Math Ncert Exemplar 2019 Solutions for Class 7 Maths Chapter 8 - Rationals Numbers

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Page No 242:

Question 1:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$ , where p and q are integers and
(a) q = 0
(b) q = 1
(c) q ≠ 1
(d) q ≠ 0

Answer:

By definition, a number that can be expressed in the form of $\frac{p}{q}$ , where p and q are integers and q ≠ 0, is called a rational number.

Hence, the correct answer is option (d).

Page No 243:

Question 2:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
Which of the following rational numbers is positive?

(a) $\frac{- 8}{7}$

(b) $\frac{19}{- 13}$

(c) $\frac{- 3}{- 4}$

(d) $\frac{- 21}{13}$

Answer:

We know that, when the numerator and denominator of a rational number, both are negative, it is a positive rational numbers.

$\frac{- 3}{- 4} = \frac{3}{4}$

Hence, the correct answer is option (c).

Page No 243:

Question 3:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
Which of the following rational numbers is negative?
(a) $- (\frac{- 3}{7})$

(b) $\frac{- 5}{- 8}$

(c) $\frac{9}{8}$

(d) $\frac{3}{- 7}$

Answer:

(a) $- (\frac{- 3}{7}) = \frac{3}{7}$

(b) $\frac{- 5}{- 8} = \frac{5}{8}$

(c) $\frac{9}{8}$

(d) $\frac{3}{- 7} = - \frac{3}{7}$

Hence, the correct answer is option (d).

Page No 243:

Question 4:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
In the standard form of a rational number, the common factor of numerator and denominator is always:
(a) 0
(b) 1
(c) – 2
(d) 2

Answer:

By definition, in the standard form of a rational number, the common factor of numerator and denominator is always1

Hence, the correct answer is option (b).

Page No 243:

Question 5:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
Which of the following rational numbers is equal to its reciprocal?
(a) 1
(b) 2
(c) $\frac{1}{2}$
(d) 0

Answer:

(a) $Reciprocal of 1 = \frac{1}{1} = 1$
(b) $Reciprocal of 2 = \frac{1}{2}$
(c) $Reciprocal of \frac{1}{2} = \frac{1}{\frac{1}{2}} = 2$
(d) $Reciprocal of 0 = \frac{1}{0}$

Hence, the correct answer is option (a).

Page No 243:

Question 6:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
The reciproal of $\frac{1}{2}$ is
(a) 3
(b) 2
(c) – 1
(d) 0

Answer:

Reciproal of $\frac{1}{2}$ $= \frac{1}{\frac{1}{2}} = 2$

Hence, the correct answer is option (b).

Page No 243:

Question 7:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
The standard form of $\frac{- 48}{60} is$
(a) $\frac{48}{60}$

(b) $\frac{- 60}{48}$

(c) $\frac{- 4}{5}$

(d) $\frac{- 4}{- 5}$

Answer:

$\frac{- 48}{60} = \frac{- 4 \times 12}{5 \times 12} = - \frac{4}{5}$

Hence, the correct answer is option (c).

Page No 244:

Question 8:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
Which of the following is equivalent to $\frac{4}{5}$ ?
(a) $\frac{5}{4}$

(b) $\frac{16}{25}$

(c) $\frac{16}{20}$

(d) $\frac{15}{25}$

Answer:

$\frac{16}{20} = \frac{4 \times 4}{4 \times 5} = \frac{4}{5}$

Hence, the correct answer is option (c).

Page No 244:

Question 9:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
How many rational numbers are there between two rational numbers?
(a) 1
(b) 0
(c) unlimited
(d) 100

Answer:

There are unlimited numbers between two rational numbers.

Hence, the correct answer is option (c).

Page No 244:

Question 10:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
In the standard form of a rational number, the denominator is always a
(a) 0
(b) negative integer
(c) positive integer
(d) 1

Answer:

By definition, a rational number is said to be in the standard form, if its denominator is a positive integer.

Hence, the correct answer is option (c).

Page No 244:

Question 11:

In the given question, there are four options, out of which, only one is correct. Write the correct one.
To reduce a rational number to its standard form, we divide its numerator and denominator by their
(a) LCM
(b) HCF
(c) product
(d) multiple

Answer:

To reduce a rational number to its standard form, we divide its numerator and denominator by their HCF.

Hence, the correct answer is option (b).

Page No 244:

Question 12:

Which is greater number in the following:
(a) $\frac{- 1}{2}$

(b) 0

(c) $\frac{1}{2}$

(d) –2

Answer:

$\frac{1}{2}$ is the rightmost number on the number line.

Hence, the correct answer is option (c).

Page No 244:

Question 13:

Fill in the blanks to make the statement true.
$- \frac{3}{8}$ is a ______ rational number.

Answer:

Because its numerator is a negative integer.
So, $- \frac{3}{8}$ is a negative rational number.

Page No 244:

Question 14:

Fill in the blanks to make the statement true.
1 is a ______ rational number.

Answer:

The given rational number 1 is positive number, because its numerator and denominator are positive integer.
So,, 1 is a positive rational number.

Page No 245:

Question 15:

Fill in the blanks to make the statement true.
The standard form of $\frac{- 8}{- 36}$ is ______.

Answer:

$\frac{- 8}{- 36} = \frac{- 2 \times 4}{- 9 \times 4} = \frac{2}{9}$

So, the standard form of $\frac{- 8}{- 36}$ is $\frac{2}{9}$ .

Page No 245:

Question 16:

Fill in the blanks to make the statement true.
The standard form of $\frac{18}{- 24}$ is ______.

Answer:

$\frac{18}{- 24} = - \frac{3 \times 6}{4 \times 6} = - \frac{3}{4}$
Hence, the standard form of $\frac{18}{- 24}$ is $- \frac{3}{4}$ .

Page No 245:

Question 17:

Fill in the blanks to make the statement true.
On a number line, $\frac{- 1}{2}$ is to the ______ of zero (0).

Answer:

On a number line, $\frac{- 1}{2}$ is to the left of zero (0).

Page No 245:

Question 18:

Fill in the blanks to make the statement true.
On a number line, $\frac{4}{3}$ is to the ______ of zero (0).

Answer:

On a number line, $\frac{4}{3}$ is to the right of zero (0).

Page No 245:

Question 19:

Fill in the blanks to make the statement true.
$- \frac{1}{2}$ is ______ than $\frac{1}{5} .$

Answer:

$- \frac{1}{2}$ is a negative rational number and $\frac{1}{5}$ is a positive rational number.
So, $- \frac{1}{2}$ is less than $\frac{1}{5} .$

Page No 245:

Question 20:

Fill in the blanks to make the statement true.
$- \frac{3}{5}$ is ______ than 0.

Answer:

$- \frac{3}{5}$ is a negative rational number and left to the zero.
So, $- \frac{3}{5}$ is less than 0

Page No 245:

Question 21:

Fill in the blanks to make the statement true.
$\frac{- 16}{24} and \frac{20}{- 16}$ represent ______ rational numbers.

Answer:

$\frac{- 16}{24} = \frac{- 2 \times 8}{3 \times 8} = - \frac{2}{3}$

$\frac{20}{- 16} = \frac{4 \times 5}{- 4 \times 4} = - \frac{5}{4} - \frac{2}{3} \neq - \frac{5}{4}$

So, $\frac{- 16}{24} and \frac{20}{- 16}$ represent different rational numbers.

Page No 245:

Question 22:

Fill in the blanks to make the statement true.
$\frac{- 27}{45} and \frac{- 3}{5}$ represent ______ rational numbers.

Answer:

$\frac{- 27}{45} = \frac{- 3 \times 9}{5 \times 9} = - \frac{3}{5}$

So, $\frac{- 27}{45} and \frac{- 3}{5}$ represent same rational numbers.

Page No 245:

Question 23:

Fill in the blanks to make the statement true.
Additive inverse of $\frac{2}{3}$ is ______.

Answer:

Since, additive inverse is the negative of a number.

So, the additive inverse of $\frac{2}{3}$ is $- \frac{2}{3}$ .

Page No 245:

Question 24:

Fill in the blanks to make the statement true.
$\frac{- 3}{5} + \frac{2}{5} =__________.$

Answer:

$\frac{- 3}{5} + \frac{2}{5} = \frac{- 3 + 2}{5} = - \frac{1}{5}$

Page No 245:

Question 25:

Fill in the blanks to make the statement true.
$\frac{- 5}{6} + \frac{- 1}{6} =______________.$

Answer:

$\frac{- 5}{6} + \frac{- 1}{6} = \frac{- 5 - 1}{6} = \frac{- 6}{6} = - 1$

Page No 245:

Question 26:

Fill in the blanks to make the statement true.
$\frac{3}{4} \times (\frac{- 2}{3}) =_______.$

Answer:

$\frac{3}{4} \times (\frac{- 2}{3}) = \frac{3 \times (- 2)}{4 \times 3} = - \frac{6}{12} = \frac{- 1}{2}$

Page No 245:

Question 27:

Fill in the blanks to make the statement true.
$\frac{- 5}{3} \times (\frac{- 3}{5}) =_____________.$

Answer:

$\frac{- 5}{3} \times (\frac{- 3}{5}) = \frac{(- 5) \times (- 3)}{3 \times 5} = \frac{15}{15} = 1$

Page No 245:

Question 28:

Fill in the blanks to make the statement true.
$\frac{- 6}{7} = \frac{__}{42}$

Answer:

$\frac{- 6}{7} = \frac{__}{42} 42 = 7 \times 6 ∵ \frac{- 6 \times 6}{7 \times 6} = \frac{- 36}{42}$

Page No 245:

Question 29:

Fill in the blanks to make the statement true.
$\frac{1}{2} = \frac{6}{__}$

Answer:

ans

Page No 245:

Question 30:

Fill in the blanks to make the statement true.
$\frac{- 2}{9} - \frac{7}{9} =______________.$

Answer:

$\frac{- 2}{9} - \frac{7}{9} = \frac{- 2 - 7}{9} = \frac{- 9}{9} = - 1 ∵ \frac{- 2}{9} - \frac{7}{9} = - 1$

Page No 246:

Question 31:

Fill in the blanks to make the statement true.
$\frac{7}{- 8} \frac{8}{9}$

Answer:

$\frac{7}{- 8}$ is a negative rational number and $\frac{8}{9}$ is a positive rational number.

So, $\frac{7}{- 8} < \frac{8}{9}$

Page No 246:

Question 32:

Fill in the blanks to make the statement true.
$\frac{3}{7} \frac{- 5}{6}$

Answer:

$\frac{- 5}{6}$ is a negative rational number and $\frac{3}{7}$ is a positive rational number.

So, $\frac{3}{7} > \frac{- 5}{6}$ .

Page No 246:

Question 33:

Fill in the blanks to make the statement true.
$\frac{5}{6} \frac{8}{4}$

Answer:

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \frac{8}{4} = \frac{8 \times 3}{4 \times 3} = \frac{24}{12} 24 > 10 ∴ \frac{24}{12} > \frac{10}{12} \Rightarrow \frac{5}{6} > \frac{8}{4}$

Page No 246:

Question 34:

Fill in the blanks to make the statement true.
$\frac{- 9}{7} \frac{4}{- 7}$

Answer:

$\frac{4}{- 7} = \frac{- 4}{7} - 9 < - 4 ∴ \frac{- 9}{7} < \frac{- 4}{7}$

Page No 246:

Question 35:

Fill in the blanks to make the statement true.
$\frac{8}{8} \frac{2}{2}$

Answer:

$\frac{8}{8} = 1 \frac{2}{2} = 1 ∴ \frac{8}{8} = \frac{2}{2}$

Page No 246:

Question 36:

Fill in the blanks to make the statement true.
The reciprocal of ______ does not exist.

Answer:

The reciprocal of zero does not exist, as the reciprocal of 0 is $\frac{1}{0}$ , which is not defined.

Page No 246:

Question 37:

Fill in the blanks to make the statement true.
The reciprocal of 1 is ______.

Answer:

The reciprocal of 1 is 1.

Page No 246:

Question 38:

Fill in the blanks to make the statement true.
$\frac{- 3}{7} \div (\frac{- 7}{3}) =______________.$

Answer:

$\frac{- 3}{7} \div (\frac{- 7}{3}) = \frac{- 3}{7} \times \frac{3}{- 7} = \frac{9}{49}$

Hence, $\frac{- 3}{7} \div (\frac{- 7}{3}) = \frac{9}{49}$ .

Page No 246:

Question 39:

Fill in the blanks to make the statement true.
$0 \div (\frac{- 5}{6}) =____________.$

Answer:

$0 \div (\frac{- 5}{6}) = 0$

Because 0 divided by any number is 0.

Page No 246:

Question 40:

Fill in the blanks to make the statement true.
$0 \times (\frac{- 5}{6}) =____________.$

Answer:

$0 \times (\frac{- 5}{6}) = 0$

Because 0 multiplied by any number is 0.

Page No 246:

Question 41:

Fill in the blanks to make the statement true.
$________\times (\frac{- 2}{5}) = 1 .$

Answer:

ans

Page No 246:

Question 42:

Fill in the blanks to make the statement true.
The standard form of rational number –1 is ______.

Answer:

The standard form of rational number –1 is –1.

Page No 246:

Question 43:

Fill in the blanks to make the statement true.
If m is a common divisor of a and b, then $\frac{a}{q} = \frac{a \div m}{___}$ .

Answer:

If m is a common divisor of a and b, then $\frac{a}{b} = \frac{a + m}{b + m}$ .

Page No 246:

Question 44:

Fill in the blanks to make the statement true.
If p and q are positive integers, then $\frac{p}{q}$ is a ______ rational number and $\frac{p}{- q}$ is a ______ rational number.

Answer:

If p and q are positive integers, then p/q is a positive rational number, because both the numerator and denominator are positive and p−q $\frac{p}{- q}$ is a negative rational number, because the denominator is negative.

Page No 246:

Question 45:

Fill in the blanks to make the statement true.
Two rational numbers are said to be equivalent or equal, if they have the same ______ form.

Answer:

ans

Page No 246:

Question 46:

Fill in the blanks to make the statement true.
If $\frac{p}{q}$ is a rational number, then q cannot be ______.

Answer:

If $\frac{p}{q}$ is a rational number, then q cannot be zero(0).

Page No 247:

Question 47:

State whether the statement given in question is True or False.
Every natural number is a rational number but every rational number need not be a natural number.

Answer:

True; Every natural number is a rational number but every rational number need not be a natural number.
Beacause $\frac{1}{3}$ is a rational number, but not a natural number.

Page No 247:

Question 48:

State whether the statement given in question is True or False.
Zero is a rational number.

Answer:

Zero can be written as $0 = \frac{0}{1}$ . We know that, a number of the form $\frac{p}{q}$ ,
where p, q are integers and q ≠ 0 is a rational number.
So, zero is a rational number., where p, q are integers and q ≠ 0 is a rational number. So, zero is a rational number.

Page No 247:

Question 49:

State whether the statement given in question is True or False.
Every integer is a rational number but every rational number need not be an integer.

Answer:

Integers…. – 3, –2, –1, 0, 1, 2, 3,…
Rational numbers: $1, - 1, \frac{1}{2}, \frac{- 3}{2}, 1 \frac{1}{2}, . . . .$
Hence, every integer is rational number, but every rational number is not an integer.

Page No 247:

Question 50:

State whether the statement given in question is True or False.
Every negative integer is not a negative rational number.

Answer:

False; because all the integers are rational numbers, whether it is negative/positive but vice-versa is not true.

Page No 247:

Question 51:

State whether the statement given in question is True or False.
If $\frac{p}{q}$ is a rational number and m is a non-zero integer, then $\frac{p}{q} = \frac{p \times m}{q \times m}$

Answer:

True;

Let m = 1,2, 3,…

$When, m = 2, \frac{p}{q} = \frac{p \times 2}{q \times 2} = \frac{p}{q} When, m = 3, \frac{p}{q} = \frac{p \times 3}{q \times 3} = \frac{p}{q} Hence, \frac{p}{q} = \frac{p \times m}{q \times m}$

Note: When both the numerator and denominator of a rational number are multiplied/divide by the same non-zero number, then we get the same rational number.

Page No 247:

Question 52:

State whether the statement given in question is True or False.
If $\frac{p}{q}$ is a rational number and m is a non-zero common divisor of p and q, then $\frac{p}{q} = \frac{p \div m}{p \div m} .$

Answer:

True;

Let m = 1, 2, 3, 4, ...

When, m = 2, $\frac{p}{q} = \frac{p \div 2}{q \div 2} \Rightarrow \frac{p}{q} = \frac{p}{2} \div \frac{q}{2} \Rightarrow \frac{p}{q} = \frac{p}{2} \times \frac{2}{q} \Rightarrow \frac{p}{q} = \frac{p}{q}$

When, m = 3, $\frac{p}{q} = \frac{p \div 3}{q \div 3} \Rightarrow \frac{p}{q} = \frac{p}{3} \div \frac{q}{3} \Rightarrow \frac{p}{q} = \frac{p}{3} \times \frac{3}{q} \Rightarrow \frac{p}{q} = \frac{p}{q}$

Hence, $\frac{p}{q} = \frac{p \div m}{q \div m}$ .

Page No 247:

Question 53:

State whether the statement given in question is True or False.
In a rational number, denominator always has to be a non-zero integer.

Answer:

True;
The basic definition of the rational number is that it is in the form of $\frac{p}{q}$ , where q ≠ 0.
It is because any number divided by zero is not defined.

Page No 247:

Question 54:

State whether the statement given in question is True or False.
If $\frac{p}{q}$ is a rational number and m is a non-zero integer, then $\frac{p \times m}{q \times m}$ is a rational number not equivalent to $\frac{p}{q} .$

Answer:

False;

Let m = 1,2, 3,…

$When, m = 2, \frac{p \times 2}{q \times 2} = \frac{p}{q} When, m = 3, \frac{p \times 3}{q \times 3} = \frac{p}{q} Hence, \frac{p \times m}{q \times m} = \frac{p}{q} .$

If m is a non-zero integer, then $\frac{p \times m}{q \times m}$ is a rational number equivalent to $\frac{p}{q} .$

Page No 247:

Question 55:

State whether the statement given in question is True or False.
Sum of two rational numbers is always a rational number.

Answer:

True;
The sum of two rational numbers is always a rational number.

$\frac{1}{3} + \frac{2}{5} = \frac{5 + 6}{15} = \frac{11}{15}$

Page No 247:

Question 56:

State whether the statement given in question is True or False.
All decimal numbers are also rational numbers.

Answer:

True; All decimal numbers are also rational numbers.
$0.4 = \frac{4}{10} = \frac{2}{5}$

Page No 247:

Question 57:

State whether the statement given in question is True or False.
The quotient of two rationals is always a rational number.

Answer:

False;
The quotient of two rationals is not always a rational number.
e.g. $\frac{1}{0}$

Page No 247:

Question 58:

State whether the statement given in question is True or False.
Every fraction is a rational number.

Answer:

True; Every fraction is a rational number but vice-versa is not true.

Page No 247:

Question 59:

State whether the statement given in question is True or False.
Two rationals with different numerators can never be equal.

Answer:

False;
Let $\frac{3}{5} and \frac{6}{10}$ are two rational numbers with different denominators.

$\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

Hence, two rationals with different numerators can be equal.

Page No 247:

Question 60:

State whether the statement given in question is True or False.
8 can be written as a rational number with any integer as denominator.

Answer:

False; because 8 can be written as a rational number with 1 as denominator i.e. $\frac{8}{1}$ .

Page No 247:

Question 61:

State whether the statement given in question is True or False.
$\frac{4}{6} is equivalent to \frac{2}{3} .$

Answer:

True;

$\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$

Hence, $\frac{4}{6} is equivalent to \frac{2}{3} .$

Page No 247:

Question 62:

State whether the statement given in question is True or False.
The rational number $\frac{- 3}{4}$ lies to the right of zero on the number line.

Answer:

False;
Every negative ratioanl number lies to the left on the number.
Hence, the rational number $\frac{- 3}{4}$ lies to the right of zero on the number line.

Page No 247:

Question 63:

State whether the statement given in question is True or False.
The rational numbers $\frac{- 12}{- 5} and \frac{- 7}{17}$ are on the opposite sides of zero on the number line.

Answer:

True;

$\frac{- 12}{- 5} = \frac{12}{5}$
The rational numbers $\frac{- 12}{- 5} and \frac{- 7}{17}$ are on the opposite sides of zero on the number line beacuse $\frac{12}{5} and \frac{- 7}{17}$ are positive and negative rational number.

Page No 248:

Question 64:

State whether the statement given in question is True or False.
Every rational number is a whole number.

Answer:

False; because $\frac{- 8}{9}$ is a rational number, but it is not a whole number, because whole numbers are 0,1,2….

Page No 248:

Question 65:

State whether the statement given in question is True or False.
Zero is the smallest rational number

Answer:

False;
Rational numbers can be negative and negative rational numbers are smaller than zero.

Page No 248:

Question 66:

Match the following:
Column I Column II
$(i) \frac{a}{b} \div \frac{a}{b} (a) \frac{- a}{b} (ii) \frac{a}{b} \div \frac{c}{d} (b) - 1 (iii) \frac{a}{b} \div (- 1) (c) 1 (iv) \frac{a}{b} \div \frac{- a}{b} (d) \frac{b c}{a d} (v) \frac{b}{a} \div (\frac{d}{c}) (e) \frac{a d}{b c}$

Answer:

$(i) \frac{a}{b} \div \frac{a}{b} = \frac{a}{b} \times \frac{b}{a} = 1 \to (c) 1 (ii) \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c} \to (e) \frac{a d}{b c} (iii) \frac{a}{b} \div (- 1) = \frac{- a}{b} \to (a) \frac{- a}{b} (iv) \frac{a}{b} \div \frac{- a}{b} = \frac{a}{b} \times \frac{b}{- a} = - 1 \to (b) - 1 (v) \frac{b}{a} \div (\frac{d}{c}) = \frac{b}{a} \times \frac{c}{d} = \frac{b c}{a d} \to (d) \frac{b c}{a d}$

Page No 248:

Question 67:

Write each of the following rational numbers with positive denominators: $\frac{5}{- 8}, \frac{15}{- 28}, \frac{- 17}{- 13} .$
â€‹

Answer:

(a) $\frac{5}{- 8} = \frac{- 5}{8}$

(b) $\frac{15}{- 28} = \frac{- 15}{28}$

(c) $\frac{- 17}{- 13} = \frac{17}{13}$

Page No 248:

Question 68:

â€‹Express $\frac{3}{4}$ as a rational number with denominator:
(i) 36 (ii) – 80

Answer:

(i) 36 = 4 × 9

$∴ \frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}$

(ii) −80 = 4 × (−20)

$∴ \frac{3}{4} = \frac{3 \times (- 20)}{4 \times (- 20)} = \frac{- 60}{- 80}$

Page No 248:

Question 69:

Reduce each of the following rational numbers in its lowest form:
â€‹ $(i) \frac{- 60}{72} (ii) \frac{91}{- 364}$

Answer:

$(i) \frac{- 60}{72} = \frac{(- 5) \times 12}{6 \times 12} = \frac{- 5}{6} (ii) \frac{91}{- 364} = \frac{91}{(- 4) \times 91} = \frac{- 1}{4}$

Page No 248:

Question 70:

Express each of the following rational numbers in its standard form:
â€‹ $(i) \frac{- 12}{- 30} (ii) \frac{14}{- 49} (iii) \frac{- 15}{35} (iv) \frac{299}{- 161}$

Answer:

$(i) \frac{- 12}{- 30} = \frac{(- 12) \div 6}{(- 30) \div 6} = \frac{2}{5} (ii) \frac{14}{- 49} = \frac{14 \div 7}{(- 49) \div 7} = \frac{- 2}{7} (iii) \frac{- 15}{35} = \frac{(- 15) \div 5}{35 \div 5} = \frac{- 3}{7} (iv) \frac{299}{- 161} = \frac{- 299 \div 23}{161 \div 23} = \frac{- 13}{7}$

Page No 248:

Question 71:

â€‹Are the rational numbers $\frac{- 8}{18} and \frac{32}{- 112}$ equivalent? Give reason.

Answer:

Given: rational numbers $\frac{- 8}{28} and \frac{32}{- 112}$

$\frac{- 8}{28} = \frac{- 8 \div 4}{28 \div 4} = \frac{- 2}{7}$

$\frac{32}{- 112} = \frac{- 32}{112} = \frac{- 32 \div 16}{112 \div 16} = \frac{- 2}{7}$

Their standard forms are equal.

Hence, they are equal.

Page No 248:

Question 72:

Arrange the rational numbers $\frac{- 7}{10}, \frac{5}{- 8}, \frac{2}{- 3}, \frac{- 1}{4}, \frac{- 3}{5}$ â€‹in ascending order.

Answer:

Page No 248:

Question 73:

Represent the following rational numbers on a number line:
â€‹ $\frac{3}{8}, \frac{- 7}{3}, \frac{22}{- 6} .$

Answer:

(Recreate)

Page No 249:

Question 74:

If $\frac{- 5}{7} = \frac{x}{28},$ â€‹find the value of x.

Answer:

ans

Page No 249:

Question 75:

Give three rational numbers equivalent to:
â€‹ $(i) \frac{- 3}{4} (ii) \frac{7}{11}$

Answer:

ans

Page No 249:

Question 76:

Write the next three rational numbers to complete the pattern:
â€‹ $(i) \frac{4}{- 5}, \frac{8}{- 10}, \frac{12}{- 15}, \frac{16}{- 20},_____,_____,_____. (ii) \frac{- 8}{7}, \frac{- 16}{14}, \frac{- 24}{21}, \frac{- 32}{28},____,____,_____.$

Answer:

ans

Page No 249:

Question 77:

â€‹List four rational numbers between $\frac{5}{7} and \frac{7}{8} .$

Answer:

ans

Page No 249:

Question 78:

Find the sum of
â€‹ $(i) \frac{8}{13} and \frac{3}{11} (ii) \frac{7}{3} and \frac{- 4}{3}$

Answer:

ans

Page No 249:

Question 79:

Solve:
â€‹ $(i) \frac{29}{4} - \frac{30}{7} (ii) \frac{5}{13} - \frac{- 8}{26}$

Answer:

ans

Page No 249:

Question 80:

Find the product of:
$(i) \frac{- 4}{5} and \frac{- 5}{12} (ii) \frac{- 22}{11} and \frac{- 21}{11}$

Answer:

ans

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Question 81:

Simplify
â€‹(i) $\frac{13}{11} \times \frac{- 14}{5} + \frac{13}{11} \times \frac{- 7}{5} + \frac{- 13}{11} \times \frac{34}{5}$ (ii) $\frac{6}{5} \times \frac{3}{7} - \frac{1}{5} \times \frac{3}{7}$

Answer:

ans

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Question 82:

Simplify:
â€‹(i) $\frac{3}{7} \div (\frac{21}{- 55})$ (ii) $1 \div (- \frac{1}{2})$

Answer:

ans

Page No 249:

Question 83:

Which is greater in the following?
â€‹(i) $\frac{3}{4}, \frac{7}{8}$ (ii) $- 3 \frac{5}{7}, 3 \frac{1}{9}$

Answer:

ans

Page No 249:

Question 84:

â€‹Write a rational number in which the numerator is less than ‘–7 × 11’ and the denominator is greater than ‘12 + 4’.

Answer:

ans

Page No 249:

Question 85:

â€‹If x = $\frac{1}{10}$ and y = $\frac{- 3}{8}$ , then
evaluate x + y, x – y, x × y and x ÷ y.

Answer:

ans

Page No 250:

Question 86:

Find the reciprocal of the following:
â€‹ $(i) (\frac{1}{2} \times \frac{1}{4}) + (\frac{1}{2} \times 6) (ii) \frac{20}{51} \times \frac{4}{91} (iii) \frac{3}{13} \div \frac{- 4}{65} (iv) (- 5 \times \frac{12}{15}) - (- 3 \times \frac{2}{9})$

Answer:

ans

Page No 250:

Question 87:

Complete the following table by finding the sums:

+	$- \frac{1}{9}$	$\frac{4}{11}$	$\frac{- 5}{6}$
$\frac{2}{3}$
$- \frac{5}{4}$		$\frac{- 39}{44}$
$- \frac{1}{3}$

Answer:

ans

Page No 250:

Question 88:

â€‹Write each of the following numbers in the form $\frac{p}{q}$ , where p and q are integers:
(a) six-eighths
(b) three and half
(c) opposite of 1
(d) one-fourth
(e) zero
(f) opposite of three-fifths

Answer:

ans

Page No 250:

Question 89:

â€‹If p = m × t and q = n × t, then $\frac{p}{q} = \frac{}{}$

Answer:

ans

Page No 250:

Question 90:

Given that $\frac{p}{q}$ and $\frac{r}{s}$ are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers we say that:
â€‹ $(a) \frac{}{} < \frac{}{}, if p \times s < r \times q (b) \frac{p}{q} = \frac{r}{s}, if_______=________(c) \frac{}{} > \frac{}{}, if p \times s > r \times q$

Answer:

ans

Page No 251:

Question 91:

â€‹In each of the following cases, write the rational number whose numerator and denominator are respectively as under:
(a) 5 – 39 and 54 – 6
(b) (–4) × 6 and 8 ÷ 2
(c) 35 ÷ (–7) and 35 –18
(d) 25 + 15 and 81 ÷ 40

Answer:

ans

Page No 251:

Question 92:

â€‹Write the following as rational numbers in their standard forms:
(a) 35% (b) 1.2 (c) $- 6 \frac{3}{7}$
(d) 240 ÷ (– 840) (e) 115 ÷ 207

Answer:

ans

Page No 251:

Question 93:

Find a rational number exactly halfway between:
â€‹ $(a) \frac{- 1}{3} and \frac{1}{3} (b) \frac{1}{6} and \frac{1}{9} (c) \frac{5}{- 13} and \frac{- 7}{9} (d) \frac{1}{15} and \frac{1}{12}$

Answer:

ans

Page No 251:

Question 94:

Taking $x = \frac{- 4}{9}, y = \frac{5}{12} and z = \frac{7}{18},$ find:
â€‹(a) the rational number which when added to x gives y.
(b) the rational number which subtracted from y gives z.
(c) the rational number which when added to z gives us x.
(d) the rational number which when multiplied by y to get x.
(e) the reciprocal of x + y.
(f) the sum of reciprocals of x and y.
(g) (x ÷ y) × z (h) (x – y) + z
(i) x + (y + z) (j) x ÷ (y ÷ z)
(k) x – (y + z)

Answer:

ans

Page No 251:

Question 95:

â€‹What should be added to $\frac{- 1}{2}$ to obtain the nearest natural number?

Answer:

ans

Page No 251:

Question 96:

â€‹What should be subtracted from $\frac{- 2}{3}$ to obtain the nearest integer?

Answer:

ans

Page No 251:

Question 97:

â€‹What should be multiplied with $\frac{- 5}{8}$ to obtain the nearest integer?

Answer:

ans

Page No 251:

Question 98:

â€‹What should be divided by $\frac{1}{2}$ to obtain the greatest negative integer?

Answer:

ans

Page No 252:

Question 99:

â€‹From a rope 68 m long, pieces of equal size are cut. If length of one piece is $4 \frac{1}{4}$ m, find the number of such pieces.

Answer:

ans

Page No 252:

Question 100:

â€‹If 12 shirts of equal size can be prepared from 27m cloth, what is length of cloth required for each shirt?

Answer:

ans

Page No 252:

Question 101:

â€‹Insert 3 equivalent rational numbers between
(i) $\frac{- 1}{2} and \frac{1}{5}$ (ii) 0 and –10

Answer:

ans

Page No 252:

Question 102:

Put the (âœ“), wherever applicable

Number	Natural Number	Whole Number	Integer	Fraction	Rational â€‹Number
(a) – 114
(b) $\frac{19}{27}$
(c) $\frac{623}{1}$
(d) $- 19 \frac{3}{4}$
(e) $\frac{73}{71}$
(f) 0

Answer:

ans

Page No 252:

Question 103:

‘a’ and ‘b’ are two different numbers taken from the numbers 1 – 50. What is the largest value that $\frac{a - b}{a + b}$ can have? What is the largest
value that $\frac{a + b}{a - b}$ can have?

Answer:

ans

Page No 252:

Question 104:

150 students are studying English, Maths or both. 62 per cent of the students are studying English and 68 per cent are studying Maths. How many students are studying both?

Answer:

ans

Page No 253:

Question 105:

A body floats $\frac{2}{9}$ of its volume above the surface. What is the ratio of the body submerged volume to its exposed volume? Re-write it as a rational number.

Answer:

ans

Page No 253:

Question 106:

In the given question find the odd one and give reason.

$(a) \frac{4}{3} \times \frac{3}{4} (b) \frac{- 3}{2} \times \frac{- 2}{3} (c) 2 \times \frac{1}{2} (d) \frac{- 1}{3} \times \frac{3}{1}$

Answer:

ans

Page No 253:

Question 107:

In the given question find the odd one and give reason.

$(a) \frac{4}{- 9} (b) \frac{- 16}{36} (c) \frac{- 20}{- 45} (d) \frac{28}{- 63}$

Answer:

ans

Page No 253:

Question 108:

In the given question find the odd one and give reason.

$(a) \frac{- 4}{3} (b) \frac{- 7}{6} (c) \frac{- 10}{3} (d) \frac{- 8}{7}$

Answer:

ans

Page No 253:

Question 109:

In the given question find the odd one and give reason.
$(a) \frac{- 3}{7} (b) \frac{- 9}{15} (c) \frac{+ 24}{20} (d) \frac{+ 35}{25}$

Answer:

ans

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Question 110:

In the given question find the odd one and give reason.
What’s the Error? Chhaya simplified a rational number in this manner $\frac{- 25}{- 30} = \frac{- 5}{6}$ . What error did the student make?

Answer:

If a negative (−) sign comes in both numerator and denominator, then it will be cancelled.
So, the resulting fraction will be positive.
$∴ \frac{- 25}{- 30} = \frac{25}{30} = \frac{5}{6}$
Here, Chhaya divided the numerator by 5 but denominator by −5.

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