Liz And Jay Each Have A Piece Of String Liz’s String Is 4/6 Yards Long And Jay’s String Is 5/7 Yards Long Whose String Is Longer?

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Introduction

In mathematics, fractions are used to represent parts of a whole. When comparing the lengths of two fractions, we need to determine which one is longer. In this article, we will compare the lengths of two fractions, 4/6 and 5/7, to determine whose string is longer.

Understanding Fractions

A fraction is a way of expressing a part of a whole as a ratio of two numbers. The top number, called the numerator, represents the number of equal parts we have, and the bottom number, called the denominator, represents the total number of parts the whole is divided into. For example, the fraction 3/4 means we have 3 equal parts out of a total of 4 parts.

Comparing Fractions

To compare two fractions, we need to find a common denominator, which is the least common multiple (LCM) of the two denominators. The LCM of 6 and 7 is 42. We can then convert both fractions to have a denominator of 42.

Converting 4/6 to Have a Denominator of 42

To convert 4/6 to have a denominator of 42, we need to multiply both the numerator and the denominator by 7.

4/6 = (4 × 7) / (6 × 7) = 28/42

Converting 5/7 to Have a Denominator of 42

To convert 5/7 to have a denominator of 42, we need to multiply both the numerator and the denominator by 6.

5/7 = (5 × 6) / (7 × 6) = 30/42

Determining Whose String is Longer

Now that we have both fractions with a common denominator of 42, we can compare them to determine whose string is longer. We can see that 30/42 is greater than 28/42, so Jay's string is longer.

Conclusion

In conclusion, when comparing the lengths of two fractions, 4/6 and 5/7, we need to find a common denominator and convert both fractions to have that denominator. We can then compare the fractions to determine whose string is longer. In this case, Jay's string is longer.

Real-World Applications

Comparing fractions is an important skill in mathematics, and it has many real-world applications. For example, in cooking, we may need to compare the amounts of ingredients in a recipe. In construction, we may need to compare the lengths of materials to determine how much is needed for a project. In finance, we may need to compare the interest rates of different investments.

Tips and Tricks

Here are some tips and tricks for comparing fractions:

  • Find a common denominator: To compare two fractions, we need to find a common denominator. We can do this by finding the least common multiple (LCM) of the two denominators.
  • Convert both fractions: Once we have found a common denominator, we need to convert both fractions to have that denominator.
  • Compare the numerators: Once we have both fractions with a common denominator, we can compare the numerators to determine whose string is longer.

Practice Problems

Here are some practice problems to help you practice comparing fractions:

  • Compare the lengths of 2/3 and 3/4.
  • Compare the lengths of 1/2 and 2/3.
  • Compare the lengths of 3/5 and 4/5.

Answer Key

Here are the answers to the practice problems:

  • 3/4 is longer than 2/3.
  • 2/3 is longer than 1/2.
  • 4/5 is longer than 3/5.

Conclusion

Q: What is the first step in comparing two fractions?

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

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that is common to both. 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) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I convert a fraction to have a common denominator?

A: To convert a fraction to have a common denominator, you need to multiply both the numerator and the denominator by the same number. This number is the multiple of the denominator that is needed to make the denominator equal to the common denominator.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number in a fraction, and it represents the number of equal parts we have. The denominator is the bottom number in a fraction, and it represents the total number of parts the whole is divided into.

Q: Can I compare fractions with different denominators?

A: Yes, you can compare fractions with different denominators. However, you need to find a common denominator first. This is the least common multiple (LCM) of the two denominators.

Q: How do I compare two fractions with the same denominator?

A: To compare two fractions with the same denominator, you can simply compare the numerators. The fraction with the larger numerator is the larger fraction.

Q: What is the importance of comparing fractions in real-life situations?

A: Comparing fractions is an important skill in mathematics, and it has many real-world applications. For example, in cooking, we may need to compare the amounts of ingredients in a recipe. In construction, we may need to compare the lengths of materials to determine how much is needed for a project. In finance, we may need to compare the interest rates of different investments.

Q: Can I use a calculator to compare fractions?

A: Yes, you can use a calculator to compare fractions. However, it is also important to understand the concept of comparing fractions and to be able to do it manually.

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 converting both fractions to have the same denominator
  • Comparing the denominators instead of the numerators
  • Not considering the sign of the fractions (e.g. positive or negative)

Q: How can I practice comparing fractions?

A: You can practice comparing fractions by working through examples and exercises. You can also try comparing fractions in real-life situations, such as comparing the amounts of ingredients in a recipe or the lengths of materials in a construction project.

Conclusion

In conclusion, comparing fractions is an important skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can compare the lengths of two fractions and determine whose string is longer. With practice, you can become proficient in comparing fractions and apply this skill to real-world problems.