Rational Numbers - Class 8 Mathematics - Chapter 1 - Notes, NCERT Solutions & Extra Questions
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Extra Questions - Rational Numbers | NCERT Exemplar | Mathematics | Class 8
Solve the following: Select the rational numbers from the list which are also the integers.
$$\frac{9}{4}, \frac{8}{4}, \frac{7}{4}, \frac{6}{4}, \frac{9}{3}, \frac{8}{3}, \frac{7}{3}, \frac{6}{3}, \frac{5}{2}, \frac{4}{2}, \frac{3}{1}, \frac{3}{2}, \frac{1}{1}, \frac{0}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-3}{2}, \frac{-4}{2}, \frac{-5}{2}, \frac{-6}{2}$$
To solve the given problem, we determine which rational numbers from the list are also integers. A rational number represented as a fraction $$\frac{a}{b}$$ is an integer if the division (a \div b) results in a whole number (i.e., no remainder).
Let's calculate the division for each fraction in the list and check which results are integers:
$$\frac{9}{4} = 2.25$$ (not an integer)
$$\frac{8}{4} = 2$$ (integer)
$$\frac{7}{4} = 1.75$$ (not an integer)
$$\frac{6}{4} = 1.5$$ (not an integer)
$$\frac{9}{3} = 3$$ (integer)
$$\frac{8}{3} \approx 2.67$$ (not an integer)
$$\frac{7}{3} \approx 2.33$$ (not an integer)
$$\frac{6}{3} = 2$$ (integer)
$$\frac{5}{2} = 2.5$$ (not an integer)
$$\frac{4}{2} = 2$$ (integer)
$$\frac{3}{1} = 3$$ (integer)
$$\frac{3}{2} = 1.5$$ (not an integer)
$$\frac{1}{1} = 1$$ (integer)
$$\frac{0}{1} = 0$$ (integer)
$$\frac{-1}{1} = -1$$ (integer)
$$\frac{-2}{1} = -2$$ (integer)
$$\frac{-3}{2} = -1.5$$ (not an integer)
$$\frac{-4}{2} = -2$$ (integer)
$$\frac{-5}{2} = -2.5$$ (not an integer)
$$\frac{-6}{2} = -3$$ (integer)
The rational numbers from the list that are also integers are:
$$ \frac{8}{4}, \frac{9}{3}, \frac{6}{3}, \frac{4}{2}, \frac{3}{1}, \frac{1}{1}, \frac{0}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-4}{2}, \frac{-6}{2} $$
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Verify the property $x \times y=y \times z$ of rational numbers by using
(a) $x=7$ and $y=\frac{1}{2}$
(b) $x=\frac{2}{3}$ and $y=\frac{9}{4}$
(c) $x=\frac{-5}{7}$ and $y=\frac{14}{15}$
(d) $x=\frac{-3}{8}$ and $y=\frac{-4}{9}$
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If 16 shirts of equal size can be made out of $24 \mathrm{~m}$ of cloth, how much cloth is needed for making one shirt?
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Name the property used in each of the following.
(i) $-\frac{7}{11} \times \frac{-3}{5}=\frac{-3}{5} \times \frac{-7}{11}$
(ii) $-\frac{2}{3} \times\left[\frac{3}{4}+\frac{-1}{2}\right]=\left[\frac{-2}{3} \times \frac{3}{4}\right]+\left[\frac{-2}{3} \times \frac{-1}{2}\right]$
(iii) $\frac{1}{3}+\left[\frac{4}{9}+\left(\frac{-4}{3}\right)\right]=\left[\frac{1}{3}+\frac{4}{9}\right]+\left[\frac{-4}{3}\right]$
(iv) $\frac{-2}{7}+0=0+\frac{-2}{7}=-\frac{2}{7}$
(v) $\frac{3}{8} \times 1=1 \times \frac{3}{8}=\frac{3}{8}$
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Find five rational numbers between 0 and 1 .
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Find the product of additive inverse and multiplicative inverse of $-\frac{1}{3}$.
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Roller Coaster at an amusement park is $\frac{2}{3} \mathrm{~m}$ high. If a new roller coaster is built that is $\frac{3}{5}$ times the height of the existing coaster, what will be the height of the new roller coaster?
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Ask Chatterbot AIFill ups - Rational Numbers | NCERT Exemplar | Mathematics | Class 8
The equivalent of $\frac{5}{7}$, whose numerator is 45 is ______________.
To find the equivalent fraction of $ \frac{5}{7} $ with a numerator of 45, we use the property of equivalent fractions where both the numerator and denominator are multiplied by the same number to keep the value of the fraction unchanged.
Given: $$ \frac{5}{7} \quad \text{and numerator} = 45 $$
Let's determine the number that 5 is multiplied by to get 45: $$ 45 = 5 \times k $$ Solving for ( k ): $$ k = \frac{45}{5} = 9 $$
Now, we multiply both the numerator and the denominator of $ \frac{5}{7} $ by 9: $$ \frac{5 \times 9}{7 \times 9} = \frac{45}{63} $$
Thus, the equivalent fraction of $\frac{5}{7} $ with a numerator of 45 is: $$ \frac{45}{63} $$
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The reciprocal of $\frac{2}{5} \times\left(\frac{-4}{9}\right)$ is ______________________.
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Rational numbers can be added or multiplied in any ______________________.
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Ask Chatterbot AIMCQ - Rational Numbers | NCERT Exemplar | Mathematics | Class 8
A number which can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$ is
(a) natural number.
(b) whole number.
(c) integer.
(d) rational number.
A number which can be expressed in the form $$\frac{p}{q}$$ where $ p $ and $ q$ are integers and $ q \neq 0$, is defined as a rational number. The definition includes any case where both $ p $ and $ q $ are integers, which allows for the representation of fractions, whole numbers, and integers, depending on the values of $ p$ and $ q$. For example:
If $ p = 2 $ and $ q = 1 $, then $\frac{p}{q} = 2 $, which is a whole number and an integer.
If $ p = 1 $ and $ q = 2 $, then $\frac{p}{q} = 0.5 $, which is neither a whole number nor an integer but is a rational number.
Therefore, the correct answer to the question is: (d) rational number.
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The additive inverse of $\frac{-7}{19}$ is
(a) $\frac{-7}{19}$
(b) $\frac{7}{19}$
(c) $\frac{19}{7}$
(d) $\frac{-19}{7}$
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The reciprocal of -1 is
(a) 1
(b) -1
(c) 0
(d) Not defined
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Which of the following statements is always true?
(a) $\frac{x-y}{2}$ is a rational number between $x$ and $y$.
(b) $\frac{x+y}{2}$ is a rational number between $x$ and $y$.
(c) $\frac{x \times y}{2}$ is a rational number between $x$ and $y$.
(d) $\frac{x \div y}{2}$ is a rational number between $x$ and $y$.
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Ask Chatterbot AITrue/False - Rational Numbers | NCERT Exemplar | Mathematics | Class 8
If $\frac{x}{y}$ is a rational number, then $y$ is always a whole number. (True/False)
False.
The statement, "If $\frac{x}{y}$ is a rational number, then $y$ is always a whole number," is false. A rational number is defined as a number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ (numerator) is an integer, $q$ (denominator) is a nonzero integer. The denominator $y$ in $\frac{x}{y}$ only needs to be a nonzero integer, not necessarily a whole number. For example, $y$ could be negative, and as long as $x$ is also an integer, the fraction $\frac{x}{y}$ will be rational. Thus, $y$ does not have to be a whole number; it can be any nonzero integer.
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The multiplicative inverse of $\frac{-3}{5}$ is $\frac{5}{3}$. (True/False)
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The negative of 0 does not exist. (True/False)
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-1 is not the reciprocal of any rational number. (True/False)
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0 is a rational number. (True/False)
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The rational numbers $\frac{1}{2}$ and -1 are on the opposite sides of zero on the number line. (True/False)
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Every whole number is a rational number. (True/False)
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