Question 13 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{9}{-13}\text{ or }\frac{7}{-12}$
Answer$\frac{9}{-13}\text{ or }\frac{7}{-12}$
$\Rightarrow\frac{9\times(-1)}{-13\times(-1)}\text{ or }\frac{7\times(-1)}{-12\times(-1)}$
$\Rightarrow\frac{-9}{13}\text{ or }\frac{-7}{12}$ (Making denominator positive)
$LCM$ of $13$ and $12 = 156$
$\therefore\frac{-9}{13}=\frac{-9\times12}{13\times12}=\frac{-108}{156}$
$\frac{-7}{12}=\frac{-7\times13}{12\times13}=\frac{-91}{156}$
It is clear that $\frac{-91}{156}\text{ or }\frac{-7}{12}\text{ or }\frac{7}{-12}$ is greater.
View full question & answer→Question 23 Marks
Evaluate: $\frac{-5}{-8}-\frac{-3}{4}$
Answer$\frac{-5}{-8}-\frac{-3}{4}$
$\frac{-5}{-8}=\frac{-5\times(-1)}{-8\times(-1)}=\frac{5}{8}$
$LCM$ of $8$ and $4 = 8$
$\therefore\frac{-3}{4}=\frac{-3\times2}{4\times2}=\frac{-6}{8}$
$\therefore\frac{-5}{-8}-\frac{-3}{4}$
$=\frac{5}{8}-\Big(\frac{-6}{8}\Big)$
$=\frac{5}{8}+\frac{6}{8}$
$=\frac{5+6}{8}$
$=\frac{11}{8}$
View full question & answer→Question 33 Marks
What should be added to $\Big(\frac{-13}{4}+\frac{-3}{8}\Big)$
Answer$1-\Big(\frac{-13}{4}+\frac{-3}{8}\Big)$
$LCM$ of $4$ and $8 = 8$
$\therefore1=\frac{1\times8}{1\times8}=\frac{8}{8}$
$\frac{-13}{4}=\frac{-13\times2}{4\times2}=\frac{-26}{8}$
$\therefore1-\Big(\frac{-13}{4}+\frac{-3}{8}\Big)$
$=\frac{8}{8}-\Big(\frac{-26}{8}+\frac{-3}{8}\Big)$
$=\frac{8}{8}+\frac{26}{8}+\frac{3}{8}$
$=\frac{8+26+3}{8}=\frac{37}{8}$
View full question & answer→Question 43 Marks
By what number should $\frac{-44}{9}$ be divided to get $\frac{-11}{3}?$
AnswerLet the required number be x$\frac{-44}{9}\div\text{x}=\frac{-11}{3}$
$\Rightarrow\text{x}=\frac{44}{9}\times\frac{3}{-11}$
$\Rightarrow\text{x}=\frac{4}{3}$
View full question & answer→Question 53 Marks
Find five rational numbers between $-3$ and $-2.$
Answer5 rational numbers between $-3$ and $-2$
We can write $-3=\frac{-3\times6}{6}=\frac{-18}{6}$
$\Rightarrow-2=\frac{-2\times6}{6}=\frac{-12}{6}$
$\Rightarrow-2=\frac{-2\times6}{6}=\frac{-12}{6}$
$\Rightarrow\frac{-18}{6}<\frac{-12}{6}$
$\Rightarrow\frac{-18}{6}<\frac{-17}{6}<\frac{-16}{6}<\frac{-15}{6}$
$<\frac{-14}{6}<\frac{-13}{6}<\frac{-12}{6}$
$\therefore$ Five rational number between $-3$ and $-2$ will be $\frac{-17}{6},\frac{16}{6},\frac{-15}{6},\frac{-14}{6}\frac{-13}{6}$
View full question & answer→Question 63 Marks
Add the following rational numbers: $\frac{-9}{24}\text{and}\frac{-1}{18}$
Answer$\frac{-9}{24}\text{and}\frac{-1}{18}$
$LCM$ of $24$ and $18 = 72$
$\therefore\frac{-9}{24}=\frac{-9\times3}{24\times3}=\frac{-27}{72}$
$\frac{-1}{18}=\frac{-1\times4}{18\times4}=\frac{-4}{72}$
$\therefore\frac{-9}{24}+\frac{-1}{18}=\frac{-27}{72}+\frac{-4}{72}$
$=\frac{-27+(-4)}{72}=\frac{-27-4}{72}$
$=\frac{-31}{72}$
View full question & answer→Question 73 Marks
Subtract:
$\frac{-5}{6}\text{ from }\frac{1}{3}$
Answer$\frac{-5}{6}\text{ from }\frac{1}{3}$
$Lcm$ of $6$ and $3 = 6$
$\therefore\frac{1}{3}=\frac{1\times2}{3\times2}=\frac{2}{6}$
$\therefore\frac{1}{3}-\Big(\frac{-5}{6}\Big)=\frac{1}{3}+\frac{5}{6}$
$=\frac{2}{6}+\frac{5}{6}=\frac{2+5}{6}=\frac{7}{6}$
View full question & answer→Question 83 Marks
The sum of two rational numbers is $\frac{-3}{8}$ If one of them is $\frac{3}{16}$ find the other.
AnswerSum of two numbers $=\frac{-3}{8}$
One number $=\frac{3}{16}$
$\therefore$ second number $=\frac{-3}{8}-\frac{3}{16}$
$(LCM$ of $8$ and $16 = 16)$
$=\frac{-6-3}{16}=\frac{-9}{16}$
View full question & answer→Question 93 Marks
Add the following rational numbers: $-4\text{ and }\frac{1}{2}$
Answer$-4\text{ and }\frac{1}{2}$
$\frac{-4}{1}=\frac{-4\times2}{1\times2}=\frac{-8}{2}$
$\therefore \ -4+\frac{1}{2}=\frac{-8}{2}+\frac{1}{2}$
$=\frac{-8+1}{2}=\frac{-7}{2}$
View full question & answer→Question 103 Marks
A bus is moving at an average speed of $46\frac{2}{3}\text{km/h}$ How much distance will it cover in $2\frac{2}{5}\text{hours?}$
AnswerSpeed of bus $46\frac{2}{3}\text{km/h}$
$\frac{140}{3}\text{km/h}$
Distance covered in $2\frac{2}{5}\text{hours}$
$=\frac{140}{3}\times2\frac{2}{5}\text{km}$
$=\frac{140}{3}\times\frac{12}{5}\text{km}=112\text{km}.$
View full question & answer→Question 113 Marks
The sum of two rational numbers is $\frac{4}{21}$ If one of them is $\frac{5}{7}$ find the other.
AnswerSum of two numbers $=\frac{4}{21}$
One number $=\frac{5}{7}$
$\therefore$ second number $=\frac{4}{21}-\frac{5}{7}$
$(LCM$ of $21$ and $7 = 21)$
$=\frac{4-15}{21}=\frac{-11}{21}$
View full question & answer→Question 123 Marks
Simplify: $2+\frac{-1}{2}+\frac{-3}{4}$
Answer$2+\frac{-1}{2}+\frac{-3}{4}$
$LCM$ of $2, 4 = 4$
$\therefore\frac{2}{1}=\frac{2\times4}{1\times4}=\frac{8}{4}$
$\frac{-1}{2}=\frac{-1\times2}{2\times2}=\frac{-2}{4}$
$\frac{-3}{4}=\frac{-3}{4}$
$\therefore \ 2+\frac{-1}{2}+\frac{-3}{4}$
$=\frac{8}{4}+\frac{-2}{4}+\frac{-3}{4}$
$=\frac{8+(-2)+(-3)}{4}$
$=\frac{8-2-3}{4}=\frac{8-5}{4}$
$=\frac{3}{4}$
View full question & answer→Question 133 Marks
Simplify:
$\Big(\frac{16}{15}\times\frac{-25}{8}\Big)+\Big(\frac{-14}{27}\times\frac{6}{7}\Big)$
Answer$\Big(\frac{16}{15}\times\frac{-25}{8}\Big)+\Big(\frac{-14}{27}\times\frac{6}{7}\Big)$
$=\frac{16\times(-25)}{15\times8}+\frac{-14\times6}{27\times7}$
$=\frac{2\times(-5)}{1\times1}+\frac{-2\times2}{9\times1}$
$=\frac{-10}{3}+\frac{-4}{9}=\frac{-30+(-4)}{9}$
$=\frac{-30-4}{9}$
$(LCM$ of $3, 9 = 9)$
$=\frac{-34}{9}$
View full question & answer→Question 143 Marks
Simplify: $\frac{16}{-21}\times\frac{-14}{5}$
Answer$\frac{16}{-21}\times\frac{-14}{5}$$\frac{16}{-21}=\frac{16\times(-1)}{-21\times(-1)}$
$=\frac{-16}{21}$
$\therefore\frac{-16}{21}\times\frac{14}{5}$
$=\frac{(-16)\times(-14)}{21\times5}$
$=\frac{(-16)\times(-2)}{3\times5}$
$=\frac{32}{15}$
View full question & answer→Question 153 Marks
Evaluate: $\frac{4}{9}-\frac{2}{-3}$
Answer$\frac{4}{9}-\frac{2}{-3}$ $LCM$ of $9$ and $3 = 9$
$\therefore\frac{2}{-3}=\frac{2\times(-1)}{-3\times(-1)}=\frac{-2}{3}$
$=\frac{-2\times3}{3\times3}=\frac{-6}{9}$
$\therefore\frac{4}{9}-\frac{2}{-3}=\frac{4}{9}-\Big(\frac{-6}{9}\Big)$
$=\frac{4}{9}+\frac{6}{9}$ $=\frac{4+6}{9}$
$=\frac{10}{9}$
View full question & answer→Question 163 Marks
Subtract:
$5\text{ from }\frac{-3}{5}$
Answer $5\text{ from }\frac{-3}{5}$
$\frac{5}{1}=\frac{5\times5}{1\times5}=\frac{25}{5}$
$\therefore\frac{-3}{5}-\frac{5}{1}=\frac{-3}{5}-\frac{25}{5}$
$=\frac{-3-25}{5}=\frac{-28}{5}$
View full question & answer→Question 173 Marks
Simplify: $\frac{-8}{15}+\frac{2}{3}$
Answer$\frac{-8}{15}+\frac{2}{3}$
$=\frac{-8}{15}+\frac{2\times(-1)}{-3\times(-1)}$
$=\frac{-8}{15}+\frac{-2}{3}$
$LCM$ of $15, 3 = 15$
$\frac{-2}{3}=\frac{-2\times5}{3\times5}=\frac{-10}{15}$
$\therefore \ \frac{-8}{15}+\frac{-2}{3}=\frac{-8}{15}+\frac{-10}{15}$
$=\frac{-8+(-10)}{15}=\frac{-8-10}{15}$
$=\frac{-18}{15}=\frac{-18\div3}{15\div3}=\frac{-6}{5}$
View full question & answer→Question 183 Marks
Subtract: $\frac{-13}{9}\text{ from }0$
Answer$\frac{-9}{7}\text{ from }-1$$0-\Big(\frac{-13}{9}\Big)=0+\frac{13}{9}$
$=\frac{13}{9}$
View full question & answer→Question 193 Marks
Simplify: $\Big(\frac{13}{8}\times\frac{12}{13}\Big)+\Big(\frac{-4}{9}\times\frac{3}{-2}\Big)$
Answer$\Big(\frac{13}{8}\times\frac{12}{13}\Big)+\Big(\frac{-4}{9}\times\frac{3}{-2}\Big)$
$\Big\{\frac{3}{-2}=\frac{3\times(-1)}{-2\times(-1)}=\frac{-3}{2}\Big\}$
$\Big(\frac{13}{8}\times\frac{12}{13}\Big)+\Big(\frac{-4}{9}\times\frac{-3}{2}\Big)$
$=\frac{13\times12}{8\times12}+\frac{(-4)\times(-3)}{9\times2}$
$=\frac{3}{2}+\frac{(-2)+(-1)}{3\times1}=\frac{3}{2}+\frac{2}{3}$ $(LCM$ of $2$ and $3 = 6)$ $=\frac{9+4}{6}=\frac{13}{6}$
View full question & answer→Question 203 Marks
Subtract: $\frac{5}{9}\text{ from }\frac{-2}{3}$
Answer$\frac{5}{9}\text{ from }\frac{-2}{3}$$LCM$ of $9$ and $3 = 9$
$\therefore\frac{-2}{3}=\frac{-2\times3}{3\times3}=\frac{-6}{9}$
$\frac{-2}{5}-\frac{5}{9}=\frac{-6}{9}-\frac{5}{9}$
$=\frac{-6-5}{9}=\frac{-11}{9}$
View full question & answer→Question 213 Marks
Multiply: $\frac{25}{-9}\text{ by }\frac{3}{-10}$
Answer$\frac{25}{-9}\text{ by }\frac{3}{-10}$
$=\frac{25}{-9}\times\frac{3}{-10}=\frac{25}{-9}$
$=\frac{25\times(-1)}{-9\times(-1)}=\frac{-25}{9}$
$\frac{3}{-10}=\frac{3\times(-1)}{-10\times(-1)}$
$=\frac{-3}{10}=\frac{3\times(-1)}{-10\times(-1)}$
$=\frac{-3}{10}=\frac{(-25)\times(-3)}{9\times10}$
$=\frac{(-5)\times(-1)}{3\times2}=\frac{5}{6}$
View full question & answer→Question 223 Marks
Simplify: $\Big(\frac{6}{55}\times\frac{-22}{9}\Big)+\Big(\frac{-26}{125}\times\frac{-10}{39}\Big)$
Answer$\Big(\frac{6}{55}\times\frac{-22}{9}\Big)+\Big(\frac{-26}{125}\times\frac{-10}{39}\Big)$
$=\frac{6\times(-22)}{55\times9}+\frac{-26\times(-10)}{125\times39}$
$=\frac{2\times(-2)}{5\times3}+\frac{2\times(-2)}{25\times3}$
$=\frac{-4}{15}-\frac{-4}{75}=\frac{-4}{15}+\frac{4}{75}$ $(LCM$ of $15$ and $75 = 75)$
$=\frac{-20+4}{75}$$=\frac{-16}{75}$
View full question & answer→Question 233 Marks
Evaluate: $\frac{3}{4}-\frac{4}{5}$
Answer$\frac{3}{4}-\frac{4}{5}$
$LCM$ of $4$ and $5 = 20$
$\frac{3}{4}=\frac{3\times5}{4\times5}=\frac{15}{20}$
$\frac{4}{5}=\frac{4\times4}{5\times4}=\frac{16}{20}$
$\therefore\frac{3}{4}-\frac{4}{5}$
$=\frac{15}{20}-\frac{16}{20}$
$=\frac{15-16}{20}$$=\frac{-1}{20}$
View full question & answer→Question 243 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{5}{9}\text{ or }\frac{-3}{-8}$
Answer$\frac{5}{9}\text{ or }\frac{-3}{-8}$
$\Rightarrow\frac{5}{9}\text{ or }\frac{-3\times(-1)}{-8\times(-1)}$
$\Rightarrow\frac{5}{9}\text{ or }\frac{3}{8}$ (Making denominator positive)
$LCM$ of $9$ and $8 = 72$
$\therefore\frac{5}{9}=\frac{5\times8}{9\times8}=\frac{40}{72}$
$\frac{3}{8}=\frac{3\times9}{8\times9}=\frac{27}{72}$
It is clear that $40 > 27$
$\therefore\frac{40}{72}\text{ or }\frac{5}{9}\text{ is greater}.$
View full question & answer→Question 253 Marks
What should be added to $\frac{-7}{8}$ to get $\frac{5}{9}?$
AnswerLet the required number be $x$ $\text{x}+\Big(\frac{-7}{8}\Big)$
$\Rightarrow\text{x}=\frac{5}{9}-\Big(\frac{-7}{8}\Big)$
$=\frac{5}{9}+\frac{-7}{8}$
$LCM$ of $9$ and $8$ is $72$ $=\frac{40+63}{72}$
$\frac{103}{72}$
Hence, the other number is $\frac{103}{72}$
View full question & answer→Question 263 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{4}{-3}\text{ or }\frac{-8}{7}$
Answer$\frac{4}{-3}\text{ or }\frac{-8}{7}$
$\Rightarrow\frac{4\times(-1)}{-3\times(-1)}\text{ or }\frac{-8}{7}$
$\Rightarrow\frac{-4}{3}\text{ or }\frac{-8}{7}$ $LCM$ of $3$ and $7 = 21$
$\therefore\frac{-4}{3}=\frac{-4\times7}{3\times7}=\frac{-28}{21}$
$\frac{-8}{7}=\frac{-8\times3}{7\times3}=\frac{-24}{21}$
It is clear that $\frac{-24}{21}\text{ or }\frac{-8}{7}$ is greater.
View full question & answer→Question 273 Marks
Evaluate: $\frac{7}{11}-\frac{-4}{-11}$
Answer$\frac{7}{11}-\frac{-4}{-11}$
$\frac{-4}{-11}=\frac{-4\times(-1)}{-11\times(-1)}=\frac{4}{11}$
$\therefore\frac{7}{11}-\frac{4}{11}$
$=\frac{7-4}{11}$
$=\frac{3}{11}$
View full question & answer→Question 283 Marks
Evaluate: $\frac{7}{24}-\frac{19}{36}$
Answer$\frac{7}{24}-\frac{19}{36}$
$LCM$ of $24$ and $36 = 72$
$\therefore\frac{7}{24}=\frac{7\times3}{24\times3}=\frac{21}{72}$
$\frac{19}{36}=\frac{19\times2}{36\times2}=\frac{38}{72}$
$\therefore\frac{7}{24}-\frac{19}{36}$
$=\frac{21}{72}-\frac{38}{72}$
$=\frac{21-38}{72}$
$=\frac{-17}{72}$
View full question & answer→Question 293 Marks
By what rational number should $\frac{-8}{39}$ be multiplied to obtain $\frac{5}{26}?$
Answer Product of two numbers $=\frac{5}{26}$
One number $=\frac{-8}{39}$
$\therefore$ Second number $\frac{5}{26}\div\Big(\frac{-8}{39}\Big)$
$=\frac{5}{26}\times\frac{39}{8}$
$=\frac{5\times3}{2\times(-8)}$
$=\frac{15\times(-1)}{(-16)\times(-1)}$
$=\frac{-15}{16}$
View full question & answer→Question 303 Marks
Subtract:
$\frac{-32}{13}\text{ from }\frac{-6}{5}$
Answer$\frac{-32}{13}\text{ from }\frac{-6}{5}$
$LCM$ of $13$ and $5 = 65$
$\frac{-6}{5}=\frac{-6\times13}{5\times13}=\frac{-78}{65}$
$\frac{-32}{13}=\frac{-32\times5}{13\times5}=\frac{-160}{65}$
$\therefore\frac{-6}{5}-\Big(\frac{-32}{13}\Big)=\frac{-6}{5}+\frac{32}{13}$
$=\frac{-78}{65}+\frac{160}{65}=\frac{-78+160}{65}$
$=\frac{82}{65}$
View full question & answer→Question 313 Marks
Express the following rational numbers in standard form: $\frac{-46}{115}$
Answer$\frac{-46}{115}$ $H.C.F.$ of $46$ and $115$ is $23$ Dividing both the numerator and the denominator by $23$ $=\frac{-46\div23}{115\div23}$ $=\frac{-2}{5}$
View full question & answer→Question 323 Marks
Evaluate: $-3-\frac{4}{7}$
Answer$-3-\frac{4}{7}$ $=\frac{-3}{1}-\frac{4}{7}$ $LCM$ of $1$ and $7 = 7$
$\therefore\frac{-3}{1}=\frac{-3\times7}{1\times7}=\frac{-21}{7}$
$\therefore\frac{-21}{7}-\frac{4}{7}$
$=\frac{-21-4}{7}$ $=\frac{-25}{7}$
View full question & answer→Question 333 Marks
Add the following rational numbers: $\frac{27}{-4}\text{ and }\frac{-15}{8}$
Answer$\frac{27}{-4}\text{ and }\frac{-15}{8}$
$\frac{27}{-4}=\frac{27\times(-1)}{-4\times(-1)}=\frac{-27}{4}$
$\Rightarrow\frac{-27}{4}\times\frac{2}{2}=\frac{-54}{8}$
$\therefore \frac{-27}{4}+\frac{-15}{8}=\frac{-54}{8}+\frac{-15}{8}$
$=\frac{-54+(-15)}{8}$
$=\frac{-54-15}{8}=\frac{-69}{8}$
View full question & answer→Question 343 Marks
How many pieces, each of length $3\frac{3}{4}\text{m}$ can be cut from a rope of length $30$ metres?
AnswerTotal length of rape $= 30m$ Lenghth of one piece $=3\frac{3}{4}$
$=\frac{15}{4}\text{m}$
$\therefore$ No. of pieces of rope $=30\div\frac{15}{4}$
$=30\times\frac{4}{15}=2\times4$
$=8\text{ pieces}$
View full question & answer→Question 353 Marks
Express the following rational numbers in standard form: $\frac{-209}{247}$
Answer$\frac{-209}{247}$ $H.C.F.$ of $209$ and $247$ is $19$
Dividing both the numerator and the denominator by $19$ $=\frac{-209\div19}{247\div19}$ $=\frac{-11}{13}$
View full question & answer→Question 363 Marks
List five rational numbers between $-2$ and $-1.$
Answer$-2=\frac{-2\times6}{1\times6}=\frac{12}{6}$
$-1=\frac{-1\times6}{1\times6}=\frac{-6}{6}$
The interger between $-12$ and $-6$ are $-11, -10, -9, -8, -7$
Hence, fiver rational numbers between $-2$ and $-1$ are $\frac{-11}{6},\frac{10}{6},\frac{-9}{6},\frac{-8}{6},\frac{-7}{6}$
View full question & answer→Question 373 Marks
Subtract: $\frac{3}{4}\text{ from }\frac{1}{3}$
Answer$\frac{3}{4}\text{ from }\frac{1}{3}$ $Lcm$ of $4$ and $3 = 12$
$\therefore\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}$
$\frac{1}{3}=\frac{1\times4}{3\times4}=\frac{4}{12}$
$\therefore\frac{1}{3}-\frac{3}{4}=\frac{4}{12}-\frac{9}{12}$
$=\frac{4-9}{12}=\frac{-5}{12}$
View full question & answer→Question 383 Marks
Subtract: $\frac{-8}{9}\text{ from }\frac{-3}{5}$
Answer$\frac{-8}{9}\text{ from }\frac{-3}{5}$$Lcm$ of $9$ and $5 = 45$
$\therefore\frac{-8}{9}=\frac{-8\times5}{9\times5}=\frac{-40}{45}$
$\frac{-3}{5}=\frac{-3\times9}{5\times9}=\frac{-27}{45}$
$\therefore\frac{-3}{5}-\Big(\frac{-8}{9}\Big)=\frac{-3}{5}+\frac{8}{9}$
$=\frac{27}{45}+\frac{-27+40}{45}=\frac{13}{45}$
View full question & answer→Question 393 Marks
Find the cost of $3\frac{1}{3}\text{metre}$ metres of cloth at $\text{Rs. }40\frac{1}{2}\text{\metre.}$
AnswerCost of $1$ metre of cloth $=\text{Rs. }40\frac{1}{2}$
$=\text{Rs. }\frac{81}{2}$
$\therefore\text{ cost of }3\frac{1}{3}\text{m}\text{ cloth }$
$=\text{Rs. }\frac{81}{2}\times3\frac{1}{3}$
$=\text{Rs. }\frac{81}{2}\times\frac{10}{3}$
$=\text{Rs. }135$
View full question & answer→Question 403 Marks
A car is moving at an average speed of $56\frac{3}{5}\text{km/h}$ How much distance will it cover in $7\frac{1}{2}$ hours?
AnswerAverage speed $=56\frac{3}{5}\text{km/h}$
$=\frac{283}{5}\text{km/h}$ Time $=7\frac{1}{2}\text{h}=\frac{15}{2}\text{h}$
Distance = Speed $\times$ time $=\frac{283}{5}\times\frac{1.5}{2}$
$=424\frac{1}{2}\text{km}$
The car will cover $424\frac{1}{2}\text{km in }7\frac{1}{2}\text{h}$
View full question & answer→Question 413 Marks
Find five rational numbers between $\frac{-3}{5}\text{ and }\frac{-1}{2}$
Answer$LCM$ of $5$ and $2 = 10$
Now $\frac{-3}{5}=\frac{-3\times2}{5\times2}=\frac{-6}{10}$
$=\frac{-6\times6}{10\times6}=\frac{-36}{60}$
and $\frac{-1}{2}=\frac{-1\times5}{2\times5}=\frac{-5}{10}$
$=\frac{-5\times6}{10\times6}=\frac{-30}{60}$
and $\frac{-36}{60}<\frac{-35}{60}<\frac{-34}{60}<\frac{-33}{60}$
$<\frac{-32}{60}<\frac{-31}{60}<\frac{-30}{60}$
$\therefore$ five rational numbers between $\frac{-3}{5}$ and $\frac{-1}{2}$ are $\frac{-36}{60}<\frac{-34}{60}<\frac{-33}{60}<\frac{-32}{60}<\frac{-31}{60}$
View full question & answer→Question 423 Marks
Subtract: $\frac{-9}{7}\text{ from }-1$
Answer$\frac{-9}{7}\text{ from }-1$$\frac{-1}{1}=\frac{-1\times7}{1\times7}=\frac{-7}{7}$
$\therefore-1-\Big(\frac{-9}{7}\Big)=-1+\frac{9}{7}$
$=\frac{-7}{7}+\frac{9}{7}=\frac{-7+9}{7}=\frac{2}{7}$
View full question & answer→Question 433 Marks
Multiply: $\frac{-36}{5}\text{ by }\frac{20}{-3}$
Answer$\frac{-36}{5}\text{ by }\frac{20}{-3}$
$=\frac{20}{-3}=\frac{20\times(-1)}{-3\times(-1)}$
$=\frac{-20}{3}=\frac{20\times(-1)}{-3\times(-1)}$
$=\frac{-20}{3}$
$\therefore\frac{-36}{5}\times\frac{-20}{3}$
$=\frac{-36}{5}\times\frac{-20}{3}$
$=\frac{(-36)\times(-20)}{5\times3}$
$=\frac{(-12)\times(-4)}{1\times1}=\frac{48}{1}=48$
View full question & answer→Question 443 Marks
Simplify: $\frac{5}{-18}\times\frac{-9}{20}$
Answer$\frac{5}{-18}\times\frac{-9}{20}$
$\frac{5}{-18}=\frac{5\times(-1)}{-18\times(-1)}$
$=\frac{-5}{18}$
$\therefore\frac{-5}{18}\times\frac{-9}{20}$
$=\frac{(5)\times(-9)}{18\times20}$
$=\frac{(-1)\times(-1)}{2\times4}$
$=\frac{1}{8}$
View full question & answer→Question 453 Marks
Add the following rational numbers: $\frac{-2}{5}\text{ and }\frac{3}{4}$
Answer$\frac{-2}{5}\text{ and }\frac{3}{4}$
$\frac{-2}{5}=\frac{-2\times4}{5\times4}=\frac{-8}{20}$
$\frac{3}{4}=\frac{3\times5}{4\times5}=\frac{15}{20}$
$\therefore \ \frac{-2}{5}+\frac{3}{4}=\frac{-8}{20}+\frac{15}{20}$
$=\frac{-8+15}{20}=\frac{7}{20}$
View full question & answer→Question 463 Marks
Simplify: $\Big(\frac{-12}{7}\times\frac{-14}{27}\Big)+\Big(\frac{-8}{45}\times\frac{9}{16}\Big)$
Answer$\Big(\frac{-12}{7}\times\frac{-14}{27}\Big)+\Big(\frac{-8}{45}\times\frac{9}{16}\Big)$
$=\frac{(-12)\times(-14)}{7\times27}+\frac{-8\times9}{45\times16}$
$=\frac{-4\times(-2)}{1\times9}+\frac{-1\times1}{5\times2}$
$=\frac{8}{9}-\frac{-1}{10}=\frac{8}{9}+\frac{1}{10}$ $(LCM$ of $9$ and $10 = 90)$ $=\frac{80+9}{90}$
$=\frac{89}{90}$
View full question & answer→Question 473 Marks
Add the following rational numbers: $\frac{-5}{36}\text{ and }\frac{-7}{12}$
Answer$\frac{-5}{36}\text{ and }\frac{-7}{12}$
$=\frac{-7}{12}=\frac{-7\times3}{12\times3}=\frac{-21}{36}$
$( \because\text{LCM }\text{of }36,12=36)$
$\therefore\frac{-5}{36}+\frac{-5}{36}=\frac{-5}{36}+\frac{-21}{36}$
$=\frac{-5+(-21)}{36}=\frac{-5-21}{36}$
$=\frac{-26}{36}$ $($Dividing by $2)$ $=\frac{-13}{18}$
View full question & answer→Question 483 Marks
Subtract: $-7\text{ from }\frac{-4}{7}$
Answer$-7\text{ from }\frac{-4}{7}$
$\frac{-7}{1}=\frac{-7\times7}{1\times7}=\frac{-49}{7}$
$\therefore\frac{-4}{7}-\Big(\frac{-7}{1}\Big)=\frac{-4}{7}+\frac{7}{1}$
$=\frac{-4}{7}+\frac{49}{7}=\frac{-4+49}{7}$
$=\frac{45}{7}$
View full question & answer→Question 493 Marks
The product of two rational numbers is $\frac{-16}{2}.$ If one of the numbers is $\frac{-4}{3},$ find the other.
AnswerProduct of two numbers $=\frac{-16}{9}$ One number $=\frac{-16}{9}$
$\therefore$ Second number $\Big(\frac{-16}{9}\Big)\div\Big(\frac{-4}{3}\Big)$
$=\frac{-16}{9}\times\frac{3}{-4}$
$=\frac{-16\times3}{9\times(-4)}$
$=\frac{-4\times1}{3\times(-1)}$
$=\frac{-4}{-3}=\frac{4}{3}$
View full question & answer→Question 503 Marks
The sum of two rational numbers is $\frac{-4}{3}$ If one of them is $-5$ find the other.
AnswerSum of two numbers $=\frac{-4}{3}$
One number $-5$
$\therefore$ second number $=\frac{-4}{3}-(-5)$
$=\frac{-4}{3}+\frac{5}{1}$
$=\frac{-4+15}{3}=\frac{11}{3}$
View full question & answer→Question 513 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{7}{-9}\text{ or }\frac{-5}{8}$
Answer$\frac{7}{-9}\text{ or }\frac{-5}{8}$
$\Rightarrow\frac{7\times(-1)}{-9\times(-1)}\text{ or }\frac{-5}{8}$
$\Rightarrow\frac{-7}{9}\text{ or }\frac{-5}{8}$ (Making denominator positive)
$LCM$ of $9$ and $8 = 72$
$\therefore\frac{-7}{9}=\frac{-7\times8}{9\times8}=\frac{-56}{72}$
$\frac{-5}{8}=\frac{-5\times9}{8\times9}=\frac{-45}{72}$
It is clear that $\frac{-45}{72}\text{ or }\frac{-5}{8}$ is greater.
View full question & answer→Question 523 Marks
By what number should $\frac{-8}{15}$ be multiplied to get $24$?
AnswerThe required number $=24\div\Big(\frac{-8}{15}\Big)$
$=24\times\frac{15}{-8}$
$=24\times\frac{15\times(-1)}{-8\times(-1)}$
$=24\times\frac{(-15)}{-8}$
$=3\times(-15)$
$=-45$
View full question & answer→Question 533 Marks
What should be added to $\frac{-3}{8}$ to get $\frac{5}{12}?$
AnswerRequired number $=\frac{5}{12}-\Big(\frac{-3}{8}\Big)$
$=\frac{5}{12}+\frac{3}{8}=\frac{10+9}{24}$ $(LCM$ of $12, 8 = 24)$ $=\frac{19}{24}$
View full question & answer→Question 543 Marks
Subtract: $\frac{-18}{11}\text{ from }1$
Answer$\frac{-18}{11}\text{ from }1$$\frac{1}{1}=\frac{1\times11}{1\times11}=\frac{11}{11}$
$\therefore1-\Big(\frac{-18}{11}\Big)=1+\frac{18}{11}$
$\frac{11}{11}+\frac{18}{11}=\frac{11+18}{11}=\frac{29}{11}$
View full question & answer→Question 553 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{-12}{5}\text{ or }-3$
Answer$\frac{-12}{5}\text{ or }-3$
$\frac{-12}{5}\text{ or }\frac{-12}{5}$
$LCM$ of $5$ and $1 = 5$
$\frac{-3}{1}=\frac{-3\times5}{1\times5}=\frac{-15}{5}$
It is clear that $\frac{-12}{5}$ is greater.
View full question & answer→Question 563 Marks
The cost of $2\frac{1}{2}\text{ metres}$ of cloth is $\text{Rs. }78\frac{3}{4}.$ Find the cost of cloth per metre.
AnswerCost of $2\frac{1}{2}\text{m or }\frac{5}{2}\text{m}$ of Cloth $\text{Rs. }78\frac{3}{4}$
$=\text{Rs. }\frac{315}{4}$
$\therefore$ Cost of one metre of cloth $=\text{Rs. }\frac{315}{4}\div\frac{5}{2}$
$=\text{Rs. }\frac{315}{4}\times\frac{2}{5}$
$=\text{Rs. }\frac{63}{2}$
$=\text{Rs. }31\frac{1}{2}$
View full question & answer→Question 573 Marks
Which of the two rational numbers is greater in the following pairs? $\frac{4}{-5}\text{ or }\frac{-7}{8}$
Answer$\frac{4}{-5}\text{ or }\frac{-7}{8}$
$\Rightarrow\frac{4\times(-1)}{-5\times(-1)}\text{ or }\frac{-7}{8}$
$\Rightarrow\frac{-4}{5}\text{ or }\frac{-7}{8}$ (Making denominator positive) $LCM$ of $5$ and $8 = 40$
$\therefore\frac{-4}{5}=\frac{-4\times8}{5\times8}=\frac{-32}{40}$ and $\frac{-4}{5}=\frac{-7\times5}{5\times8}=\frac{-32}{40}$
It is clear that $\frac{-32}{40}\text{ or }\frac{4}{-5}\text{ or }\frac{4}{-5}$ is greater.
View full question & answer→Question 583 Marks
Evaluate:
$\frac{14}{15}-\frac{13}{20}$
Answer$\frac{14}{15}-\frac{13}{20}$
$LCM$ of $15$ and $20 = 60$
$\therefore\frac{14}{15}=\frac{14\times4}{15\times4}=\frac{56}{60}$
$\frac{13}{20}=\frac{13\times3}{20\times3}=\frac{39}{60}$
$\therefore\frac{14}{15}-\frac{13}{20}$
$=\frac{56}{60}-\frac{39}{60}$
$=\frac{56-39}{60}$
$=\frac{17}{60}$
View full question & answer→Question 593 Marks
The sum of two rational numbers is $-4$. If one of them is $\frac{-11}{6}$, find other.
AnswerLet the required number be $x$ $\text{x}+\Big(\frac{-11}{6}\Big)=-4$
$\Rightarrow\text{x}=(-4)-\Big(\frac{-11}{6}\Big)$
$=-4+\frac{11}{6}$
$=\frac{-24+11}{6}$
$=\frac{-13}{6}$
Hence, the other number is $\frac{-13}{6}$
View full question & answer→Question 603 Marks
Evaluate: $\frac{-5}{14}-\frac{-2}{7}$
Answer$\frac{-5}{14}-\frac{-2}{7}$
$LCM$ of $14$ and $7 = 14$
$\therefore\frac{-2}{7}=\frac{-2\times2}{7\times2}=\frac{-4}{14}$
$\therefore\frac{-5}{14}-\frac{-2}{7}$
$=\frac{-5}{14}-\Big(\frac{-4}{14}\Big)$
$=\frac{-5}{14}+\frac{4}{14}$
$=\frac{-5+4}{14}$
$=\frac{-1}{14}$
View full question & answer→Question 613 Marks
The product of two rational numbers is $-9$. If one of the numbers is $-12$, find the other.
AnswerProduct of two rational numbers $= -9$
One number $= -12$
Second number $= (-9) ÷ (-12)$
$=-9\times\frac{-1}{12}\Big\{\because\frac{1}{-12}=\frac{1\times(-1)}{-12\times(-1)}=\frac{-1}{12}\Big\}$
$=\frac{3}{4}$
View full question & answer→Question 623 Marks
If $24$ pairs of trousers of equal size can be prepared with $54m$ of cloth, what length of cloth is required for each pair of trousers?
AnswerCloth required for $24$ pairs of trousers $= 54m$
Cloth required for one pair $= (54 ÷ 24)m$ $=54\times\frac{1}{24}=\frac{9}{4}\text{m}$ $=2\frac{1}{4}\text{m}$
View full question & answer→Question 633 Marks
Add the following rational numbers:
$\frac{-5}{9}\text{ and }\frac{2}{3}$
Answer $\frac{-5}{9}\text{ and }\frac{2}{3}$
$\frac{2}{3}=\frac{2\times3}{3\times3}=\frac{6}{9}$
$\therefore\frac{-5}{9}+\frac{2}{3}=\frac{-5}{9}+\frac{6}{9}$
$=\frac{-5+6}{9}=\frac{1}{9}$
View full question & answer→Question 643 Marks
Simplify: $\frac{-9}{11}+\frac{2}{3}+\frac{-3}{4}$
Answer$\frac{-9}{11}+\frac{2}{3}+\frac{-3}{4}$
$LCM$ of $11, 3, 4 = 11 \times 3 \times 4 = 132$
$\therefore\frac{-9}{11}=\frac{-9\times12}{11\times12}=\frac{-108}{132}$
$\frac{2}{3}=\frac{2\times44}{3\times44}=\frac{88}{132}$
$\frac{-3}{4}=\frac{-3\times33}{4\times33}=\frac{-99}{132}$
$\therefore\frac{-9}{11}+\frac{2}{3}+\frac{-3}{4}$
$=\frac{-108}{132}+\frac{88}{132}+\frac{-99}{132}$
$=\frac{-108+88+(-99)}{132}$
$=\frac{-108+88-99}{132}$
$=\frac{-207+88}{132}$$=\frac{-119}{132}$
View full question & answer→Question 653 Marks
The product of two rational numbers is $10$. If one of the numbers is $-8$, find the other.
AnswerProduct of two number $= 10$ One number $= -8$
Second number $= 10 ÷ (-8)$ $=10\times\frac{1\times(-1)}{-8}$
$=10\times\frac{1\times(-1)}{8}$
$=\frac{-10}{8}-\frac{-5}{4}$
View full question & answer→Question 663 Marks
Express the following rational numbers in standard form:
$\frac{84}{-147}$
Answer$\frac{84}{-147}$Converting the number to a positive denominator:
$=\frac{84\times(-1)}{-147\times(-1)}=\frac{-84}{147}$
$H.C.F.$ of $84$ and $147$ is $21$
Dividing both the numerator and the denominator by $21$
$=\frac{-84\div(21)}{147\div(21)}$
$=\frac{-4}{7}$
View full question & answer→Question 673 Marks
Add the following rational numbers: $\frac{-7}{27}\text{ and }\frac{5}{18}$
Answer$\frac{-7}{27}\text{ and }\frac{5}{18}$
$\frac{-7}{27}=\frac{-7\times2}{27\times2}=\frac{-14}{54}$
$(\because\text{LCM }\text{of }27,18=54)$
$\frac{5}{18}=\frac{5\times3}{18\times3}=\frac{15}{54}$
$\therefore\frac{-7}{27}+\frac{5}{8}=\frac{-14}{54}+\frac{15}{54}$
$=\frac{-14+15}{54}=\frac{1}{54}$
View full question & answer→Question 683 Marks
Add the following rational numbers: $\frac{1}{9}\text{ and }\frac{4}{-27}$
Answer$\frac{1}{9}\text{ and }\frac{4}{-27}$
$\frac{-1}{9}=\frac{1\times(-1)}{-9\times(-1)}=\frac{-1}{9}$
$=\frac{-1\times3}{9\times3}=\frac{-3}{27}$
$\frac{4}{-27}=\frac{4\times(-1)}{-27\times(-1)}=\frac{-4}{27}$
$=\frac{-3+(-4)}{27}=\frac{-3-4}{27}$
$=\frac{-7}{27}$
View full question & answer→Question 693 Marks
What are rational numbers? Give examples of five positive and five negative rational numbers. Is there any rational number which is neither positive nor negative? Name it.
Answer$i.$ Rational numbers: The numbers of the form $\frac{\text{p}}{\text{q}}$ where $p$ and $q$ are integers and $\text{q}\neq0$ are called rational numbers.
$ii.$ Positive rational numbers: $\frac{3}{4},\frac{7}{8},\frac{15}{11}$$\frac{-3}{-5},\frac{-9}{-4}$
$iii.$ Negative rational numbers: $\frac{-5}{7},\frac{-3}{8}$ $\frac{11}{-5},\frac{13}{-7},\frac{-8}{3}$
Yes, there is one rational number $(0)$ which is neither positive nor negative.
View full question & answer→Question 703 Marks
The sum of two rational numbers is $-3$ If one of them is $\frac{-15}{6}$ find the other.
AnswerSum of two numbers $= -3$ One number $=\frac{-15}{7}$
$\therefore$ second number $=-3-\Big(\frac{-15}{7}\Big)$
$=\frac{-3}{1}+\frac{15}{7}$
$=\frac{-21+15}{7}=\frac{-6}{7}$
View full question & answer→Question 713 Marks
Divide the sum $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.
AnswerSum $=\frac{65}{12}+\frac{8}{3}=\frac{65+32}{12}=\frac{97}{12}$Difference $=\frac{65}{12}-\frac{8}{3}=\frac{65-32}{12}=\frac{33}{12}$
$=\frac{97}{12}\div=\frac{33}{12}$
$=\frac{97}{12}\times=\frac{12}{33}$
$=\frac{97}{33}$
View full question & answer→