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Question 1:

What are rational numbers? Give examples of five positive and five negative rational numbers. Is there any rational number which is neither positive nor negative? Name it.

The numbers that are in the form of  $\frac{p}{q}$, where p and q are integers and q ≠0, are called rational numbers.

For example:

Five positive rational numbers:

$\frac{5}{7},\frac{-3}{-4},\frac{7}{8},\frac{-14}{-15},\frac{5}{9}$

Five negative rational numbers:

$\frac{-3}{7},\frac{-3}{8},\frac{8}{-9},\frac{-19}{25},\frac{8}{-25}$

Yes, there is a rational number that is neither positive nor negative, i.e. zero (0).

Question 2:

Which of the following are rational numbers?

(i) $\frac{5}{-8}$
(ii) $\frac{-6}{11}$
(iii) $\frac{7}{15}$
(iv) $\frac{-8}{-12}$
(v) 6
(vi) −3
(vii) 0
(viii) $\frac{0}{1}$
(ix) $\frac{1}{0}$
(x) $\frac{0}{0}$

Question 3:

Write down the numerator and the denominator of each of the following rational numbers:

(i) $\frac{8}{19}$
(ii) $\frac{5}{-8}$
(iii) $\frac{-13}{15}$
(iv) $\frac{-8}{-11}$
(v) 9

(i) $\frac{8}{19}$
Numerator = 8

Denominator =19

(ii)$\frac{5}{-8}$
Numerator  = 5

Denominator = −8

(iii) $\frac{-13}{5}$

Numerator = −13
Denominator =15

(iv)$\frac{-8}{-11}$

Numerator = −8
Denominator = −11

(v) 9
i.e $\frac{9}{1}$
Numerator = 9
Denominator = 1

Question 4:

Write each of the following integers as a rational number. Write the numerator and the denominator in each case.

(i) 5
(ii) −3
(iii) 1
(iv) 0
(v) −23

(i) 5
The rational number will be $\frac{5}{1}$.
Numerator = 5
Denominator = 1

(ii) -3
The rational number will be $\frac{-3}{1}$.
Numerator   = -3
Denominator = 1

(iii)1
The rational number will be $\frac{1}{1}$.
Numerator = 1
Denominator = 1

(iv) 0
The rational number will be $\frac{0}{1}$.
Numerator =0
Denominator = 1

(v) -23
The rational number will be $\frac{-23}{1}$.
Numerator = -23
Denominator = 1

Question 5:

Which of the following are positive rational numbers?

(i) $\frac{3}{-5}$
(ii) $\frac{-11}{15}$
(iii) $\frac{-5}{-8}$
(iv) $\frac{37}{53}$
(v) $\frac{0}{3}$

Positive rational numbers:
(iii) $\frac{-5}{-8}$

(iv) $\frac{37}{53}$
(vi) 8 because 8 can be written as .

0 is neither positive nor negative.

Question 6:

Which of the following are negative rational numbers?

(i) $\frac{-15}{-14}$
(ii) 0
(iii) $\frac{-5}{7}$
(iv) $\frac{4}{-9}$
(v) −6
(vi) $\frac{1}{-2}$

Negative rational numbers:

(iii) $\frac{-5}{7}$

(iv) $\frac{4}{-9}$

(v)  -6

(vi) $\frac{1}{-2}$

Question 7:

Find four rational numbers equivalent to each of the following.

(i) $\frac{6}{11}$
(ii) $\frac{-3}{8}$
(iii) $\frac{7}{-15}$
(iv) 8
(v) 1
(vi) −1

(i) Following are the four rational numbers that are equivalent to $\frac{6}{11}$.

(ii) Following are the four rational numbers that are equivalent to $\frac{-3}{8}$.
$\frac{-3×2}{8×2}$,
$\frac{-3×3}{8×3}$$\frac{-3×4}{8×4}$ and $\frac{-3×5}{8×5}$

i.e. $\frac{-6}{16}$$\frac{-9}{24}$$\frac{-12}{32}$ and $\frac{-15}{40}$

(iii) Following are the four rational numbers that are equivalent to $\frac{7}{-15}$.

(iv) Following are the four rational numbers that are equivalent to 8, i.e. $\frac{8}{1}$.

(v) Following are the four rational numbers that are equivalent to ­­-1, i.e. $\frac{1}{1}$.

(vi)
Following are the four rational numbers that are equivalent to ­­-1, i.e. $\frac{-1}{1}$.

Question 8:

Write each of the following as a rational number with positive denominator.

(i) $\frac{12}{-17}$
(ii) $\frac{1}{-2}$
(iii) $\frac{-8}{-19}$
(iv) $\frac{11}{-6}$

(i) $\frac{12×\left(-1\right)}{\left(-17\right)×\left(-1\right)}=\frac{-12}{17}$

(ii) $\frac{1×\left(-1\right)}{\left(-2\right)×\left(-1\right)}=\frac{-1}{2}$

(iii) $\frac{-8}{-19}=\frac{-8×\left(-1\right)}{\left(-19\right)×\left(-1\right)}=\frac{8}{19}$

(iv) $\frac{11×\left(-1\right)}{-6×\left(-1\right)}=\frac{-11}{6}$

Question 9:

Express $\frac{5}{8}$ as a rational number with numerator

(i) 15
(ii) −10

(i) Numerator of  $\frac{5}{8}$  is 5.
5 should be multiplied by 3 to get 15.
Multiplying both the numerator and the denominator by 3:

(ii)  Numerator of  $\frac{5}{8}$  is 5.
5 should be multiplied by −2 to get −10.
Multiplying both the numerator and the denominator by −2:

Question 10:

Express $\frac{4}{7}$ as a rational number with denominator
(i) 21
(ii) −35

(i) Denominator of $\frac{4}{7}$ is 7.
7 should be multiplied by 3 to get 21.
Multiplying both the numerator and the denominator by 3:

$\frac{4×3}{7×3}$$\frac{12}{21}$

$\frac{4×3}{7×3}$$\frac{4}{7}$

(ii)
Denominator of $\frac{4}{7}$ is 7.
7 should be multiplied by -5 to get −35.
Multiplying both the numerator and the denominator by 5:

Question 11:

Express $\frac{-12}{13}$ as a rational number with numerator
(i) −48
(ii) 60

(i) Numerator of $\frac{-12}{13}$ is −12.
−12 should be multiplied by 4 to get 48.
Multiplying both the numerator and the denominator by 4
:

$\begin{array}{l}\frac{-12×4}{13×4}=\frac{-48}{52}\\ \\ \frac{-12}{13}=\frac{-48}{52}\end{array}$

(ii)
Numerator of $\frac{-12}{13}$ is −12.​
12 should be multiplied by 5 to get 60

Multiplying its numerator and denominator by -5:

$\begin{array}{l}\frac{-12×\left(-5\right)}{13×\left(-5\right)}=\frac{60}{-65}\\ \\ \frac{-12}{13}=\frac{60}{-65}\end{array}$

Question 12:

Express $\frac{-8}{11}$ as a rational number with denominator

(i) 22
(ii) −55

(i) Denominator of$\frac{-8}{11}$  is 11.
Clearly, 11×2= 22

Multiplying both the numerator and the denominator by 2:

$\begin{array}{l}\frac{-8×2}{11×2}=\frac{-16}{22}\\ \\ \frac{-8}{11}=\frac{-16}{22}\end{array}$

(ii)
Denominator of$\frac{-8}{11}$  is 11.
Clearly, 11×5=55

Multiplying both the numerator and the denominator by 5:

Question 13:

Express $\frac{14}{-5}$ as a rational number with numerator

(i) 56
(ii) −70

(i) Numerator of $\frac{14}{-5}$ is 14.
Clearly, 14×4=56

Multiplying both the numerator and the denominator by 4:

$\frac{14×4}{-5×4}$=$\frac{56}{-20}$

$\frac{14}{-5}$=$\frac{56}{-20}$

(ii) −70
Numerator of $\frac{14}{-5}$ is 14.​
Clearly, 14×(−5)=−70
Multiplying both the numerator and the denominator by -5:

$\frac{14×\left(-5\right)}{\left(-5\right)×\left(-5\right)}$=$\frac{-70}{25}$

$\frac{14}{-5}$=​$\frac{-70}{25}$

Question 14:

Express $\frac{13}{-8}$ as a rational number with denominator

(i) −40
(ii) 32

(i) Denominator of $\frac{13}{-8}$ is −8.
Clearly, (
−8)×5= −40
Multiplying both the numerator and the denominator by 5:
$\begin{array}{l}\frac{13×5}{-8×5}=\frac{65}{-40}\\ \\ \frac{13}{-8}=\frac{65}{-40}\end{array}$

(ii) Denominator of $\frac{13}{-8}$ is −8.
Clearly, (−8)×(
−4)= 32

Multiplying both the numerator and the denominator by −4:

$\frac{13×\left(-4\right)}{-8×\left(-4\right)}=\frac{-52}{32}$

$\frac{13}{-8}$=$\frac{-52}{32}$

Question 15:

Express $\frac{-36}{24}$ as a rational number with numerator

(i) −9
(ii) 6

(i) Numerator of  $\frac{-36}{24}$ is -36.

Clearly, (−36) ÷ 4= (−9)
Dividing both the numerator and the denominator by 4:

$\frac{-36÷4}{24÷4}=\frac{-9}{6}$

(ii)
Numerator of  $\frac{-36}{24}$ is −36.​
Clearly, (−36) ÷ ( −6) = 6
Dividing both the numerator and the denominator by -6:

$\frac{-36÷\left(-6\right)}{24÷\left(-6\right)}=\frac{6}{-4}$

$\frac{-36}{24}$$\frac{6}{-4}$

Question 16:

Express $\frac{84}{-147}$ as a rational number with denominator

(i) 7
(ii) −49

(i) Denominator of $\frac{84}{-147}$ is 147.
∴ −147 ÷(−21)=7
Dividing both the numerator and the denominator by -21:

$\begin{array}{l}\frac{84÷\left(-21\right)}{-147÷\left(-21\right)}=\frac{-4}{7}\\ \\ \frac{84}{-147}=\frac{-4}{7}\end{array}$

(ii)
Denominator of
$\frac{84}{-147}$ is 147.
−147÷3=−49
Dividing both the numerator and the denominator by 3:

$\begin{array}{l}\frac{84÷3}{-147÷3}=\frac{28}{-49}\\ \end{array}$

$\frac{84}{-147}=\frac{28}{-49}$

Question 17:

Write each of the following rational numbers in standard form:

(i) $\frac{35}{49}$
(ii) $\frac{8}{-36}$
(iii) $\frac{-27}{45}$
(iv) $\frac{-14}{-49}$
(v) $\frac{91}{-78}$
(vi) $\frac{-68}{119}$
(vii) $\frac{-87}{116}$
(viii) $\frac{299}{-161}$

(i) $\frac{35}{49}$
H.C.F. of 35 and 49 is 7. Dividing the numerator and the denominator by 7:

$\frac{35÷7}{49÷7}=\frac{5}{7}$
So,
in the standard form.

(ii)$\frac{8}{-36}$
Denominator is -36, which is negative.
Multiplying both the numerator and the denominator by -1:

$\frac{8×\left(-1\right)}{-36×\left(-1\right)}=\frac{-8}{36}$ H.C.F. of 8 and 36 is 4
.
Dividing its numerator and denominator by 4:

$\frac{-8÷4}{36÷4}=\frac{-2}{9}$

So,  in the standard form.

(iii) $\frac{-27}{45}$ H.C.F. of 27 and 45 is 9.

Dividing its numerator and denominator by 9:
$\frac{-27÷9}{45÷9}=\frac{-3}{5}$
Hence,  in the standard form. H.C.F. of 14 and 49 is 7.
Dividing both the numerator and the denominator by 7. H.C.F. of 91 and 78 is 13.
Dividing both the numerator and the denominator by 13: H.C.F. of 68 and 119 is 17.
Dividing both the numerator and the denominator by 17: H.C.F. of 87 and 116 is 29.
Dividing both the numerator and the denominator by 29:

The denominator is negative.
Multiplying both the numerator and denominator by -1: H.C.F. of 299 and 161 is 23.
Dividing both the numerator and the denominator by 23:

Question 18:

Fill in the blanks:

(i) $\frac{-9}{5}=\frac{......}{20}=\frac{27}{......}=\frac{-45}{......}$
(ii) $\frac{-6}{11}=\frac{-18}{......}=\frac{......}{44}$

(i)

$\frac{-9×4}{5×4}=\frac{-36}{20}\phantom{\rule{0ex}{0ex}}\frac{-9×\left(-3\right)}{5×\left(-3\right)}=\frac{27}{-15}\phantom{\rule{0ex}{0ex}}\frac{-9×5}{5×5}=\frac{-45}{25}\phantom{\rule{0ex}{0ex}}\therefore \frac{-9}{5}=\frac{-36}{20}=\frac{27}{-15}=\frac{-45}{25}$

(ii)

Question 19:

Which of the following are pairs of equivalent rational numbers?

(i) $\frac{-13}{7},\frac{39}{-21}$
(ii) $\frac{3}{-8},\frac{-6}{16}$
(iii) $\frac{9}{4},\frac{-36}{-16}$
(iv) $\frac{7}{15},\frac{-28}{60}$
(v) $\frac{3}{12},\frac{-1}{4}$
(vi) $\frac{2}{3},\frac{3}{2}$

(i) $\frac{-13}{7},\frac{39}{-21}$
We have:
(−13)×(−21) = 273

And 7×39=273

(ii)
We have:

3×16=48

And (−8) ×(−6) =48

∴ 3×16 =(−8)×(−6)

(iii)

We have:

9×(−16)= −144

And 4×(-36)= −144

9×(−16) = 4×(−36)

$\frac{9}{4}=\frac{-36}{-16}$
Therefore, they are equivalent rational numbers.

(iv)

We have:

7×60 =420
And 15×(-28)= −420

∴ 7×60 ≠15×(−28)
Therefore, the rational numbers are not equivalent.

(v)

We have:
3 ×4=12
And 12×(−1)= −12

12 ≠ −12
Therefore, the rational numbers are not equivalent.

(vi) $\frac{2}{3},\frac{3}{2}$

We have:

2×2=4

And 3×3=9

2×2≠3×3

Therefore, the rational numbers are not equivalent.

Question 20:

Find x such that:

(i) $\frac{-1}{5}=\frac{8}{x}$
(ii) $\frac{7}{-3}=\frac{x}{6}$
(iii) $\frac{3}{5}=\frac{x}{-25}$
(iv) $\frac{13}{6}=\frac{-65}{x}$
(v) $\frac{16}{x}=-4$
(vi) $\frac{-48}{x}=2$

(i)$\frac{-1}{5}=\frac{8}{x}$

=> −x =5×8
=> x= −40

(ii)$\frac{7}{-3}=\frac{x}{6}$
=> (
−3)x=7×6

=>
x=$\frac{\left(7×6\right)}{\left(-3\right)}$
=>  x=−14

(iii) $\frac{3}{5}=\frac{x}{-25}$
=>    5x=3×(−25)

=>   x=$\frac{3×\left(-25\right)}{5}$
=>x  = (−15)

(iv)$\frac{13}{6}=\frac{-65}{x}$

=> 13x=6×(−65)

=>  x=$\frac{6×\left(-65\right)}{13}$

=>  x= 6×(−​5)

=>  x = −​30

(v)$\frac{16}{x}=-4$
=> $\mathrm{x =}\frac{16}{\left(-4\right)}$
=>  x= (−4)

vi)$\frac{-48}{x}=2$
=>
$\frac{-48}{2}=\frac{x}{1}$
=>$2x=\left(-48\right)×1$
=>$x=\frac{-48}{2}$
x= (−24)

Question 21:

Which of the following rational numbers are equal?

(i) $\frac{8}{-12}\mathrm{and}\frac{-10}{15}$
(ii) $\frac{-3}{9}\mathrm{and}\frac{7}{-21}$
(iii) $\frac{-8}{-14}\mathrm{and}\frac{15}{21}$

(i)

8×15 =120
And ( −10)×(−12)=120

8×15 =(−10) ×(−12)

Therefore, the rational numbers are equal.

ii)

(−3)×(−21) =63
And 7× 9=63

∴ (−3)×(−21) =7×9

Therefore, the rational numbers are equal.

(iii) $\frac{-8}{-14},\frac{15}{21}$

(−8) × 21 = −168
And 15 ×(
−​14) = − ​210

(−8) × 21 ≠ 15 × 14

Therefore, the rational numbers are not equal.

Question 22:

State whether the given statement is true of false:

(i) Zero is the smallest rational number.
(ii) Every integer is a rational number.
(iii) The quotient of two integers is always a rational number.
(iv) Every fraction is a rational number.
(v) Every rational number is a fraction.

(i) False
For example,
1 is smaller than zero and is a rational number.
(ii)True
All integers can be written with the denominator 1.

(iii) False
Though 0 is an integer, when the denominator is 0, it is not a rational number.
For example, $\frac{1}{0}$ is not a rational number.

(iv)True
(v) False
−1 is a rational number but not a fraction.

Question 1:

Represent each of the following rational numbers on the number line:

(i) $\frac{1}{3}$
(ii) $\frac{2}{7}$
(iii) $\frac{7}{3}$
(iv) $\frac{22}{7}$
(v) $\frac{37}{8}$
(vi) $\frac{-1}{3}$
(vii) $\frac{-3}{4}$
(viii) $\frac{-12}{7}$
(ix) $\frac{36}{-5}$
(x) $\frac{-43}{9}$

(i) (ii) (iii) (7/3)=2+(1/3) (iv)

$\frac{22}{7}$ can be written as . So, we need to move to the right of point 3. Then, we need to move $\frac{1}{7}$ distance more to the right. (v) $\frac{37}{8}$ can be written as 4+$\frac{5}{8}$.  So, we need to move to the right of point 4. Then, we need to move $\frac{5}{8}$ distance more to the right. (vi) (vii) (viii)

$\frac{-12}{7}$ can be written as $-1-\frac{5}{7}$. So, we need to move to the left of point -1. Then, we need to move $\frac{5}{7}$ distance more to the left. (ix)

$\frac{36}{-5}$ can be written as $-7-\frac{1}{5}$. So, we need to move to the left of point -7. Then, we need to move $\frac{1}{5}$ distance more to the left. (x) $\frac{-43}{9}$ can be written as $-4-\frac{7}{9}$. So, we need to move to the left of point -4. Then, we need to move $\frac{7}{9}$ distance more to the left. Question 2:

Which of the two rational numbers is greater in each of the following pairs?

(i)
(ii)
(iii)
(iv)
(v)
(vi)

Question 3:

Which of the two rational numbers is greater in each of the following pairs?

(i)
(ii)
(iii)
(iv)
(v)
(vi)   Question 4:

Fill in the blanks with the correct symbol out of >, = and <:

(i) $\frac{-3}{7}......\frac{6}{-13}$
(ii) $\frac{5}{-13}......\frac{-35}{91}$
(iii) $-2......\frac{-13}{5}$
(iv) $\frac{-2}{3}......\frac{5}{-8}$
(v) $0......\frac{-3}{-5}$
(vi) $\frac{-8}{9}......\frac{-9}{10}$   Question 5:

Arrange the following rational numbers in ascending order:

(i) $\frac{2}{5},\frac{7}{10},\frac{8}{15},\frac{13}{30}$
(ii) $\frac{-3}{4},\frac{5}{-12},\frac{-7}{16},\frac{9}{-24}$
(iii) $\frac{-3}{10},\frac{7}{-15},\frac{-11}{20},\frac{17}{-30}$
(iv) $\frac{2}{3},\frac{3}{4},\frac{5}{-6},\frac{-7}{12}$    Question 6:

Arrange the following rational numbers in descending order:

(i) $\frac{-2}{5},\frac{7}{-10},\frac{-11}{15},\frac{19}{-30}$
(ii) $-2,\frac{-13}{6},\frac{8}{-3},\frac{1}{3}$
(iii) $\frac{-4}{9},\frac{5}{-12},\frac{-7}{18},\frac{2}{-3}$
(iv) $\frac{17}{-30},\frac{11}{-15},\frac{-7}{10},\frac{3}{5}$    Question 7:

Which of the following statements are true?

(i) $\frac{-3}{5}$ lies to the left of 0 on the number line.
(ii) $\frac{-12}{7}$ lies to the right of 0 on the number line.
(iii) lie on opposite sides of 0 on the number line.
(iv) $\frac{-18}{-13}$ lies to the left of 0 on the number line.
(v) $\frac{-5}{-8}$ lies on the right of $\frac{-5}{7}$ on the number line.

Question 8:

Find five rational numbers between −3 and −2.

Question 9:

Find five rational numbers between −1 and 1.

Question 10:

Find five rational numbers between $\frac{-3}{5}$ and $\frac{-1}{2}$.

Question 1:

(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)

(i)
$\frac{12}{7}+\frac{3}{7}=\frac{12+3}{7}=\frac{15}{7}$

(ii)
$\frac{-2}{5}+\frac{1}{5}=\frac{-2+1}{5}=\frac{-1}{5}$

(iii)

$\frac{3}{-8}×\frac{-1}{-1}=\frac{-3}{8}$

$\frac{-3}{8}+\frac{1}{8}=\frac{-3+1}{8}=\frac{-2}{8}$

(iv)

$\frac{7}{-11}×\frac{-1}{-1}=\frac{-7}{11}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{-5}{11}+\frac{-7}{11}=\frac{-5+\left(-7\right)}{11}=\frac{-5-7}{11}=\frac{-12}{11}$

(v)

$\frac{-11}{-13}×\frac{-1}{-1}=\frac{11}{13}$

=$\frac{-9}{13}+\frac{11}{13}=\frac{-9+11}{13}=\frac{2}{13}$

(vi)

$\frac{-2}{9}+\frac{-5}{9}=\frac{-2-5}{9}=\frac{-7}{9}$

(vii)

$\frac{\left(-17\right)}{9}+\frac{\left(-11\right)}{9}=\frac{-17-11}{9}=\frac{-28}{9}$

(viii)
$\frac{5}{-7}×\frac{-1}{-1}=\frac{-5}{7}$

$\frac{-3}{7}+\frac{\left(-5\right)}{7}=\frac{-3-5}{7}=\frac{-8}{7}$

Question 2:

(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)

(i)$\frac{-2}{5}+\frac{3}{4}$

(ii)$\frac{-5}{9}+\frac{2}{3}$ (iii)$-4+\frac{1}{2}$

(iv)
$\frac{-7}{27}+\frac{5}{18}$

The denominators of the given rational numbers are 27 and 18. L.C.M. of 27 and 18 is 54.

(v)$\frac{-5}{36}+\left(\frac{-7}{12}\right)$ The denominators of the given rational numbers are 36 and 12.
L.C.M. of 36 and 12 is 36.

(vi)
$\frac{1}{-9}+\left(\frac{4}{-27}\right)$ (vii) (viii)$\frac{27}{-4}+\left(\frac{-15}{8}\right)$ Question 3:

Evaluate:

(i) $\frac{-3}{5}+\frac{7}{5}+\frac{-1}{5}$
(ii) $\frac{-12}{7}+\frac{3}{7}+\frac{-2}{7}$
(iii) $\frac{11}{-12}+\frac{3}{-8}+\frac{1}{4}$
(iv) $\frac{-16}{9}+\frac{-5}{12}+\frac{7}{18}$
(v) $-3+\frac{1}{8}+\frac{-2}{5}$
(vi) $\frac{-13}{8}+\frac{5}{16}+\frac{-1}{4}$

(ii)
$\begin{array}{l}\frac{\begin{array}{l}\\ -12\end{array}}{7}+\frac{3}{7}+\frac{-2}{7}\\ =\frac{\left(-12\right)}{7}+\frac{3}{7}+\frac{\left(-2\right)}{7}\\ =\frac{-12+3-2}{7}\\ =\frac{-14+3}{7}\\ =\frac{-11}{7}\end{array}$   Question 4:

Simplify:

(i) $\frac{-8}{15}+\frac{2}{-3}$
(ii) $\frac{-7}{10}+\frac{13}{-15}+\frac{27}{20}$
(iii) $-1+\frac{7}{-9}+\frac{11}{12}$
(iv) $\frac{-11}{39}+\frac{5}{26}+2$
(v) $2+\frac{-1}{2}+\frac{-3}{4}$
(vi) $\frac{-9}{11}+\frac{2}{3}+\frac{-3}{4}$    $\left(v\right)\phantom{\rule{0ex}{0ex}}2+\frac{-1}{2}+\frac{-3}{4}$
2+12+3  Question 5:

Express each of the following rational numbers as the sum of an integer and a rational number:

(i) $\frac{12}{5}$
(ii) $\frac{-11}{7}$
(iii) $\frac{-25}{9}$
(iv) $\frac{-103}{20}$

Question 1:

(i) 5
(ii) −9
(iii) $\frac{3}{14}$
(iv) $\frac{-11}{15}$
(v) $\frac{15}{-4}$
(vi) $\frac{-18}{-13}$
(vii) 0
(viii) $\frac{1}{-6}$

(i) Additive inverse of 5 is −5.

(ii) Additive inverse of −9 is 9.

(v) Additive inverse of $\frac{15}{-4}=\frac{15×\left(-1\right)}{\left(-4\right)×\left(-1\right)}$

(vi) Additive inverse of $\frac{-18}{-13}=\frac{-18×\left(-1\right)}{\left(-13\right)×\left(-1\right)}$

(vii) Additive inverse of​ 0 is 0.

(viii) Additive inverse of   $\frac{1}{-6}=\frac{1×\left(-1\right)}{\left(-6\right)×\left(-1\right)}$

Subtract:

(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)

(vi)

.

Question 3:

Evaluate:

(i) $\frac{3}{4}-\frac{4}{5}$
(ii) $-3-\frac{4}{7}$
(iii) $\frac{7}{24}-\frac{19}{36}$
(iv) $\frac{14}{15}-\frac{13}{20}$
(v) $\frac{4}{9}-\frac{2}{-3}$
(vi) $\frac{7}{11}-\frac{-4}{-11}$
(vii) $\frac{-5}{14}-\frac{-2}{7}$
(viii) $\frac{-5}{-8}-\frac{-3}{4}$

[L.C.M. of 8 and 4 is 8.]

Question 4:

Subtract the sum of $\frac{-36}{11}$ and $\frac{49}{22}$ from the sum of $\frac{33}{8}$ and $\frac{-19}{4}$.

Question 5:

The sum of two rational numbers is $\frac{4}{21}$. If one of them is $\frac{5}{7}$, find the other.

Question 6:

The sum of two rational numbers is $\frac{-3}{8}$. If one of them is $\frac{3}{16}$, find the other.

Question 7:

The sum of two rational numbers is −3. If one of them is $\frac{-15}{7}$, find the other.

Question 8:

The sum of two rational numbers is $\frac{-4}{3}$. If one of them is −5, find the other.

Question 9:

What should be added to $\frac{-3}{8}$ to get $\frac{5}{12}$?

Let the required number be x.

Question 10:

What should be added to $\frac{-12}{5}$ to get 3?

Let the number that is to be added be x.

Question 11:

What should be added to $\frac{-5}{7}$ to get $\frac{-2}{3}$?

Let the number that is to be added be x.

Question 12:

What should be added to $\frac{2}{9}$ to get −1?

Let the number that is to be added be x.

Question 13:

What should be added to $\left(\frac{-13}{4}+\frac{-3}{8}\right)$ to get 1?

Question 14:

What should be subtracted from $\frac{-3}{4}$ to get $\frac{5}{6}$?

Question 15:

What should be subtracted from $\frac{-2}{3}$ to get $\frac{-5}{6}$?

Question 16:

What should be subtracted from $\frac{-3}{4}$ to get 1?

Question 1:

Multiply:

(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)

$\begin{array}{l}\left(\mathrm{vi}\right)\frac{\overline{)2}{\overline{)5}}^{5}}{-{\overline{)9}}_{3}}×\frac{{\overline{)3}}^{1}}{-\overline{)1}{\overline{)0}}_{2}}=\frac{5}{3}×\frac{1}{2}=\frac{5}{6}\\ \left(\mathrm{vii}\right)\frac{-{\overline{)7}}^{1}}{\overline{)1}{\overline{)0}}_{1}}×\frac{-\overline{)4}{\overline{)0}}^{4}}{\overline{)2}{\overline{)1}}_{3}}=\frac{4}{3}\\ \left(\mathrm{viii}\right)\frac{-\overline{)3}{\overline{)6}}^{12}}{{\overline{)5}}_{1}}×\frac{\overline{)2}{\overline{)0}}^{4}}{-{\overline{)3}}_{1}}=12×4=48\\ \left(\mathrm{ix}\right)\frac{-\overline{)1}{\overline{)3}}^{1}}{\overline{)1}{\overline{)5}}_{3}}×\frac{-\overline{)2}{\overline{)5}}^{5}}{\overline{)2}{\overline{)6}}_{2}}=\frac{-1}{3}×\frac{-5}{2}=\frac{5}{6}\end{array}$

Question 2:

Simplify:

(i) $\frac{3}{20}×\frac{4}{5}$
(ii) $\frac{-7}{30}×\frac{5}{14}$
(iii) $\frac{5}{-18}×\frac{-9}{20}$
(iv) $\frac{-9}{8}×\frac{-16}{3}$
(v) $-32×\frac{-7}{36}$
(vi) $\frac{16}{-21}×\frac{-14}{5}$

Question 3:

Simplify:

(i) $\frac{7}{24}×-48$
(ii) $\frac{-19}{36}×16$
(iii) $\frac{-3}{4}×\frac{4}{3}$
(iv) $-13×\frac{17}{26}$
(v) $\frac{-13}{5}×-10$
(vi) $\frac{-9}{16}×\frac{-64}{27}$

$\begin{array}{l}\text{(i)}\\ \frac{7}{\overline{)2}{\overline{)4}}_{1}}×\left(-\overline{)4}{\overline{)8}}^{2}\right)\\ =7×\left(-2\right)\\ =-14\\ \left(ii\right)\\ \frac{-19}{\overline{)3}{\overline{)6}}_{9}}×\overline{)1}{\overline{)6}}^{4}\\ =\frac{-19}{9}×4\\ =\frac{-76}{9}\\ \\ \left(iii\right)\\ \frac{-{\overline{)3}}^{1}}{{\overline{)4}}_{1}}×\frac{{\overline{)4}}^{1}}{{\overline{)3}}_{1}}\\ =-1\\ \\ \left(iv\right)\\ -13×\frac{17}{26}\\ =\frac{-\overline{)1}{\overline{)3}}^{1}×17}{\overline{)2}{\overline{)6}}^{2}}\\ =\frac{-17}{2}\\ \\ \left(v\right)\\ \frac{-13}{{\overline{)5}}_{1}}×\left(-\overline{)1}{\overline{)0}}^{2}\right)\\ =26\\ \\ \left(vi\right)\\ \frac{\left(-{\overline{)9}}^{1}\right)}{\overline{)1}{\overline{)6}}_{1}}×\frac{\left(-\overline{)6}{\overline{)4}}^{4}\right)}{\overline{)2}{\overline{)7}}_{3}}\\ =\frac{4}{3}\\ \end{array}$

Question 4:

Simplify:

(i) $\left(\frac{13}{8}×\frac{12}{13}\right)+\left(\frac{-4}{9}×\frac{3}{-2}\right)$
(ii) $\left(\frac{16}{15}×\frac{-25}{8}\right)+\left(\frac{-14}{27}×\frac{6}{7}\right)$
(iii) $\left(\frac{6}{55}×\frac{-22}{9}\right)-\left(\frac{26}{125}×\frac{-10}{39}\right)$
(iv) $\left(\frac{-12}{7}×\frac{-14}{27}\right)-\left(\frac{-8}{45}×\frac{9}{16}\right)$

Question 5:

Find the cost of $3\frac{1}{3}$ metres of cloth at Rs $40\frac{1}{2}$ per metre.

Question 6:

A bus is moving at an average speed of $46\frac{2}{3}$ km/h. How much distance will it cover in $2\frac{2}{5}$ huors?

Question 1:

Find the multiplicative inverse of reciprocal of each of the following:

(i) 18
(ii) −16
(iii) $\frac{13}{25}$
(iv) $\frac{-17}{12}$
(v) $\frac{-6}{19}$
(vi) $\frac{-3}{-5}$
(vii) −1
(viii) 0

Question 2:

Simplify:

(i) $\frac{4}{9}÷\left(\frac{-5}{12}\right)$
(ii) $-8÷\left(\frac{-5}{16}\right)$
(iii) $\frac{-12}{7}÷\left(-18\right)$
(iv) $\left(\frac{-1}{10}\right)÷\left(\frac{-8}{5}\right)$
(v) $\left(\frac{-16}{35}\right)÷\left(\frac{-15}{14}\right)$
(vi) $\left(\frac{-65}{14}\right)÷\left(\frac{13}{-7}\right)$

$\begin{array}{l}\text{(i}\right)\frac{4}{9}÷\left(\frac{-5}{12}\right)\\ =\frac{4}{{\overline{)9}}_{3}}×\frac{\overline{)1}{\overline{)2}}^{4}}{\left(-5\right)}\\ =\frac{4×4}{3×\left(-5\right)}\\ =\frac{-16}{15}\\ \text{(ii}\right)-8÷\left(\frac{-5}{16}\right)\\ =-8×\frac{-16}{5}\\ =\frac{128}{5}\\ \text{(iii}\right)\frac{-12}{7}÷\left(-18\right)\\ =\frac{-\overline{)1}{\overline{)2}}^{2}}{7}×\left(\frac{-1}{\overline{)1}{\overline{)8}}_{3}}\right)\\ =\frac{2}{21}\\ \text{(iv}\right)\frac{-1}{10}÷\left(\frac{-8}{5}\right)\\ =\frac{-1}{\overline{)1}{\overline{)0}}^{2}}×\left(\frac{\overline{)5}}{-8}\right)\\ =\frac{-1}{2}×\frac{1}{\left(-8\right)}\\ =\frac{-1}{-16}\\ =\frac{1}{16}\\ \text{(v}\right)\frac{-16}{35}÷\left(\frac{-15}{14}\right)\\ =\frac{-16}{\overline{)3}{\overline{)5}}_{5}}×\frac{\overline{)1}{\overline{)4}}^{2}}{\left(-15\right)}\\ =\frac{-32}{-75}\\ =\frac{32}{75}\\ \text{(vi}\right)\left(\frac{-65}{14}\right)÷\left(\frac{13}{-7}\right)\\ =\left(\frac{-\overline{)6}{\overline{)5}}^{5}}{\overline{)1}{\overline{)4}}_{2}}\right)×\frac{\left(-\overline{)7}\right)}{\overline{)1}\overline{)3}}\\ =\left(\frac{-5}{2}\right)×\left(\frac{-1}{1}\right)\\ =\frac{5}{2}\end{array}$

Question 3:

Fill in the blanks:

(i) (......) ÷ $\left(\frac{-7}{5}\right)=\frac{10}{19}$
(ii) (......) ÷ $\left(-3\right)=\frac{-4}{15}$
(iii) $\frac{9}{8}÷\left(......\right)=\frac{-3}{2}$
(iv) $\left(-12\right)÷\left(......\right)=\frac{-6}{5}$

Question 4:

Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.

Question 5:

By what number should $\frac{-44}{9}$ be divided to get $\frac{-11}{3}$?

Question 6:

By what number should $\frac{-8}{15}$ be multiplied to get 24?

Question 7:

The product of two rational numbers is 10. If one of the numbers is −8, find the other.

Question 8:

The product of two rational numbers is −9. If one of the numbers is −12, find the other.

Question 9:

The product of two rational numbers is $\frac{-16}{9}$. If one of the numbers is $\frac{-4}{3}$, find the other.

Question 10:

By what rational number should $\frac{-8}{39}$ be multiplied to obtain $\frac{5}{26}$?

Question 11:

If 24 pairs of trousers of equal size can be prepared with 54 m of cloth, what length of cloth is required for each pair of trousers?

Question 12:

How many pieces, each of length $3\frac{3}{4}$ m, can be cut from a rope of length 30 m?

Question 13:

The cost of $2\frac{1}{2}$ metres of cloth is Rs $78\frac{3}{4}$. Find the cost of cloth per metre.

Question 1:

Mark (✓) against the correct answer
$\frac{33}{-55}$ in standard form is

(a) $\frac{3}{-5}$
(b) $\frac{-3}{5}$
(c) $\frac{-33}{55}$
(d) none of these Question 2:

Mark (✓) against the correct answer
$\frac{-102}{119}$ in standard form is

(a) $\frac{-4}{7}$
(b) $\frac{-6}{7}$
(c) $\frac{-6}{17}$
(d) none of these Question 3:

Mark (✓) against the correct answer
If $\frac{x}{6}=\frac{7}{-3},$  then the value of x is

(a) −14
(b) 14
(c) 21
(d) −21

Question 4:

What should be added to $\frac{-5}{9}$ to get 1?
(a) $\frac{4}{9}$
(b) $\frac{-4}{9}$
(c) $\frac{14}{9}$
(d) $\frac{-14}{9}$

Question 5:

What should be subtracted from $\frac{-3}{4}$ to get $\frac{5}{6}$?
(a) $\frac{19}{12}$
(b) $\frac{-19}{12}$
(c) $\frac{1}{12}$
(d) $\frac{-1}{12}$

Question 6:

Mark (✓) against the correct answer
Which is smaller out of ?
(a) $\frac{5}{-6}$
(b) $\frac{-7}{12}$
(c) cannot be compared

Question 7:

Mark (✓) against the correct answer
Which is larger out of $\frac{2}{-3}$ and $\frac{-4}{5}$?
(a) $\frac{2}{-3}$
(b) $\frac{-4}{5}$
(c) cannot be compared

Question 8:

Mark (✓) against the correct answer
Reciprocal of −6 is

(a) 6
(b) $\frac{1}{6}$
(c) $\frac{-1}{6}$
(d) none of these

Question 9:

Mark (✓) against the correct answer
Multiplicative inverse of $\frac{-2}{3}$ is
(a) $\frac{2}{3}$
(b) $\frac{-3}{2}$
(c) $\frac{3}{2}$
(d) none of these

Question 10:

Mark (✓) against the correct answer
$-2\frac{1}{9}-6=?$

(a) $-8\frac{1}{9}$
(b) $8\frac{1}{9}$
(c) $4\frac{1}{9}$
(d) $-4\frac{1}{9}$

Question 11:

Mark (✓) against the correct answer
$\frac{-6}{13}-\left(\frac{-7}{15}\right)=?$

(a) $\frac{-181}{195}$
(b) $\frac{181}{195}$
(c) $\frac{1}{195}$
(d) $\frac{-1}{195}$

Question 12:

Mark (✓) against the correct answer
$-2\frac{1}{3}+4\frac{3}{5}=?$

(a) $-2\frac{4}{15}$
(b)  $2\frac{4}{15}$
(c) $-2\frac{1}{5}$
(d) $2\frac{2}{15}$

Question 13:

Mark (✓) against the correct answer
$\frac{2}{3}-1\frac{5}{7}=?$

(a) $1\frac{1}{21}$
(b)  $-1\frac{1}{21}$
(c) $\frac{5}{21}$
(d) $\frac{-5}{21}$

Question 14:

Which is greater between $\frac{-4}{9}$ and $\frac{-5}{12}$?
(a) $\frac{-4}{9}$
(b) $\frac{-5}{12}$
(c) both are equal

The correct option is (b).

Question 15:

Mark (✓) against the correct answer
$\frac{-9}{14}+?=-1$

(a) $\frac{5}{14}$
(b) $\frac{-5}{14}$
(c) $\frac{1}{7}$
(d) $\frac{-1}{7}$

Question 16:

Mark (✓) against the correct answer
$\frac{5}{4}-\frac{7}{6}-\frac{-2}{3}=?$

(a) $\frac{3}{4}$
(b) $\frac{-3}{4}$
(c) $\frac{-7}{12}$
(d) $\frac{7}{12}$

Question 17:

Mark (✓) against the correct answer
$1÷\frac{1}{2}=?$

(a) $\frac{1}{2}$
(b) 2
(c) $2\frac{1}{2}$
(d) $1\frac{1}{2}$

Question 18:

Mark (✓) against the correct answer
$\frac{-3}{14}×?=\frac{5}{12}$

(a) $\frac{-35}{18}$
(b) $\frac{35}{18}$
(c) $\frac{7}{3}$
(d) $\frac{-7}{3}$

Question 19:

Mark (✓) against the correct answer
$0÷\frac{-7}{5}=?$

(a) not defined
(b) $\frac{-5}{7}$
(c) 0
(d) $\frac{5}{7}$

Question 20:

Mark (✓) against the correct answer
$\frac{-3}{8}÷0=?$

(a) $\frac{-3}{8}$
(b) 0
(c) $\frac{-8}{3}$
(d) not defined

Question 1:

Express each of the following rational numbers in standard form:

(i) $\frac{-209}{247}$
(ii) $\frac{-46}{115}$
(iii) $\frac{84}{-147}$   Question 2:

List five rational numbers between −2 and −1.

Question 3:

The sum of two rational numbers is −4. If one of them is $\frac{-11}{6}$, find the other.

Question 4:

What should be added to $\frac{-7}{8}$ to get $\frac{5}{9}$?

Hence , the other number is $\frac{103}{72}$

Question 5:

A car is moving at an average speed of $56\frac{3}{5}$ km per hour. How much distance will it cover in $7\frac{1}{2}$ hours?

Question 6:

By what number should $-4\frac{3}{8}$ be divided to obtain $-3\frac{1}{2}$?

Question 7:

How many pieces, each of length $3\frac{3}{4}$ m, can be cut from a rope of length 45 m?

Question 8:

Find the cost of $3\frac{1}{3}$ m of cloth at Rs $121\frac{1}{2}$ per metre.

Question 9:

Mark (✓) against the correct answer
$\frac{55}{-66}$ in standard form is

(a) $\frac{5}{-6}$
(b) $\frac{-5}{6}$
(c) $\frac{-55}{66}$
(d) none of these Question 10:

Mark (✓) against the correct answer
What should be subtracted from $\frac{-2}{3}$ to get $\frac{3}{4}$?
(a) $\frac{-17}{12}$
(b) $\frac{17}{12}$
(c) $\frac{-12}{17}$
(d) $\frac{12}{17}$

Question 11:

Mark (✓) against the correct answer
The product of two numbers is $\frac{-1}{6}$. If one of them is $\frac{-5}{8}$, the other number is
(a) $\frac{-4}{15}$
(b) $\frac{4}{15}$
(c) $\frac{15}{4}$
(d) $\frac{-15}{4}$

Question 12:

Mark (✓) against the correct answer
The multiplicative inverse of $\frac{-3}{4}$ is
(a) $\frac{3}{4}$
(b) $\frac{4}{3}$
(c) $\frac{-4}{3}$
(d) none of these

Question 13:

Mark (✓) against the correct answer
$\frac{-9}{14}+?=-1$
(a) $\frac{5}{14}$
(b) $\frac{-5}{14}$
(c) $\frac{1}{7}$
(d) $\frac{-1}{7}$

Question 14:

Mark (✓) against the correct answer
$78\frac{3}{4}÷2\frac{1}{2}=?$
(a) $31\frac{1}{2}$
(b) $39\frac{3}{8}$
(c) $40\frac{1}{3}$
(d) none of these

Question 15:

Mark (✓) against the correct answer
Which is smaller between $\frac{-5}{6}$ and $\frac{-7}{12}$?
(a) $\frac{-5}{6}$
(b) $\frac{-7}{12}$
(c) cannot be compared

Question 16:

Fill in the blanks.

(i)$\left(......\right)÷\left(\frac{-7}{5}\right)=\frac{-2}{3}$
(ii) $\left(\frac{-65}{14}\right)÷\left(......\right)=2\frac{1}{2}$
(iii) $\left(\frac{-3}{8}\right)+\left(......\right)=\frac{5}{12}$
(iv) Multiplicative inverse of $-1\frac{3}{4}$is ...... .

Question 17:

Write 'T' for true and 'F' for false for each of the following:

(i) $\frac{-15}{-11}$ lies to the left of 0 on the number line.
(ii) lie on opposite side of 0 on the number line.
(iii) $\frac{-8}{13}$ lies to the left of 0 on the number line.
(iv) $\frac{-4}{5}>\frac{-2}{3}$.
(v) $\frac{-3}{5}$ is the largest among .