Rational Numbers Test

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Rational Numbers Test - 9

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Weekly Quiz Competition

Question 1
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Two rational numbers between $$\dfrac{2}{3}$$ and $$\dfrac{5}{3}$$ are :

A
$$\dfrac{1}{6}$$ and $$\dfrac{2}{6}$$

B
$$\dfrac{1}{2}$$ and $$\dfrac{2}{1}$$

C
$$\dfrac{5}{6}$$ and $$\dfrac{7}{6}$$

D
$$\dfrac{2}{3}$$ and $$\dfrac{4}{3}$$

Solution

Changing the denominators of both numbers to 6, we get
$$\dfrac { 2 }{ 3 } =\dfrac { 4 }{ 6 } \quad \& \quad \dfrac { 5 }{ 3 } =\dfrac { 10 }{ 6 }$$
Numbers between the given rational numbers from the options are
$$\dfrac { 5 }{ 6 } \quad \& \quad \dfrac { 7 }{ 6 } $$
So, correct answer is option C.
Question 2
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Between any two rational numbers,

A
there is no rational number

B
there is exactly one rational number

C
there are infinitely many rational numbers

D
there are only rational numbers and no irrational numbers

Solution

Recall that to find a rational number between $$r$$ and $$s,$$ you can add

$$r$$ and $$s$$ and divide the sum by $$2,$$ that is $$\displaystyle \frac { r+s }{ 2 }$$ lies between r and s.
For example, $$\displaystyle \frac { 5 }{ 2 }$$ is a number between $$2$$ and

$$3.$$ We can proceed in this manner to find many more rational numbers between $$2$$ and $$3.$$
Hence, we can conclude that there are infinitely many rational numbers between any two given rational numbers.
Question 3
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Two rational numbers between $$\dfrac{1}{5}$$ and $$\dfrac{4}{5}$$ are :

A
1 and $$\dfrac{3}{5}$$

B
$$\dfrac{2}{5}$$ and $$\dfrac{3}{5}$$

C
$$\dfrac{1}{2}$$ and $$\dfrac{2}{1}$$

D
$$\dfrac{3}{5}$$ and $$\dfrac{6}{5}$$

Solution

Since the denominator of both rational numbers are same. So, for getting the rational numbers between the given rational numbers, we only have to consider the numerators of the rational numbers.
Two numbers between 1 & 4 are 2 and 3.
So, two rational numbers between the given rational numbers will be $$\dfrac { 2 }{ 5 }$$ and $$ \dfrac { 3 }{ 5 } $$
So, correct answer is option B.
Question 4
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Every rational number is

A
A natural number

B
An integer

C
A real number

D
A whole number

Solution

Real number is a value that represents a quantity along the number line.
Real number includes all rational and irrational numbers.
Rational numbers are numbers which can be represented in the form $$ \dfrac { p }{ q } $$ where, $$ q\neq 0 $$ and $$ p,q $$ are integers.
Therefore, rational number is a subset of real number.

We know that rational and irrational numbers taken together are known as real numbers. Therefore, every real number is either a rational number or an irrational number. Hence, every rational number is a real number. Therefore, (c) is the correct answer.
Question 5
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Which of the following order is correct for rational numbers between $$\displaystyle \frac{4}{11}$$ and $$\displaystyle \frac{9}{16}?$$

A
$$\dfrac{4}{11}$$, $$\dfrac{67}{176}$$, $$\dfrac{79}{176}$$, $$\dfrac{9}{16}$$

B
$$\dfrac{4}{11}$$, $$\dfrac{67}{276}$$, $$\dfrac{79}{176}$$, $$\dfrac{9}{16}$$

C
$$\displaystyle \frac{4}{11}, \frac{17}{38}, \frac{13}{27}, \frac{22}{43}, \frac{9}{16}$$

D
$$\displaystyle \frac{4}{11}, \frac{27}{38}, \frac{13}{27}, \frac{22}{43}, \frac{9}{16}$$

Solution

$$L.C.M$$ of $$11$$ and $$16$$ is $$11\times 16 =176$$

$$\dfrac{4}{11}=\dfrac{4\times 16}{11\times 16}=\dfrac{64}{176}$$

$$\dfrac{9}{16}=\dfrac{9\times 11}{16\times 11}=\dfrac{99}{176}$$

rational numbers between $$\dfrac{4}{11}$$ and $$\dfrac{9}{16}$$ are

$$\dfrac{67}{176}$$, $$\dfrac{79}{176}$$
Question 6
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Find the five rational numbers between $$\displaystyle \frac{1}{2}$$ and $$\displaystyle \frac{3}{2}$$

A
$$0.5 < 0.6 < 0.7 < 0.8 ... < 1.1 < ... < 1.15 < 1.50$$

B
$$0.5 < 0.6 < 1.7 < 3.8 ... < 1.8< ... < 1.15 < 1.50$$

C
$$0.5 < 0.6 < 0.7 < 2.8 ... < 1.1 < ... < 1.15 < 1.50$$

D
$$0.5 < 0.6 < 0.7 < 0.8 ... < 3.1 < ... < 1.15 < 1.50$$
Question 7
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Which of the following numbers are rational ?

A
$$1$$

B
$$ -6$$

C
$$3\dfrac{1}{2}$$

D
All above are rational

Solution

$$\Rightarrow$$ A rational number is a type of real numbers which can be expressed in the form of $$\dfrac pq, $$ where $$ q \neq0.$$
$$\Rightarrow$$ All the numbers are rational as they are in the form of $$\dfrac pq, $$ where $$ q \neq0.$$

i.e, $$\dfrac{1}{1}\ , \dfrac{-6}{1}\ ,\dfrac{7}{2}$$
Question 8
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Find the nine rational numbers between $$0$$ and $$1$$.

A
$$0.1,0.2,0.3, ... ,0.9$$

B
$$ 1.1,0.2 ,10.3, ... ,0.9$$

C
$$0.1,0.2,0.3, ... ,20.9 $$

D
$$0.1 , 0.2 , 10.3 , ... , 0.9 $$

Solution

$$0 < (0+0.1)=0.1 <(0.1+0.1)= 0.2 < (0.2+0.1)=0.3 < ... < (0.8+1)=0.9 <(0.9+0.1)= 1$$
$$0 < 0.1 < 0.2 < 0.3 < ... < 0.9 < 1$$

$$\therefore$$ The nine rational numbers between $$0$$ and $$1$$ are
$$0.1,0.2,0.3, ... ,0.9$$
Question 9
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Compare $$\displaystyle \frac{19}{20}$$ and $$\displaystyle \frac{14}{20}$$

A
$$\displaystyle \frac{19}{20} = \displaystyle \frac{14}{20}$$

B
$$\displaystyle \frac{19}{20} > \displaystyle \frac{14}{20}$$

C
$$\displaystyle \frac{19}{20} \geq \displaystyle \frac{14}{20}$$

D
$$\displaystyle \frac{19}{20} \leq \displaystyle \frac{14}{20}$$

Solution

As $$19\gt14$$
$$\dfrac{19}{20}\gt\dfrac{14}{20}$$

$$\therefore$$ Option B is correct.
Question 10
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Which of the rational number on the number line below is in the wrong place?

A
$$\dfrac {1}{4}$$

B
$$1$$

C
$$\dfrac {1}{2}$$

D
$$\dfrac {3}{4}$$

E
None of the above

Solution

Right order is $$0,\displaystyle \frac{1}{4},\dfrac{1}{3},\frac{1}{2},\frac{3}{4},1$$

as $$0< \dfrac{1}{4} < \dfrac{1}{3} < \dfrac{1}{2} < \dfrac{3}{4} <1$$

Hence, all the given fraction are correctly placed

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Rational Numbers Test - 9

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Rational Numbers Test - 9

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Question 1

$$\dfrac{1}{6}$$ and $$\dfrac{2}{6}$$

$$\dfrac{1}{2}$$ and $$\dfrac{2}{1}$$

$$\dfrac{5}{6}$$ and $$\dfrac{7}{6}$$

$$\dfrac{2}{3}$$ and $$\dfrac{4}{3}$$

Solution

Question 2

there is no rational number

there is exactly one rational number

there are infinitely many rational numbers

there are only rational numbers and no irrational numbers

Solution

Question 3

1 and $$\dfrac{3}{5}$$

$$\dfrac{2}{5}$$ and $$\dfrac{3}{5}$$

$$\dfrac{1}{2}$$ and $$\dfrac{2}{1}$$

$$\dfrac{3}{5}$$ and $$\dfrac{6}{5}$$

Solution

Question 4

A natural number

An integer

A real number

A whole number

Solution

Question 5

$$\dfrac{4}{11}$$, $$\dfrac{67}{176}$$, $$\dfrac{79}{176}$$, $$\dfrac{9}{16}$$

$$\dfrac{4}{11}$$, $$\dfrac{67}{276}$$, $$\dfrac{79}{176}$$, $$\dfrac{9}{16}$$

$$\displaystyle \frac{4}{11}, \frac{17}{38}, \frac{13}{27}, \frac{22}{43}, \frac{9}{16}$$

$$\displaystyle \frac{4}{11}, \frac{27}{38}, \frac{13}{27}, \frac{22}{43}, \frac{9}{16}$$

Solution

Question 6

$$0.5 < 0.6 < 0.7 < 0.8 ... < 1.1 < ... < 1.15 < 1.50$$

$$0.5 < 0.6 < 1.7 < 3.8 ... < 1.8< ... < 1.15 < 1.50$$

$$0.5 < 0.6 < 0.7 < 2.8 ... < 1.1 < ... < 1.15 < 1.50$$

$$0.5 < 0.6 < 0.7 < 0.8 ... < 3.1 < ... < 1.15 < 1.50$$

Question 7

$$1$$

$$ -6$$

$$3\dfrac{1}{2}$$

All above are rational

Solution

Question 8

$$0.1,0.2,0.3, ... ,0.9$$

$$ 1.1,0.2 ,10.3, ... ,0.9$$

$$0.1,0.2,0.3, ... ,20.9 $$

$$0.1 , 0.2 , 10.3 , ... , 0.9 $$

Solution

Question 9

$$\displaystyle \frac{19}{20} = \displaystyle \frac{14}{20}$$

$$\displaystyle \frac{19}{20} > \displaystyle \frac{14}{20}$$

$$\displaystyle \frac{19}{20} \geq \displaystyle \frac{14}{20}$$

$$\displaystyle \frac{19}{20} \leq \displaystyle \frac{14}{20}$$

Solution

Question 10

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

$$1$$

$$\dfrac {1}{2}$$

$$\dfrac {3}{4}$$

None of the above

Solution

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