Exercise 1(A)
Q1 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 1
(i) Find a rational numbers between $\frac{1}{2}\text{ and } \frac{3}{4}$
Sol :
$=\frac{1}{2}\left[\frac{1}{2} + \frac{3}{4}\right]$
$=\frac{1}{2}\left[\frac{2+3}{4}\right]$
$=\frac{1}{2} \times \frac{5}{4}$
$=\frac{5}{8}$
(ii) Find two rational numbers between 0.1 and 0.2
Sol :First$=\frac{1}{2}[0.1+0.2]$
$=\frac{1}{2} \times 0 \cdot 3$
=0.15
$=\frac{3}{20}$
Second$=\frac{1}{2}\left[\frac{3}{20}+\frac{2}{10}\right]$
$=\frac{1}{2}\left[\frac{3+4}{20}\right]$
$=\frac{1}{2} \times \frac{7}{20}$
$=\frac{7}{40}$
Numbers are $\frac{3}{20} \text{ and } \frac{7}{40}$
(iii) How many rational numbers can you find between two given rational numbers?
Sol :
There are infinite numbers of rational numbers between two rational numbers
Q2 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 2
Find two rational numbers between
(i) $\frac{4}{5}$ and $\frac{7}{13}$
Sol :
First$=\frac{1}{2}\left[\frac{4}{5}+\frac{7}{13}\right]$
$=\frac{1}{2}\left[\frac{52+35}{65}\right]$
$=\frac{87}{130}$
Second
$=\frac{1}{2}\left[\frac{87}{130}+\frac{7}{13}\right]$
$=\frac{1}{2}\left[\frac{87+10}{130}\right]$
$=\frac{1}{2}\left[\frac{157}{130}\right]$
$=\frac{157}{260}$
(ii) $\frac{3}{4}$ and $1 \frac{1}{5}$
Sol :
First$=\frac{1}{2}\left[\frac{3}{4}+\frac{6}{5}\right]$
$=\frac{1}{2}\left[\frac{15+24}{20}\right]$
$=\frac{1}{2}\left[\frac{39}{20}\right]$
$=\frac{39}{40}$
Second$\Rightarrow \frac{1}{2}\left[\frac{39}{40}+\frac{6}{5}\right]$
Rational number between 3 and $\frac{7}{2}$
$\Rightarrow \frac{1}{2}\left[\frac{39+118}{40}\right]$
$=\frac{1}{2} \times \frac{87}{40}$
$=\frac{87}{80}$
Q3 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 3
Find three rational numbers between 0 and 0.2
Sol :
(i) Rational number between 0 and 0.2
First$=\frac{1}{2}[0+0.2]$
=0.1
Second
$=\frac{1}{2}[0+0.1]$
$=\frac{1}{2}[0.1]$
=0.05
Third
$=\frac{1}{2}[0.1+0.2]$
$=\frac{1}{2}[0.3]$
=0.15
∴Three rational numbers are 0.05 , 0.1 and 0.15
Q4 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 4
Find three rational numbers between 3 and 4.
Sol :
Rational number between 3 and 4
$=\frac{1}{9}[3+4]$
$=\frac{1}{2} \times 7$
$=\frac{7}{2}$
$=\frac{1}{2}\left[3+\frac{7}{2}\right]$
$=\frac{1}{2} \times \frac{6+7}{2}$
$=\frac{13}{4}$
Rational number between $\frac{7}{2}$ and 4
$=\frac{1}{2}\left[\frac{2}{2}+4\right]$
$=\frac{1}{2}\left[\frac{7+8}{2}\right]$
$=\frac{1}{2} \times \frac{15}{2}$
$=\frac{15}{4}$
∴Three numbers are
$\frac{13}{4}, \frac{7}{2}$ and $\frac{15}{4}$
Q5 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 5
Find the rational number that is one seventh of the way from $1\frac{3}{4}$ to $4\frac{3}{8}$
Sol :
$=1 \frac{3}{4}$ to $4 \frac{3}{8}$
$=\frac{7}{4}$ to $\frac{35}{8}$
$=\frac{14}{8}$ to $\frac{35}{8}$
There are 21 terms between 14 and 35
i.e. $\frac{15}{8}, \frac{16}{8}, \frac{17}{8}, \frac{18}{8}\dots \frac{38}{8}$
$\Rightarrow \frac{1}{7}$ th of 21 terms
$=21 \times \frac{1}{7}$
=3rd
3rd term
$=\frac{17}{8} \text{ or } 2\frac{1}{8}$
Q6 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 6
Find four rational numbers between -1 and $-\frac{1}{2}$
Sol :
Rational number between $-1 \text{ and } \frac{1}{-2}$
$\left.=\frac{1}{2}\left[-1+\frac{1}{-2}\right)\right]$
$=\frac{1}{2}\left[ \frac{2+1}{-2} \right]$
$=\frac{1}{2}\left(-\frac{3}{2}\right)$
$=\frac{-3}{4}$
Rational number between -1 and $\left[\frac{-3}{4}\right]$
$=\frac{1}{2}\left[-1+\frac{(-3)}{4}\right]$
$=\frac{1}{2}\left[-1\frac{-3}{4}\right]$
$=\frac{1}{2}\left(\frac{-7}{4}\right)$
$=\frac{-7}{8}$
Rational number between $\frac{-3}{4}$ and $\frac{-1}{2}$
$=\frac{1}{2}\left[\frac{-3}{4}+\left(\frac{-1}{2}\right)\right]$
$=\frac{1}{2}\left[\frac{-3}{4}-\frac{1}{2}\right]$
$=\frac{1}{2}\left[\frac{-5}{4}\right]$
$=\frac{-5}{8}$
Rational number between $\frac{-5}{8}$ and $\frac{-1}{2}$
$=\frac{1}{2}\left[\frac{-5}{8}+\left(\frac{-1}{2}\right)\right]$
$=\frac{1}{2}\left[-\frac{5}{8}-\frac{1}{2}\right]$
$=\frac{1}{2}\left[\frac{-5-4}{8}\right]$
$=\frac{1}{2}\left[-\frac{9}{8}\right]$
$=\frac{-9}{16}$
Four numbers are
$\frac{-3}{4}, \frac{-7}{8}, \frac{-5}{8}, \frac{-9}{16}$
Q7 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 7
Express $\frac{12}{125}$ as decimal fraction.
Sol :
$\frac{12}{125}$
Divide 12 by 125
Q8 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 8
Find a vulgar fraction equivalent to 0.03.
Sol :
$0.0\overline{3}$
=0.033333
Let x=0.033333...
10x=0.33333...
100x=3.3333...
Subtract (1) and (2)
90x=3.00...
$\begin{aligned} x &=\frac{3}{90} \\ &=\frac{1}{30} \end{aligned}$
Q9 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 9
Express the following rational numbers in the form $\frac{p}{q},$p , q are integers , q≠0
(i) $6 . \overline{46}$=6.464646..
Sol :
Let x=6.4646464..(i)
100x=646.4646...(ii)
Subtract (i) from (ii)
99x=646-6
99x=640
$x=\frac{640}{99}$
(ii) $0 . 1\overline{36}$=0.136136363
Sol :
Let x=0.13636363...
10x=1.363636..(i)
100x=136.363636...(ii)
Subtracting (i) from (ii)
90x=135
$\begin{aligned} x &=\frac{135}{90} \\ &=\frac{27}{18} \end{aligned}$
(iii) $3.\overline{146}$
Sol :
Let x=$3.\overline{146}$
x=3.146146146...(i)
1000x=3146.146146.....(ii)
Subtracting (i) from (ii)
999x=3143
$x=\frac{3143}{999}$
(iv) $-5.\overline{12}$
Sol :
Let x=$-5.\overline{12}$
x=-5.121212....(i)
100x=-512.121212...(ii)
Subtracting (i) from (ii)
99x=-507
$x=\frac{-507}{99}=\frac{-169}{33}$
Q10 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 10
Write the terminating decimal numeral for the given rational numbers.
(i) $=\frac{7}{4}$
Sol :
(ii) $\frac{29}{50}$
Sol :
(iii) $\frac{17}{32}$
Sol :
Q11 | Ex-1A |Class 8 |Rational Numbers and Irrational Number | S.Chand | New Learning | Composite mathematics | myhelper
Question 11
Write the repeating decimal for each of the following and use a bar to shoe the repetend.
Sol :
$0.1111....=0.\overline{1}$
(ii) $\frac{-4}{3}$
Sol :
$-1.3333....=-1.\overline{3}$
(iii) $\frac{1}{6}$
Sol :
$0.1666....=0.1\overline{6}$
No comments:
Post a Comment