ML Aggarwal Solutions Class 9 Math 1st Chapter Rational and Irrational Numbers Exercise 1.1
ML Aggarwal Understanding ICSE Mathematics Class 9 Solutions First Chapter Rational and Irrational Numbers Exercise 1.1. APC Solution Class 9 Exercise 1.1.
(1) Insert a rational number between 2/9 and 3/8, and arrange in descending order.
Solution:
LCM of 9 & 8 is 72.
2/9 = 2×8/9×8,
= 16/72
3/8 = 3×9/8×9
= 27/72
Since, 16 < 27, 2/9 < 3/8
A rational number between 2/9 & 3/8 =
∴ In descending order:
3/8, 43/144, 2/9
(2) Insert two rational numbers between 1/3 and 1/4, and arrange in ascending order.
Solution:
LCM of 4 & 3 is 12
1/2 = 1×4/3×4, 1/4 = 1×3/4×3
= 4/12 , 3/12
Since, 3 < 4, 1/4 < 1/3
One rational between 1/3 & 1/4 =
Now, since 1/4 < 1/3
∴ another rational number
∴ In ascending order: 1/4, 13/48, 7/24, 1/3
(3) Insert two rational numbers between – 1/3 and – 1/2 and arrange in ascending order.
Solution:
LCM of 3 & 2 is 6
∴ -1/2 = – 1×3/2×3 = -3/6, -1×2/3×2 = -2/6
Since, -3 < -2, -1/32 < – 1/3
(4) Insert three rational numbers between 1/3 and 4/5, and arrange in descending order.
Solution:
LCM of 3 & 5 is 15
∴ 1/3 = 1×5/3×5 = 5/15,
4/5 = 4×3/5×3 = 12/15
Now, since we want 4 rational numbers therefore, we multiply 4+1 = 5 to both numerator and denominator of the above fractions we get,
5/15 = 5×5/15×5 = 25/75, 12/15 = 12×5/15×5 = 60/75
Now, Since, 25 < 30 < 35 < 40 < 45 < 60
∴ 25/75 < 30/75 < 35/75 < 40/75 < 45/75 < 60/75
∴ 4 rational numbers between 1/3 & 4/5 are in descending order,
4/5, 45/75, 40/75, 35/75, 30/75, 1/3
(5) Insert three rational numbers between 4 and 4-5.
Solution:
A rational number between 4 & 4.6 is
= 4 – 4.5/2 = 8.5/2 = 4.25
A Rational number between 4 & 4.15 = 4+4.25/2 = 8.25/2
= 4.125
A rational number between 4 & 4.125 = 4+4.125/2 = 8.125/2
= 4.0625
∴ 3 rational number between 4 & 4.5 are 4, 4.0625, 4.125, 4.25, 4.5
(6) Find six rational numbers between 3 and 4.
Solution:
A rational number between 3 & 4 is = 3+4/2 = 7/2 = 3.5
A rational number between 3 & 3.5 = 3+3.5/2 = 6.5/2 = 3.25
A rational number between 3 & 3.25 = 3+3.25/2 = 6.25/2 = 3.125
A rational number between 3 & 3.125 = 3+3.125/2 = 6.125/2 = 3.0625
A rational number between 3 & 3.0625 = 3+3.0625/2 = 6.0625/2 = 3.03125
A rational number between 3 & 3.03125 = 3+3.03125/2 = 6.03125/2 = 3.015625
∴ Six rational number between 3 & 4 are
3, 3.015625, 3.03125, 3.0625, 3.125, 3.25, 3.5, 4.
(7) Find five rational numbers between 3/5 and 4/5.
Solution:
Since we want 5 rational numbers between 3/5 & 4/5
We multiply both numerator and denominator of the above fractions by 5+1 = 6 we get,
3/5 = 3×6/5×6 = 18/30;
4/5 = 4×6/5×6 = 24/3
∵ 18 < 19 < 20 < 21 < 22 < 23 < 24
∴ 18/30 < 19/30 < 20/30 < 21/30 < 22/30 < 23/30 < 24/30
∴ 5 rational numbers between 3/5 & 4/5 are,
3/5, 19/30, 20/30, 21/30, 22/30, 23/30, 4/5
(8) Find ten rational numbers between – 2/5 and 1/7
Solution:
LCM of 5 & 7 is 35.
∴ – 2/5 = 2×7/5×7 = -14/35; 1/7 = 1×5/7×5 = 5/35
∵ -14 < -10 < -8 < -6 < -4 < -2 < 0 < 1
1 < 2 < 3 < 4 < 5
∴ -14/35 < -10/35 < -8/35 < -6/35 < -4/35 < -2/35 <0
0 < 1/35 < 2/35 < 3/35 < 4/35 < 5/35
∴ 10 rational numbers between -2/5 & 1/7 are,
-2/5, -10/35, -3/35, -6/35, -4/35, -2/35, 0, 1/35, 2/35, 3/35, 4/35, 1/7
(9) Find six rational number between 1/2 and 2/3.
Solution:
LCM of 2 & 3 is 6.
∴ 1/2 = 1×3/2×3 = 3/6,
2/3 = 2×2/3×2 = 4/6
∵ we want 6 rational numbers between 3/6 & 4/6 we multiply both fraction with 6+1 = 7
On both numerators and denominator we get,
3/6 = 3×7/6×7 = 21/42, 4/6 = 4×7/6×7 = 28/42
∵ 21 < 22 < 23 < 25 < 26 < 27 < 28
∴21/42 <22/42 < 23/42 < 25/42 < 26/42 < 27/42 < 28/42
∴ 6 rational numbers between 1/2 & 2/3 are 1/2, 22/42, 23/42 25/42, 26/42, 27/42, 2/3