Rational Numbers

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Questions:

Multiple Choice Question

5 x 1 = 5

Which of the following is the product of \(\frac { 7 }{ 8 } \) and \(\frac { -2 }{ 21 } ?\)

(a)

\(\frac { -1 }{ 12 } \)

(b)

\(\frac { 1 }{ 12 } \)

(c)

\(\frac { -16 }{ 63 } \)

(d)

\(\frac { -147 }{ 16 } \)

Which of the following is the reciprocal of p?

(a)

-p

(b)

(c)

\(\frac { 1 }{ p } \)

(d)

\(\frac { -1 }{ p } \)

\(\frac{x+y}{2}\) is a rational number

(a)

between x and y

(b)

less than x and y both

(c)

greater than x and y both

(d)

less than x but greater than y

Three rational numbers lying between\(\frac{-5}{4}\) and \(\frac{1}{2}\)are

(a)

-1, 0 ,\(\frac{4}{3}\)

(b)

\(\frac{-3}{4},\frac{-1}{2},\frac{1}{4}\)

(c)

\(\frac{-3}{4},\frac{4}{3},\frac{1}{4}\)

(d)

\(\frac{-7}{4},-1,0\)

Which of the following is not true?

(a)

\(\frac{10}{11}+\frac{11}{12}=\frac{11}{12}+\frac{10}{11}\)

(b)

\(\frac{10}{11}\times \frac{11}{12}=\frac{11}{12}\times \frac{10}{11}\)

(c)

\(\frac{10}{11}+ \frac{11}{12}=\frac{11}{12}\div \frac{10}{11}\)

(d)

\(\frac{10}{11}\div \frac{11}{12}=\frac{11}{12}\times \frac{10}{11}\)

True or False

5 x 1 = 5

Zero is a rational number

(a)

False

(b)

True

All the fractions are not rational numbers but all the rational numbers are fractions

(a)

True

(b)

False

All the integers are rational numbers.

(a)

False

(b)

True

All the whole numbers are integers

(a)

False

(b)

True

10.

All the natural numbers are whole numbers.

(a)

False

(b)

True

Generic 3

4 x 3 = 12

11.

Find three rational numbers between -3 and - 4

12.

Find using distributivity. \( \{{7\over5}\times ({-3\over12}) \}+ \{{7\over16}\times {5\over12} \}\)

13.

If a property holds for rational number, will it also hold for integers? For whole numbers? Which will? Which will not?

14.

Using appropriate properties, find \({2\over 3}\times{-5\over 7}+{7\over3}+{2\over3}\times{-2\over7}\)

Fill up / 1 Marks

5 x 1 = 5

15.

______________is the only rational number which is equal to its additive inverse.

16.

The rational number _________is neither positive nor negative.

17.

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Whole numbers	___________	___________	Yes	__________

18.

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Integers	___________	Yes	__________	No

19.

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Rational numbers	Yes	Yes	__________	No

Generic 2

3 x 2 = 6

20.

Multiply the reciprocal of \(\frac { 7 }{ 8 } \) by the reciprocal of \(\frac { -2 }{ 21 } \)

21.

Verify that -(-x) = x,for x = \(-13\over17\)

22.

Verify that -(-x) = x, for x = \(11\over15\)

Generic 5

3 x 5 = 15

23.

Find three rational numbers between \(\frac { 1 }{ 2 } \) and (-2)

24.

Use the distributivity of multiplication of rational numbers over addition to simplify:
\({2\over7}\times[{7\over16}-{21\over4}]\)

25.

Find the value of \({x\over y }+xy\) , using appropriate property and name it \(x={3\over4} \ y=2\)

Match the following

5 x 1 = 5

26.

On a number line the rational number equidistant from a and -1 is:

27.

The product of (-3) and the negative reciprocal of 1^1/2, is:

28.

Additive inverse of -\({1\over13}\)is

29.

Multiplicative inverse of-13 is

30.

Rational numbers are not associative for

1 Marks

3 x 1 = 3

31.

Is there a rational number that is equal to its negative? If yes, write it.

32.

Which rational numbers are their own reciprocals?

33.

Is there any rational number which when multiplied by 0 gives 1?

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Answers Key:

Multiple Choice Question

5 x 1 = 5

(a)
\(\frac { -1 }{ 12 } \)

(a)
between x and y

(b)
\(\frac{-3}{4},\frac{-1}{2},\frac{1}{4}\)

True or False

5 x 1 = 5

(a)
False

(b)
False

(a)
False

(a)
False

10.

(a)
False

Generic 3

4 x 3 = 12

11.

We have
A rational number between -3 and -4
= \(\frac { (-3)+(-4) }{ 2 } =\frac { -7 }{ 2 } \)
A rational number between (-3) and \(\frac { -7 }{ 2 } \)
\(\left[ \left( -3 \right) +\left( \frac { -7 }{ 2 } \right) \right] \div 2=\left[ \frac { -6+(-7) }{ 2 } \right] \times \frac { 1 }{ 2 } \)
= \(\frac { -13 }{ 2 } \times \frac { 1 }{ 2 } =\frac { -13 }{ 4 } \)
A rational number between \(\left( \frac { -7 }{ 2 } \right) \) and (-4)
= \(\left[ \frac { -7 }{ 2 } +(-4) \right] \div 2=\left[ \frac { -7+(-8) }{ 2 } \right] \div 2\)
= \(\frac { -15 }{ 2 } \times \frac { 1 }{ 2 } =\frac { -15 }{ 4 } \)
Thus, the three rational numbers
\(\left( \frac { -7 }{ 2 } \right) ,\left( \frac { -13 }{ 4 } \right) \) and \(\left( \frac { -15 }{ 4 } \right) \) are between (-3) and (-4).

12.

We have, \( \{{7\over5}\times ({-3\over12}) \}+ \{{7\over16}\times {5\over12} \}={7\over5}\times[{-3\over12}+{5\over12}]\)
[by distributivity, taking \({7\over5}\) as common factor]
\(={7\over5}\times[{-3+5 \over 12}]= {7\over5}\times {2\over 12}={7\over30}\)

13.

All properties of operations on rational numbers also hold in case of integers except the following property:
a \(\div\) b is a rational number if b \(\neq\) 0 but a \(\div\)b is not necessarily an integer in case ab \(\in\) J,
All properties of operations on rational numbers also hold in case of whole numbers except the following properties:
(i) If a and b are rational numbers, then (a - b) may or may not be a whole number,
(ii) If a and b are rational numbers, then a\(\div\)b (where b \(\neq\) 0) is not necessarily a whole number,

14.

We have,
\({2\over3}\times({-5\over7})+{7\over3}+{2\over3}\times({-2\over7})={-5\over 7}\times{2\over3}-{2\over7}\times{2\over3}+{7\over3} \\=({-5\over7}-{2\over7})\times {2\over3}+{7\over3}=({-5-2\over7}) \times {2\over3}+{7\over3}={-7\over7}\times{2\over3}+{7\over3} \\-{2\over3}+{7\over3}={5\over3}\)

Fill up / 1 Marks

5 x 1 = 5

15.

()
0

16.

()
0

17.

()

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Whole numbers	yes e.g (0+5=5)Whole numbers	No e.g (5-7=-2)Not a Whole numbers	Yes e.g (3x7=21)Whole numbers	No e.g (5\(\div\)8) Not a Whole numbers

18.

()

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Integers	Yes e.g (-6+5=-1) Integers	Yes(7-5=2) Integers	yes (5x8=40) Integers	No (5\(\div\)8)\(\neq\)Integers

19.

()

Numbers	Closed under
Numbers	Addition	Subtraction	Multiplication	Division
Rational numbers	Yes e.g \(({3\over8}+{(-5)\over7})\\=3\times7+(-5)\times8\over56\\ {21+(-40)\over56}={-19\over56},Rational \ number\)	Yes\(({(-5)\over 7}+{2\over3}) \\= (-5)\times3-2\times7\over21 \\{-15-14\over21}={-29\over21}, \\Rational \ number\)	yes \(e.g ({-4\over5}\times{-6\over11})={24\over55} \\ Rational \ number\)	No \(e.g ({4\over5}\div0) Not \ defined\)

Generic 2

3 x 2 = 6

20.

\(\therefore\) Reciprocal of \(\frac { 7 }{ 8 } \) is \(\frac { 8 }{ 7 } \)
Reciprocal of \(\frac { -2 }{ 21 } \) is \(\frac { -21 }{ 2 } \)
\(\therefore\) \(\left[ Reciprocal\ of\frac { 7 }{ 8 } \right] \times \left[ Reciprocal\ of\left( \frac { -2 }{ 21 } \right) \right] \)
= \(\frac { 8 }{ 7 } \times \left( \frac { -21 }{ 2 } \right) =\frac { 4\times (-3) }{ 1\times 1 } =-12\)

21.

We have, x = \(-13\over17\)
\(LHS =-(-x)=-\{-({-13\over17})\}=-\{{13\over17}\}\)
=-\({13\over17}\) = x = RHS
So, - (-x) = x is verified for x = \({13\over17}\).

22.

We have, x =\(11\over15\)
LHS = -(-x) = -\(({11\over15})={11\over15}=X=RHS\)
S0, - (-x) = x is verified for x =\(11\over15\)

Generic 5

3 x 5 = 15

23.

We have
A rational number between \(\frac { 1 }{ 2 } \)
= \(\left[ \frac { 1 }{ 2 } +(-2) \right] \div 2=\left[ \frac { 1-4 }{ 2 } \right] \div 2\)
= \(\left[ \frac { -3 }{ 2 } \right] \times \frac { 1 }{ 2 } =\frac { -3 }{ 4 } \)
A rational number between \(\frac { 1 }{ 2 } \) and \(\left( \frac { -3 }{ 4 } \right) \)
= \(\left[ \frac { 1 }{ 2 } +\left( \frac { -3 }{ 4 } \right) \right] \div 2\)
\(\left[ \frac { 2-3 }{ 4 } \right] \times \frac { 1 }{ 2 } =\frac { -1 }{ 4 } \times \frac { 1 }{ 2 } =\frac { -1 }{ 8 } \)
A rational number between \(\left( \frac { -3 }{ 4 } \right) \) and (-2)
= \(\left[ \left( \frac { -3 }{ 4 } \right) +(-2) \right] \div 2=\left[ \frac { (-3)+(-8) }{ 4 } \right] \times \frac { 1 }{ 2 } \)
= \(\frac { -11 }{ 4 } \times \frac { 1 }{ 2 } =\frac { -11 }{ 8 } \)
Thus, the three rational numbers
\(\left( \frac { -3 }{ 4 } \right) ,\left( \frac { -1 }{ 8 } \right) \) and \(\left( \frac { -11 }{ 8 } \right) \) are between \(\frac { 1 }{ 2 } \) and (-2)

24.

\(-{11\over8}\)

25.

\({15\over 8}\)

Match the following

5 x 1 = 5

26.

()
\(-\frac { 1 }{ 2 } \)

27.

()
2

28.

()
180

29.

()
-\({1\over13}\)

30.

()
Division

1 Marks

3 x 1 = 3

31.

()
Yes, 0

32.

()
1 and - 1

33.

()
No

8th Standard CBSE Mathematics Unit 1 Rational Numbers Important Question Paper

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Answers Key:

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