InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
find three rational numbers between (i) –1 and –2 (ii) 0.1 and 0.11(iii)5/7 and6/ 7 (iv)1/4 and1/ 5A. `-1 and -2`B. 0.1 and 0.11C. `5/7 and 6/7`D. `1/4 and 1/5` |
|
Answer» Let y =-1 and x = -2 Here, ` x gt y` and we have to find three ratonal numbers, so n=3 `d= (y-x)/(n+1) = (-1 + 2)/(3+1) = 1/ 4 ` Since, the three, rational numbers x and y are x+ d, x+2d and x+ 3d now , ` x=d =-2 + 1/4 = (-8+1)/4 = -7/4 ` `x+ 2d =-2 + 2/ 4 = (-8 + 2)/4 = -6/4 = (-3)/2` and ` x+ 3d = 2-3/4 = (-8+3)/4 = (-5)/4 ` Hene, three rational numbers between -1 and -2 are `(-7)/4 , (-3) / 2 and (-5)/ 4 ` Alternate method let x=-1 and y =-2 we know, a rational number between x and y `= (x+y)/2 ` A rational number between -1 and -2 `= (-1-2)/2 = -3/2 ` and a rational number between -1 and `- 3/2 = (-1-3/2)/2 = (-2-3)/4 = 5/4 ` Similarly, `-7/4` is a rational number between - 1and -2 Hence, required solution `=3/2,-5/4,-7/4` (ii) let x= 0.1 and y= 0.11 Here, `x lt y ` and we have to find three rational numbers, so consider, n=3 `d= (y-x)/(n+1) = (0.11-0.1)/(3+1) = 0.01/4` Since, the three rational numbers between x and y are (x + d) , ( x + 2d) and ( x+ 3d) now, `x + d = 0.01 + (0.01)/4 = (0.4+0.01)/4 = (0.41)/4 = (0.41)/4 = 0.1025` ` x+ 2d = 0.1 + (0.02)/4 = (0.4+ 0.02)/4 = (0.42)/4 = 0.105` `and x + 3d = 0.1 + 0.03/4 = (0.4+0.03)/4 = (0.43)/4 = 0.1075` Hence , three rational numbers between, 0.1 and 0.11 and 0.1025, 0.105, 0.1075 Also, without using above formula the three rational numbers between 0.1 and 0.11 are 0.101, 0.102, 0.103. (iii) Let , `x= 5/7 and y=6/7` Here, ` x lt y ` Here, we have to find three rational numbers. consider, n=3, `d=(y-x)/(n+1)` `d=(6/7-5/7)/4 = (1/7)/4= 1/28` Since, the three rational number between x and y are ( x+d) , (x+2d) and (x+3d) now, `x+d=5/7+1/28=(20+1)/28= 21/28` `x+2d=5/7+2/28= (20+2)/28= 22/28` ` x+3d=5/7=3/28=(20+3)/28=23/28` Hence, three rational numbers between ` 5/7and 6/7 are 21/28, 22/28,23/28` Also, without using formula, the three rational numbers between `5/7 and 6/7 are 51/70 , 52/70, 53/70` Let ` x=1/5 and y1/4` Here, `xlty` Here, we have to find three rational numbers. consider, , n=3 `d=(y-x)/(n+1)= (1/4-1/5)/(3+1)=((5-4)/20)/4= 1/80` since, the three rational numbers between x and y are x+d , x+2d adn x+3d. `now x+d= 1/5+1/80=(16+1)/80 = 17/80` `x+2d=1/5+2/80=(16+2)/80 = 18/80=9/40` `x+3d = 1/5+3/80=(16+3)/80=19/80` Hence, three rational numbers between ` 1/4 and 1/5 are 17/80 , 9/40,19/80` Alternate method Let ` x=1/4 and y=1/5` so, a rational number between x and `y= (x + y)/2` A rational number between ` 1/4 and 1/5 = (1/4+1/5)/2=((5+4)/20)/2` `9/(2xx20)=9/40` Again, a rational number between `1/4 and 9/40= (1/4 + 9/40)/2= ((10+9)/40)/2=19/(2xx40)=19/80` Again , a rational number between `1/5 and 9/40= (1/5+9/40)/2= ((8+9)/40)/2=(17/40)/2=17/(40xx2)=17/80` Hence, three rational numbers between `1/4 and 1/5 are ,9/40, 19/80, 17/80` |
|