Solution 1.1:5c

From Förberedande kurs i matematik 1

It is quite easy to see that

21=0532=231=066653=351=302=06

which means that

123523 .

Because it is more difficult to evaluate the decimal expansion of 58 and 2134, we compare instead 58 and 2134 with 12  35 and 23 by rewriting the fractions so that they have a common denominator. We start by comparing 58 with 12  35 and 23

• We have

21=2414=84 and thus 2185 .

• Then, we have

53=5838=4024 and 85=8555=4025 , which gives that 5385.

• Finally,

32=3828=2416 and 85=8353=2415, and this gives that 8532.

Thus, 12355823.

When we compare 2134 with 12  35 and 23, we obtain:

• because

21=217117=3417, so 213421

• furthermore,

53=534334=170102 and 3421=345215=170105, i.e. 533421.

• we have

85=817517=85136 and 3421=344214=84136, which gives that 342185.

The answer is \displaystyle {1}/{2}\;<{3}/{5}\;<{21}/{34}\;<{5}/{8}\;<{2}/{3}\;.