Sort the Common Fractions String 205/307, 233/350, 211/325, 198/367, 204/416 in Ascending Order. Online Calculator

Multiple fractions 205/307, 233/350, 211/325, 198/367, 204/416 compared and then sorted in ascending order

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

The operation of sorting fractions in ascending order:
205/307, 233/350, 211/325, 198/367, 204/416

Analyze the fractions to be compared and ordered, by category:

positive proper fractions: 205/307, 233/350, 211/325, 198/367, 204/416

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


The fraction: 205/307

205/307 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 205 = 5 × 41
  • 307 is a prime number.
  • GCF (205; 307) = 1


The fraction: 233/350

233/350 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 233 is a prime number.
  • 350 = 2 × 52 × 7
  • GCF (233; 350) = 1


The fraction: 211/325

211/325 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 211 is a prime number.
  • 325 = 52 × 13
  • GCF (211; 325) = 1


The fraction: 198/367

198/367 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 198 = 2 × 32 × 11
  • 367 is a prime number.
  • GCF (198; 367) = 1


The fraction: 204/416

  • The prime factorizations of the numerator and denominator:
  • 204 = 22 × 3 × 17
  • 416 = 25 × 13
  • Multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).
  • GCF (204; 416) = 22 = 4

204/416 = (204 ÷ 4)/(416 ÷ 4) = 51/104


The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:


204/416 = (22 × 3 × 17)/(25 × 13) = ((22 × 3 × 17) ÷ 22)/((25 × 13) ÷ 22) = 51/104




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:

307 is a prime number.

350 = 2 × 52 × 7

325 = 52 × 13

367 is a prime number.

104 = 23 × 13


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 (307, 350, 325, 367, 104) = 23 × 52 × 7 × 13 × 307 × 367 = 2,050,575,800



Calculate the expanding number of each fraction:

Divide the LCM by the denominator of each fraction.

205/307 ⟶ 2,050,575,800 ÷ 307 = (23 × 52 × 7 × 13 × 307 × 367) ÷ 307 = 6,679,400

233/350 ⟶ 2,050,575,800 ÷ 350 = (23 × 52 × 7 × 13 × 307 × 367) ÷ (2 × 52 × 7) = 5,858,788

211/325 ⟶ 2,050,575,800 ÷ 325 = (23 × 52 × 7 × 13 × 307 × 367) ÷ (52 × 13) = 6,309,464

198/367 ⟶ 2,050,575,800 ÷ 367 = (23 × 52 × 7 × 13 × 307 × 367) ÷ 367 = 5,587,400

51/104 ⟶ 2,050,575,800 ÷ 104 = (23 × 52 × 7 × 13 × 307 × 367) ÷ (23 × 13) = 19,717,075



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:

205/307 = (6,679,400 × 205)/(6,679,400 × 307) = 1,369,277,000/2,050,575,800

233/350 = (5,858,788 × 233)/(5,858,788 × 350) = 1,365,097,604/2,050,575,800

211/325 = (6,309,464 × 211)/(6,309,464 × 325) = 1,331,296,904/2,050,575,800

198/367 = (5,587,400 × 198)/(5,587,400 × 367) = 1,106,305,200/2,050,575,800

51/104 = (19,717,075 × 51)/(19,717,075 × 104) = 1,005,570,825/2,050,575,800



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:
1,005,570,825/2,050,575,800 < 1,106,305,200/2,050,575,800 < 1,331,296,904/2,050,575,800 < 1,365,097,604/2,050,575,800 < 1,369,277,000/2,050,575,800

The initial fractions sorted in ascending order:
204/416 < 198/367 < 211/325 < 233/350 < 205/307

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:
- 214/319, - 237/360, - 214/331, - 201/372, - 207/423

» 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: