Question #155090

find rational number r=m/n . Such that √5 < r < √6


1
Expert's answer
2021-01-12T17:16:26-0500

The distance between the numbers 5\sqrt{5} and 6\sqrt{6} is equal to

65=(65)(6+5)6+5=16+5>15\sqrt{6}-\sqrt{5} = \frac{(\sqrt{6}-\sqrt{5})(\sqrt{6}+\sqrt{5})}{\sqrt{6}+\sqrt{5}} = \frac{1}{\sqrt{6}+\sqrt{5}}>\frac15

Therefore, if n5n\geq 5 , then the length of the interval (5n,6n\sqrt{5}n,\sqrt{6}n ) is equal to 6n5n>1\sqrt{6}n-\sqrt{5}n >1 and this interval must contain an integer m.

We have 5n<m<6n\sqrt{5}n < m <\sqrt{6}n and, therefore, the ratio m/n belongs to the interval (5,6)(\sqrt{5},\sqrt{6}).

To find the ratio m/n in the interval (5,6)(\sqrt{5},\sqrt{6}) we should seek a full-squared integer m2 in the interval (5n2, 6n2).

Let's take n=5.

5n2=552=1255n^2 = 5\cdot 5^2 = 125

6n2=652=1506n^2 = 6\cdot 5^2 = 150

The only full-squared integer between 125 and 150 is 144=122=m2.

Therefore, r=m/n = 12/5 = 2.4 satisfies the original condition.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS