write all the rational numbers between 1/4,11/2
Answers
Answer:
The rational number between a and b is (a + b)/2
Hence rational number between 1 and 2 is (1+2)/2 = 3/2 = 1.5
Rational number between 1 and 3/2 is [1+(3/2)]/2 = 5/4 = 1.25
Rational number between 1 and 5/4 is [1+(5/4)]/2 = 9/8 = 1.125
Rational number between 3/2 and 2 is [(3/2) + 2]/2 = 7/4= 1.75
Therefore four rational numbers between 1 and 2 are 9/8, 5/4, 3/2, and 7/4
Answer:
Answer:
(i) -2/3 * 3/5 + 5/2 – 3/5 * 1/6
= -2/3 * 3/5 – 3/5 * 1/6 + 5/2 [Using associative property]
= 3/5 * (-2/3 – 1/6) + 5/2 [Using distributive property]
= 3/5 * {(-4 - 1)/6} + 5/2 [LCM (3, 2) = 6]
= 3/5 * (-5/6) + 5/2
= -3/6 + 5/2
= -1/2 + 5/2
= (-1 + 5)/2
= 4/2
= 2
(ii) 2/5 * (3/-7) – 1/6 * 3/2 + 1/14 * 2/5
= 2/5 * (-3/7) + 1/14 * 2/5 – 1/6 * 3/2 [Using associative property]
= 2/5 * (-3/7 + 1/14) – 1/2 * 1/2 [Using distributive property]
= 2/5 * {(-6 + 1)/14} – 1/4 [LCM (7, 14) = 14]
= 2/5 * (-5/14) – 1/4
= -1/7 – 1/4
= (-4 -7)/28 [LCM (7, 4) = 28]
= -11/28
Question 2:
Write the additive inverse of each of the following:
(i) 2/8 (ii) -5/9 (iii) -6/-5 (iv) 2/-9 (v) 19/-6
Answer:
We know that additive inverse of a rational number a/b is (-a/b) such that a/b + (-a/b) = 0
(i) Additive inverse of 2/8 = -2/8
(ii) Additive inverse of -5/9 = 5/9
(iii) -6/-5 = 6/5
Additive inverse of 6/5 = -6/5
(iv) 2/-9 = -2/9
Additive inverse of -2/9 = 2/9
(v) 19/-6 = -19/6
Additive inverse of -19/6 = 19/6
Question 3:
Verify that -(-x) = x for:
(i) x = 11/15 (ii) x = -13/17
Answer:
(i) Putting x = 11/15 in -(-x) = x, we get
=> -(-11/15) = 11/15
=> 11/15 = 11/15
=> LHS = RHS
Hence, verified.
(i) Putting x = -13/17 in -(-x) = x, we get
=> -{-(-13/17)} = -13/17
=> -(13/17) = -13/17
=> -13/17 = -13/17
=> LHS = RHS
Hence, verified.
Question 4:
Find the multiplicative inverse of the following:
(i) -13 (ii) -13/19 (iii) 1/5 (iv) (-5/8)*(-3/7) (v) -1 * (-2/5) (vi) -1
Answer:
We know that multiplicative inverse of a rational number a is 1/a such that a * 1/a = 1
(i) Multiplicative inverse of -13 = -1/13
(ii) Multiplicative inverse of -13/19 = -19/13
(iii) Multiplicative inverse of 1/5 = 5
(iv) (-5/8)*(-3/7) = (5 * 3)/(8 * 7) = 15/56
Multiplicative inverse of 15/56 = 56/15
(v) -1 * (-2/5) = 2/5
Multiplicative inverse of 2/5 = 5/2
(vi) Multiplicative inverse of -1 = 1/-1 = -1
Question 5:
Name the property under multiplication used in each of the following:
(i) -4/5 * 1 = 1 * -4/5
(ii) -13/17 * -2/7 = -2/7 * -13/17
(iii) -19/29 * 29/-19 = 1
Answer:
(i) 1 is the multiplicative identity.
(ii) Commutative property.
(iii) Multiplicative Inverse property.
Question 6:
Multiply 6/13 by the reciprocal of -7/16
Answer:
The reciprocal of -7/16 = -16/7
Now, 6/13 * (-16/7) = -(6 * 16)/(13 * 7) = -96/91
Question 7:
Tell what property allows you to compute 1/3 * (6 * 4/3) as (1/3 * 6) * 4/3
Answer:
By using associative property of multiplication, a * (b * c) = (a * b) * c
Question 8:
Is 8/9 the multiplicative inverse of -1? Why or why not?
Answer:
Since multiplicative inverse of a rational number a is 1/a if a * 1/a = 1
Therefore, 8/9 * (-1) = 8/9 * -9/8 = -1
But its product must be positive.
So, 8/9 is not multiplicative inverse of (-1)
Question 9:
Is 0.3 the multiplicative inverse of 3? Why or why not?
Answer:
Since multiplicative inverse of a rational number a is 1/a if a * 1/a = 1
Therefore, 0.3 * 3 = 3/10 * 10/3 = (3 * 10)/(10 * 3) = 30/30 = 1
So, 0.3 is the multiplicative inverse of 3
Question 10:
Write:
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Answer:
(i) 0 (ii) 1 and -1 (iii) 0