Question 14 Marks
Verify the property: x × (y × z) = (x × y) × z by taking:
$\text{x}=\frac{1}{2},\text{y}=\frac{5}{-4},\text{z}=\frac{-7}{5}$
Answer$\text{x}\times(\text{x}\times\text{z})=(\text{x}\times\text{y})\times\text{z}$
$\text{x}=\frac{1}{2},\text{y}=\frac{5}{-4},\text{z}=\frac{-7}{5}$
$\text{L.H.S.}=\text{x}\times(\text{y}\times\text{z})=\frac{1}{2}\times\Big(\frac{5}{-4}\times\frac{-7}{5}\Big)$
$=\frac{1}{2}\times\Big(\frac{5\times(-7)}{-4\times5}\Big)$
$=\frac{1}{2}\times\Big(\frac{-7}{-4}\Big)=\frac{1}{2}\times\frac{7}{4}=\frac{1\times7}{2\times4}=\frac{7}{8}$
$\text{R.H.S}=(\text{x}\times\text{y})\times\text{z}$
$=\Big(\frac{1}{2}\times\frac{5}{-4}\Big)\times\frac{-7}{5}$
$=\frac{1\times5}{2\times(-4)}\times\frac{-7}{5}=\frac{5}{-8}\times\frac{-7}{5}$
View full question & answer→Question 24 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$\frac{4}{9}$ and $\frac{7}{-12}$
Answer$\frac{4}{9}$ and $\frac{7}{-12}$
$\frac{7\times(-1)}{-12\times(-1)}=\frac{-7}{12}$
Now $\frac{4}{9}+\frac{-7}{12}$
LCM of 9, 12 = 36
$\therefore\frac{4}{9}=\frac{4\times4}{9\times4}=\frac{16}{36}$
$\frac{-7}{12}=\frac{-7\times3}{12\times3}=\frac{-21}{36}$
$\therefore\frac{4}{9}+\frac{-7}{12}=\frac{16}{36}+\frac{-21}{36}$
$=\frac{16-21}{36}=\frac{-5}{36}$
and $\frac{-7}{12}+\frac{4}{9}=\frac{-21}{36}+\frac{16}{36}$
$=\frac{-21+16}{36}=\frac{-5}{36}$
$\therefore\frac{4}{9}+\frac{-7}{12}=\frac{-7}{12}+\frac{4}{9}$
View full question & answer→Question 34 Marks
Find (x + y) ÷ (x − y), if
$\text{x}=\frac{5}{4},\text{y}=\frac{-1}{3}$
Answer$\text{x}=\frac{5}{4},\text{y}=\frac{-1}{3}$
$\therefore\text{x}+\text{y}=\frac{5}{4}+\frac{-1}{3}$
$=\frac{15-4}{12}=\frac{11}{12}$
$\text{x}-\text{y}=\frac{5}{4}-\frac{-1}{3}$
$=\frac{15+4}{12}=\frac{19}{12}$
$\therefore (\text{x}+\text{y})\div(\text{x}-\text{y})=\frac{11}{12}\div\frac{19}{12}$
$=\frac{11}{12}\times\frac{12}{19}=\frac{11}{19}$
View full question & answer→Question 44 Marks
Verify the property: x × (y × z) = (x × y) × z by taking:
$\text{x}=\frac{-7}{3},\text{y}=\frac{12}{5},\text{z}=\frac{4}{9}$
Answer$\text{x}\times(\text{x}\times\text{z})=(\text{x}\times\text{y})\times\text{z}$
$\text{x}=\frac{-7}{3},\text{y}=\frac{12}{5},\text{z}=\frac{4}{9}$
$=\text{L.H.S.}=\text{x}\times(\text{y}\times\text{z})$
$=\frac{-7}{3}\times\Big(\frac{12}{5}\times\frac{4}{9}\Big)=\frac{-7}{3}\times\Big(\frac{4\times4}{5\times3}\Big)$
$=\frac{-7}{3}\times\frac{16}{15}=\frac{-7\times16}{3\times15}=\frac{-112}{45}$
$=\text{R.H.S.}=(\text{x}\times\text{y})\times\text{z}=\frac{-7}{3}\times\Big(\frac{4\times4}{5\times3}\Big)$
$=\frac{-7\times4}{1\times5}\times\frac{4}{9}=\frac{-28}{5}\times\frac{4}{9}$
$=\frac{-7\times4}{1\times5}\times\frac{4}{9}=\frac{-28}{5}\times\frac{4}{9}$
$=\frac{-28\times4}{5\times9}=\frac{-112}{45}$
$\therefore\text{L.H.S.}=\text{R.H.S.}$
View full question & answer→Question 54 Marks
Find (x + y) ÷ (x − y), if
$\text{x}=\frac{2}{5},\text{y}=\frac{1}{2}$
Answer$\text{x}=\frac{2}{5},\text{y}=\frac{1}{2}$
$\therefore\text{x}+\text{y}=\frac{2}{5}+\frac{1}{2}$
$=\frac{4+5}{10}=\frac{9}{10}$
$\text{x}-\text{y}=\frac{2}{5}-\frac{1}{2}$
$=\frac{4-5}{10}=\frac{-1}{10}$
$\therefore(\text{x}+\text{y})\div(\text{x}-\text{y})=\frac{9}{10}\div\frac{-1}{10}$
$=\frac{9}{10}\times\frac{10}{-1}$
$=\frac{9}{-1}=-9$
View full question & answer→Question 64 Marks
Find (x + y) ÷ (x − y), if
$\text{x}=\frac{2}{3},\text{y}=\frac{3}{2}$
Answer$\text{x}=\frac{2}{3},\text{y}=\frac{3}{2}$
$\therefore\text{x}+\text{y}=\frac{2}{3}+\frac{3}{2}$
$=\frac{4+9}{6}=\frac{13}{6}$
and $\text{x}-\text{y}=\frac{2}{3}-\frac{3}{2}$
$=\frac{4-9}{6}=\frac{-5}{6}$
Now $(\text{x}+\text{y})\div(\text{x}-\text{y})=\frac{13}{6}\div\frac{-5}{6}$
$=\frac{13}{6}\times\frac{6}{-5}$
$=\frac{13}{-5}=\frac{13\times(-1)}{-5(-1)}=\frac{-13}{5}$
View full question & answer→Question 74 Marks
Divide the sum of $\frac{65}{12}$ and $\frac{12}{7}$ by their difference.
AnswerSum of $\frac{65}{12}$ and $\frac{12}{7}=\frac{65}{12}\div\frac{12}{7}$
$=\frac{455+144}{84}=\frac{599}{84}$
and diffierence of $\frac{65}{12}\ \text{and}\ \frac{12}{7}=\frac{65}{12}-\frac{12}{7}$
$=\frac{455-144}{84}=\frac{311}{84}$
$\therefore$ Required number $=\frac{599}{84}\div\frac{311}{84}$
$=\frac{599}{84}\times\frac{84}{311}=\frac{599}{311}$
View full question & answer→Question 84 Marks
Find (x + y) ÷ (x − y), if
$\text{x}=\frac{1}{4},\text{y}=\frac{3}{2}$
Answer$\text{x}=\frac{1}{4},\text{y}=\frac{3}{2}$
$\text{x}+\text{y}=\frac{1}{4}+\frac{3}{2}$
$=\frac{1+6}{4}=\frac{7}{4}$
$\text{x}-\text{y}=\frac{1}{4}-\frac{3}{2}$
$=\frac{1-6}{4}=\frac{-5}{4}$
$\therefore(\text{x}+\text{y})\div(\text{x}-\text{y})=\frac{7}{4}\div\Big(\frac{-5}{4}\Big)$
$=\frac{7}{4}\times\frac{4}{-5}=\frac{7}{-5}$
$=\frac{7\times(-1)}{-5\times(-1)}=\frac{-7}{5}$
View full question & answer→Question 94 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$\frac{2}{-7}$ and $\frac{12}{-35}$
Answer$\frac{2}{-7}$ and $\frac{12}{-35}$
LCM of 7, 35 = 35
$\frac{2}{-7}=\frac{2\times(-5)}{-7\times(-5)}=\frac{-12}{35}$
$\frac{12}{-35}=\frac{12\times(-1)}{-35\times(-1)}=\frac{-12}{35}$
Now $\frac{-2}{7}+\frac{-12}{35}=\frac{-10}{35}+\frac{-12}{35}$
$=\frac{-10-12}{35}=\frac{-22}{35}$
and $\frac{-12}{35}+\frac{-2}{7}=\frac{-12}{35}+\frac{-10}{35}$
$=\frac{-12-10}{35}=\frac{-22}{35}$
$\therefore\frac{-2}{7}+\frac{-12}{35}=\frac{-12}{35}+\frac{-2}{7}$
View full question & answer→Question 104 Marks
Find (x + y) ÷ (x − y), if
$\text{x}=\frac{2}{7},\text{y}=\frac{4}{3}$
Answer$\text{x}=\frac{2}{7},\text{y}=\frac{4}{3}$
$\therefore\text{x}+\text{y}=\frac{2}{7}+\frac{4}{3}$
$=\frac{6+28}{21}=\frac{34}{21}$
$\text{x}-\text{y}=\frac{2}{7}-\frac{4}{3}$
$=\frac{6-28}{21}=\frac{-22}{21}$
$\therefore(\text{x}+\text{y})\div(\text{x}-\text{y})=\frac{34}{21}\div\frac{-22}{21}$
$=\frac{34}{21}\times\frac{21}{-22}$
$=\frac{17\times1}{1\times(-11)}=\frac{17}{-11}$
$=\frac{17\times(-1)}{-11\times(-1)}=\frac{-17}{11}$
View full question & answer→Question 114 Marks
Simplify:
$\frac{7}{9}+\frac{3}{-4}$
Answer$\frac{7}{9}+\frac{3}{-4}$
$\frac{3}{-7}=\frac{-3}{4}$
The LCM of the denominator 9 and 4 is 36.
Now,
We will express $\frac{7}{9}$ and $\frac{-3}{4}$ in the form in which it takes denominator as 36.
$\frac{7}{9}=\frac{7\times4}{9\times4}=\frac{28}{36}$
$\frac{-3}{4}=\frac{-3\times9}{4\times9}=\frac{-27}{24}$
So,
$\frac{28}{36}+\frac{-27}{36}$
$=\frac{28-27}{36}=\frac{1}{36}$
View full question & answer→Question 124 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$\frac{-11}{5}$ and $\frac{4}{7}$
Answer$\frac{-11}{5}$ and $\frac{4}{7}$
$\frac{-11}{5}+\frac{4}{7}$
LCM of the 5 and 7 = 35
$\therefore\frac{-11}{5}=\frac{-11\times7}{5\times7}=\frac{-77}{35}$
and $\frac{4}{7}=\frac{4\times5}{7\times5}=\frac{20}{35}$
$\therefore\frac{-11}{5}+\frac{4}{7}=\frac{-77}{35}+\frac{20}{35}$
$=\frac{-77+20}{35}=\frac{-57}{35}$
and $\frac{4}{7}+\frac{-11}{5}=\frac{20}{35}-\frac{77}{35}$
$=\frac{20-77}{35}=\frac{-57}{35}$
$\therefore\frac{-11}{5}+\frac{4}{7}=\frac{4}{7}+\frac{-11}{5}$
View full question & answer→Question 134 Marks
Verify the property: x × y = y × x by taking:
$\text{x}=0,\text{y=}\frac{-15}{8}$
Answer$\text{x}\times\text{y}=\text{y}\times\text{x}$
$\text{x}=0,\text{y=}\frac{-15}{8}$
$\text{L.H.S.}=\text{x}\times\text{y}=0\times\frac{-15}{8}=0$
$\text{R.H.S.}=\text{y}\times\text{x}=\frac{-15}{8}\times0=0$
$\therefore\text{L.H.S.}=\text{R.H.S.}$
View full question & answer→Question 144 Marks
Divide the sum of $\frac{-13}{5}$ and $\frac{12}{7}$ by the product of $\frac{-31}{7}$and$\frac{-1}{2}$.
AnswerSum of $\frac{-13}{5}$ and $\frac{12}{7}=\frac{-13}{5}\div\frac{12}{7}$
$=\frac{-91+60}{35}=\frac{-31}{35}$
Product of $\frac{-31}{7}$ and $\frac{-1}{2}=\frac{-31}{7}\times\frac{-1}{2}$
$\frac{-31\times(-1)}{7\times2}=\frac{31}{14}$
$\therefore$ Required number $=\frac{-31}{35}\div\frac{31}{14}$
$=\frac{-31}{35}\times\frac{14}{31}=\frac{-1\times2}{5\times1}=\frac{-2}{5}$
View full question & answer→Question 154 Marks
Find ten rational numbers between $\frac{3}{5}$ and $\frac{3}{4}.$
AnswerThe LCM of the denominators 5 and 4 of both the fraction is 20.
We can write:
$\frac{3}{5}=\frac{3\times4}{5\times4}=\frac{12}{20}$
$\frac{3}{4}=\frac{3\times5}{4\times5}=\frac{15}{20}$
Since the intergers between the numerators 12 and 15 are not sufficient, we will multiply both the fraction by 5.
$\frac{12}{20}=\frac{12\times5}{200\times5}=\frac{60}{100}$
$\frac{15}{20}=\frac{15\times5}{20\times5}=\frac{75}{100}$
There are 14 integers between 60 and 75. They are 61, 62, 63,........... 73 and 74.
Therefore, $\frac{60}{100},\frac{61}{100},\frac{62}{100},...........\frac{73}{100}$ and $\frac{74}{100}$ are the 14 fractions.
We can take any 10 of these.
View full question & answer→Question 164 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$-4$ and $\frac{4}{-7}$
Answer$-4$ and $\frac{4}{-7}$
$\frac{-4}{1}$ and $\frac{4}{-7}$
$\begin{pmatrix}\because\frac{4}{-7}=\frac{4\times(-1)}{-7\times(-1)}=\frac{-4}{7 }\end{pmatrix}$
LCM of 1, 7= 7
$\therefore\frac{-4}{1}=\frac{-4\times7}{1\times7}=\frac{-28}{7}$
and $\frac{-4}{7}$
$\therefore-4+\frac{-4}{7}=\frac{-28}{7}+\frac{-4}{7}$
$=\frac{-28-4}{7}=\frac{-32}{7}$
and $\frac{-4}{7}+(-4)=\frac{-4}{7}+\frac{-28}{7}$
$=\frac{-4-28}{7}=\frac{-32}{7}$
$\therefore-4+\frac{-4}{7}=\frac{-4}{7}+(-4)$
View full question & answer→Question 174 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$4$ and $\frac{-3}{5}$
Answer4 and $\frac{-3}{5}$
LCM of 1, 5 = 5
$\therefore\frac{4}{1}=\frac{4\times5}{1\times5}=\frac{20}{5}$
Now $\frac{4}{1}+\frac{-3}{5}=\frac{20}{5}+\frac{-3}{5}$
$=\frac{20-3}{5}=\frac{17}{5}$
and $\frac{-3}{5}+\frac{4}{1}=\frac{-3}{5}+\frac{20}{5}$
$=\frac{-3+20}{5}=\frac{17}{5}$
$\therefore4+\frac{-3}{5}=\frac{-3}{5}+4$
View full question & answer→Question 184 Marks
Simplify:
$\frac{1}{-12}+\frac{2}{-15}$
Answer$\frac{1}{-12}=\frac{2}{-15}$
$\frac{1}{-12}=\frac{-1}{12}$
$\frac{2}{-15}=\frac{-2}{15}$
The LCM of the denominators 12 and 15 is 60.
Now,
We will express $\frac{-1}{12}$ and $\frac{-2}{15}$ in the form in which it taken denominator as 60.
$\frac{-1}{12}=\frac{-1\times5}{12\times5}=\frac{-5}{60}$
$\frac{-2}{15}=\frac{-2\times4}{15\times4}=\frac{-8}{60}$
So,
$\frac{-5}{60}+\frac{-8}{60}$
$=\frac{-5-8}{60}=\frac{-13}{60}$
View full question & answer→Question 194 Marks
Verify commutativty of addition of rational numbers for the following pairs of rotional number:
$\frac{-3}{5}$ and $\frac{-2}{-15}$
Answer$\frac{-3}{5}$ and $\frac{-2}{-15}$
$\frac{-2}{-15}=\frac{-2\times(-1)}{-15\times(-1)}=\frac{2}{15}$
LCM of 5 and 15 - 15
$\therefore\frac{-3}{5}=\frac{-3\times3}{5\times3}=\frac{-9}{15}$
Now $\frac{-3}{5}+\frac{2}{15}=\frac{-9}{15}+\frac{2}{15}$
$\frac{-9+2}{15}=\frac{-7}{15}$
and $\frac{2}{15}+\frac{-3}{5}=\frac{2}{15}+\frac{-9}{15}$
$=\frac{2-9}{15}=\frac{-7}{15}$
$\therefore\frac{-3}{5}+\frac{2}{15}=\frac{2}{15}+\frac{-3}{5}$
View full question & answer→