Rational Numbers Test

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Rational Numbers Test - 10

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Weekly Quiz Competition

Question 1
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$$\displaystyle \frac{5}{8} - \frac{2}{8} =$$ _________

A

B

C

D

Solution

$$\displaystyle \frac{5}{8} - \frac{2}{8} = \frac{5-2}{8} = \frac{3}{8}$$
It was represented by
Question 2
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If we divide a positive integer by another positive integer, what is the resulting number ?

A
Always a natural number

B
Always an integer

C
A rational number

D
An irrational number

Solution

If we divide a positive integer by another positive integer, the resulting number is always a rational number.
Though it can be a natural number and an integer only if the denominator is $$1.$$
Question 3
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A number that can be expressed in the form $$\displaystyle\frac{a}{b}$$, where $$a$$ and $$b$$ are integers and $$b$$ is not equal to zero is called

A
a fraction

B
an integer

C
a rational number

D
a real number

Solution

A rational number : A number that can be in the form of $$\dfrac{p}{q},$$ where $$p$$ and $$q$$ are integers and $$q$$ is not equal to zero.
Question 4
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A rational number between $$\displaystyle \frac{1}{4}$$ and $$\displaystyle \frac{1}{3}$$ is

A
$$\displaystyle \frac{7}{24}$$

B
$$\displaystyle \frac{16}{48}$$

C
$$\displaystyle \frac{13}{48}$$

D
All the above

Solution

$$\dfrac{1}{4} = \dfrac{6}{24} = \dfrac{12}{48} $$

$$\dfrac{1}{3} = \dfrac{8}{24} = \dfrac{16}{48}$$

From this, we can see $$\dfrac{7}{24} , \dfrac{13}{48}, \dfrac{16}{48}$$ all lie between $$\dfrac{1}{4}$$ and $$\dfrac{1}{3}$$
Question 5
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If p: every fraction is a rational number
q: every rational number is a fraction
then which of the following is correct?

A
p is true and q is false.

B
p is false and q is true.

C
Both p and q are true.

D
Both p and q are false.

Solution

Fraction is defined as a part of a whole thing. Every fraction is of the form $$\dfrac{m}{n}$$ where $$m$$ is a whole number and $$n $$ is a natural number.

Rational number is defined as the number of the form $$\dfrac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b\neq 0$$.

Since all whole numbers and natural numbers are present in set of integers every fraction is a rational number. Examples : $$\dfrac{4}{5}, \dfrac{4}{7}$$

So, the statement $$p$$ is true.

But every rational number is not a fraction. Examples : $$\dfrac{3}{-4}, \dfrac{17}{-16}$$. Here the denominators are not natural numbers and hence they are not fractions.

So, the statement $$q$$ is false.
Question 6
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$$\displaystyle \frac{-2}{-19}$$ is a

A
Positive rational number.

B
Negative rational number.

C
Either positive or negative number.

D
Has neither a positive numerator nor a negative number

Solution

$$\because$$ Both numerator and denominator are negative (i.e., same sign)
Question 7
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A rational number equivalent to $$\displaystyle \frac{-5}{-3}$$ is

A
$$\displaystyle \frac{-25}{15}$$

B
$$\displaystyle \frac{25}{-15}$$

C
$$\displaystyle \frac{25}{15}$$

D
None of these

Solution

$$\because\, \displaystyle {\frac{-5}{-3}\, =\, \frac{-5}{-3}\, \times\, \frac{5}{5}\, =\, \frac{25}{15}}$$
Question 8
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Which of the following statement is false?

A
Every fraction is a rational number

B
Every rational number is a fraction

C
Every integer is a rational number

D
All the above

Solution

$${\textbf{Step 1: Consider, option A.}}$$

$${\text{Every fraction is a rational number}}.$$

$${\text{As we know that,}}$$ $${\text{ A rational number is a type of real numbers, }}$$
$${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$
$${\text{Any fraction with non - zero denominators is rational number}}{\text{.}}$$

$${\text{Thus, option A}}{\text{. is true}}{\text{.}}$$

$${\textbf{Step 2: Consider, option B}}{\textbf{.}}$$

$${\text{Every rational number is a fraction}}.$$

$${\text{As we know that,}}$$ $${\text{ A rational number is a type of real numbers, }}$$
$${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$

$${\text{The given statement is not true as 10 is a rational number but it is not a fraction}}{\text{.}}$$

$${\text{Thus, option B}}{\text{. is false}}{\text{.}}$$

$${\textbf{Step 3: Consider, option C}}{\textbf{.}}$$

$${\text{Every integer is a rational number}}.$$

$${\text{A rational number is a type of real numbers, }}$$
$${\text{which is in the form of }}\dfrac{p}{q}{\text{ where }}q{\text{ is not equal to zero}}{\text{. }}$$

$${\text{And we know that an integer can be represented as }}\dfrac{p}{q}$$
$${\text{form where denominator will be always 1}}{\text{.}}$$

$${\text{For example: }}10 = \dfrac{{10}}{1}$$

$${\text{Thus, option C}}{\text{. is true}}{\text{.}}$$

$${\textbf{Hence, option B is correct answer.}}$$
Question 9
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Which one of the following is the rational number lying between $$\displaystyle \frac{6}{7} \ and \ \frac{7}{8}?$$

A
$$\displaystyle \frac{3}{4}$$

B
$$\displaystyle \frac{99}{122}$$

C
$$\displaystyle \frac{95}{112}$$

D
$$\displaystyle \frac{97}{112}$$

Solution

Required rational number $$\displaystyle =\frac{1}{2}\left ( \frac{6}{7}+\frac{7}{8} \right )=\frac{1}{2}\left ( \frac{48+49}{56} \right )=\frac{97}{112}$$
Hence option (d) is correct
Question 10
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Which of the following is a rational number (s)?

A
$$\displaystyle \frac{-2}{9}$$

B
$$\displaystyle \frac{4}{-7}$$

C
$$\displaystyle \frac{-3}{17}$$

D
All the three given numbers

Solution

A rational number is a number that can be expressed as a fraction $$\dfrac{p}{q} $$ where $$p$$ and $$q$$ are integers and $$q\neq0$$.
A rational number $$\dfrac{p}{q}$$ is said to have numerator $$p$$ and denominator $$q.$$

In the given options, both numerator and denominator are integers.
Hence, all the three given numbers are rational number.

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Rational Numbers Test - 10

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Rational Numbers Test - 10

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Question 1

Solution

Question 2

Always a natural number

Always an integer

A rational number

An irrational number

Solution

Question 3

a fraction

an integer

a rational number

a real number

Solution

Question 4

$$\displaystyle \frac{7}{24}$$

$$\displaystyle \frac{16}{48}$$

$$\displaystyle \frac{13}{48}$$

All the above

Solution

Question 5

p is true and q is false.

p is false and q is true.

Both p and q are true.

Both p and q are false.

Solution

Question 6

Positive rational number.

Negative rational number.

Either positive or negative number.

Has neither a positive numerator nor a negative number

Solution

Question 7

$$\displaystyle \frac{-25}{15}$$

$$\displaystyle \frac{25}{-15}$$

$$\displaystyle \frac{25}{15}$$

None of these

Solution

Question 8

Every fraction is a rational number

Every rational number is a fraction

Every integer is a rational number

All the above

Solution

Question 9

$$\displaystyle \frac{3}{4}$$

$$\displaystyle \frac{99}{122}$$

$$\displaystyle \frac{95}{112}$$

$$\displaystyle \frac{97}{112}$$

Solution

Question 10

$$\displaystyle \frac{-2}{9}$$

$$\displaystyle \frac{4}{-7}$$

$$\displaystyle \frac{-3}{17}$$

All the three given numbers

Solution

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