Compare the Two Common Fractions 513/689 and 522/691, Which One is Larger? Online Calculator

Fractions 513/689 and 522/691 are compared by building equivalent fractions, which have either equal denominators or equal numerators

To compare and sort multiple fractions, they should either have the same denominator or the same numerator.

The operation of comparing fractions:
513/689 and 522/691

Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:


The fraction: 513/689

513/689 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 513 = 33 × 19
  • 689 = 13 × 53
  • GCF (513; 689) = 1


The fraction: 522/691

522/691 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 522 = 2 × 32 × 29
  • 691 is a prime number.
  • GCF (522; 691) = 1



To compare and sort the fractions, make their numerators the same.

To make the fractions' numerators the same - we have to:

  • 1) calculate their common numerator
  • 2) calculate the expanding number of each fraction
  • 3) expand the fractions to equivalent forms having equal numerators

Calculate the common numerator

The common numerator is nothing else than the least common multiple (LCM) of the numerators of the fractions.


To calculate the LCM, we need the prime factorization of the numerators:

513 = 33 × 19

522 = 2 × 32 × 29


Multiply all the unique prime factors: if there are repeating prime factors we only take them once, and only the ones having the highest exponent (the highest powers).


External link » Calculate LCM, the least common multiple of numbers, online calculator

LCM (513, 522) = 2 × 33 × 19 × 29 = 29,754



Calculate the expanding number of each fraction:

Divide the LCM by the numerator of each fraction.

513/689 ⟶ 29,754 ÷ 513 = (2 × 33 × 19 × 29) ÷ (33 × 19) = 58

522/691 ⟶ 29,754 ÷ 522 = (2 × 33 × 19 × 29) ÷ (2 × 32 × 29) = 57



Make the numerators of the fractions the same:

  • Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated above.
  • This way all the fractions will have the same numerator:

513/689 = (58 × 513)/(58 × 689) = 29,754/39,962

522/691 = (57 × 522)/(57 × 691) = 29,754/39,387



The fractions have the same numerator, compare their denominators.

The larger the denominator the smaller the positive fraction.

The larger the denominator the larger the negative fraction.


::: The operation of comparing fractions :::
The final answer:

The fractions sorted in ascending order:
29,754/39,962 < 29,754/39,387

The initial fractions sorted in ascending order:
513/689 < 522/691

How are the numbers written: comma ',' used as a thousands separator; point '.' as a decimal separator; numbers rounded off to max. 12 decimals (if the case). Used symbols: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.

Other similar operations

Compare and sort the fractions in ascending order:
522/691 and 527/698

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Compare and sort common fractions, online calculator:

Tutoring: Comparing ordinary fractions

How to compare two fractions?

1. Fractions that have different signs:

  • Any positive fraction is larger than any negative fraction:
  • ie: 4/25 > - 19/2

2. A proper and an improper fraction:

  • Any positive improper fraction is larger than any positive proper fraction:
  • ie: 44/25 > 1 > 19/200
  • Any negative improper fraction is smaller than any negative proper fraction:
  • ie: - 44/25 < -1 < - 19/200

3. Fractions that have both like numerators and denominators:

  • The fractions are equal:
  • ie: 89/50 = 89/50

4. Fractions that have unlike (different) numerators but like (equal) denominators.

  • Positive fractions: compare the numerators, the larger fraction is the one with the larger numerator:
  • ie: 24/25 > 19/25
  • Negative fractions: compare the numerators, the larger fraction is the one with the smaller numerator:
  • ie: - 19/25 < - 17/25

5. Fractions that have unlike (different) denominators but like (equal) numerators.

  • Positive fractions: compare the denominators, the larger fraction is the one with the smaller denominator:
  • ie: 24/25 > 24/26
  • Negative fractions: compare the denominators, the larger fraction is the one with the larger denominator:
  • ie: - 17/25 < - 17/29

6. Fractions that have different denominators and numerators (unlike denominators and numerators).

More on ordinary (common) fractions / theory: