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Question 13 Marks
Simplify: $\frac{5}{3}-\frac{7}{6}+\frac{-2}{3}$
Answer
$\Big(\frac{5}{3}-\frac{7}{6}+\frac{-2}{3}\Big)$
$=\Big(​​\frac{5\times2}{3\times2}-\frac{7}{6}+\frac{-2\times2}{3\times2}\Big)$
$=\Big(\frac{10}{7}-\frac{7}{6}+\frac{-4}{6}\Big)$
$=\Big(\frac{10}{6}-\frac{7}{6}+\frac{-4}{6}\Big)$
$=\Big(\frac{-10-7-4}{6}\Big)$
$=\Big(\frac{10-11}{6}\Big)$
$=\Big(\frac{-1}{6}\Big)$
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Question 23 Marks
Simplifiy: $\frac{-5}{16}+\frac{7}{24}$
Answer
$\frac{-5}{16}+\frac{7}{24}$ $LCM$ of $16$ and $24$ is $48$
$\frac{-5}{16}=\frac{-5\times3}{16\times3}=\frac{-15}{48}$
$\frac{7}{24}=\frac{7\times2}{24\times2}=\frac{14}{48}$
$\frac{-5}{16}+\frac{7}{24}=\frac{-15}{48}+\frac{14}{48}$
$=\frac{14-15}{48}$
$=\frac{-1}{48}$
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Question 33 Marks
The sum of two numbers is $-8$ If one of the numbers is $\frac{-15}{7}$ find the other.
Answer
Let the required number be $x$. $\frac{-15}{7}+\text{x}=-8$
$\text{x}=-8-\frac{-15}{7}$
$\text{x}=-8+\frac{15}{7}$
$\text{x}=\frac{8\times7}{1\times7}+\frac{15}{7}$
$\text{x}=\frac{15-56}{7}$
$\text{x}=-\frac{41}{7}$
The required number is $-\frac{41}{7}$
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Question 43 Marks
What should be added to $\frac{-5}{11}$ so as to get $\frac{26}{33}?$
Answer
Let the required number be $x$ $-\frac{5}{11}+\text{x}=\frac{26}{33}$
$\text{x}=\frac{26}{33}-\frac{-5}{11}$
$\text{x}=\frac{26}{33}-\frac{5\times3}{11\times3}$
$\text{x}=\frac{26}{33}-\frac{-15}{33}$
$\text{x}=\frac{26+15}{33}$
$\text{x}=\frac{41}{33}$
The required number is $\frac{41}{33}$
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Question 53 Marks
Simplifiy: $\frac{8}{9}+\frac{-11}{6}$
Answer
$\frac{8}{9}+\frac{-11}{6}$
$\frac{8}{9}-\frac{11}{6}$
$LCM$ of $5$ and $10$ is $10$ $\frac{8}{9}=\frac{8\times2}{9\times2}=\frac{16}{18}$
$\frac{11}{6}=\frac{11\times3}{6\times3}=\frac{33}{18}$
$\frac{8}{9}+\frac{-11}{6}=\frac{16}{18}-\frac{33}{18}$
$=\frac{16-33}{18}$
$=\frac{-17}{18}$
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Question 63 Marks
Add and express the sum as a mixed fraction:
$\frac{24}{7}+\frac{-11}{4}$
Answer
$\frac{24}{7}+\frac{-11}{4}$
$LCM$ of $7$ and $4$ is $28$
$\frac{24}{7}=\frac{24\times4}{7\times4}=\frac{96}{28}$
$\frac{-11}{4}=\frac{-11\times7}{4\times7}=-\frac{77}{28}$
$\frac{24}{7}+\frac{-11}{4}=\frac{96}{28}+\frac{-77}{28}$
$=\frac{96}{28}-\frac{77}{28}$
$=\frac{96-77}{28}$
$=\frac{19}{28}$
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Question 73 Marks
The product of two rational numbers is $\frac{-8}{9}.$ If one of the numbers is $\frac{-4}{15}$ find the other.
Answer
Let the number to be found be $x$. $\text{x}\times\frac{-4}{15}=\frac{-8}{9}$
$\text{x}=\frac{-8}{9}\div\frac{-4}{15}$
$\Rightarrow\text{x}=\frac{-8}{9}\times\frac{15}{-4}$
$\Rightarrow\text{x}=\frac{-8\times15}{9\times-4}$
$\Rightarrow\text{x}=\frac{-120}{-36}$
$\Rightarrow\frac{120}{36}$
$\Rightarrow\frac{10}{3}$
Hence the number is $\text{x}=\frac{10}{3}$
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Question 83 Marks
Find $(x + y) + (x - y)$, if$\text{x}=\frac{2}{3},\text{y}=\frac{3}{2}$
Answer
$\text{x}=\frac{2}{3},\text{y}=\frac{3}{2}$
$(\text{x}+\text{y})\div(\text{x}-\text{y})$
$=\Big(\frac{2}{3}+\frac{3}{2}\Big)\div\Big(\frac{2}{3}-\frac{3}{2}\Big)$
$=\Big(\frac{4+9}{6}\Big)\div\Big(\frac{4-9}{6}\Big)$
$=\Big(\frac{4+9}{6}\Big)\times\Big(\frac{6}{4-9}\Big)$
$=\Big(\frac{4+9}{4-9}\Big)$
$=\Big(\frac{13}{-5}\Big)$
$=\Big(\frac{-13}{5}\Big)$
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Question 93 Marks
The cost of $2\frac{1}{3}$ metres of cloth is $75\frac{1}{4}$ Find the cost of cloth per metre.
Answer
$\text{Rs. } \frac{301}{4}$
$2\frac{1}{3}=\frac{7}{3}$
Cost per metre $=\frac{301}{4}\div\frac{7}{3}$
$=\frac{301}{4}\times\frac{3}{7}$
$=\frac{43\times3}{4}$
$=\frac{129}{4}$
$\text{Rs. }32 \frac{1}{4}$
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Question 103 Marks
By what number should we multiply $\frac{-8}{13}$ so that the product may be $24$?
Answer
Let the number to be found be $x$.
$\text{x}\times\frac{-8}{13}=24$
$\Rightarrow\text{x}=24\div\frac{-8}{13}$
$\Rightarrow=24\times\frac{13}{-8}$
$\text{x}=-3\times13$
$​​\text{x}=-39$
Hence the number is $x = - 39$
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Question 113 Marks
Add the following rational numbers:
$\frac{3}{4}\text{ and }\frac{-3}{5}$
Answer
$\frac{3}{4}\text{ and }\frac{-3}{5}$
If $\frac{\text{p}}{\text{q}}$ and $\frac{\text{r}}{\text{s}}$ are two rational numbers such that q and s do not have a common factor.
$\frac{\text{p}}{\text{q}}+\frac{\text{r}}{\text{s}}=\frac{\text{p}\times\text{s}+\text{r}\times\text{q}}{\text{q}\times\text{s}}$
$\frac{3}{4}+\frac{-3}{5}=\frac{3\times(5)+(-3)\times4}{4\times5}$
$=\frac{15-12}{20}=\frac{3}{20}$
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Question 123 Marks
Add the following rational numbers: $\frac{-7}{27}\text{ and }\frac{11}{18}$
Answer
$\frac{-7}{27}\text{ and }\frac{11}{18}$
$LCM$ of $27$ and $18$ is $54$ $\frac{-7}{27}=\frac{-7\times2}{27\times2}=\frac{-14}{54}$
$\frac{11}{18}=\frac{11\times3}{18\times3}=\frac{33}{54}$
$\frac{-7}{27}+\frac{11}{18}=\frac{-14}{54}+\frac{33}{54}$
$=\frac{33-14}{54}$ $=\frac{19}{54}$
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Question 133 Marks
By what number should we multiply $\frac{-15}{28}$ so that the product may be $\frac{-5}{7}?$
Answer
Let the number to be found be $x$.
$\text{x}\times\frac{-15}{28}=\frac{-5}{7}$
$\Rightarrow-\text{x}=\frac{-5}{7}\div\frac{-15}{28}$
$\Rightarrow\text{-x}=\frac{-5}{7}\times\frac{-28}{15}$
$\Rightarrow\text{-x}=\frac{-8}{9}\times\frac{15}{-4}$
$\Rightarrow\text{x}=\frac{4}{3}$
Hence the number is $\frac{4}{3}$
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Question 143 Marks
Find (x + y) + (x - y), if

$\text{x}=\frac{5}{4},\text{y}=\frac{-1}{3}$

Answer
$\text{x}=\frac{5}{4},\text{y}=\frac{-1}{3}$
$(\text{x}+\text{y})\div(\text{x}-\text{y})$
$=\Big(\frac{5}{4}+\frac{-1}{3}\Big)\div\Big(\frac{5}{4}-\frac{-1}{3}\Big)$
$=\Big(\frac{5\times3-1\times4}{12}\Big)\div\Big(\frac{12}{5\times3+1\times4}\Big)$
$=\Big(\frac{5\times3-1\times4}{{5\times3+1\times4}}\Big)$
$=\Big(\frac{11}{19}\Big)$
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Question 153 Marks
Simplifiy: $\frac{-8}{19}+\frac{-4}{57}$
Answer
$\frac{-8}{19}+\frac{-4}{57}$
$LCM$ of $19$ and $57$ is $57$ $\frac{-8}{19}=\frac{-8\times3}{19\times3}=\frac{-24}{57}$
$\frac{-8}{19}+\frac{-4}{57}=\frac{-24}{57}+\frac{-4}{57}$
$=\frac{-24-4}{57}=\frac{-28}{57}$
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Question 163 Marks
Add and express the sum as a mixed fraction:
$\frac{-31}{6}+\frac{-27}{8}$
Answer
$\frac{-31}{6}+\frac{-27}{8}$
$\frac{-31}{6}=\frac{-31\times4}{6\times4}=\frac{-214}{24}$
$\frac{-27}{8}=\frac{-27\times3}{8\times3}=\frac{-81}{24}$
$\frac{-31}{6}+\frac{-27}{8}=\frac{-124}{24}+\frac{-81}{24}$
$=\frac{-124}{24}-\frac{81}{24}$
$=\frac{-124-81}{24}$
$=\frac{-205}{24}$
$=-8\frac{13}{24}$
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Question 173 Marks
Evaluate the following:
$-\frac{4}{7}-\frac{2}{-3}$
Answer
$-\frac{4}{7}-\frac{2}{-3}$
$LCM$ of $3$ and $7$ is $21$
$\frac{-4}{7}=\frac{-4\times3}{7\times3}=-\frac{12}{21}$
$\frac{2}{-3}=\frac{2\times7}{-3\times7}=\frac{14}{21}$
$\frac{-4}{7}-\frac{2}{-3}=-\frac{12}{21}-\frac{-14}{21}$
$\frac{-14}{21}-\frac{12}{21}-\frac{-14}{21}$
$=\frac{14}{21}-\frac{12}{21}$
$=\frac{14-12}{21}$
$=\frac{2}{21}$
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Question 183 Marks
Find $(x + y) + (x - y)$, if$\text{x}=\frac{2}{5},\text{y}=\frac{1}{2}$
Answer
$\text{x}=\frac{2}{5},\text{y}=\frac{1}{2}$
$(\text{x}+\text{y})\div(\text{x}-\text{y})$
$=\Big(\frac{2}{5}+\frac{1}{2}\Big)\div\Big(\frac{2}{5}-\frac{1}{2}\Big)$
$=\Big(\frac{4+5}{10}\Big)\div\Big(\frac{4-5}{10}\Big)$
$=\Big(\frac{4+5}{10}\Big)\times\Big(\frac{10}{4-5}\Big)$
$=\Big(\frac{4+5}{4-5}\Big)$
$=\Big(\frac{9}{-1}\Big)$
$=\Big(-\frac{9}{1}\Big)$
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Question 193 Marks
What number should be subtracted from $\frac{3}{7}$ so as to get $\frac{5}{4}?$
Answer
Let the required number be $x$ $\frac{3}{7}-\text{x}=\frac{5}{4}$
$-\text{x}=\frac{5}{4}-\frac{3}{7}$
$-\text{x}=\frac{5\times7}{4\times7}-\frac{3\times4}{7\times4}$
$-\text{x}=\frac{35}{18}-\frac{12}{18}$
$-\text{x}=\frac{35-12}{28}$
$-\text{x}=\frac{23}{28}$
$\text{x}=-\frac{23}{28}$
The required number is $\frac{23}{28}$
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Question 203 Marks
The sum of two numbers is $\frac{-4}{3},$ If one of the numbers is $-5$ find the other.
Answer
Let the required number be $x$ $-5+\text{x}=\frac{-4}{3}$
$\text{x}=\frac{-4}{3}+5$
$\text{x}=\frac{-4}{3}+\frac{5\times3}{1\times3}$
$\text{x}=\frac{-4}{3}+\frac{15}{3}$
$\text{x}=\frac{-4+15}{3}$
$\text{x}=\frac{11}{3}$
The required number is $\frac{11}{3}$
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Question 213 Marks
Simplifiy: $\frac{1}{-12}+\frac{2}{-15}$
Answer
$\frac{1}{-12}+\frac{2}{-15}$
$\frac{1}{-12}+\frac{2}{-15}$
$LCM$ of $12$ and $15$ is $60$ $\frac{-1}{12}=\frac{-1\times5}{12\times5}=\frac{-5}{60}$
$\frac{-1}{12}=\frac{-2\times4}{15\times4}=\frac{-8}{60}$
$\frac{-1}{12}-\frac{2}{15}=\frac{-5}{60}+\frac{-8}{60}$
$=\frac{-5-8}{60}$
$=\frac{-13}{60}$
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Question 223 Marks
Simplify: $\Big(\frac{-3}{2}+\frac{5}4{}-\frac{7}{4}\Big)$
Answer
$\Big(\frac{-3}{2}+\frac{5}4{}-\frac{7}{4}\Big)$
$=\Big(​​\frac{-3\times2}{2\times2}+\frac{5}{4}-\frac{7}{4}\Big)$
$=\Big(\frac{-6}{4}+\frac{5}{4}-\frac{7}{4}\Big)$
$=\Big(\frac{-6+5-7}{4}\Big)$
$=\Big(\frac{-13+5}{4}\Big)$
$=\Big(-\frac{8}{4}\Big)$
$=-2$
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Question 233 Marks
Evaluate the following:
$-2-\frac{5}{9}$
Answer
$-2-\frac{5}{9}$
$\frac{2}{1}-\frac{-5}{-9}$
$=\frac{-2\times9-5\times1}{9\times1}$
$=\frac{-18-5}{9}$
$=-\frac{23}{9}$
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Question 243 Marks
What should be added to $\frac{-7}{8}$ so as to get $\frac{5}{9}?$
Answer
Let the required number be $x$
$\frac{-7}{8}+\text{x}=\frac{5}{9}$
$\text{x}=\frac{5}{9}-\frac{7}{9}$
$\text{x}=\frac{5\times8}{9\times8}-\frac{-7\times9}{8\times9}$
$\text{x}=\frac{40}{72}-\frac{63}{72}$
$\text{x}=\frac{40+63}{72}$
$\text{x}=\frac{103}{72}$
The required number is $\frac{103}{72}$
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Question 253 Marks
Add and express the sum as a mixed fraction: $\frac{-12}{5}+\frac{43}{10}$
Answer
$\frac{-12}{5}+\frac{43}{10}$
$LCM$ of $5$ and $10$ is $10$ $\frac{-12}{5}=\frac{-12\times2}{5\times2}=\frac{-24}{10}$
$\frac{-12}{5}+\frac{43}{10}=\frac{-24}{10}+\frac{43}{10}$
$=\frac{-24+43}{10}$
$=\frac{19}{10}$
$=1\frac{9}{10}$
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Question 263 Marks
What should be added to $\frac{-5}{7}$ so as to get $\frac{-2}{3}?$
Answer
Let the required number be $x$ $\frac{-5}{7}+\text{x}=\frac{-2}{3}$
$\text{x}=\frac{-2}{3}-\frac{-5}{7}$
$\text{x}=\frac{-2\times7}{3\times7}-\frac{5\times3}{7\times3}$
$\text{x}=-\frac{14}{21}-\frac{-15}{21}$
$\text{x}=\frac{1}{21}$
The required number is $\frac{1}{21}$
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Question 273 Marks
Simplify: $\frac{-2}{5}-\frac{-3}{10}-\frac{-4}{7}$
Answer
$\frac{-2}{5}-\frac{-3}{10}-\frac{-4}{7}$
$=\Big(​​\frac{-2\times14}{5\times14}-\frac{-3\times7}{10\times7}-\frac{-4\times10}{7\times10}\Big)$
$=\Big(\frac{-28}{70}-\frac{-21}{70}-\frac{-40}{70}\Big)$
$=\Big(\frac{-28+21+40}{70}\Big)$
$=\Big(\frac{33}{70}\Big)$
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Question 283 Marks
Simplify: $\frac{5}{4}-\frac{7}{6}-\frac{-2}{3}$
Answer
$\frac{5}{4}-\frac{7}{6}-\frac{-2}{3}$
$=\Big(​​\frac{5\times3}{4\times3}-\frac{7\times2}{6\times2}-\frac{-2\times4}{3\times4}\Big)$
$=\Big(\frac{15}{12}-\frac{14}{12}-\frac{-8}{12}\Big)$
$=\Big(\frac{15-14+8}{12}\Big)$
$=\Big(\frac{9}{12}\Big)$
$=\Big(\frac{3}{4}\Big)$
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Question 293 Marks
The sum of two numbers is $\frac{-1}{3},$ If one of the numbers is $\frac{-12}{3}$ find the other.
Answer
Let the required number be x $\frac{-12}{3}+\text{x}=-\frac{1}{3}$
$\text{x}=-\frac{1}{3}-\frac{-12}{3}$
$\text{x}=\frac{-1+12}{3}$
$\text{x}=\frac{11}{3}$ The required number is $\frac{11}{3}$
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Question 303 Marks
Add the following rational numbers: $-3\text{ and }\frac{3}{5}$
Answer
$\frac{-3}{1}\text{ and }\frac{3}{5}$
If $\frac{\text{p}}{\text{q}}$ and $\frac{\text{r}}{\text{s}}$ are two rational numbers such that $q$ and $s$ do not have a common factor.
$\frac{\text{p}}{\text{q}}+\frac{\text{r}}{\text{s}}=\frac{\text{p}\times\text{s}+\text{r}\times\text{q}}{\text{q}\times\text{s}}$
$\frac{3}{1}+\frac{3}{5}=\frac{5\times(-3)+(3)\times1}{5}$
$=\frac{-15+3}{5}$
$=\frac{-12}{5}$
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Question 313 Marks
Add the following rational numbers: $\frac{31}{-4}\text{ and }\frac{-5}{8}$
Answer
$\frac{31}{-4}\text{ and }\frac{-5}{8}$
$LCM$ of $4$ and $8$ is $4$
$\frac{31}{-4}=\frac{31\times2}{-4\times2}=\frac{62}{-8}$
$\frac{31}{-4}+\frac{-5}{8}=\frac{62}{-8}+\frac{-5}{8}$
$=\frac{-62-5}{8}$
$=\frac{-67}{8}$
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Question 323 Marks
Evaluate the following: $\frac{4}{7}-\frac{-5}{-7}$
Answer
$\frac{4}{7}-\frac{-5}{-7}$
$\frac{4}{7}-\frac{5}{7}=\frac{4-5}{7}$
$=​​\frac{-1}{7}$
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Question 333 Marks
The cost of $7\frac{2}{3}$ metres of rope is $12\frac{3}{4}$ Find its cost per metre.
Answer
$\text{Rs. } \frac{51}{4}$ $7\frac{2}{3}=\frac{23}2{}$ Cost per metre $=\frac{51}{4}\div\frac{23}{3}$ $=\frac{51}{4}\times\frac{3}{23}$ $=\frac{153}{92}$ $\text{Rs. }1 \frac{61}{92}$
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Question 343 Marks
Evaluate the following: $\frac{2}{3}-\frac{3}{5} $
Answer
$\frac{2}{3}-\frac{3}{5} $
$​​​​​​​LCM$ of $3$ and $5$ is $15$ $\frac{2}{3}=\frac{2\times5}{3\times5}=\frac{10}{15}$
$\frac{3}{5}=\frac{3\times3}{5\times3}=\frac{9}{15}$
$\frac{2}{5}-\frac{3}{5}=\frac{10}{15}-\frac{9}{15}$
$=\frac{1}{15}$
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Question 353 Marks
What number should be subtracted from $\frac{-5}{3}$ so as to get $\frac{5}{6}?$
Answer
The required number be $x$ Let $\frac{-5}{3}-\text{x}=\frac{5}{6}$
$\Rightarrow-\text{x}=\frac{5}{6}-​​\frac{5}{3}$
$\Rightarrow-\text{x}=​​\frac{5}{6}-\frac{-5\times2}{3\times2}$
$\Rightarrow-\text{x}=\frac{5}{6}-\frac{-5\times2}{3\times2}$
$\Rightarrow-\text{x}=​​\frac{5}{6}-\frac{-10}{6}$
$\Rightarrow-\text{x}=\frac{5+10}{6}$
$\Rightarrow-\text{x}=\frac{15}{6}$
$\text{x}-\frac{15}{6}$
The required number is $\frac{15}{6}$
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Question 363 Marks
By what number should we multiply $\frac{-1}{6}$ so that the product may be $\frac{-23}{9}?$
Answer
Let the number to be found be $x$.
$\text{x}\times\frac{-1}{6}=\frac{-23}{9}$
$\Rightarrow-\text{x}=\frac{-23}{9}\times6$
$\Rightarrow\text{-x}=\frac{-23\times6}{9}$
$\Rightarrow\text{-x}=\frac{-138}{9}$
$\Rightarrow\text{x}=\frac{138}{9}$
$\text{x}=\frac{46}{3}$
Hence the number is $\frac{46}{3}$
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Question 373 Marks
Find 10 rational numbers between $\frac{7}{13}$ and $\frac{-4}{13}$.
Answer
$\frac{-3}{13}, \frac{-2}{13}, \frac{-1}{13}, \frac{0}{13}, \frac{1}{13}, \frac{2}{13}, \frac{3}{13}, \frac{4}{13}, \frac{5}{13}, \frac{6}{13}$
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Question 383 Marks
Find six rational numbers between $\frac{-4}{8}$ and $\frac{3}{8}$.
Answer
$\frac{-3}{8}, \frac{-2}{8}, \frac{-1}{8}, \frac{0}{8}, \frac{1}{8}, \frac{2}{8}$
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Question 423 Marks
The cost of $2 \frac{1}{3}$ metres of cloth is $₹ 75 \frac{1}{4}$. Find the cost of cloth per metre.
Answer
₹ $32 \frac{1}{4}$
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Question 433 Marks
The cost of $7 \frac{2}{3}$ metres of rope is $₹ 12 \frac{3}{4}$. Find its cost per metre.
Answer
₹ $1 \frac{61}{92}$
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Question 443 Marks
Simplify:
$\left(\frac{3}{11} \times \frac{5}{6}\right)-\left(\frac{9}{12} \times \frac{4}{3}\right)+\left(\frac{5}{13} \times \frac{6}{15}\right)$
Answer
$\frac{-177}{286}$
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Question 453 Marks
Simplify:
$\left(\frac{13}{9} \times \frac{-15}{2}\right)+\left(\frac{7}{3} \times \frac{8}{5}\right)+\left(\frac{3}{5} \times \frac{1}{2}\right)$
Answer
$\frac{-34}{5}$
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Question 463 Marks
Simplify the following and express the result as a rational number in standard form:
$\frac{-13}{9} \times \frac{27}{-26}$
Answer
$\frac{3}{2}$
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Question 473 Marks
Simplify the following and express the result as a rational number in standard form:
$\frac{-19}{36} \times 16$
Answer
$\frac{-76}{9}$
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Question 483 Marks
Simplify the following and express the result as a rational number in standard form:
$\frac{7}{6} \times \frac{-3}{28}$
Answer
$\frac{-1}{8}$
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Question 493 Marks
Simplify the following and express the result as a rational number in standard form:
$\frac{-16}{21} \times \frac{14}{5}$
Answer
$\frac{-32}{15}$
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Question 503 Marks
What should be subtracted from $\left(\frac{3}{4}-\frac{2}{3}\right)$ to get $\frac{-1}{6}$ ?
Answer
$\frac{1}{4}$
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