Brianna Is Practicing The Piano. She Spends 2 6 \frac{2}{6} 6 2 ​ Hour Practicing Scales And 3 4 \frac{3}{4} 4 3 ​ Hour Practicing The Song For Her Recital. Which Of The Following Statements Are True? Select All The Correct Answers.(A) 12 Is A Common

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Introduction

When it comes to comparing fractions, it's essential to understand the concept of equivalent ratios. In this article, we'll explore the world of fractions and help you determine which statements are true regarding Brianna's piano practice.

Understanding the Problem

Brianna spends 26\frac{2}{6} hour practicing scales and 34\frac{3}{4} hour practicing the song for her recital. To compare these fractions, we need to find a common denominator. The least common multiple (LCM) of 6 and 4 is 12.

Finding a Common Denominator

To find a common denominator, we can multiply the numerator and denominator of each fraction by the necessary multiples to get the LCM.

  • 26\frac{2}{6} = 2×26×2\frac{2 \times 2}{6 \times 2} = 412\frac{4}{12}
  • 34\frac{3}{4} = 3×34×3\frac{3 \times 3}{4 \times 3} = 912\frac{9}{12}

Now that we have a common denominator, we can compare the fractions.

Comparing Fractions

Since both fractions have the same denominator (12), we can compare the numerators.

  • 412\frac{4}{12} < 912\frac{9}{12}

This means that Brianna spends less time practicing scales (412\frac{4}{12} hour) than practicing the song for her recital (912\frac{9}{12} hour).

Analyzing the Statements

Now that we've compared the fractions, let's analyze the statements.

  • Statement A: 12 is a common denominator for both fractions.
    • True: We found that the LCM of 6 and 4 is 12, making it a common denominator for both fractions.
  • Statement B: Brianna spends more time practicing scales than the song for her recital.
    • False: We found that Brianna spends less time practicing scales (412\frac{4}{12} hour) than practicing the song for her recital (912\frac{9}{12} hour).
  • Statement C: The time Brianna spends practicing scales is equivalent to the time she spends practicing the song for her recital.
    • False: We found that the time Brianna spends practicing scales (412\frac{4}{12} hour) is less than the time she spends practicing the song for her recital (912\frac{9}{12} hour).

Conclusion

In conclusion, the only true statement is that 12 is a common denominator for both fractions. The other statements are false. By understanding the concept of equivalent ratios and finding a common denominator, we can compare fractions and determine which statements are true.

Key Takeaways

  • To compare fractions, find a common denominator by determining the least common multiple (LCM) of the denominators.
  • Once you have a common denominator, compare the numerators to determine which fraction is larger or smaller.
  • Be careful when analyzing statements, as they may be true or false depending on the comparison.

Additional Examples

To further reinforce your understanding of comparing fractions, try the following examples:

  • Compare 12\frac{1}{2} and 34\frac{3}{4}.
  • Compare 23\frac{2}{3} and 56\frac{5}{6}.
  • Compare 35\frac{3}{5} and 710\frac{7}{10}.

Introduction

In our previous article, we explored the concept of comparing fractions and how to determine which statements are true. In this article, we'll answer some frequently asked questions (FAQs) about comparing fractions.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 6 and 4 is 12.

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 appears in both lists. Alternatively, you can use the following formula:

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 greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 6 and 4 is 2.

Q: How do I compare fractions with different denominators?

A: To compare fractions with different denominators, find a common denominator by determining the LCM of the denominators. Then, compare the numerators.

Q: What if the denominators have a common factor?

A: If the denominators have a common factor, you can simplify the fractions by dividing both the numerator and denominator by that factor. For example, if you have the fraction 68\frac{6}{8}, you can simplify it by dividing both the numerator and denominator by 2, resulting in 34\frac{3}{4}.

Q: Can I compare fractions with unlike signs?

A: Yes, you can compare fractions with unlike signs. To do this, convert the fractions to equivalent fractions with like signs. For example, if you have the fractions 34\frac{3}{4} and −23-\frac{2}{3}, you can convert them to equivalent fractions with like signs by finding a common denominator and then changing the sign of one of the fractions.

Q: What if the fractions have different units?

A: If the fractions have different units, you cannot compare them directly. You need to convert the fractions to equivalent fractions with the same unit. For example, if you have the fractions 34\frac{3}{4} of a pizza and 23\frac{2}{3} of a pizza, you can convert them to equivalent fractions with the same unit by finding a common denominator and then converting the fractions to the same unit.

Q: Can I compare mixed numbers?

A: Yes, you can compare mixed numbers. To do this, convert the mixed numbers to improper fractions. For example, if you have the mixed number 2142\frac{1}{4}, you can convert it to an improper fraction by multiplying the whole number part by the denominator and then adding the numerator: 214=(2×4)+14=942\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}.

Conclusion

In conclusion, comparing fractions requires finding a common denominator and then comparing the numerators. By understanding the concept of equivalent ratios and finding a common denominator, you can compare fractions and determine which statements are true. We hope this Q&A article has helped you better understand comparing fractions.

Key Takeaways

  • To compare fractions, find a common denominator by determining the LCM of the denominators.
  • Once you have a common denominator, compare the numerators to determine which fraction is larger or smaller.
  • Be careful when analyzing statements, as they may be true or false depending on the comparison.
  • You can compare fractions with unlike signs by converting them to equivalent fractions with like signs.
  • You can compare fractions with different units by converting them to equivalent fractions with the same unit.
  • You can compare mixed numbers by converting them to improper fractions.