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 multiplicative identity for rational numbers?

(a)

(b)

-1

(c)

(d)

None of these

Which of the following is the identity element?

(a)

(b)

-1

(c)

(d)

None of these

The reciprocal of \({-3\over8 } \times {-24\over 13}\) is

(a)

\(9\over 13\)

(b)

\(-9\over 13\)

(c)

\(-13 \over 9\)

(d)

\(13\over9\)

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)

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

Fill up / 1 Marks

5 x 1 = 5

The reciprocal of the reciprocal of a number is __________.

The rational number ______________has no reciprocal.

\({1\over15}\times[{27\over31}+{32\over37}]=[{1\over15}\times{27\over31}]+\) ______________________

Numbers	Associative for
Numbers	Addition	Subtraction	Multiplication	Division
Whole numbers	yes	________	_________	________

10.

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

Generic 3

5 x 3 = 15

11.

\((-\frac{2}{3})+\frac{2}{3}=0\).How?

12.

Take some more rational numbers, add them as above and see if the two sums are equal.

13.

Find three rational numbers between -3 and - 4

14.

Write any five non-negative rational numbers less than 1

15.

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

Generic 2

5 x 2 = 10

16.

The product of two rational numbers is \(\frac { -28 }{ 75 } \) if one of the numbers is \(\frac { 14 }{ 25 } \) find the other

17.

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

18.

Simplify : \({7\over8}+{1\over16}-{1\over12}\)

19.

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

20.

Write the additive inverse of the following \(19\over-6\)

Generic 5

3 x 5 = 15

21.

Recall the associativity of the four operations for whole numbers through this table:

Operation	Numbers	Remarks
Addition	_________	Addition is associative.
Subtraction	_________	Subtraction is not associative.
Multiplication	Is \(7 \times (2 \times 5)=(7 \times 2) \times 5 ?\) Is \(4 \times (6 \times 0)=(4 \times 6) \times 0?\) For any three whole numbers a, b and c \(a \times (b \times c)=(a \times b) \times c?\)	Multiplication is associative.
Division	_________	Division is not associative.

Fill in this table and verify the remarks given in the last column.
Check for yourself the associativity of different operations for natural numbers.

22.

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

23.

Rearrange suitably and find the sum in each of the following:
\({2\over 3}+{9\over2}+{7\over4}+{-6\over3}+{-3\over2}\)

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Answers Key:

Multiple Choice Question

5 x 1 = 5

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

(a)
1

(d)
\(13\over9\)

Fill up / 1 Marks

5 x 1 = 5

()
The number itself

()
0

()
\([{1\over15}\times{32\over37}]\)[using distributive property over addition]

()

Numbers	Associative for
Numbers	Addition	Subtraction	Multiplication	Division
Whole numbers	Yes, e.g. 2 +(0+5) =(2 +0)+5 \(\Rightarrow\)7=7, which is true	\(No \ e.g \\ 2-(0-5)\\ \neq (2-0)-5\\ \Rightarrow 7 \neq -3\) which is not true	Yes, e.g.2 x[3 x (-5)] =(2 x3)x(-5) \(\Rightarrow\)2 x[-15]=(6)x(-5) \(\Rightarrow\)-30=- 30, which is true.	No eg \((2\div 3)\div 5\neq 2 \div (3\div5) \\ \Rightarrow ({2\over 3})\div 5\neq2 \div ({3\over 5}) \\ \Rightarrow {2\over 3}\times {1\over 5} \neq 2 \times {5\over 3}\\ \Rightarrow {2\over 15} \neq {10\over 3}\) which is not true

10.

()

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

Generic 3

5 x 3 = 15

11.

\((-\frac{2}{3})+\frac{2}{3}= \frac{(-2)+2}{3}=0\)
Note 1. The negative of a rational number is also called its 'additive inverse'.
Note 2. The number 0 is the negative (additive inverse) of itself.
Note 3. 0 is the only rational number which itself is its own negative.
Note 4. The negative of the negative of a rational number is that rational number itself.

12.

Example 1. We have,
\(\frac{-3}{4}+[\frac{2}{3}+(\frac{-6}{7})]=\frac{-3}{4}+(\frac{-4}{21})=\frac{-79}{84}\)
\([\frac{-3}{4}+\frac{2}{3}]+(\frac{-6}{7})=\frac{-1}{12}+(\frac{-6}{7})=\frac{-79}{84}\)
So, Yes; \(\frac{-3}{4}+[\frac{2}{3}+(\frac{-6}{7})]=[\frac{-3}{4}+\frac{2}{3}]+(\frac{-6}{7})\).
Example 2.
\(\frac{-1}{4}+[\frac{2}{9}+(\frac{-5}{11})] = \frac{-1}{4}+(\frac{-23}{99})=\frac{-191}{396}\)
\([\frac{-1}{4}+\frac{2}{9}]+(\frac{-5}{11})=\frac{-1}{36}+(\frac{-5}{11})=\frac{-191}{396}\)
So, Yes; \(\frac{-1}{4}+[\frac{2}{9}+(\frac{-5}{11})] = [\frac{-1}{4}+\frac{2}{9}]+(\frac{-5}{11})\)

13.

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).

14.

\(0,{1\over5},{2\over5},{3\over5},{4\over5}\)

15.

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]
\(={7\over5}\times[{-3+5 \over 12}]= {7\over5}\times {2\over 12}={7\over30}\)

Generic 2

5 x 2 = 10

16.

\(\therefore\) product of two rational numbers = \(\frac { -28 }{ 75 } \)
Any one of the ratioanl numbers = \(\frac { 14 }{ 25 } \)
\(\therefore\) the other number = \(\left[ \frac { -28 }{ 75 } \right] \div \frac { 14 }{ 25 } \)
= \(\frac { -28 }{ 75 } \times \frac { 25 }{ 14 } =\frac { -2\times 1 }{ 3\times 1 } =\frac { -2 }{ 3 } \)
Thus, the required rational number is \(\left( \frac { -2 }{ 3 } \right) \)

17.

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

18.

\(41\over48\)

19.

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

20.

we have, \(19\over-6\) so, additive inverse of \(19\over-6\) is \(19\over6\)

Generic 5

3 x 5 = 15

21.

Operation	Numbers	Remarks
Addition	0 + (2 + 6) = (0 + 2) + 6 = 8 ; 3 + (0 + 5) = (3 + 0) + 5 = 8. For any three whole numbers a, b and c, a + (b + c) = (a + b) + c.	Addition is associative.
Subtraction	(0 - 2) - 6 = - 2 - 6 = - 8 ; 0- (2 - 6) = 0 - (- 4) = 0 + 4 = 4 So, (0 - 2) - 6 ≠ 0 - (2 - 6).	Subtraction is not associative.
Multiplication	Is \(7 \times (2 \times 5)=(7 \times 2) \times 5 ?\) Yes ; \(7 \times (2 \times 5)=(7 \times 2) \times 5 = 70\). Is \(4 \times (6 \times 0)=(4 \times 6) \times 0?\) Yes ; \(4 \times (6 \times 0)=(4 \times 6) \times 0=0\). For any three whole numbers a, b and c \(a \times (b \times c)=(a \times b) \times c\)	Multiplication is associative.
Division	\(2 \div (6\div 3)=2 \div 2 =1\) \((2\div 6)\div 3=\frac{2}{6} \div 3= \frac{1}{3}\div 3 = \frac{1}{3} \times \frac{1}{3}= \frac{1}{9}\) \(\because 1 \ne \frac{1}{9}\) \(\therefore 2 \div (6 \div 3) \ne (2\div 6) \div 3\).	Division is not associative.

Check for the associativity of different operations for natural numbers.

Operation	Numbers	Remarks
Addition	1 + (2 + 3) = (1 + 2) + 3 = 6; 2 + (5 + 7) = (2 + 5) + 7 = 14. For any three whole numbers a, b and c, a + (b + c) = (a + b) + c.	Addition is associative.
Subtraction	(4 - 6) - 5 = - 2 - 5 = - 7 and 4 - (6 - 5) = 4 - 1 = 3 So, (4 - 6) - 5 ≠ 4 - (6 - 5)	Subtraction is not associative.
Multiplication	\(2 \times (3 \times 4)=(2 \times 3) \times 4 = 24\); \(3 \times (5 \times 7)=(3 \times 5) \times 7=105\). Yes ; \(4 \times (6 \times 0)=(4 \times 6) \times 0=0\). For any three whole numbers a, b and c \(a \times (b \times c)=(a \times b) \times c\)	Multiplication is associative.
Division	\(9 \div (6\div 2)=9 \div 3 =3\) and \((9\div 6)\div 2=\frac{9}{6} \div 2= \frac{3}{2}\div 2 = \frac{3}{2} \times \frac{1}{2}= \frac{3}{4}\) .So, \(9 \div (6\div 2)=(9 \div 6) \div 2\)	Division is not associative.

22.

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)