Haryana Board Class 7 Maths Solutions For Chapter 8 Rational Numbers

Haryana Board Class 7 Maths Solutions For Chapter 8 Rational Numbers

Key Concepts

  1. 1. Introduction :
    1. The numbers used for counting objects around us are called Counting numbers (or) Natural numbers They are 1, 2, 3, 4,……,…
    2. Including ‘0’ to natural numbers we get the whole numbers i.e. 0,1,2,3,4, ………
    3. The negatives of natural numbers were put together with whole numbers to make up integers. They are……..- 3,-2,-1, 0,1,2,3,……….
    4. The numbers of the form \( \frac{\text { numerator }}{\text { denominator }} [latex] where the numerator is either 0 or a positive integer and the denominator, a positive integer are called fractions.
      Need for rational numbers: We know that integers can be. used to denote opposite situations involving numbers.
  2. Definition of a rational number:
    A rational number is defined as number that can be expressed in the form of [latex] \frac{p}{q} \) where p and q are integers and q ≠ 0.

Example :

\( \frac{4}{5} ; \frac{-3}{4} ; \frac{3}{8} ; 1 \frac{2}{3} \text { etc. } \) \( 0.5=\frac{5}{10} ; 0.333=\frac{333}{1000} \text { etc. } \)

1. Is the number \( \frac{2}{-3} \) rational ? Think about it.

Solution: Yes; \( \frac{2}{-3} \) is a rational number.

It is in the form of latex] \frac{p}{q} [/latex], where p = 2; q = -3 both are integers.

2. List ten rational numbers.

Solution:

\( \frac{3}{8}, \frac{2}{3}, \frac{-3}{2}, \frac{4}{-9}, \frac{1}{2}, \frac{3}{4}, \frac{-6}{11}, 2 \frac{3}{5}, \) \( 0.75=\frac{75}{100}, \frac{17}{79} . \)

HBSE Class 7 Rational Numbers Solutions

Fill in the boxes:

1)

Fill in the boxes

Solution:

\( \begin{aligned}
& \frac{5}{4}=\frac{5 \times 4}{4 \times 4}=\frac{20}{16} \\
& \frac{5}{4}=\frac{5 \times 5}{4 \times 5}=\frac{25}{20} \\
& \frac{5}{4}=\frac{5 \times(-3)}{4 \times(-3)}=\frac{-15}{-12} \\
& \frac{5}{4}=\frac{20}{16}=\frac{25}{20}=\frac{-15}{-12}
\end{aligned} \)

2)

Fill in the boxes 2

Solution:

\( \begin{aligned}
& \frac{-3}{7}=\frac{-3 \times 2}{7 \times 2}=\frac{-6}{14} \\
& \frac{-3}{7}=\frac{-3 \times(-3)}{7 \times(-3)}=\frac{9}{-21} \\
& \frac{-3}{7}=\frac{-3 \times 2}{7 \times 2}=\frac{-6}{14} \\
& \frac{-3}{7}=\frac{-6}{14}=\frac{9}{-21}=\frac{-6}{14}
\end{aligned} \)

Solutions To Try These

1. Is 5 a positive rational number?

Solution:

Yes,5 is a positive rational number. It can be written as \( \frac{5}{1} \) . The numerator is 5 and denominator is 1.

2. List five more positive rational numbers.

Solution:

\( \frac{3}{7}, \frac{5}{12}, \frac{4}{19}, \frac{6}{13}, \frac{17}{9} \)

Solutions To Try These

1. Is – 8 a negative rational number?

Solution:

Yes,- 8 is a negative rational number. It can be written as \( \frac{-8}{1} \) . The numerator is a negative integer and the denominator is
a positive integer.

2. List five more negative rational numbers.

Solution:

\( \frac{-4}{9}, \frac{-7}{11}, \frac{-5}{11}, \frac{-15}{22}, \frac{-3}{10} \)

Solutions To Try These

Which of these are negative rational numbers?

  1. \( \frac{-2}{3} \)
  2. \( \frac{5}{7} \)
  3. \( \frac{3}{-5} \)
  4. 0
  5. \( \frac{6}{11} \)
  6. \( \frac{-2}{-9} \)

Solution:

1) \( \frac{-2}{3} \) and 3. \( \frac{3}{-5} \) are negative rational numbers.

Solutions To Try These

Find the standard form of

1) \( \frac{-18}{45} \)

Solution: The HCF of 18 and 45 is 9.

\( \frac{-18}{45}=\frac{-18 \div 9}{45 \div 9}=\frac{-2}{5} \)

2) \( \frac{-12}{18} \)

Solution: The HCF of 12 and 18 is 6.

\( \frac{-12}{18}=\frac{-12 \div 6}{18 \div 6}=\frac{-2}{3} \)

Solutions To Try These

Find five rational numbers between

\( \frac{-5}{7}\) and \( \frac{-3}{8}\).

Solution:

\( \frac{-5}{7}=\frac{-5 \times 8}{7 \times 8}=\frac{-40}{56} \) \( \frac{-3}{8}=\frac{-3 \times 7}{8 \times 7}=\frac{-21}{56} \)

\( \frac{-40}{56}<\frac{-39}{56}<\frac{-38}{56}<\frac{-29}{56} \) \( <\frac{-27}{56}<\frac{-22}{56}<\frac{-21}{56} \)

\( \frac{-5}{7}<\frac{-39}{56}<\frac{-38}{56}<\frac{-29}{56}<\frac{-27}{56}<\frac{-22}{56} \) \( <\frac{-3}{8} \)

The five rational numbers between \( <\frac{-5}{7} \) and \( <\frac{-3}{8} \) are

\( \frac{-39}{56}, \frac{-38}{56}, \frac{-29}{56}, \frac{-27}{56}, \frac{-22}{56} . \)

Haryana Board Class 7 Maths Solutions For Chapter 8  Exercise-8.1

1. List five rational numbers between: (1) -1 and 0 . (2) -2 and -1

(3) \( \frac{-4}{5} \text { and } \frac{-2}{3} \)

4. \( \frac{-1}{2} \text { and } \frac{2}{3} \)

Solution:

First we find equivalent rational numbers having same denominator.

1) \( -1=\frac{-1}{1}=\frac{-1 \times 10}{1 \times 10}=\frac{-10}{10} \)

\( 0=\frac{0}{1}=\frac{0 \times 10}{1 \times 10}=\frac{0}{10} \) \( \Rightarrow \frac{-10}{10}<\frac{-9}{10}<\frac{-8}{10}<\frac{-7}{10}<\frac{-6}{10}<\frac{-5}{10}<\frac{0}{10} \) \( \Rightarrow-1<\frac{-9}{10}<\frac{-8}{10}<\frac{-7}{10}<\frac{-6}{10}<\frac{-5}{10}<0 \)

The five rational numbers between -1 and 0 are

\( \frac{-9}{10}, \frac{-8}{10}, \frac{-7}{10}, \frac{-6}{10}, \frac{-5}{10} \)

Haryana Board Class 7 Maths Rational Numbers Solutions

2) – 2 and -1

Solution:

\( -2=\frac{-2}{1}=\frac{-2 \times 10}{1 \times 10}=\frac{-20}{10} \) \( -1=\frac{-1}{1}=\frac{-1 \times 10}{1 \times 10}=\frac{-10}{10} \) \( \begin{aligned}
\Rightarrow \frac{-20}{10} & <\frac{-19}{10}<\frac{-18}{10}<\frac{-17}{10} \\
& <\frac{-16}{10}<\frac{-15}{10}<\frac{-10}{10}
\end{aligned} \) \( \begin{aligned}
\Rightarrow-2<\frac{-19}{10}<\frac{-18}{10}<\frac{-17}{10} & <\frac{-16}{10} \\
& <\frac{-15}{10}<-1
\end{aligned} \)

The five rational numbers between -2 and -1 are

\( \frac{-19}{10}, \frac{-18}{10}, \frac{-17}{10}, \frac{-16}{10}, \frac{-15}{10} \)

HBSE 7th Class Rational Number Word Problems

3) \( \frac{-4}{5} \text { and } \frac{-2}{3} \)

Solution:

\( \frac{-4}{5}=\frac{-4 \times 9}{5 \times 9}=\frac{-36}{45} \) \( \frac{-2}{3}=\frac{-2 \times 15}{3 \times 15}=\frac{-30}{45} \) \( \begin{aligned}
\Rightarrow \frac{-36}{45}<\frac{-35}{45} & <\frac{-34}{45}<\frac{-33}{45} \\
& <\frac{-32}{45}<\frac{-31}{45}<\frac{-30}{45}
\end{aligned} \) \( \begin{aligned}
\Rightarrow \frac{-4}{5}<\frac{-35}{45}<\frac{-34}{45}<\frac{-33}{45} & <\frac{-32}{45} \\
& <\frac{-31}{45}<\frac{-2}{3}
\end{aligned} \)

The five rational numbers between \( \frac{-4}{5} \) and \( \frac{-2}{3} \text { are } \) are

\( \frac{-35}{45}<\frac{-34}{45}<\frac{-33}{45}<\frac{-32}{45}<\frac{-31}{45} \)

Class 7 Maths Chapter 8 Rational Numbers Haryana Board

4) \( \frac{-1}{2} \text { and } \frac{2}{3} \)

Solution:

\( \begin{aligned}
& \frac{-1}{2}=\frac{-1 \times 3}{2 \times 3}=\frac{-3}{6} \\
& \frac{2}{3}=\frac{2 \times 2}{3 \times 2}=\frac{4}{6}
\end{aligned} \) \( \begin{aligned}
& \Rightarrow \frac{-3}{6}<\frac{-2}{6}<\frac{-1}{6}<0<\frac{1}{6}<\frac{2}{6}<\frac{4}{6} \\
& \Rightarrow-\frac{1}{2}<\frac{-2}{6}<\frac{-1}{6}<0<\frac{1}{6}<\frac{2}{6}<\frac{2}{3}
\end{aligned} \)

The five rational numbers between \( \frac{-1}{2} \text { and } \frac{2}{3} \text { are } \)

\( \frac{-2}{6}, \frac{-1}{6}, 0, \frac{1}{6}, \frac{2}{6} \)

2. Write four more rational numbers in each of the following patterns :

1) \( \frac{-3}{5}, \frac{-6}{10}, \frac{-9}{15}, \frac{-12}{20} \ldots . . \)

Solution:

\( \frac{-3}{5}=\frac{-3 \times 1}{5 \times 1} ; \frac{-6}{10}=\frac{-3 \times 2}{5 \times 2} \) \( \frac{-9}{15}=\frac{-3 \times 3}{5 \times 3} ; \frac{-12}{20}=\frac{-3 \times 4}{5 \times 4} \)

Thus, we observe a pattern in these numbers.

The next four numbers would be

\( \frac{-3 \times 5}{5 \times 5}=\frac{-15}{25} ; \frac{-3 \times 6}{5 \times 6}=\frac{-18}{30} \) \( \frac{-3 \times 7}{5 \times 7}=\frac{-21}{35} ; \frac{-3 \times 8}{5 \times 8}=\frac{-24}{40} \)

The required four rational numbers are

\( \frac{-15}{25}, \frac{-18}{30}, \frac{-21}{35}, \frac{-24}{40} \)

2) \( \frac{-1}{4}, \frac{-2}{8}, \frac{-3}{12}, \ldots . \)

Solution:

\( \frac{-1}{4}=\frac{-1 \times 1}{4 \times 1} ; \frac{-2}{8}=\frac{-1 \times 2}{4 \times 2} \) \( \frac{-3}{12}=\frac{-1 \times 3}{4 \times 3} \)

Thus we observe a pattern in these numbers.

The next four numbers would be

\( \begin{aligned}
& \frac{-1 \times 4}{4 \times 4}=\frac{-4}{16} ; \frac{-1 \times 5}{4 \times 5}=\frac{-5}{20} \\
& \frac{-1 \times 6}{4 \times 6}=\frac{-6}{24} ; \frac{-1 \times 7}{4 \times 7}=\frac{-7}{28}
\end{aligned} \)

The required four rational numbers are

\( \frac{-4}{16}, \frac{-5}{20}, \frac{-6}{24}, \frac{-7}{28} \)

Haryana Board 7th Class Maths Rational Numbers Questions and Answers

3) \( \frac{-1}{6}, \frac{2}{-12}, \frac{3}{-18}, \frac{4}{-24}, \ldots \ldots \)

Solution:

\( \frac{2}{-12}=\frac{-1 \times(-2)}{6 \times(-2)} ; \frac{3}{-18}=\frac{-1 \times(-3)}{6 \times(-3)} \) \( \frac{4}{-24}=\frac{-1 \times(-4)}{6 \times(-4)} \)

Thus we observe a pattem in these numbers.

\( \begin{aligned}
&\frac{-1 \times(-5)}{6 \times(-5)}=\frac{5}{-30} ; \frac{(-1) \times(-6)}{6 \times(-6)}=\frac{6}{-36} ;\\
&\frac{(-1) \times(-7)}{6 \times(-7)}=\frac{7}{-42} ; \frac{(-1) \times(-8)}{6 \times(-8)}=\frac{8}{-48}
\end{aligned} \)

The required four numbers are \( \frac{5}{-30} ; \frac{6}{-36} \)

\( \frac{7}{-42}, \frac{8}{-48} \)

4) \( \frac{-2}{3}, \frac{2}{-3}, \frac{4}{-6}, \frac{6}{-9} \ldots . \)

Solution:

\( \begin{aligned}
&\frac{2}{-3}=\frac{-2 \times(-1)}{(-3) \times(-1)} ; \frac{4}{-6}=\frac{(-2) \times(-2)}{3 \times(-2)}\\
&\frac{6}{-9}=\frac{(-2) \times(-3)}{(3) \times(-3)}
\end{aligned} \)

Thus we observe a pattern in these numbers.

The next four numbers would be

\( \begin{aligned}
&\frac{-2 \times(-4)}{3 \times(-4)}=\frac{8}{-12} ; \frac{(-2) \times(-5)}{3 \times(-5)}=\frac{10}{-15}\\
&\frac{(-2) \times(-6)}{3 \times(-6)}=\frac{12}{-18} ; \frac{(-2) \times(-7)}{3 \times(-7)}=\frac{14}{-21}
\end{aligned} \)

The required four numbers are

\( \frac{8}{-12} ; \frac{10}{-15} ; \frac{12}{-18} ; \frac{14}{-21} \)

3. Give four rational numbers equivalent to:

(1)\( \frac{-2}{7} \)
(2)\( \frac{5}{-3} \)
(3)\( \frac{4}{9} \)

Solution:

(1)

\( \begin{aligned}
& \frac{-2}{7}=\frac{-2 \times 2}{7 \times 2}=\frac{-4}{14} \\
& \frac{-2}{7}=\frac{-2 \times 3}{7 \times 3}=\frac{-6}{21} \\
& \frac{-2}{7}=\frac{-2 \times 4}{7 \times 4}=\frac{-8}{28} \\
& \frac{-2}{7}=\frac{-2 \times 5}{7 \times 5}=\frac{-10}{35}
\end{aligned} \)

Thefourrational numbers equivalent to \( \frac{-2}{7} \text { are } \frac{-4}{14}, \frac{-6}{21}, \frac{-8}{28}, \frac{-10}{35} \)

(2)\( \frac{5}{-3} \)

Solution:

\( \frac{5}{-3}=\frac{5 \times 2}{-3 \times 2}=\frac{10}{-6} \) \( \begin{aligned}
& \frac{5}{-3}=\frac{5 \times 3}{-3 \times 3}=\frac{15}{-9} \\
& \frac{5}{-3}=\frac{5 \times 4}{-3 \times 4}=\frac{20}{-12} \\
& \frac{5}{-3}=\frac{5 \times 5}{-3 \times 5}=\frac{25}{-15}
\end{aligned} \)

The four rational numbers equivalent

\( \text { to } \frac{5}{-3} \text { are } \frac{10}{-6}, \frac{15}{-9}, \frac{20}{-12}, \frac{25}{-15} \)

Sample Problems Rational Numbers Haryana Board Class 7

(3)\( \frac{4}{9} \)

Solution:

\( \begin{aligned}
& \frac{4}{9}=\frac{4 \times 2}{9 \times 2}=\frac{8}{18} \\
& \frac{4}{9}=\frac{4 \times 3}{9 \times 3}=\frac{12}{27} \\
& \frac{4}{9}=\frac{4 \times 4}{9 \times 4}=\frac{16}{36} \\
& \frac{4}{9}=\frac{4 \times 5}{9 \times 5}=\frac{20}{45}
\end{aligned} \)

The four rational numbers equivalent to \( \frac{4}{9} \) are

\( \frac{8}{18}, \frac{12}{27}, \frac{16}{36}, \frac{20}{45} \)

4. Draw the number line and represent the following rational numbers on it:

1) \( \frac{3}{4} \)

Solution:

Draw the number line and represent the following rational numbers on it

2) \( \frac{-5}{8} \)

Solution:

Draw the number line and represent the following rational numbers on it 2

3) \( \frac{-7}{4} \)

Solution:

Draw the number line and represent the following rational numbers on it 3

\( \frac{-7}{4}=\frac{-4-3}{4}=\frac{-4}{4}-\frac{3}{4}=-1-\frac{3}{4} \)

4) \( \frac{7}{8} \)

Solution:

Draw the number line and represent the following rational numbers on it 4

5. The points P, Q, R, S, T, U, A, and B on the number line are such that, TR = RS = SU and AP= PQ = QB. Name the rational numbers represented by P, Q, R, and S.

Solution:

Name the rational numbers represented by P, Q, R and S

The points P, Q, R,S on the number line such that TR = RS = SU

\( \mathrm{TR}=\mathrm{RS}=\mathrm{SU}=\frac{1}{3} \mathrm{TU} \) \( \begin{aligned}
& \mathrm{TR}=\frac{1}{3} \text { unit } \\
& \text { and } \mathrm{AP}=\mathrm{PQ}=\mathrm{QB} \\
& \mathrm{AP}=\mathrm{PQ}=\mathrm{QB}=\frac{1}{3} \mathrm{AB}
\end{aligned} \) \( \mathrm{AP}=\frac{1}{3} \text { unit. } \)

The rational number represented by P

\( P=2+\frac{1}{3}=\frac{6+1}{3}=\frac{7}{3} \)

The rational number represented by Q,

\( \mathrm{Q}=2+\frac{1}{3}+\frac{1}{3}=\frac{6+1+1}{3}=\frac{8}{3} \)

The rational number represented by R,

\( R=(-1)+\left(\frac{-1}{3}\right)=\frac{-1-3}{3}=\frac{-4}{3} \)

The rational number represented by S,

\( S=(-1)+\left(\frac{-1}{3}\right)+\left(\frac{-1}{3}\right)=\frac{-3-1-1}{3}=\frac{-5}{3} \)

6. Which of the following pairs represent the same rational number ?

1) \( \frac{-7}{21} \text { and } \frac{3}{9} \)

Solution:

\( \frac{-7}{21} \) is a negative rational number.

\( \frac{3}{9} \) is a positive rational number.

The given pair does not represent the same rational number.

2)

\( \frac{-16}{20} \text { and } \frac{20}{-25} \)

Solution:

\( \frac{-16}{20}=\frac{-16 \div 4}{20 \div 4}=\frac{-4}{5}=\frac{(-4) \times(-1)}{5 \times(-1)}=\frac{4}{-5} \) \( \frac{20}{-25}=\frac{20 \div 5}{-25 \div 5}=\frac{4}{-5} \)

The given pair represents the same rational number

3) \( \frac{-2}{-3} \text { and } \frac{2}{3} \)

Solution:

\( \frac{-2}{-3}=\frac{(-2) \times(-1)}{(-3) \times(-1)}=\frac{2}{3} \)

The given pair represents the same rational number.

4) \( \frac{-3}{5} \text { and } \frac{-12}{20} \)

Solution:

\( \frac{-3}{5}=\frac{-3 \times 4}{5 \times 4}=\frac{-12}{20} \)

The given pair represents the same rational number.

5) \( \frac{8}{-5} \text { and } \frac{-24}{15} \)

\(  \frac{8}{-5} \text { and } \frac{-24}{15} \) \( \frac{8}{-5}=\frac{8 \times 3}{-5 \times 3}=\frac{24}{-15}=\frac{24 \times(-1)}{(-15) \times(-1)}=\frac{-24}{15} \)

The given pair represents the same rational number.

6) \( \frac{1}{3} \text { and } \frac{-1}{9} \)

Solution:

\( \frac{1}{3} \) is a positive rational number.

\( \frac{-1}{9} \) is a negative rational number.

The given pair does not represent the same rational number.

7) \( \frac{-5}{-9} \text { and } \frac{5}{-9} \)

Solution:

\( \frac{-5}{-9}=\frac{-5 \times(-1)}{-9 \times(-1)}=\frac{5}{9} \)

\( \frac{5}{9} \) is a positive rational number.

\( \frac{5}{-9} \) is a negative rational number.

The given pair does not represent the same rational number.

Operations on Rational Numbers Class 7 HBSE

7. Rewrite the following rational numbers in the simplest form:

1) \( \frac{-8}{6} \)

Solution:

\( \frac{-8}{6} \)

HCF of 8 and 6 is 2.

\( \frac{-8}{6}=\frac{-8 \div 2}{6 \div 2}=\frac{-4}{3} \)

2) \( \frac{25}{45} \)

Solution:

\( \frac{25}{45} \)

HCF of 25 and 45 is 5.

\( \frac{25}{45}=\frac{25 \div 5}{45 \div 5}=\frac{5}{9} \)

Chapter 8 Rational Numbers Class 7 Solutions in Hindi Haryana Board

3) \( \frac{-44}{7 \cdot 2} \)

Solution:

\( \frac{-44}{7 \cdot 2} \)

HCF of 44 and 72 is 4.

\( \frac{-44}{72}=\frac{-44 \div 4}{72 \div 4}=\frac{-11}{18} \)

4) \( \frac{-8}{10} \)

Solution:

\( \frac{-8}{10} \)

HCF of 8 and 10 is 2.

\( \frac{-8}{10}=\frac{-8 \div 2}{10 \div 2}=\frac{-4}{5} \)

8. Fill in the boxes with the correct symbol out of >, < and =

1) \( \frac{-5}{7}\) Fill in the boxes with the correct symbol out of \( \frac{2}{3}\)

Solution:

LCM of 7 and 3 is 21.

\( \begin{aligned}
& \frac{-5}{7}=\frac{-5 \times 3}{7 \times 3}=\frac{-15}{21} \\
& \frac{2}{3}=\frac{2 \times 7}{3 \times 7}=\frac{14}{21}
\end{aligned} \) \( \text { Hence } \frac{-5}{7}<\frac{2}{3} \)

2) \( \frac{-4}{5} \) Fill in the boxes with the correct symbol out of \( \frac{-5}{7} \)

Solution:

LCM of 5 and 7 is 35

\( \frac{-4}{5}=\frac{-4 \times 7}{5 \times 7}=\frac{-28}{35} \) \( \begin{aligned}
&\frac{-5}{7}=\frac{-5 \times 5}{7 \times 5}=\frac{-25}{35}\\
&\text { Hence } \frac{-4}{5}<\frac{-5}{7}
\end{aligned} \)

3) \( \frac{-7}{8} \) Fill in the boxes with the correct symbol out of \( \frac{14}{-16} \)

Solution:

\( \begin{aligned}
&\frac{-7}{8}=\frac{-7 \times(-2)}{8 \times(-2)}=\frac{14}{-16}\\
&\text { Hence } \frac{-7}{8}=\frac{14}{-16}
\end{aligned} \)

4) \( \frac{-8}{5} \) Fill in the boxes with the correct symbol out of \( \frac{-7}{4} \)

Solution:

LCM of 5 and 4 is 20

\( \begin{aligned}
&\begin{aligned}
& \frac{-8}{5}=\frac{-8 \times 4}{5 \times 4}=\frac{-32}{20} \\
& \frac{-7}{4}=\frac{-7 \times 5}{4 \times 5}=\frac{-35}{20}
\end{aligned}\\
&\text { Hence } \frac{-8}{5}>\frac{-7}{4}
\end{aligned} \)

5) \( \frac{1}{-3} \) Fill in the boxes with the correct symbol out of \( \frac{-1}{4} \)

Solution:

LCM of3 and 4 is 12

\( \begin{aligned}
&\begin{aligned}
& \frac{1}{-3}=\frac{1 \times 4}{-3 \times 4}=\frac{4}{-12}=\frac{4 \times(-1)}{(-12) \times(-1)}=\frac{-4}{12} \\
& \frac{-1}{4}=\frac{-1 \times 3}{4 \times 3}=\frac{-3}{12}
\end{aligned}\\
&\text { Hence } \frac{1}{-3}<\frac{-1}{4}
\end{aligned} \)

6) \( \frac{5}{-11} \) Fill in the boxes with the correct symbol out of \( \frac{-5}{11} \)

Solution:

\( \begin{aligned}
&\frac{5}{-11}=\frac{5 \times(-1)}{(-11) \times(-1)}=\frac{-5}{11}\\
&\text { Hence } \frac{5}{-11}=\frac{-5}{11}
\end{aligned} \)

Equivalent Rational Numbers Class 7 Haryana Board

7) 0 rec \( \frac{-7}{6} \)

Solution: \begin{aligned}
& 0=\frac{0}{6} \\
& \text { Hence } 0>\frac{-7}{6}
\end{aligned}

9. Which is greater in each of the following:

1) \( \frac{2}{3}, \frac{5}{2} \)

Solution:

LCM of 3 and 2 is 6

\( \begin{aligned}
& \frac{2}{3}=\frac{2 \times 2}{3 \times 2}=\frac{4}{6} \\
& \frac{5}{2}=\frac{5 \times 3}{2 \times 3}=\frac{15}{6} \\
& \frac{15}{6}>\frac{4}{6}
\end{aligned} \) \( \frac{5}{2}>\frac{2}{3} \)

Important Questions for Class 7 Maths Chapter 8 Haryana Board

2) \( \frac{-5}{6}, \frac{-4}{3} \)

Solution: LCM of 6 and 3 is 6.

\( \begin{aligned}
& \frac{-5}{6}=\frac{-5 \times 1}{6 \times 1}=\frac{-5}{6} \\
& \frac{-4}{3}=\frac{-4 \times 2}{3 \times 2}=\frac{-8}{6} \\
& \frac{-5}{6}>\frac{-8}{6}
\end{aligned} \) \( \frac{-5}{6}>\frac{-4}{3} . \)

3) \( \frac{-3}{4}, \frac{2}{-3} \)

LCM of 4 and 3 is 12.

Solution:

\( \begin{aligned}
& \frac{-3}{4}=\frac{-3 \times 3}{4 \times 3}=\frac{-9}{12} \\
& \frac{2}{-3}=\frac{2 \times 4}{-3 \times 4}=\frac{8}{-12}=\frac{8 \times(-1)}{(-12) \times(-1)}=\frac{-8}{12}
\end{aligned} \) \( \begin{aligned}
&\frac{-8}{12}>\frac{-9}{12}\\
&\frac{-2}{3}>\frac{-3}{4}
\end{aligned} \) \( \frac{2}{-3}>\frac{-3}{4} \)

4) \( \frac{-1}{4}, \frac{1}{4} \)

Solution: \( \frac{1}{4}>\frac{-1}{4} \)

5) \( -3 \frac{2}{7},-3 \frac{4}{5} \)

Solution:

\( -3 \frac{2}{7}=\frac{-23}{7} \quad ; \quad-3 \frac{4}{5}=\frac{-19}{5} \)

LCM of 7 and 5 is 35.

\( \begin{aligned}
& \frac{-23}{7}=\frac{-23 \times 5}{7 \times 5}=\frac{-115}{35} \\
& \frac{-19}{5}=\frac{-19 \times 7}{5 \times 7}=\frac{-133}{35}
\end{aligned} \) \( \begin{aligned}
\frac{-115}{35} & >\frac{-133}{35} \\
\frac{-23}{7} & >\frac{-19}{5}
\end{aligned} \) \( \text { Hence }-3 \frac{2}{7}>-3 \frac{4}{5} \)

10. Write the following rational numbers in ascending order :

1) \( \frac{-3}{5}, \frac{-2}{5}, \frac{-1}{5} \)

Solution:

Denominators of each rational number is 5.

-3<-2<-l

\( \frac{-3}{5}<\frac{-2}{5}<\frac{-1}{5} \) \( \text { Ascending order is } \frac{-3}{5}, \frac{-2}{5}, \frac{-1}{5} \)

2) \( \frac{-1}{3}, \frac{-2}{9}, \frac{-4}{3} \)

Solution: LCM of 3, 9, 3 is 9.

\( \frac{-1}{3}=\frac{-1 \times 3}{3 \times 3}=\frac{-3}{9} ; \frac{-2 \times 1}{9 \times 1}=\frac{-2}{9} \) \( \frac{-4}{3}=\frac{-4 \times 3}{3 \times 3}=\frac{-12}{9} \)

-12 < -3 < -2

LCM of 3, 9, 3 is 9

\( \frac{-12}{9}<\frac{-3}{9}<\frac{-2}{9} \) \( \frac{-4}{3}<\frac{-1}{3}<\frac{-2}{9} . \) \( \text { Ascending order is } \frac{-4}{3}, \frac{-1}{3}, \frac{-2}{9} \)

HBSE Class 7 Maths Chapter 8/9 Guide Rational Numbers

3) \( \frac{-3}{7}, \frac{-3}{2}, \frac{-3}{4} \)

Solution: LCM of 7, 2, 4 is 28.

\( \begin{aligned}
& \frac{-3}{7}=\frac{-3 \times 4}{7 \times 4}=\frac{-12}{28} \\
& \frac{-3}{2}=\frac{-3 \times 14}{2 \times 14}=\frac{-42}{28}
\end{aligned} \) \( \begin{aligned}
\frac{-3}{4} & =\frac{-3 \times 7}{4 \times 7} \\
& =\frac{-21}{28}
\end{aligned} \)

– 42 < -21 < -12

LCM of 7, 2, 4 is 28

\( \begin{aligned}
& \frac{-42}{28}<\frac{-21}{28}<\frac{-12}{28} \\
& \frac{-3}{2}<\frac{-3}{4}<\frac{-3}{7}
\end{aligned} \) \( \text { Ascending order is } \frac{-3}{2}, \frac{-3}{4}, \frac{-3}{7} \)

Solutions To Try These

Find (1) \( \frac{-13}{7}+\frac{6}{7} \)

Solution: \( \frac{-13}{7}+\frac{6}{7}=\frac{-13+6}{7}=\frac{-7}{7}=-1 \)

2) \( \frac{19}{5}+\left(\frac{-7}{5}\right) \)

Solution: \( \frac{19}{5}+\frac{-7}{5}=\frac{19+(-7)}{5}=\frac{19-7}{5}=\frac{12}{5} \)

Solutions To Try These

Find:

(1) \( \frac{-3}{7}+\frac{2}{3} \)

Solution: LCM of 7 and 3 is 21.

\( \begin{aligned}
& \frac{-3}{7}=\frac{-3 \times 3}{7 \times 3}=\frac{-9}{21} \\
& \text { and } \frac{2}{3}=\frac{2 \times 7}{3 \times 7}=\frac{14}{21}
\end{aligned} \) \( \frac{-3}{7}+\frac{2}{3}=\frac{-9}{21}+\frac{14}{21}=\frac{-9+14}{21}=\frac{5}{21} \)

2) \( \frac{-5}{6}+\frac{-3}{11} \)

Solution: LCM of 6 and 11 is 66

\( \frac{-5}{6}=\frac{-5 \times 11}{6 \times 11}=\frac{-55}{66} \text { and } \) \( \begin{aligned}
&\frac{-3}{11}=\frac{-3 \times 6}{11 \times 6}=\frac{-18}{66}\\
&\frac{-5}{6}+\frac{(-3)}{11}=\frac{-55}{66}+\frac{(-18)}{66}=\frac{-55-18}{66}=\frac{-73}{66}
\end{aligned} \)

Solutions To Try These

What will be the additive inverse of \( \frac{-3}{9}, \frac{-9}{11}, \frac{5}{7} ? \)

Solution:

\( \text { Additive inverse of } \frac{-3}{9} \text { is } \frac{3}{9} \) \( \text { Additive inverse of } \frac{-9}{11} \text { is } \frac{9}{11} \) \( \text { Additive inverse of } \frac{5}{7} \text { is } \frac{-5}{7} \)

Solutions To Try These

Find 1) \( \frac{7}{9}-\frac{2}{5} \)

Solution:

\( \frac{7}{9}+\frac{(-2)}{5}=\frac{35 \div(-18)}{45}=\frac{35-18}{45}=\frac{17}{45} \)

2) \( 2 \frac{1}{5}-\frac{(-1)}{3} \)

Solution:

\( \begin{aligned}
&2 \frac{1}{5}-\frac{(-1)}{3}=\frac{11}{5}+\text { Additive inverse of } \frac{-1}{3}\\
&\frac{11}{5}+\frac{1}{3}=\frac{33 \div 5}{15}=\frac{38}{15}=2 \frac{8}{15}
\end{aligned} \)

Solutions To Try These

What will be

1) \( \frac{-3}{5} \times 7 ? \)

Solution: \( \frac{-3}{5} \times 7=\frac{(-3) \times 7}{5}=\frac{-21}{5}=-4 \frac{1}{5} \)

2) \( \frac{-6}{5} \times(-2) ? \)

Solution: \( \frac{-6}{5} \times(-2)=\frac{(-6) \times(-2)}{5}=\frac{12}{5}=2 \frac{2}{5} \)

Important Concepts Rational Numbers Class 7 HBSE

Solutions To Try These

Find:

(1) \( \frac{-3}{4} \times \frac{1}{7} \)

Solution: \( \frac{-3}{4} \times \frac{1}{7}=\frac{(-3) \times 1}{4 \times 7}=\frac{-3}{28} \)

(2) \( \frac{2}{3} \times \frac{-5}{9} \)

Solution: \( \frac{2}{3} \times \frac{-5}{9}=\frac{2 \times(-5)}{3 \times 9}=\frac{-10}{27} \)

Solutions To Try These

What will be the reciprocal of \( \frac{-6}{11} \) and \( \frac{-8}{5} \) ?

Solution:

\( \text { The reciprocal of } \frac{-6}{11} \text { is } \frac{-11}{6} \) \( \text { The reciprocal of } \frac{-8}{5} \text { is } \frac{-5}{8} \)

Solutions To Try These

Find:

(1) \( \frac{2}{3} \times \frac{-7}{8} \)

Solution:

\( \begin{aligned}
& \frac{2 \times(-7)}{3 \times 8}=\frac{-14}{24} \\
& =\frac{-14 \div 2}{24 \div 2}=\frac{-7}{12}
\end{aligned} \)

2) \( \frac{-6}{7} \times \frac{5}{7} \)

Solution: \( \begin{aligned}
&\frac{-6}{7} \times \frac{5}{7}=\frac{(-6) \times 5}{7 \times 7}\\
&=\frac{-30}{49}
\end{aligned} \)

Step-by-Step Solutions for Rational Numbers Class 7 Haryana Board

Haryana Board Class 7 Maths Solutions For Chapter 8  Exercise-8.2

1. Find the sum:

1) \( \frac{5}{4}+\left(\frac{-11}{4}\right) \)

Solution:

\( \begin{aligned}
& \frac{5}{4}+\left(\frac{-11}{4}\right)=\frac{5+(-11)}{4} \\
& =\frac{5-11}{4}=\frac{-6}{4}=\frac{-6 \div 2}{4 \div 2}=\frac{-3}{2}
\end{aligned} \)

2) \( \frac{5}{3}+\frac{3}{5} \)

\(\)

Solution: \( \frac{5}{3}+\frac{3}{5} \)

LCM of 3 and 5 is 15.

\( \frac{5}{3}=\frac{5 \times 5}{3 \times 5}=\frac{25}{15} \) \( \frac{3}{5}=\frac{3 \times 3}{5 \times 3}=\frac{9}{15} \) \( \frac{5}{3}+\frac{3}{5}=\frac{25}{15}+\frac{9}{15}=\frac{25+9}{15} \) \( =\frac{34}{15}=2 \frac{4}{15} \)

3) \( \frac{-9}{10}+\frac{22}{15} \)

Solution: \( \frac{-9}{10}+\frac{22}{15} \)

LCM of 10 and 15 is 30.

\( \frac{-9}{10}=\frac{-9 \times 3}{10 \times 3}=\frac{-27}{30} \) \( \frac{22}{15}=\frac{22 \times 2}{15 \times 2}=\frac{44}{30} \) \( \frac{-9}{10}+\frac{22}{5}=\frac{-27}{30}+\frac{44}{30} \) \( =\frac{-27+44}{30}=\frac{17}{30} \)

4) \( \frac{-3}{-11}+\frac{5}{9} \)

Solution:

LCM of 11 and 9 is 99.

\( \frac{-3}{-11}=\frac{3}{11}=\frac{3 \times 9}{11 \times 9}=\frac{27}{99} \) \( \frac{5}{9}=\frac{5 \times 11}{9 \times 11}=\frac{55}{99} \) \( \frac{-3}{-11}+\frac{5}{9}=\frac{27}{99}+\frac{55}{99}=\frac{27+55}{99}=\frac{82}{99} \)

5) \( \frac{-8}{19}+\frac{(-2)}{57} \)

Solution:

LCM of 19 and 57 is 57

\( \frac{-8}{19}=\frac{-8 \times 3}{19 \times 3}=\frac{-24}{57} \) \( \begin{aligned}
& \text { and } \frac{-2}{57}=\frac{-2 \times 1}{57 \times 1}=\frac{-2}{57} \\
& \frac{-8}{19}+\frac{(-2)}{57}=\frac{-24}{57}+\frac{(-2)}{57} \\
& =\frac{-24-2}{57}=\frac{-26}{57}
\end{aligned} \)

6) \( \frac{-2}{3}+0 \)

Solution: \( \frac{-2}{3}+\frac{0}{3}=\frac{-2+0}{3}=\frac{-2}{3} \)

7) \( -2 \frac{1}{3}+4 \frac{3}{5} \)

Solution: \( -2 \frac{1}{3}+4 \frac{3}{5}=\frac{-7}{3}+\frac{23}{5} \)

LCM of 3 and 5 is 15.

\( \begin{aligned}
& \frac{-7}{3}=\frac{-7 \times 5}{3 \times 5}=\frac{-35}{15} \\
& \frac{23}{5}=\frac{23 \times 3}{5 \times 3}=\frac{69}{15} \\
& \frac{-7}{3}+\frac{23}{5}=\frac{-35}{15}+\frac{69}{15}
\end{aligned} \) \( =\frac{-35+69}{15}=\frac{34}{15}=2 \frac{4}{15} \)

2. Find:

1) \( \frac{7}{24}-\frac{17}{36} \)

Solution: \( \frac{7}{24}-\frac{17}{36}=\frac{7}{24}=\left(\frac{-17}{36}\right) \)

LCM of 24 and 36 is 72.

\( \frac{7}{24}=\frac{7 \times 3}{24 \times 3}=\frac{21}{72} \text { and } \frac{17}{36}=\frac{17 \times 2}{36 \times 2}=\frac{34}{72} \) \( \begin{aligned}
\frac{7}{24}+\frac{(-17)}{36} & =\frac{21}{72}+\frac{(-34)}{72} \\
& =\frac{21+(-34)}{72}=\frac{-13}{72}
\end{aligned} \)

(2) \( \frac{5}{63}-\left(\frac{-6}{21}\right) \)

Solution:

\( \frac{5}{63}-\left(\frac{-6}{21}\right)=\frac{5}{63}+\frac{6}{21} \)

LCM of 63 and 21 is 63.

\( \begin{aligned}
\frac{5}{63} & =\frac{5 \times 1}{63 \times 1}=\frac{5}{63} \\
\frac{6}{21} & =\frac{6 \times 3}{21 \times 3}=\frac{18}{63} \\
& =\frac{5+18}{63}=\frac{23}{63}
\end{aligned} \)

Practice Problems Rational Numbers Class 7 Haryana Board

3) \( \frac{-6}{13}-\frac{(-7)}{15} \)

Solution: \( \frac{-6}{13}-\frac{(-7)}{15}=\frac{-6}{13}+\frac{7}{15} \)

LCM of 13 and 15 is 195.

\( \begin{aligned}
& \frac{-6}{13}=\frac{-6 \times 15}{13 \times 15}=\frac{-90}{195} \\
& \frac{7}{15}=\frac{7 \times 13}{15 \times 13}=\frac{91}{195} \\
& \frac{-6}{13}+\frac{7}{15}=\frac{-90}{195}+\frac{91}{195}
\end{aligned} \) \( =\frac{-90+91}{195}=\frac{1}{195} \)

4) \( \frac{-3}{8}-\frac{7}{11} \)

Solution: \( \frac{-3}{8}-\frac{7}{11}=\frac{-3}{11}+\left(\frac{-7}{11}\right) \)

LCM of 8 and 11 is 88.

\( \begin{aligned}
& \frac{3}{8}=\frac{3 \times 11}{8 \times 11}=\frac{33}{88} \text { and } \frac{7}{11}=\frac{7 \times 8}{11 \times 8}=\frac{56}{88} \\
& \frac{-3}{8}+\left(\frac{-7}{11}\right)=\frac{-33}{88}+\left(\frac{-56}{88}\right)
\end{aligned} \) \( \begin{aligned}
& =\frac{-33+(-56)}{88}=\frac{-89}{88} \\
& =-1 \frac{1}{88}
\end{aligned} \)

5) \( -2 \frac{1}{9}-6 \)

Solution: \( -2 \frac{1}{9}-6=\frac{-19}{9}-6=\frac{-19}{9}+\frac{(-6)}{1} \)

LCM of 9 and 1 is 9.

\( \begin{aligned}
& \frac{19}{9}=\frac{19 \times 1}{9 \times 1}=\frac{19}{9} \text { and } \frac{6}{1}=\frac{6 \times 9}{1 \times 9}=\frac{54}{9} \\
& -2 \frac{1}{9}-6=\frac{-19}{9}+\frac{(-6)}{1}
\end{aligned} \) \( \begin{aligned}
& =\frac{-19}{9}+\left(\frac{-54}{9}\right)=\frac{-19+(-54)}{9} \\
& =\frac{-73}{9}=-8 \frac{1}{9}
\end{aligned} \)

3) Find the product

1) \( \frac{9}{2} \times\left(\frac{-7}{4}\right) \)

Solution: \( \frac{9}{2} \times\left(\frac{-7}{4}\right) \)

\( \begin{aligned}
& =\frac{9 \times(-7)}{2 \times 4} \\
& =\frac{-63}{8} \\
& =-7 \frac{7}{8}
\end{aligned} \)

Haryana Board Class 7 Maths Exercise 8.1 Solutions

2) \( \frac{3}{10} \times(-9) \)

Solution: \( \frac{3}{10} \times(-9) \)

\( \begin{aligned}
& =\frac{3 \times(-9)}{10} \\
& =\frac{-27}{10} \\
& =-2 \frac{7}{10}
\end{aligned} \)

3) \( \frac{-6}{5} \times \frac{9}{11} \)

Solution: \( \frac{-6}{5} \times \frac{9}{11} \)

\( \begin{aligned}
& =\frac{-6 \times 9}{5 \times 11} \\
& =\frac{-54}{55}
\end{aligned} \)

4) \( \frac{3}{7} \times\left(\frac{-2}{5}\right) \)

Solution: \( \frac{3}{7} \times\left(\frac{-2}{5}\right) \)

\( \begin{aligned}
& =\frac{3 \times(-2)}{7 \times 5} \\
& =\frac{-6}{35}
\end{aligned} \)

5) \( \frac{3}{11} \times \frac{2}{5} \)

Solution: \( \frac{3}{11} \times \frac{2}{5} \)

\( \begin{aligned}
& =\frac{3 \times 2}{11 \times 5} \\
& =\frac{6}{55}
\end{aligned} \)

Key Questions in Rational Numbers for Class 7 HBSE

6) \( \frac{3}{-5} \times \frac{-5}{3} \)

Solution: \( \frac{3}{-5} \times \frac{-5}{3} \)

\( \begin{aligned}
& =\frac{3 \times(-5)}{(-5) \times 3} \\
& =\frac{-15}{-15}=1
\end{aligned} \)

4. Find the value of:

1) \( (-4) \div \frac{2}{3} \)

Solution: \( (-4) \div \frac{2}{3} \)

\( =\frac{-4}{1} \div \frac{2}{3} \) \( \begin{aligned}
& =\frac{-4}{1} \times \frac{3}{2} \\
& =\frac{(-4) \times 3}{1 \times 2} \\
& =\frac{-12}{2}
\end{aligned} \)

=-6

2) \( \frac{-3}{5} \div 2 \)

Solution:

\( \begin{aligned}
& \frac{-3}{5} \div 2=\frac{-3}{5} \div \frac{2}{1} \\
& =\frac{-3}{5} \times \frac{1}{2} \\
& =\frac{-3 \times 1}{5 \times 2} \\
& =\frac{-3}{10}
\end{aligned} \)

3) \( \frac{-4}{5} \div(-3) \)

Solution:

\( \begin{aligned}
& \frac{-4}{5} \div(-3)=\frac{-4}{5} \div \frac{(-3)}{1} \\
& =\frac{-4}{5} \times \frac{1}{-3}=\frac{-4 \times(-1)}{5 \times 3}=\frac{4}{15}
\end{aligned} \)

4) \( \frac{-1}{8} \div \frac{3}{4} \)

Solution:

\( \begin{aligned}
& \frac{-1}{8} \div \frac{3}{4}=\frac{-1}{8} \times \frac{4}{3} \\
& =\frac{-1 \times 4}{8 \times 3} \\
& =\frac{-4}{24} \\
& =\frac{-4 \div 4}{24 \div 4} \\
& =\frac{-1}{6}
\end{aligned} \)

5) \( \frac{-2}{13} \div \frac{1}{7} \)

Solution:

\( \begin{aligned}
& \frac{-2}{13} \div \frac{1}{7}=\frac{-2}{13} \times \frac{7}{1} \\
& =\frac{-2 \times 7}{13 \times 1} \\
& =\frac{-14}{13} \\
& =-1 \frac{1}{13}
\end{aligned} \)

6) \( \frac{-7}{12} \div\left(\frac{-2}{13}\right) \)

Solution:

\( \begin{aligned}
& \frac{-7}{12} \div\left(\frac{-2}{13}\right) \\
& =\frac{-7}{12} \times\left(\frac{-13}{2}\right) . \\
& =\frac{-7 \times 13}{12 \times(-2)} \\
& =\frac{-91}{-24} \\
& =\frac{91}{24}=3 \frac{19}{24}
\end{aligned} \)

7) \( \frac{3}{13} \div\left(\frac{-4}{65}\right) \)

Solution:

\( \begin{aligned}
& \frac{3}{13} \div\left(\frac{-4}{65}\right) \\
& =\frac{3}{13} \times \frac{65}{-4} \\
& =\frac{3 \times 65}{13 \times(-4)}=\frac{3 \times 5}{-4} \\
& =\frac{-15}{4}=-3 \frac{3}{4}
\end{aligned} \)

Additional Questions

Very Short Answer Questions

1. What is meant by a rational number?

Solution:

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.

2. How’ to write equivalent rational numbers?

Solution: If the numerator and denominator of a rational number are multiplied or divided by a non- zero integer we get a rational number which is said to be equivalent to the given rational number.

3. How to write rational numbers in the standard form?

Solution:

A rational number is said to be in the standard form if its denominator is a I positive integer and the numerator and denominator have no common factor
other than 1.

4) Reduce \( \frac{-75}{120}\) to the standard form.

Solution:

We have \( \begin{aligned}
& \frac{-75}{120}=\frac{-75+3}{120+3} \\
& =\frac{-25}{40}=\frac{-25+5}{40 \div 5}=\frac{-5}{8}
\end{aligned} \)

5. Compare \( \frac{-3}{5} \text { and } \frac{-1}{3} \)

Solution:

\( \begin{aligned}
& \frac{-3}{5}=\frac{-3 \times 3}{5 \times 3}=\frac{-9}{15} \\
& \frac{-1}{3}=\frac{-1 \times 5}{3 \times 5}=\frac{-5}{15}
\end{aligned} \) \( \begin{aligned}
& \text { we have } \frac{-9}{15}<\frac{-8}{15}<\frac{-7}{15}<\frac{-6}{15}<\frac{-5}{15} \\
& \frac{-3}{5}<\frac{-8}{15}<\frac{-7}{15}<\frac{-6}{15}<\frac{-1}{3}
\end{aligned} \) \( \frac{-3}{5}<\frac{-1}{3} \)

6. \( \text { Add } \frac{-7}{5} \text { and } \frac{-2}{3} \text {. } \)

Solution:

LCM of 5 and 3 to 15

\( \begin{aligned}
& \frac{-7}{5}=\frac{-7 \times 3}{5 \times 3}=\frac{-21}{15} \\
& \frac{-2}{3}=\frac{-2 \times 5}{3 \times 5}=\frac{-10}{15} \\
& \frac{-7}{5}+\frac{(-2)}{3}=\frac{-21}{15}+\frac{(-10)}{15} \\
& =\frac{-21 \cdot 10}{15}=\frac{-31}{15}
\end{aligned} \)

7. \( \text { Find } \frac{5}{7}-\frac{3}{8} \)

Solution: \( \frac{5}{7}-\frac{3}{8}=\frac{40-21}{56}=\frac{19}{56} \)

8. Find (1) \( \frac{-3}{5} \times 2 \)

Solution: \( \frac{-3 \times 2}{5}=\frac{-6}{5} \)

2) \( \frac{4}{9}+\frac{(-5)}{7} \)

Solution: \( \frac{4}{9}+\frac{(-5)}{7}=\frac{4}{9} \times \frac{7}{-5}=\frac{-28}{45} \)

9. Write five rational numbers which are smaller than \( \frac{5}{6} \).

Solution: \( \frac{5}{6}=\frac{50}{60} \)

We know that \( \frac{49}{60}, \frac{48}{60}, \frac{47}{60}, \frac{46}{60}, \frac{45}{60} \)………………… are smaller than \( \frac{50}{60} \).

\( \frac{49}{60}, \frac{48}{60}, \frac{47}{60}, \frac{46}{60}, \frac{45}{60} \)………… are any tive rational numbers smaller
than \( \frac{5}{6} \)

10. What number should \( \frac{-33}{16} \) by to get \( \frac{-11}{4} \)

Solution:

The number \( \frac{-33}{16} \) should be divided by to get \( \frac{-11}{4} \)

\( \begin{aligned}
& =\frac{-33}{16} \div \frac{-11}{4} \\
& =\frac{-33}{16} \times \frac{4}{-11} \\
& =\frac{3}{4}
\end{aligned} \)

Short Answer Questions

11. Subtract :

1) \( \frac{3}{4} \text { from } \frac{1}{3} \)

Solution: \( \frac{1}{3}-\frac{3}{4} \)

\( =\frac{(4 \times 1)-(3 \times 3)}{12}=\frac{4-9}{12}=\frac{-5}{12} \)

2) \( \frac{-32}{13} \text { from } 2 \)

Solution:

\( 2-\left(\frac{-32}{13}\right)=\frac{2}{1}+\frac{32}{13} \) \( =\frac{(13 \times 2)+(1 \times 32)}{13}=\frac{26+32}{13}=\frac{58}{13} \)

3) \( -7 \text { from } \frac{-4}{7} \)

Solution: \( \frac{-4}{7}-(-7)=\frac{-4}{7}+\frac{7}{1} \)

\( =\frac{(1 \times-4)+(7 \times 7)}{7}=\frac{-4+49}{7}=\frac{45}{7} \)

12. What numbers should be added to \( \frac{-5}{8} \) so as to get \( \frac{-3}{2} \) ?

Solution:

Suppose ‘x’ is the rational number to be

added to \( \frac{-5}{8} \text { to get } \frac{-3}{2} \)

Then, \( \frac{-5}{8}+x=\frac{-3}{2} \)

\( \Rightarrow x=\frac{-3}{2}-\left(\frac{-5}{8}\right) \) \( \begin{aligned}
& \Rightarrow x=\frac{-3}{2}+\frac{5}{8} \\
& \Rightarrow x=\frac{(4 \times-3) \times(1 \times 5)}{8} \\
& \Rightarrow x=\frac{-12+5}{8}=\frac{-7}{8}
\end{aligned} \) \( x=\frac{-7}{8} \)

13. The sum of two rational numbers is 8. If one of the numbers is \( \frac{-5}{6} \) then find the other.

Solution: It is given that

Sum of the two numbers = 8 and one of the numbers = \( \frac{-5}{6} \)

Suppose the other rational number is x. Since the sum is 8

\( \begin{aligned}
& \Rightarrow x+\left(\frac{-5}{6}\right)=8 \Rightarrow x=8-\left(\frac{-5}{6}\right) \\
& \Rightarrow x=\frac{8}{1}+\frac{5}{6} \\
& \Rightarrow x=\frac{(6 \times 8)+(1 \times 5)}{6} \\
& \Rightarrow x=\frac{48+5}{6}=\frac{53}{6}
\end{aligned} \)

The other number is \( \frac{53}{6} \)

14. Represent \( \frac{-13}{5} \) on the number line.

Solution:

Represent 13 of 5 on the number line

\( \frac{13}{5}=-2 \frac{3}{5}=-2-\frac{3}{5} \). This lies between – 2 and- 3 on the number line.

Divide the number line between- 2 and – 3 into 5 equal parts.

Mark 3rd part (numerator of rational part) counting from 2.

This is the place of the required rational number \( \frac{-13}{5} \)

15. Express each of the following decimal in the \( \frac{p}{q} \) form

  1. 0.57
  2. 0.176
  3. 1.00001
  4. 25.125

Solution:

1) \( 0.57=\frac{57}{100} \)

2) \( 0.176=\frac{176}{1000}=\frac{176 \div 8}{1000 \div 8}=\frac{22}{125} \)

3) \( 1.00001=\frac{100001}{100000} \)

4) \( \begin{aligned}
25.125 & =\frac{25125}{1000}=\frac{25125 \div 5}{1000 \div 5} \\
& =\frac{5025 \div 5}{200 \div 5}=\frac{1005 \div 5}{40 \div 5}=\frac{201}{8}
\end{aligned} \)

Long Answer Questions

16. Represent these numbers on the number line. (1) \( \frac{9}{7} \) (2) \( \frac{-7}{5} \)

Solution:

Represent these numbers on the number line,

(1) \( \frac{9}{7}=1 \frac{2}{7}=1+\frac{2}{7} \).This lies between 1 and 2 on the number line.

Divide the number line between1 and 2 into 7 equal parts. Mark 2nd part countingfrom1

This is the place of the required rational number \( \frac{9}{7} \) .

(2)

This is the place of the required rational number9 of 7

\( -\frac{7}{5}=-\left(1 \frac{2}{5}\right)=-\left(1+\frac{2}{5}\right)=-1+\left(\frac{-2}{5}\right) \)

This lies between -1 and -2 on the number line.

Divide the number line between -1 and -2 into 5 equal parts.. Mark 2nd part counting from -1.

This is the place of rational number \( \frac{-7}{5} \)

17. Find a rational number between \( \frac{2}{3} \text { and } \frac{3}{4} \)

Solution:

\( \frac{2}{3}=\frac{2 \times 4}{3 \times 4}=\frac{8}{12} \) [Hint: First write the rational numbers with equal denominators]

\( \frac{3}{4}=\frac{3 \times 3}{4 \times 3}=\frac{9}{12} \) (Converting them into rational numbers with same denominators)

Now

\( \frac{8}{12}=\frac{8 \times 5}{12 \times 5}=\frac{40}{60} \text { and } \quad \frac{9}{12}=\frac{9 \times 5}{12 \times 5}=\frac{45}{60} \)

Rational numbers between \( \frac{2}{3} \text { and } \frac{3}{4} \) may be taken as \( \frac{41}{60}, \frac{42}{60}, \frac{43}{60}, \frac{44}{60} \)

We can take any one of these.

(or)

We know that between two rational numbers x and y such that x < y, there is a rational \( \frac{x+y}{2} \)

i.e \( x<\frac{x+y}{2}<y \)

So, a rational number between \( \frac{2}{3} \text { and } \frac{3}{4} \text { is } \)

\( \frac{\frac{2}{3}+\frac{3}{4}}{2}=\frac{\frac{(4 \times 2)+(3 \times 3)}{12}}{2}=\frac{\frac{8+9}{12}}{2}=\frac{17}{12} \times \frac{1}{2}=\frac{17}{24} \)

Thus we have \( \frac{2}{3}<\frac{17}{24}<\frac{3}{4} \).

18. Find ten rational numbers between \( \frac{-3}{4} \text { and } \frac{5}{6} \).

Solution: \( \frac{-3}{4}=\frac{-3 \times 6}{4 \times 6}=\frac{-18}{24} \)

\( \frac{5}{6}=\frac{5 \times 4}{6 \times 4}=\frac{20}{24} \)

[Converting them to rational numbers with the same denominators]

Clearly -17, -16, -15, -14, -13, -12, -11, -10………………0,1,2,3………….are integers between numerators -18 and 20 of these equivalent rational numbers. Thus we have \( \frac{-17}{24}, \frac{-16}{24}, \frac{-15}{24}, \frac{-14}{24}, \frac{-13}{24}, \frac{-12}{24}, \frac{-11}{24}, \frac{-10}{24}, 0, \frac{1}{24} \) ……………………… as rational numbers between

\( \frac{-18}{24}\left(=\frac{-3}{4}\right) \text { and } \frac{20}{24}\left(=\frac{5}{6}\right) \)

We can take any ten of these as required rational numbers.

 

Workbook

Choose the correct answers :

1. Which of these is a negative rational number

  1. 0
  2. \( \frac{5}{7} \)
  3. \(\frac{-5}{7}\)
  4. \(\frac{-5}{-7}\)

Answer: 3

2. The HCF of 45 and 30 is 

  1. 15
  2. 30
  3. 45
  4. 1350

Answer: 1

3. \( \frac{3}{7}+\frac{(-6)}{7}= \)

  1. \( \frac{9}{7} \)
  2. \( \frac{-9}{7} \)
  3. \( \frac{3}{7} \)
  4. \( \frac{-3}{7} \)

Answer: 4

4. Additive inverse of \( \frac{-4}{7} \) is

  1. \( \frac{-7}{4} \)
  2. \( \frac{4}{7} \)
  3. \( \frac{-4}{7} \)
  4. \( \frac{-3}{7} \)

Answer: 2

5. LCM of 3 and 7 is

  1. 10
  2. 21
  3. 4
  4. 7

Answer: 2

6. How is \( \frac{7}{4} \) is expressed as a rational number with denominator 20?

  1. \( \frac{-70}{20} \)
  2. \( \frac{-35}{20} \)
  3. \( \frac{35}{20} \)
  4. B or C

Answer: 2

7. Express \( \frac{1}{4} \) and \( \frac{1}{3} \) with same denominator.

  1. \( \frac{4}{12} \text { and } \frac{3}{12} \)
  2. \( \frac{3}{12} \text { and } \frac{4}{12} \)
  3. \( \frac{4}{7} \text { and } \frac{3}{7} \)
  4. \( \frac{3}{7} \text { and } \frac{3}{7} \)

Answer: 2

8. \( -\frac{28}{84} \) can be expressed as a rational number as……..

  1. \( \frac{4}{7} \)
  2. \( \frac{-4}{12} \)
  3. \( \frac{4}{12} \)
  4. \( \frac{4}{-7} \)

Answer: 2

9. Which of the following is true?

Statement (1):

\( \frac{-9}{15}<\frac{-2}{3}<\frac{-4}{5} \)

Statement (2): \( \frac{-4}{5}<\frac{-2}{3}<\frac{-9}{15} \)

Statement (3): \( \frac{-2}{3}<\frac{-9}{15}<\frac{-4}{5} \)

  1. only (1)
  2. only (2)
  3. only (3)
  4. both (1) and (2)

Answer: 2

10. Which of the following are three rational numbers between -2 and -1?

  1. \( \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{5} \)
  2. \( \frac{-3}{2}, \frac{-7}{4}, \frac{-5}{4} \)
  3. \( \frac{-12}{5}, \frac{-22}{5}, \frac{12}{5} \)
  4. \( \frac{3}{2}, \frac{7}{4}, \frac{5}{4} \)

Answer: 2

11. A rational number between \( \frac{-2}{3} \text { and } \frac{1}{4} \) is…………

  1. \( \frac{5}{12} \)
  2. \( \frac{-5}{12} \)
  3. \( \frac{5}{24} \)
  4. \( \frac{-5}{24} \)

Answer: 4

12. If \( \frac{p}{q} \) is the fractional form of 0.36 then p + q =…………..

  1. 15
  2. 17
  3. 19
  4. 21

Answer: 1

13. The denominator of afractionwhich equals to the decimal fraction of 0.125 is………….

  1. 900
  2. 1000
  3. 999
  4. 990

Answer: 3

14. 0.9 + 9.1 =……….

  1. 9.91
  2. 9.19
  3. 10.1
  4. 10.1

Answer: 3

15. The reciprocal of 9 lies in the number system…….

  1. N
  2. W
  3. Z
  4. N and W

Answer: 3

16. The sum of two rational numbers is 8 and one of them is \( \frac{-5}{6} \).Then the second number is……………….

Answer: 1

17. Which of the rational numbers

\( \frac{-11}{28}, \frac{-5}{7}, \frac{-9}{14}, \frac{-29}{42} \) is the greatest?

  1. \( \frac{-11}{28} \)
  2. \( \frac{-5}{7} \)
  3. \( \frac{-9}{14} \)
  4. \( \frac{-29}{42} \)

Answer: 1

18. \( \frac{7}{8}-\frac{2}{3}= \) = ……………..

Answer: 2

19. \( \text { If } \frac{x}{9}=\frac{4}{x} \text { then } x= \)…………..

Answer: 4

20. Which of the following is not a rational number ?

  1. \( \frac{-2}{3} \)
  2. -0.3
  3. π
  4. 0

Answer: 3

21. Rama : \( \frac{5}{3} \) is a rational number and 5 is only a natural number.

Shyama: Both \( \frac{5}{3} \) and 5 are rational numbers.

Which of the statements are true?

  1. Both Rama and Shyama
  2. Only Rama
  3. Only Shyama
  4. Neither Rama nor Shyama

Answer: 3

22. Which of the following is different among the following rationals ?

  1. \( \frac{1}{7} \)
  2. \( \frac{2}{3} \)
  3. \( \frac{27}{8} \)
  4. \( \frac{145}{6} \)

Answer: 3

23. 0.4 + 0.3 + 0.2 =………

  1. 0.432
  2. 0.432
  3. 0.1
  4. 1

Answer: 4

24. \( \frac{2 . \overline{9}}{4 . \overline{9}}=\)……….

  1. \( \frac{1}{2} \)
  2. \( \frac{3}{5} \)
  3. 1
  4. not defined

Answer: 2

25. A bus is moving at an average speed of \( 60 \frac{2}{5} \)km/hr.How much distance it will cover in \( 7 \frac{1}{2} \)

  1. 423 km
  2. 433 km
  3. 443 km
  4. 453 km

Answer: 4

26. The area of a rectangular park whose length is \( 36 \frac{3}{5} \)m and breadth is \( 16 \frac{2}{3} \)m………

  1. 1830 m²
  2. 1220 m²
  3. 610 m²
  4. 305 m²

Answer: 3

27.

  1. 10x = 157.3232………
  2. 1000 x = 15732.3232………
  3. Subtracting we get x = \( x=\frac{15575}{990} \)
  4. Let x = 15.732
    Arrange the steps in order to express 15.732 in \( x=\frac{p}{q} \)
  1. 2, 1, 3, 4
  2. 4, 2, 1, 3
  3. 3, 1, 2, 4
  4. 4, 2, 3, 1

Answer: 2

28. Identify the rational number A marked in the following number line.

  1. \( x=\frac{3}{7} \)
  2. \( x=\frac{4}{6} \)
  3. \( x=\frac{4}{7} \)
  4. \( x=\frac{5}{7} \)

Answer: 3

29. Write the rational numbers for the points labelled with letters P, Q, R, S in order on the number line

Write the rational numbers for the points labelled with letters P, Q, R, S in order on numberline

  1. \( \frac{-3}{2}, \frac{-5}{4}, \frac{-3}{4}, \frac{-1}{4} \)
  2. \( \frac{-1}{4}, \frac{-3}{4}, \frac{-5}{4}, \frac{-3}{2} \)
  3. \( \frac{6}{4}, \frac{5}{4}, \frac{3}{4}, \frac{1}{4} \)
  4. \( \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{6}{4} \)

Answer: 1

30. Which letter of the number indicates \( \frac{17}{5} \)?

Which letter of the number indicates 17 of 5 in the folowing

  1. A
  2. B
  3. C
  4. D

Answer: 2

Fill in the blanks :

31. All integers and fractions are…………..

Answer: rational numbers

32. Equivalent rational number for \( \frac{-3}{7} \) is………..

Answer: \( \frac{-6}{14} \)

33. The number…………. is neither a positive nor a negative rational number

Answer: zero

34. There are……….. number of rational numbers between any two rational numbers.

Answer: infinite

35. Both the numerator and the denominator of a rational number are positive then it is called a……………..

Answer: positive rational number

36. Match the following:

1. Reduce to standard form \( \frac{-3}{-15} \)    (   ) A) \( \frac{1}{4} \)

2.Which is greater \( \frac{-1}{4}, \frac{1}{4} \)   (   ) B) \( \frac{10}{9} \)

3. The additive inverse of \( \frac{5}{7} \)            (   ) C) \( \frac{-5}{7} \)

4) \( \frac{-2}{9} \times(-5)= \)                             (   ) D) \( \frac{-15}{2} \)

5) \( (-5) \div \frac{2}{3}= \)                                  (   ) E) \( \frac{1}{5} \)

Answer:

1. E 2. A 3. C 4. B 5. D

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