3}{2}.

Conclusion

In conclusion, we have answered some frequently asked questions about comparing fractions and provided additional guidance on how to evaluate true statements involving the comparison of fractions. We hope that this article has provided a clear and concise guide to comparing fractions and has helped to answer any questions you may have had.