CBSE 8 maths Chapter-1: Rational Numbers

Chapter-1: Rational Numbers


  1. Using appropriate properties find the value of âˆ’23×35+52−35×16
Sol.   âˆ’23×35+52−35×16 â€“ âˆ’23×35−35×16+52
=−35(23+16)+52
=−35(4+16)+52
=−35(56)+52
=−1530+52
=−12+52
=42
=2
 therefore−23×35+52−35×16 =2

  1. Using appropriate properties find the value of 25×−37−16×32+114×25
Sol. 25×−37−16×32+114×25
−25×37−12×12+17×15
=−635−14+135
=−535−14
=−17−14
=−1128
=25×−37−16×32+114×25
=−1128

  1. Solving the additive inverse of 28
Sol.  Additive inverse of 28is−28

  1. Solving the additive inverse of âˆ’59
Sol.  Additive inverse of âˆ’59is59

  1. Solving the additive inverse of âˆ’6−5
Sol−6−5−6−5
Additive inverse of 65is−65
Additive inverse of âˆ’6−5−65

  1. Solving the additive inverse of 2−9
Sol.  Additive inverse of 2−9is29

  1. Verify that –(-x )=x for x=1115
Sol. x=1115
–x=−1115 âˆ’(−x)=−(−1115)
=1115=x
 therefore−(−x)=x

  1. Verify that –(-x )=x for x=−1317
Sol. x=−1317
–x=(−−1317)=1317 âˆ’(−x)=−(−1317)
=−1317=x
 therefore−(−x)=x

  1. Solve that multiplicative inverse of -13
Sol. Given multiplicative inverse -13 is−113

  1. Solve that multiplicative inverse of âˆ’1319
Sol. Given multiplicative inverse âˆ’1319is−1913

  1. Solve that multiplicative inverse of âˆ’58×−37
Sol. Given multiplicative inverse âˆ’58×−37is 85×73 or âˆ’85×−73

  1. What is the multiplicative inverse of -1.
Sol.  The multiplicative inverse of -1 is -1.

  1. Name the property under multiplicative used in each of the following.
 (i) âˆ’45×1=1×−45=−45
(ii) âˆ’1317×−27=−27×−1317
(iii) âˆ’1929×29−19=1
Sol. (i) ROLE OF 1
(ii) COMMUTATIVITY
(iii) MULTIPLICATIVE  INVERSE

  1. Multiply 613 by the reciprocal of âˆ’76
Sol. Reciprocal of âˆ’76 is 6−7
 therefore613×6−7
=−3691

  1. what property allows you to compute 13×(6×43)(13×6)×45
Sol.  Associativity

  1. Is 89 the multiplicative inverse of âˆ’118 ? why or why not.
Sol.
−118=−98
=89×−98
=−1≠1
 therefore89 is  not the multiplicative inverse of âˆ’118

  1. Is 0.3 the multiplicative inverse of 313 ? why or why not.
Sol.        313=103=3.3
                          3.3×0.3=0.99≠1
 therefore0.3isnotthemultiplicativeinverseof313

Exercise 1.1:
Question 1:
Solve the following using appropriate properties:
  1. ii)25∗−37–16∗32+114∗25
Answers:
  1. ii) Use commutativity of rational numbers
25∗−37–16∗32+114∗25
Using distributive property
25∗[−37+114]–14
25∗−3∗2+114–14
=25∗−514–14
=−4−728=−1128

Question 2:
(i) âˆ’5−7
(ii) âˆ’4−7
(iii)2−5
(iv)5−9
Solution:
(i) âˆ’5−7
additive inverse = 5−7
(ii)  âˆ’4−7 = 47
Additive inverse = âˆ’47
(iii)  2−5
Additive inverse = 25
(iv)  5−9
Additive inverse = 59
Question 3:
(i)  x = 1115
(ii) x= âˆ’1317
Solution:
x = 1115
The additive inverse of  is   x = âˆ’1115
This equality
x = 1115+frac−1115=0
=−(−x)=x
x= âˆ’1317
The additive inverse of  x= âˆ’1317 =  x= 1317
This equaliity
x = âˆ’1317+frac1317=0
=−(−x)=x

Question 4:
(i)  -12
(ii) âˆ’1913
(iii) 13
(iv)  âˆ’38∗−38
(v)  âˆ’1∗−35
Solutions:
  1. The multiplicative inverse of  -12 is 112
  2. The multiplicative inverse of âˆ’1913 is−1319
  3. The multiplicative inverse of 13  is 31 = 3
  4. The multiplicative inverse of  âˆ’1∗−35 is 53

Question 5:
Find out the multiplication property that is used in the following sums:
(i) âˆ’37∗1=1∗−37=−37
(ii) âˆ’1417∗−34=−34∗−1417
(iii) âˆ’1737∗37−17=1
Solutions:
  1. This use the multiplicative identity property and the multiplicative identity is 1
  2. This uses the commutativity property
  3. This uses the multiplicative inverse property
Question 6:
Find the product of  715 and the reciprocal of âˆ’417
Solution:
Reciprocal of  âˆ’417 =  âˆ’1714
715∗−1714=11960

Question 7:
  1. Which rational number does not have a reciprocal
  2. Which rational number is equal to its own reciprocal
  3. Which rational number is equal to its own negative
Solution:
  1. Zero is the rational number that has no reciprocal
  2. One and negative one are rational numbers that are equal to their own reciprocal
  3. Zero is a rational number that is also equal to its own negative.

Question 8: Answer or complete the following sentences:
  1. Does zero have a reciprocal?
  2. The two numbers _____ and _________ are their own reciprocals.
  3. What is the reciprocal of -6?
  4. What is the reciprocal of ?
  5. The product of two rational numbers is always a __________.
  6. The reciprocal of a positive rational number is ___________.

Solutions:
  1. No it doesn’t
  2. 1, -1
  3. -1/6
  4. X
  5. Rational number
  6. Positive rational number

Question 9:
Find the multiplicative inverse of the following:
  1. -12
  2. −1218
  3. 27
  4. -1
  5. −38∗−57
  6. −2∗−27
Solutions:
  1. −112
  2. −1218
  3. 72
  4. -1
  5. 5615
  6. 72

Question 10:
Is 0.3 the multiplicative inverse of   313? Why or why not?
Answer:
313=103 0.3∗313=0.3∗103=310∗103=1
As proved above the product is one.
Therefore, 0.3 is the multiplicative inverse of  313

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