Viết phân số 25/27;15/16 thành tổng 3 phân số có mẫu số khác nhau và dều có tử số là 1.

Guest Jul 7, 2017

#1**0 **

This is the Google translation:

Write 25/27, 15/16 as the sum of three fractions with different denominators, and have a numerator of 1.

But I still do not understand Why are there 2 fractions ? Anyone got any thoughts?

Melody
Jul 7, 2017

#6**+2 **

We can use the concept of "Egyptian" fractions, here

25/27

Take the ceiling function for the division of 27 by 25 = 2

Write this as the denominator of a fraction with a numerator of 1 = 1/2

Subtract this from 25/27

25/27 - 1/2 = 23/54

Take the ceiling function for the division of 54 by 23 = 3

Write this as the denominator of a fraction with a numerator of 1 = 1/3

Subtract this from 23/54 - 1/3 = 5/54

Take the ceiling function for the division of 54 by 5 = 11

Write this as the denominator of a fraction with a numerator of 1 = 1/11

Subtract this from 5/54

5/54 - 1/11 = 1/594

We can now stop the process since the last subraction results in a unit fraction

So.....25/27 can be represented as the sum of these unit fractions

1/2 + 1/3 + 1/11 + 1/594

BTW.....this is the shortest length possible...in other words.....this fraction cannot be represented by the sum of any fewer than 4 unit fractions

15/16

Using a similar process this can be represented as the sum of the following unit fractions :

1/2 + 1/3 + 1/10 + 1/240

Again....this fraction cannot be represented by the sum of any fewer than four unit fractions

Here is website devoted to Egyptian Fractions....it is rather interesting :

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

CPhill
Jul 7, 2017