Rational Numbers Test

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Rational Numbers Test - 26

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Weekly Quiz Competition

Question 1
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If $$x + 0 = 0 + x = x,$$ which is rational number, then $$0$$ is called

A
Identity for addition of rational numbers.

B
Additive inverse of x.

C
Multiplicative inverse of x.

D
Reciprocal of x.

Solution

Sum of any rational number and zero is always equal to that rational number.
Hence, $$0$$ is called the identity for addition of rational numbers.

$$\therefore$$ Option A is the correct answer.
Question 2
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Which of the following is not true ?

A
Rational numbers are closed under addition

B
Rational numbers are closed under subtraction

C
Rational numbers are closed under multiplication

D
Rational numbers are closed under division

Solution

Rational numbers are not closed under division.
Here, $$\dfrac{1}{0}$$ is not defined but 1 and 0 are rational numbers.
$$\therefore$$ Option D is the correct answer.
Question 3
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The numerical expression $$\dfrac{3}{8} + \dfrac{(-5)}{7} = \dfrac{-19}{56}$$ shows that

A
Rational numbers are not closed under addition.

B
Rational numbers are closed under addition.

C
Rational numbers are closed under multiplication.

D
Addition of rational numbers is not commutative.

Solution

The sum of any two rational numbers is always a rational number.
Here, $$\dfrac{3}{8}$$ and $$\dfrac{-5}{7}$$ are rational numbers and their sum $$\dfrac{-19}{56}$$ is also a rational number.
$$\therefore$$ Option B is the correct answer.
Question 4
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Rational numbers between $$\displaystyle \frac{3}{8}$$ and $$\displaystyle \frac{7}{12}$$ are

A
$$\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{48}, \frac{7}{12}$$

B
$$\displaystyle \frac{3}{8}, \frac{41}{196}, \frac{23}{48}, \frac{7}{12}$$

C
$$\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{148}, \frac{7}{12}$$

D
none of the above

Solution

A rational number between two numbers $$ a $$ and $$ b = \dfrac {(a +

b)}{2} $$
So, a rational number between $$\dfrac {3}{8} $$ and $$ \dfrac {7}{12}$$
$$ = \dfrac {\dfrac {3}{8} + \dfrac {7}{12}}{2} = \dfrac {23}{48} $$

Now, another rational number between $$ \dfrac {3}{8} $$ and $$ \dfrac {23}{48} $$
$$= \dfrac {\dfrac {3}{8} + \dfrac {23}{48}}{2} = \dfrac {41}{96} $$

Hence, required two rational numbers between $$\dfrac {3}{8} $$ and $$ \dfrac {7}{12} $$ are $$\dfrac {3}{8} ,\dfrac {41}{96}, \dfrac {23}{48}, \dfrac {7}{12}$$
Question 5
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Choose the rational number, which does not lie, between the rational numbers, $$-\dfrac{2}{3}$$ and $$-\dfrac{1}{5}$$

A
$$-\dfrac{3}{10}$$

B
$$\dfrac{3}{10}$$

C
$$-\dfrac{1}{4}$$

D
$$-\dfrac{7}{20}$$

Solution

The given rational numbers $$-\dfrac { 2 }{ 3 }$$ and $$-\dfrac { 1 }{ 5 }$$ are negative rational numbers because the numerator and denominator of both the rational numbers are of opposite signs that is the numerator of both the integers is negative while the denominators are positive.

Therefore, none of the positive rational number can lie between the given negative rational numbers $$-\dfrac { 2 }{ 3 }$$ and $$-\dfrac { 1 }{ 5 }$$.

Hence, $$\dfrac { 3 }{ 10 }$$ does not lie between the rational numbers $$-\dfrac { 2 }{ 3 }$$ and $$-\dfrac { 1 }{ 5 }$$.
Question 6
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________ are rational numbers between $$\displaystyle \frac{1}{3}$$ and $$\displaystyle \frac{1}{4}$$

A
$$\displaystyle \frac{1}{3}, \frac{7}{64}, \frac{13}{48}, \frac{1}{4}$$

B
$$\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{48}, \frac{1}{4}$$

C
$$\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{68}, \frac{1}{4}$$

D
none of the above

Solution

A

rational number between two numbers $$ a $$ and $$ b = \frac {(a +

b)}{2} $$

So, a

rational number between $$\frac {1}{3} $$ and $$ \frac {1}{4} = \frac {(\frac {1}{3} + \frac {1}{4})}{2} = \frac {7}{24} $$

Now, another rational number between $$ \frac {7}{24} $$ and $$ \frac {1}{4} = \frac {(\frac {7}{24} + \frac {1}{4})}{2} = \frac {13}{48} $$

Hence, required two rational numbers between $$\frac {1}{3} $$ and $$ \frac {1}{4} $$ are $$\frac {1}{3} ,\frac {7}{24}, \frac {13}{48}, \frac {1}{4}$$
Question 7
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________ are rational numbers between $$\displaystyle -\dfrac{3}{4}$$ and $$\displaystyle \dfrac{1}{2}.$$

A
$$\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{9}{16}$$

B
$$\dfrac{-15}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$$

C
$$\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$$

D
none of the above

Solution

A rational number between two numbers $$ a $$ and $$ b = \dfrac {(a + b)}{2} $$

So,
a rational number between $$ - \dfrac {3}{4} $$ and $$ \dfrac {1}{2} $$
$$= \dfrac {-\dfrac {3}{4} + \dfrac {1}{2}}{2} = - \dfrac {1}{8} $$

Now,
another rational number between $$ - \dfrac {3}{4} $$ and $$ - \dfrac {1}{8} $$
$$=\dfrac {- \dfrac {3}{4} - \dfrac {1}{8}}{2} = - \dfrac {7}{16} $$

Another rational number between $$ - \dfrac {1}{8} $$ and $$ \dfrac {1}{2} =$$

$$\dfrac { - \dfrac {1}{8} + \dfrac {1}{2}} {2} = \dfrac {3}{16} $$

Hence, required three rational numbers between $$ - \dfrac {3}{4} $$ and $$ \dfrac {1}{2} $$ are $$ - \dfrac {3}{4}, - \dfrac {7}{16}, - \dfrac {1}{8}, \dfrac {3}{16}, \dfrac {1}{2} $$
Question 8
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Choose the rational number which does not lie between rational numbers $$\dfrac{3}{5}$$ and $$\dfrac{2}{3}$$.

A
$$\dfrac{46}{75}$$

B
$$\dfrac{47}{75}$$

C
$$\dfrac{49}{75}$$

D
$$\dfrac{50}{75}$$

Solution

All the options have a denominator of $$75$$. Hence, let us convert $$\dfrac35$$ and $$\dfrac23$$ into equivalent fractions having denominator $$75$$.
$$\dfrac{3}{5} $$ $$=\dfrac{3\times 15}{5\times 15} $$ $$=\dfrac{45}{75}$$

$$\dfrac{2}{3}$$ $$=\dfrac{2\times 25}{3\times 25}$$ $$=\dfrac{50}{75}$$

Hence, $$\dfrac{50}{75}$$ does not lie between the given rational numbers.
Question 9
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__________ are rational numbers between between 5 and -2.

A
$$\displaystyle 5, \frac{33}{4},\ \frac{3}{2}, -\frac{1}{4}, -2$$

B
$$\displaystyle \frac{13}{4},\ \frac{3}{2}, -\frac{1}{4} $$

C
$$\displaystyle 5, \frac{13}{4},\ \frac{13}{2}, -\frac{1}{4}, -2$$

D
none of the above

Solution

A rational number between two numbers $$ a $$ and $$ b = \dfrac {(a + b)}{2} $$

So, a rational number between $$ 5 $$ and $$ - 2 = \dfrac

{( 5 - 2 )}{2} = \dfrac {3}{2} $$

Now, another rational number between $$ 5 $$ and $$ \dfrac {3}{2} =

\dfrac {( 5 + \dfrac {3}{2})}{2} = \dfrac {13}{4} $$

Another rational number between $$ \dfrac {3}{2} $$ and $$ -2 =

\dfrac {( \dfrac {3}{2}) - 2}{2} = -\dfrac {1}{4} $$

$$ \therefore \dfrac {13}{4}, \dfrac {3}{2}, - \dfrac {1}{4} $$ are the rational numbers between $$ 5 $$ and $$ -2 $$
Question 10
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Which property of multiplication is illustrated by : $$\displaystyle {\frac{-4}{3}\, \times\, \left (\frac{-6}{5}\, +\, \frac{8}{7} \right )\, =\, \left (\frac{-4}{3}\, \times\, \frac{-6}{5} \right )\, +\, \left (\frac{-4}{3}\, \times\, \frac{8}{7} \right )}$$

A
Associative property

B
Commutative property

C
Distributive property

D
None of these

Solution

We know that distributive property of multiplication with respect to addition is defined as:

For any real number $$a, b, c$$,

$$a\times (b + c) = a\times b + a\times c$$

Let us take $$a=\dfrac {-4}{3},b=\dfrac {-6}{5}$$ and $$c=\dfrac {8}{7}$$ and apply the distributive property stated above:

$$\dfrac { -4 }{ 3 } \times \left( \dfrac { -6 }{ 5 } +\dfrac { 8 }{ 7 } \right) =\left( \dfrac { -4 }{ 3 } \times \dfrac { -6 }{ 5 } \right) +\left( \dfrac { -4 }{ 3 } \times \dfrac { 8 }{ 7 } \right)$$

Hence, it is the distributive property of multiplication.

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Rational Numbers Test - 26

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Rational Numbers Test - 26

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Question 1

Identity for addition of rational numbers.

Additive inverse of x.

Multiplicative inverse of x.

Reciprocal of x.

Solution

Question 2

Rational numbers are closed under addition

Rational numbers are closed under subtraction

Rational numbers are closed under multiplication

Rational numbers are closed under division

Solution

Question 3

Rational numbers are not closed under addition.

Rational numbers are closed under addition.

Rational numbers are closed under multiplication.

Addition of rational numbers is not commutative.

Solution

Question 4

$$\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{48}, \frac{7}{12}$$

$$\displaystyle \frac{3}{8}, \frac{41}{196}, \frac{23}{48}, \frac{7}{12}$$

$$\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{148}, \frac{7}{12}$$

none of the above

Solution

Question 5

$$-\dfrac{3}{10}$$

$$\dfrac{3}{10}$$

$$-\dfrac{1}{4}$$

$$-\dfrac{7}{20}$$

Solution

Question 6

$$\displaystyle \frac{1}{3}, \frac{7}{64}, \frac{13}{48}, \frac{1}{4}$$

$$\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{48}, \frac{1}{4}$$

$$\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{68}, \frac{1}{4}$$

none of the above

Solution

Question 7

$$\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{9}{16}$$

$$\dfrac{-15}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$$

$$\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$$

none of the above

Solution

Question 8

$$\dfrac{46}{75}$$

$$\dfrac{47}{75}$$

$$\dfrac{49}{75}$$

$$\dfrac{50}{75}$$

Solution

Question 9

$$\displaystyle 5, \frac{33}{4},\ \frac{3}{2}, -\frac{1}{4}, -2$$

$$\displaystyle \frac{13}{4},\ \frac{3}{2}, -\frac{1}{4} $$

$$\displaystyle 5, \frac{13}{4},\ \frac{13}{2}, -\frac{1}{4}, -2$$

none of the above

Solution

Question 10

Associative property

Commutative property

Distributive property

None of these

Solution

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