The operation of sorting fractions in ascending order:
17/26, 27/18, 41/23, 39/17, 66/26, 108/17, 152/15, 269/20
Analyze the fractions to be compared and ordered, by category:
1 positive proper fraction: 17/26;
positive improper fractions: 27/18, 41/23, 39/17, 66/26, 108/17, 152/15, 269/20;
How to sort and order fractions by categories:
Any positive proper fraction is smaller than
any positive improper fraction
Sort the positive improper fractions:
27/18, 41/23, 39/17, 66/26, 108/17, 152/15, 269/20
Reduce (simplify) fractions to their lowest terms equivalents:
27/18 = 33/(2 × 32) = (33 ÷ 32)/((2 × 32) ÷ 32) = 3/2
41/23 already reduced to the lowest terms;
the numerator and denominator have no common prime factors:
41 is a prime number;
23 is a prime number;
39/17 already reduced to the lowest terms;
the numerator and denominator have no common prime factors:
39 = 3 × 13;
17 is a prime number;
66/26 = (2 × 3 × 11)/(2 × 13) = ((2 × 3 × 11) ÷ 2)/((2 × 13) ÷ 2) = 33/13
108/17 already reduced to the lowest terms;
the numerator and denominator have no common prime factors:
108 = 22 × 33;
17 is a prime number;
152/15 already reduced to the lowest terms;
the numerator and denominator have no common prime factors:
152 = 23 × 19;
15 = 3 × 5;
269/20 already reduced to the lowest terms;
the numerator and denominator have no common prime factors:
269 is a prime number;
20 = 22 × 5;
To sort fractions in ascending order, build up their denominators the same.
Calculate LCM, the least common multiple of the denominators of the fractions.
LCM will be the common denominator of the compared fractions.
In this case, LCM is also called LCD, the least common denominator.
The prime factorization of the denominators:
2 is a prime number
23 is a prime number
17 is a prime number
13 is a prime number
15 = 3 × 5
20 = 22 × 5
Multiply all the unique prime factors, by the largest exponents:
LCM (2, 23, 17, 13, 15, 20) = 22 × 3 × 5 × 13 × 17 × 23 = 304,980
Calculate the expanding number of each fraction
Divide LCM by the denominator of each fraction:
For fraction: 3/2 is 304,980 ÷ 2 = (22 × 3 × 5 × 13 × 17 × 23) ÷ 2 = 152,490
For fraction: 41/23 is 304,980 ÷ 23 = (22 × 3 × 5 × 13 × 17 × 23) ÷ 23 = 13,260
For fraction: 39/17 is 304,980 ÷ 17 = (22 × 3 × 5 × 13 × 17 × 23) ÷ 17 = 17,940
For fraction: 33/13 is 304,980 ÷ 13 = (22 × 3 × 5 × 13 × 17 × 23) ÷ 13 = 23,460
For fraction: 108/17 is 304,980 ÷ 17 = (22 × 3 × 5 × 13 × 17 × 23) ÷ 17 = 17,940
For fraction: 152/15 is 304,980 ÷ 15 = (22 × 3 × 5 × 13 × 17 × 23) ÷ (3 × 5) = 20,332
For fraction: 269/20 is 304,980 ÷ 20 = (22 × 3 × 5 × 13 × 17 × 23) ÷ (22 × 5) = 15,249
Expand the fractions
Build up all the fractions to the same denominator (which is LCM).
Multiply the numerators and denominators by their expanding number:
3/2 = (152,490 × 3)/(152,490 × 2) = 457,470/304,980
41/23 = (13,260 × 41)/(13,260 × 23) = 543,660/304,980
39/17 = (17,940 × 39)/(17,940 × 17) = 699,660/304,980
33/13 = (23,460 × 33)/(23,460 × 13) = 774,180/304,980
108/17 = (17,940 × 108)/(17,940 × 17) = 1,937,520/304,980
152/15 = (20,332 × 152)/(20,332 × 15) = 3,090,464/304,980
269/20 = (15,249 × 269)/(15,249 × 20) = 4,101,981/304,980
The fractions have the same denominator, compare their numerators.
The larger the numerator the larger the positive fraction.
The fractions sorted in ascending order:
457,470/304,980 < 543,660/304,980 < 699,660/304,980 < 774,180/304,980 < 1,937,520/304,980 < 3,090,464/304,980 < 4,101,981/304,980
The initial fractions in ascending order:
27/18 < 41/23 < 39/17 < 66/26 < 108/17 < 152/15 < 269/20
::: Comparing operation :::
The final answer: