Compare the Two Common Fractions 821/861 and 823/867, Which One is Larger? Online Calculator

Fractions 821/861 and 823/867 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:
821/861 and 823/867

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


The fraction: 821/861

821/861 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 821 is a prime number.
  • 861 = 3 × 7 × 41
  • GCF (821; 861) = 1


The fraction: 823/867

823/867 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 823 is a prime number.
  • 867 = 3 × 172
  • GCF (823; 867) = 1



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

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

  • 1) calculate their common denominator
  • 2) then calculate the expanding number of each fraction
  • 3) expand the fractions to equivalent forms having the same denominator

Calculate the common denominator

The common denominator is nothing else than the least common multiple (LCM) of the denominators of the fractions.


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

861 = 3 × 7 × 41

867 = 3 × 172


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 (861, 867) = 3 × 7 × 172 × 41 = 248,829



Calculate the expanding number of each fraction:

Divide the LCM by the denominator of each fraction.

821/861 ⟶ 248,829 ÷ 861 = (3 × 7 × 172 × 41) ÷ (3 × 7 × 41) = 289

823/867 ⟶ 248,829 ÷ 867 = (3 × 7 × 172 × 41) ÷ (3 × 172) = 287



Make the denominators 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 denominator:

821/861 = (289 × 821)/(289 × 861) = 237,269/248,829

823/867 = (287 × 823)/(287 × 867) = 236,201/248,829



The fractions have the same denominator, compare their numerators.

The larger the numerator the larger the positive fraction.

The larger the numerator the smaller the negative fraction.


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

The fractions sorted in ascending order:
236,201/248,829 < 237,269/248,829

The initial fractions sorted in ascending order:
823/867 < 821/861

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:
- 821/861 and - 827/871

» New. All the Calculations Performed by Our Visitors: Common Fractions Compared and Sorted. Data organized on a Monthly Basis

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: