find three ratioal and three irrational numbers between the following

a. 6.5 and 7.8

b. 2.3 and 3.2

c. 1/7 and 2/7

d. root 3/2 and root 5/2
asked Jun 9 in Algebra 2 Answers by kishore

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

a) First, the rational numbers. Average of 6.5 and 7.8=(6.5+7.8)/2=7.15. Now take two more averages: (6.5+7.15)/2=6.825 and (7.15+7.8)/2=7.475. So 3 rational numbers could be 6.825, 7.15 and 7.475.

Now the irrationals. Square both numbers: 42.25 and 60.84. If we take any three numbers between these limits (excluding perfect squares) and then take the square root of the numbers we will get 3 irrational numbers. Lets’s pick 43, 51 and 59. The square roots are √43, √51 and √59. These all lie between 6.5 and 7.8.

b) Using the techniques from (a), we have rationals: 2.75, 2.525, 2.975. And irrationals √6, √10, √7.

c) Rationals: 3/14, 5/28, 1/4; irrationals: (√3)/10, (√5)/10, (√7)/10. (1/7)²=1/49=0.02 approx, (2/7)²=4/49=0.08 approx. The numbers 0.03=3/100, 0.05=5/100, 0.07=7/100 lie between these squares. So the square roots must be n the range 1/7 to 2/7: (√3)/10, (√5)/10, (√7)/10.

d) We need 3 rationals whose square lies between 1.5 and 2.5. Multiply these by 100: 150 and 250. Now it’s easier to find perfect squares between these: 169=13², 196=14², 225=15². Divide each of these candidates by 10 and we get 1.3, 1.4 and 1.5. These are rational examples that can also be expressed as fractions: 13/10, 7/5, 3/2. Right between 3/2 and 5/2 lies 4/2=2, so √2 lies between √(3/2) and √5/2). We need two more irrationals. √(3/2)=√(6/4)=(√6)/2 and √(5/2)=√(10/4)=(√10)/2. Can we pick an integer between 6 and 10? We can’t pick 8 or 9 because we end up with √2 (which we’ve already chosen) and 9 is a perfect square. But √7/2 we can pick. The average of the two square roots is also irrational, so we have ((√6)/2+(√10)/2)/2=((√6)+(√10))/4 as our third irrational.

answered Jun 9 by Rod Top Rated User (569,800 points)

Related questions

1 answer
0 answers
1 answer
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
81,467 questions
85,643 answers
2,171 comments
69,130 users