2. Find the values of (-5) x (-5), (-5) x (-4), (-5) x (-3), (-5) x (-2), (-5) x (-1), (5) x (0), (-5) x1, (-5) x 2, ( 5 ) x 3, (-5) x 4and (5) x 5.
Answers
Answer:
Using appropriate properties, find:
(i) -2/3 * 3/5 + 5/2 – 3/5 * 1/6 (ii) 2/5 * (3/-7) – 1/6 * 3/2 + 1/14 * 2/5
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
Question 11:
Fill in the blanks:
(i) Zero has _______________ reciprocal.
(ii) The numbers _______________ and _______________ are their own reciprocals.
(iii) The reciprocal of -5 is _______________.
(iv) Reciprocal of 1/x where x ≠ 0 is _______________.
(v) The product of two rational numbers is always a _______________.
(vi) The reciprocal of a positive rational number is _______________.
Answer:
(i) No (ii) 1, -1 (iii) -1/5 (iv) x
(v) Rational Number (vi) Positive rational number
Exercise 1.2
Question 1:
Represent these numbers on the number line:
(i) 7/4 (ii) -5/6
Answer:
(i) 7/4 = 1
Class_8_RationalNumbers_NumberLine
Here, P is 1
(ii)- 5/6
Class_8_RationalNumbers_NumberLine1
Here, M is -5/6
Question 2:
Represent -2/11, -5/11, -9/11 on the number line.
Answer:
Here, B = -2/11, C = -5/11, D = -9/11
Class_8_RationalNumbers_NumberLine2
Question 3:
Write five rational numbers which are smaller than 2.
Answer:
2 can be represented as 14/7
Hence, five rational numbers smaller than 2 are:
13/7, 12/7, 11/7, 10/7 and 9/7
Question 4:
Find ten rational numbers between -2/5 and 1/2
Answer:
Given rational numbers are -2/5 and 1/2
Here, LCM of 5 and 2 is 10
So, -2/5 = (-2 * 2)/(5 * 2) = -4/10
and 1/2 = (1 * 5)/(2 * 5) = 5/10
Again, -4/10 = (-4 * 2)/(10 * 2) = -8/20
and 5/10 = (5 * 2)/(10 * 2) = 10/20
Hence, ten rational numbers between -2/5 and 1/2 are:
-7/20, -6/20, -5/20, -7/20, -4/20, -3/20, -2/20, -1/20, 0, 1/20, 2/20
- -5 × -5 = 25
- -5×-4=20
- -5×-3=15
- -5×-2=10
- -5×-1=5
- 5×0=0
- -5×1=(-5)
- -5×2=(-10)
- -5×3=(-15)
- -5×4=(-20)
- 5×5=25