Comparing Fractions with Large Numerators and Denominators

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In summary, a fraction is a numerical representation of a part of a whole, written in the form of a numerator over a denominator. To compare two fractions, a common denominator is needed and the numerators are compared. The size of the numerator or denominator does not determine which fraction is greater, but rather the relationship between the numerator and denominator. Two fractions can be equal if they have the same numerator and denominator. To simplify fractions before comparing them, the greatest common factor (GCF) is found and both the numerator and denominator are divided by it. This results in an equivalent fraction with smaller numbers for easier comparison.
  • #1
anemone
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Which of the fractions is greater?

\(\displaystyle A=\frac{\overbrace{333\cdots331}^{ \large{\text{2012 pieces}}}}{\underbrace{333\cdots334}_{ \large{\text{2012 pieces}}}}\)

or

\(\displaystyle B=\frac{\overbrace{222\cdots221}^{ \large{\text{2012 pieces}}}}{\underbrace{222\cdots223}_{ \large{\text{2012 pieces}}}}\)?
 
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  • #2
[tex]A=1-\frac{1}{33...34/3}=1-\frac{1}{111..11 +\frac{1}{3}}[/tex]

[tex]B=1-\frac{1}{22...23/2}=1-\frac{1}{111..11+\frac{1}{2}}[/tex]

B is greater.
 
  • #3
Very beautiful and impressive, M R!(Clapping)
 

FAQ: Comparing Fractions with Large Numerators and Denominators

What is a fraction?

A fraction is a numerical representation of a part of a whole. It is written in the form of a numerator over a denominator, where the numerator represents the number of equal parts being considered, and the denominator represents the total number of equal parts in the whole.

How do you compare two fractions?

To compare two fractions, you need to have a common denominator. If the fractions have different denominators, you can find the lowest common denominator by finding the least common multiple of the two denominators. Then, you can convert both fractions to equivalent fractions with the lowest common denominator and compare the numerators.

Which fraction is greater - a fraction with a smaller numerator or a fraction with a smaller denominator?

The size of the numerator or denominator does not determine which fraction is greater. The value of a fraction depends on the relationship between the numerator and denominator. To determine which fraction is greater, you need to find a common denominator and compare the numerators.

Can two fractions with the same numerator be equal?

Yes, two fractions can have the same numerator and still be equal if they have the same denominator. For example, 2/5 and 2/5 are equal because they both represent 2 parts out of 5 equal parts.

How can you simplify fractions before comparing them?

To simplify fractions, you need to find the greatest common factor (GCF) of the numerator and denominator. Then, divide both the numerator and denominator by the GCF. This will give you an equivalent fraction with smaller numbers, making it easier to compare with other fractions.

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