Chapter-1 RATIONAL NUMBERS

1.Using appropriate properties, find:

Explanation:

–2/3 × 3/5 + 5/2 – 3/5 × 1/6

= +5/2 – 2/3 × 3/5 – 3/5 × 1/6 (Using commutative property)

= 5/2 + 3/5 (– 2/3 – 1/6) 

= 5/2 – 3/5 × (2/3 + 1/6)

= 5/2 – 3/5 × ((4 + 1)/6)

= 5/2 – 3/5 × ((5)/6) (Using distributive property)

= 5/2 – 15/30 (simplifying)

= 5/2 – 1/2

= 4/2

= 2 (Answer)


2. 2/5 × (– 3/7) – 1/6 × 3/2 + 1/14 × 2/5

Explanation:

2/5 × (– 3/7) – 1/6 × 3/2 + 1/14 × 2/5

= – (1/6 × 3/2) + 2/5 × (– 3/7) + 1/14 × 2/5 (Using commutative property)

= – 3/12 + 2/5 × (– 3/7 + 1/14) 

= – 3/12 + 2/5 × ((– 6 + 1)/14) 

= – 1/4 + 2/5 × ((–5)/14)) 

= – 1/4 + –10/70) (simplifying)

= – 1/4 – 1/7 

= (– 4– 7)/28

= – 11/28 (Answer)


3.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

Explanation:

(i) –13

The Multiplicative Inverse of number –13 will be –1/13.

(ii) –13/19

The Multiplicative Inverse of fraction –13/19 will be –19/13.

(iii) 1/5

The Multiplicative Inverse of fraction 1/5 will be 5.

(iv) –5/8 × (–3/7) can be simplified to 15/56

The Multiplicative Inverse of fraction 15/56 will be 56/15.

(v) –1 × (–2/5) can be simplified to 2/5

The Multiplicative Inverse of faction 2/5 will be 5/2.

(vi) –1

The Multiplicative Inverse of number –1 will be –1.


4.Name the property under multiplication used in each of the following:

(i) –4/5 × 1 = 1 × (–4/5) = –4/5

(ii) –13/17 × (–2/7) = –2/7 × (–13/17)

(iii) –19/29 × 29/–19 = 1

Explanation:

(i) –4/5 × 1 = 1 × (–4/5) = –4/5

Multiplicative property is used in the above equation.

(ii) –13/17 × (–2/7) = –2/7 × (–13/17)

Commutative property is used in the above equation.

(iii) –19/29 × 29/–19 = 1

Multiplicative property is used in the above equation.


5.Multiply 6/13 by the reciprocal of –7/16.

Explanation:

The Reciprocal of –7/16 will be 16/–7. It can be further written as –16/7.

As asked in the question,

6/13 × (Reciprocal of –7/16)

6/13 × (–16/7)

–6/13 × (16/7) = –96/91


6.Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.

Explanation:

1/3 × (6 × 4/3) = (1/3 × 6) × 4/3

The Associative property allows to compute in this way. The values can be grouped in multiplication, this is the associative property.


7. Is 8/9 the multiplication inverse of –118 ? Why or why not?

Explanation:

Simplifying: –118 = –9/8

To be Multiplicatively inverse the product has to be 1

But here,

8/9 × (–9/8) = –1 ≠ 1 (not equal to 1)

That is why, 8/9 is not multiplicatively inverse to –118 .


8. If 0.3 is the multiplicative inverse of 313? Why or why not?

Explanation:

Simplifying: 313 = 10/3

0.3 can be re-written as 3/10.

To be Multiplicatively inverse the product has to be 1

And here,

10/3 × 3/10 = 1 (is equal to 1)

That is why, 0.3 is multiplicatively inverse to 313 .


9.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.

Explanation:

(i) 0 is the only rational number whose reciprocal does not exist.

Because:

0 can be written as 0/1

And reciprocal of 0/1 will be 1/0, and it is not defined.


(ii) 1 and –1 are the rational numbers whose reciprocals are equal to the number itself.

Because:

1 can be re-written as 1/1.

And reciprocal of 1/1 will be 1/1 which is again equal to 1, similar case is with –1. The reciprocal of –1 is equal to –1.


(iii) 0 is the only rational number which is also equal to its negative.

Because:

Zero is neither positive nor negative.

Negative of 0 will be –0 and it is equal to 0 itself.


10. 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 _________.

Explanation:

(i) Zero has no reciprocal. 

Reciprocal of 0 is 1/0 but this is not defined.

(ii) The numbers –1 and 1 are their own reciprocals. 

Reciprocal of 1/1 will be 1/1 which is again equal to 1, similar case is with –1. The reciprocal of –1 is equal to –1 itself.

(iii) The reciprocal of –5 is –1/5.

–5 can be written as –5/1. So, it’s reciprocal will be –1/5.

(iv) Reciprocal of 1/x, where x ≠ 0 is x.

Reciprocal of 0 is not defined.

(v) The product of two rational numbers is always a rational number.

Example- 5/2 × 3/4 = 15/8.

(vi) The reciprocal of a positive rational number is positive.

Example- 13/45 it’s reciprocal will be 45/13 that is positive.


11.Represent these numbers on the number line.

(i) 7/4

(ii) –5/6

Explanation:

Solution-

(i) 7/4

Divide the number line into 4 parts between 0 and 1 and divide the number line between 1 and 2 into 4 parts. The desired fraction lies between 1 and 2.

The rational number 7/4 will lie 7 points away from 0 and 3 points away from 1 towards the right side of the number line.




(ii) –5/6

Divide the number line into 6 parts between 0 and –1 and divide the number line between -1 and –2 into 6 parts. The desired fraction lies between 0 and –1.

The rational number –5/6 will lie 5 points away from 0 towards the left side of number line and 1 point away from –1 towards the right side of the number line.



12. Write the additive inverse of each of the following:

(i) 2/8

(ii) –5/9

(iii) –6/–5

(iv) 2/–9 

(v) 19/–16

Explanation:

(i) 2/8

Additive inverse of given term 2/8 will be –2/8

(ii) –5/9

Additive inverse of given term –5/9 will be 5/9

(iii) –6/–5 can be simplified to 6/5

Additive inverse of given term 6/5 is –6/5

(iv) 2/–9 can be simplified to –2/9

Additive inverse of given term –2/9 is 2/9

(v) 19/–16 can be simplified to –19/16

Additive inverse of given term –19/16 is 19/16


13. Write five rational numbers which are smaller than 2.

Explanation:

Number 2 can be re-written as 20/10.

The five rational numbers which are lesser than 2 are:

4/10, 0, –11/10, 17/10, 18/10.


14. Find the rational numbers between –2/5 and 1/2.

Explanation:

Making the denominators of these terms same, let 20.

–2/5 = (–2 × 4)/(5 × 4) = –8/20

1/2 = (1 × 10)/(2 × 10) = 10/20

Rational numbers between –2/5 and 1/2 will be equal to rational numbers between –8/20 and 10/20.

The ten rational numbers are as follows: –7/20, –5/20, –3/20, –2/20, 0, 1/20, 3/20, 6/20, 7/20, 9/20.


15. Find five rational numbers between:

(i) 2/3 and 4/5

(ii) –3/2 and 5/3

(iii) 1/4 and 1/2

Explanation:

(i) 2/3 and 4/5

Making the denominators of these terms same, let 45.

2/3 = (2 × 15)/(3 × 15) = 30/45

4/5 = (4 × 9)/(5 × 9) = 36/45

Rational numbers in between 2/3 and 4/5 will be equal to rational numbers between 30/45 and 36/45.

The five rational numbers are 33/45, 31/45, 32/45, 35/45, 34/45.


(ii) –3/2 and 5/3

Making the denominators of these terms same, let 18.

–3/2 = (–3 × 9)/(2× 9) = –27/18

5/3 = (5 × 6)/(3 × 6) = 30/18

Rational numbers in between –3/2 and 5/3 will be equal to rational numbers between –27/18 and 30/18.

The five rational numbers are –30/18, –40/18, 0, 2/18, 15/18.


(iii) 1/4 and 1/2

Making the denominators of these terms same, let 32.

1/4 = (1 × 8)/(4 × 8) = 8/32

1/2 = (1 × 16)/(2 × 16) = 16/32

Rational numbers in between 1/4 and 1/2 will be equal to rational numbers between 8/32 and 16/32.

The five rational numbers are 9/32, 10/32, 12/32, 14/32, 15/32.


16. Write five rational numbers greater than –2.

Explanation:

–2 can be re-written as –40/20.

Five rational numbers larger than –2 are –31/20, –27/20, –14/20, 7/20, 51/20.


17. Find ten rational numbers between 3/5 and 3/4.

Explanation:

Making the denominators of these terms same, let 100.

3/5 = (3 × 20)/(5× 20) = 60/100

3/4 = (3 × 25)/(4 × 25) = 75/100

Rational numbers in between 3/5 and 3/4 will be equal rational numbers between 60/100 and 75/100.

The five rational numbers are 62/100, 65/100, 66/100, 67/100, 68/100, 69/100, 70/100, 71/100, 72/100, 73/100.