Question 13 Marks
After reading $\frac{7}{9}$ of a book, $40$ pages are left. How many pages are there in the book$?$
AnswerLet no. of pages $= 1$
Then no. of pages read $=\frac{7}{9}$
and no. of pages left $=1-\frac{7}{9}$
$=\frac{9-7}{9}$
$=\frac{2}{9}$
If $\frac{2}{9}$ pages are left then total number of pages $= 1$
and $1$ page is left then total number of pages $=1\div\frac{2}{9}$
$=1\times\frac{9}{2}$
and if 40 pages are left then total number of pages $=\frac{9}{2}\times40$
$=\frac{360}{2}$
$=180$
View full question & answer→Question 23 Marks
Which of the two rational numbers is greater in the given pair$?$
$\frac{2}{3}\ \text{or}\ \frac{3}{4}$
Answer$\frac{2}{3}\ \text{or}\ \frac{3}{4}$
Here, denominator are different.
$\therefore$ Making denominator same $LCM$ of $3$ and $4 = 12$
$\therefore\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}$ and
$\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}$
$\therefore$ Between $\frac{8}{12}$ and $\frac{9}{12}$ and clearly $\frac{9}{12}$ is greater or $\frac{3}{4}$ is greater.
View full question & answer→Question 33 Marks
Verify whether the given statement is true or false: $\frac{-7}{24}\div\frac{3}{-16}=\frac{3}{-16}\div\frac{-7}{24}$
AnswerFalse.
Solution:
$\text{L.H.S.}=\frac{-7}{24}\div\frac{3}{-16}$
$=\frac{-7}{24}\times\frac{-16}{3}$
$=\frac{-7\times(-16)}{24\times3}$
$=\frac{112}{72}$
$=\frac{112\div8}{72\div8}$
$=\frac{14}{9}$
$\text{R.H.S.}=\frac{3}{-16}\div\frac{-7}{24}$
$=\frac{3}{-16}\times\frac{24}{-7}$
$=\frac{3\times24}{-16\times(-7)}$
$=\frac{72}{-112}$
$=\frac{72\div8}{-112\div8}$
$=\frac{9}{-14}$
$\text{L.H.S}\neq\text{R.H.S}$
$\therefore$ It is false.
View full question & answer→Question 43 Marks
Represent the following numbers on the number line. $-3$
Answer$-3$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB, BC$ on the left of $O$ representing integers $-1, -2, -3$ respectively
$OC = -3$ as shown on the number line given below,
View full question & answer→Question 53 Marks
If $\frac{3}{5}$ of a number exceeds its $\frac{2}{7}$ by $44,$ find the number.
AnswerLet number $= 1$
Then difference between $\frac{3}{5}$ and $\frac{2}{7}$
$=\frac{3}{5}-\frac{2}{7}$
$=\frac{21-10}{35}$
$=\frac{11}{35}$
$\therefore$ $\frac{11}{35}$ of a number $= 44$
$\therefore$ Number $= 44\div\frac{11}{35}$
$= 44\times\frac{35}{11}$
$= \frac{1540}{11}$
$=140$
View full question & answer→Question 63 Marks
Which of the two rational numbers is greater in the given pair$?$
$\frac{7}{-9}\ \text{or}\ \frac{-5}{8}$
Answer$\frac{7}{-9}\ \text{or}\ \frac{-5}{8}$ Making their denominators positive,
we find that $\frac{7}{-9}\ \text{=}\ \frac{-7}{9}$
Now, $LCM$ of $9$ and $8 = 72$
$\therefore\frac{7}{-9}\ \text{=}\ \frac{-7}{9}=\frac{-7\times8}{9\times8}=\frac{-56}{72}$
$\frac{-5}{8}=\frac{-5\times9}{8\times9}=\frac{-45}{72}$
Clearly $\frac{-45}{72}$ is greaterOr $\frac{-5}{8}$ is greater.
View full question & answer→Question 73 Marks
Find the cost of $3\frac{2}{5}$ metres of cloth at $\text{Rs.}\ 63\frac{3}{4}$ per metre.
AnswerCost of $1m $ cloth $=\text{Rs.}\ 63\frac{3}{4}$
$=\frac{4\times63+3}{4}$
$=\text{Rs.}\ \frac{255}{4}$
$\therefore$ Cost of $3\frac{2}{5}\text{m}=\text{Rs.}\ \frac{255}{4}\times\frac{17}{5}$
$=\text{Rs.}\ \frac{255\times17}{4\times5}$
$=\frac{51\times17}{4}$
$=\text{Rs.}\ \frac{867}{4}$
$=\text{Rs.}\ 216\frac{3}{4}$
View full question & answer→Question 83 Marks
Represent the following numbers on the number line. $\frac{-3}{4}$
Answer$\frac{-3}{4}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB$ to the left of $O,$ representing integers $-1, -2$ respectively,
Divide $OA$ into $4$ equal parts and take $3$ part.
Then,
$\text{OA}=\frac{-3}{4}$ Which shown on the number line given below,
View full question & answer→Question 93 Marks
Rita had $Rs. 300.$ She spent $\frac{1}{3}$ of her money on notebooks and $\frac{1}{4}$ of the remainder on stationery items. How much money is left with her?
AnswerTotal amount, Rita has $= Rs. 300$
Amount spent on notebooks $=\frac{1}{3}\ \text{of}\ 300$
$\text{Rs.}\ \frac{300}{3}=\text{Rs.}\ 100$
Balance $= Rs. 300 - Rs. 100 = Rs. 200$
Amount spent on stationery $=\frac{1}{4}\ \text{of}\ 200$
$=\text{Rs.}\ 200\times\frac{1}{4}$
$=\frac{200}{4}$
$=\text{Rs.}\ 50$
Amount left at the end $= Rs. 200 - Rs. 50$
$= Rs. 150$
View full question & answer→Question 103 Marks
Verify the following:
$\Big(\frac{5}{7}\times\frac{12}{13}\Big)\times\frac{7}{18}=\frac{5}{7}\times\Big(\frac{12}{13}\times\frac{7}{18}\Big)$
Answer $\text{L.H.S.}=\Big(\frac{5}{7}\times\frac{12}{13}\Big)\times\frac{7}{18}$
$=\frac{5\times12}{7\times13}\times\frac{7}{18}$
$=\frac{60}{91}\times\frac{7}{18}$
$=\frac{420}{1638}$
$=\frac{420\div21}{1638\div21}$
$=\frac{20}{78}$
$\text{R.H.S.}=\frac{5}{7}\times\Big(\frac{12}{13}\times\frac{7}{18}\Big)$
$=\frac{5\times84}{7\times234}$
$=\frac{420}{1638}$
$=\frac{420\div21}{1638\div21}$
$=\frac{20}{78}$
$\text{L.H.S.}=\text{R.H.S.}$
$\therefore\Big(\frac{5}{7}\times\frac{12}{13}\Big)\times\frac{7}{18}=\frac{5}{7}\times\Big(\frac{12}{13}\times\frac{7}{18}\Big)$
View full question & answer→Question 113 Marks
Which of the two rational numbers is greater in the given pair$?$
$\frac{-1}{3}\ \text{or}\ \frac{4}{-5}$
Answer$\frac{-1}{3}\ \text{or}\ \frac{4}{-5}$
Making their denominators positive $\frac{4}{-5}\ \text{=}\ \frac{-4}{5}$
Now, $LCM$ of $3$ and $5 = 15$
$\therefore\frac{-1}{3}=\frac{-1\times5}{3\times5}=\frac{-5}{15}$
$\frac{4}{-5}=\frac{4}{-5}=\frac{-4\times3}{5\times3}=\frac{-12}{15}$
Clearly $\frac{-5}{15}$ is greaterOr $-\frac{1}{3}$ is greater.
View full question & answer→Question 123 Marks
Represent the following numbers on the number line.
$\frac{-1}{3}$
Answer$\frac{-1}{3}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB$ to the left of $O,$ representing integers $-1, -2$ respectively,
Divide $OA$ into $3$ equal parts and take $1$ part.
Then,
$\text{OP}=\frac{-1}{3}$ Which shown on the number line given below,
View full question & answer→Question 133 Marks
Verify the following: $\frac{-16}{7}\times\Big(\frac{-8}{9}+\frac{-7}{6}\Big)=\Big(\frac{-16}{7}\times\frac{-8}{9}\Big)+\Big(\frac{-16}{7}\times\frac{-7}{6}\Big)$
Answer$\text{L.H.S.}=\frac{-16}{7}\times\Big(\frac{-8}{9}+\frac{-7}{6}\Big)$ $=\frac{-16}{7}\times\Big(\frac{-16+(-21)}{18}\Big)$ $=\frac{-16}{7}\times\frac{-37}{18}$ $=\frac{-16\times-37}{7\times18}$ $=\frac{592}{126}$ $\text{R.H.S.}=\Big(\frac{-16}{7}\times\frac{-8}{9}\Big)+\Big(\frac{-16}{7}\times\frac{-7}{6}\Big)$ $=\frac{-16\times(-8)}{7\times9}+\frac{-16\times(-7)}{7\times6}$ $=\frac{128}{63}+\frac{112}{42}$ $=\frac{256+336}{126}$ $=\frac{592}{126}$ $\text{L.H.S.}=\text{R.H.S.}$ $\therefore\frac{-16}{7}\times\Big(\frac{-8}{9}+\frac{-7}{6}\Big)=\Big(\frac{-16}{7}\times\frac{-8}{9}\Big)+\Big(\frac{-16}{7}\times\frac{-7}{6}\Big)$
View full question & answer→Question 143 Marks
By what rational number should $\frac{-8}{39}$ be multiplied to obtain $\frac{1}{26}?$
AnswerLet $x$ be multiplied
Then,
$\frac{-8}{39}\times\text{x}=\frac{1}{26}$
$\Rightarrow\text{x}=\frac{1}{26}\div\frac{-8}{39}$
$\Rightarrow\text{x}=\frac{1}{26}\times\frac{39}{-8}$
$=\frac{39}{-208}$
$=\frac{39\div13}{-208\div13}$
$=\frac{3}{-16}$
$=\frac{-3}{16}$
View full question & answer→Question 153 Marks
Using the rearrangement property find the sum: $\frac{-13}{20}+\frac{11}{14}+\frac{-5}{7}+\frac{7}{10}$
Answer$\frac{-13}{20}+\frac{11}{14}+\frac{-5}{7}+\frac{7}{10}$ $=\Big(\frac{-13}{20}+\frac{7}{10}\Big)+\Big(\frac{11}{14}+\frac{-5}{7}\Big)$ $=\frac{-13+14}{20}+\frac{11+(-10)}{14}$ $=\frac{1}{20}+\frac{1}{14}$ $=\frac{7+10}{140}$ $=\frac{17}{140}$
View full question & answer→Question 163 Marks
In a school, $\frac{5}{8}$ of the students are boys. If there are $240$ girls, find the number of boys in the school.
AnswerLet total number of students $= 1$
and no. of boys $=\frac{5}{8}$
$\therefore$ No. of girls $=1-\frac{5}{8}$
$=\frac{8-5}{8}$
$=\frac{3}{8}$
$\therefore$ If girls are $\frac{3}{8}$, Then boys $=\frac{5}{8}$
and if girls are $1,$ Then boys $=\frac{5}{8}\div\frac{3}{8}$
$=\frac{5}{8}\times\frac{8}{3}$
$=\frac{5}{3}$
and if girls are $240,$ Then boys $=\frac{5}{3}\times240$
$=\frac{1200}{3}$
$=400$
View full question & answer→Question 173 Marks
Verify the following: $\frac{-15}{4}\times\Big(\frac{3}{7}+\frac{-12}{5}\Big)=\Big(\frac{-15}{4}\times\frac{3}{7}\Big)+\Big(\frac{-15}{4}\times\frac{-12}{5}\Big)$
Answer$\text{L.H.S.}=\frac{-15}{4}\times\Big(\frac{3}{7}+\frac{-12}{5}\Big)$
$=\frac{-15}{4}\times\Big(\frac{15-84}{35}\Big)$
$=\frac{-15}{4}\times\frac{69}{35}$
$=\frac{1035}{140}$
$\text{R.H.S.}=\Big(\frac{-15}{4}\times\frac{3}{7}\Big)+\Big(\frac{-15}{4}\times\frac{-12}{5}\Big)$
$=\frac{-15\times3}{4\times7}+\frac{-15\times(-12)}{4\times5}$
$=\frac{-45}{28}+\frac{180}{20}$
$=\frac{-225+1260}{140}$
$=\frac{1035}{140}$
$\text{L.H.S.}=\text{R.H.S.}$
$\therefore\frac{-15}{4}\times\Big(\frac{3}{7}+\frac{-12}{5}\Big)=\Big(\frac{-15}{4}\times\frac{3}{7}\Big)+\Big(\frac{-15}{4}\times\frac{-12}{5}\Big)$
View full question & answer→Question 183 Marks
Which of the two rational numbers is greater in the given pair$? \frac{-12}{5}\ \text{or}\ -3$
Answer$\frac{-12}{5}\ \text{or}\ -3$
$\frac{-12}{5}\ \text{or}\ \frac{-3}{1}$
Now, $LCM$ of $5$ and $1 = 5$
$\therefore\frac{-12}{5}=\frac{-12}{5}$
$-3=\frac{-3\times5}{1\times5}=\frac{-15}{5}$
Clearly $\frac{-12}{5}$ is greater.
View full question & answer→Question 193 Marks
Find 10 rational numbers between $\frac{-3}{4}$ and $\frac{5}{6}$.
Answer$\frac{-3}{4}$ , $\frac{5}{6}$
$LCM$ of $4$ and $6 = 12$
$=\frac{-3}{4}$
$=\frac{-3\times3}{4\times3}$
$=\frac{-9}{12}$ and $=\frac{5}{6}$
$=\frac{5\times2}{6\times2}$
$=\frac{10}{12}$
$10$ rational number can be between $\frac{-3}{4},\frac{5}{6}$
$=\frac{-8}{12},\frac{-7}{12},\frac{-6}{12},\frac{-5}{12},\frac{-4}{12},\frac{-3}{12},\frac{-2}{12},\frac{-1}{12},0\ \text{and}\ \frac{1}{12}$
View full question & answer→Question 203 Marks
A car is moving at an average speed of $60\frac{2}{5}\text{km/ hr}$. How much distance will it cover in $6\frac{1}{4}$ hours$?$
AnswerDistance cover in $1$ hours $=60\frac{2}{5}\text{km/ hr}$
$=\frac{302}{5}\text{km/ hr}$
$\therefore$ Distance cover in $6\frac{1}{4}=\frac{25}{4}$ hours
$=\frac{302}{5}\times\frac{25}{4}$
$=\frac{151\times5}{2}\ \text{km}$
$=\frac{755}{2}$
$=377\frac{1}{2}$
View full question & answer→Question 213 Marks
Using the rearrangement property find the sum: $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}$
Answer$\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}$
$=\Big(\frac{-8}{3}+\frac{-11}{6}\Big)+\Big(\frac{-1}{4}+\frac{3}{8}\Big)$
$=\frac{-16+(-11)}{6}+\frac{-2+3}{8}$
$=\frac{-16-11}{6}+\frac{1}{8}$
$=\frac{-27}{6}+\frac{1}{8}$
$=\frac{-108+3}{24}$
$=\frac{-105}{24}$
$=\frac{-105\div3}{24\div3}$
$=\frac{-35}{8}$
View full question & answer→Question 223 Marks
The area of a room is $65\frac{1}{4}\text{m}^2$. If its breadth is $5\frac{7}{16}$ metres, what is its length$?$
AnswerArea of a room is $=65\frac{1}{4}\text{m}^2$ Breadth $=5\frac{7}{16}\text{m}$
$\therefore$ Length $=$ Area $÷$ Breadth $=\Big(65\frac{1}{4}\div5\frac{7}{16}\Big)\text{m}$
$=\frac{261}{4}\div\frac{87}{16}$
$=\frac{261}{4}\times\frac{16}{87}\text{m}$
$=\frac{4176}{348}$
$=12\text{m}$
View full question & answer→Question 233 Marks
Represent the following numbers on the number line. $-1\frac{2}{3}$
Answer$-1\frac{2}{3}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB$ to the left of $O,$ is representing integers $-1, -2$ respectively,
Divide $AB$ into $3$ equal parts and take $2$ parts.
Then,
$\text{AP}=\frac{2}{3}$
$\therefore\text{OP}=-\Big(1+\frac{2}{3}\Big)=-1\frac{2}{3}$ Which shown on the number line given below,
View full question & answer→Question 243 Marks
Represent the following numbers on the number line. $-2\frac{7}{8}$
Answer$-2\frac{7}{8}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB, BC$ on the left of $O$ representing integers $-1, -2, -3$ respectively,
Divide $BC$ into $8$ equal parts and take $7$ parts.
Then,
$\text{BP}=\frac{7}{8}$
$\therefore\text{OP}=-\Big(2+\frac{7}{8}\Big)=-2\frac{7}{8}$ as shown on the number line given below,
View full question & answer→Question 253 Marks
By what rational number should we multiply $\frac{-15}{56}$ to get $\frac{-5}{7}$?
AnswerLet $x$ be multiplied
Then,
$\text{x}=\frac{-15}{56}=\frac{-5}{7}$
$\Rightarrow\text{x}=\frac{-5}{7}\div\frac{-15}{56}$
$\Rightarrow\text{x}=\frac{-5}{7}\times\frac{56}{-15}$
$=\frac{-280}{-105}=\frac{280}{105}$
$=\frac{280\div35}{105\div35}$
$=\frac{8}{3}$
$\therefore$ Required rational number $=\frac{8}{3}$
View full question & answer→Question 263 Marks
From a rope $11m$ long, two pieces of lengths $2\frac{3}{5}\text{m}$ and $3\frac{3}{10}$ are cut off. What is the length of the remaining rope$?$
AnswerTotal length of rope $= 11m$
Sum of lengths of two parts $=\Big(2\frac{3}{5}+3\frac{3}{10}\Big)\text{m}$
$=\Big(\frac{13}{5}+\frac{33}{10}\Big)\text{m}$
$=\frac{26+33}{10}$
$=\frac{59}{10}\text{m}$ Length of remaining rope $=\Big(11-\frac{59}{10}\Big)\text{m}$
$=\Big(\frac{110-59}{10}\Big)$
$=\frac{51}{10}\text{m}$
$=5\frac{1}{10}\text{m}$
View full question & answer→Question 273 Marks
By what number should $\frac{-33}{8}$ be divided to get $\frac{-11}{2}?$
AnswerLet required number $= x$
Then,
$\frac{-33}{8}\div\text{x}=\frac{-11}{2}$
$\Rightarrow\frac{-33}{8}\times\frac{1}{\text{x}}=\frac{-11}{2}$
$\Rightarrow\frac{1}{\text{x}}=\frac{-11}{2}\div\frac{-33}{8}$
$\Rightarrow\frac{1}{\text{x}}=\frac{-11}{2}\times\frac{8}{-33}$
$=\frac{-88}{-66}=\frac{88}{66}$
$=\text{x}=\frac{66}{88}$
$=\frac{66\div22}{88\div22}$
$=\frac{3}{4}$
$\therefore$ Required number $=\frac{3}{4}$
View full question & answer→Question 283 Marks
The product of two rational numbers is $-9.$ If one of the numbers is $-12,$ find the other.
AnswerProduct of two number $= -9$
One number $= -12$
Let second number $= x$
Then,
$-12\times\text{x} = -9$
$\Rightarrow\text{x}=-9\div-12$
$\Rightarrow\text{x}=-9\times\frac{1}{-12}$
$=\frac{9\div3}{12\div3}$
$=\frac{3}{4}$
$\therefore$ Second number $=\frac{3}{4}$
View full question & answer→Question 293 Marks
A basket contains three types of fruits weighing $19\frac{1}{3}\text{kg}$ in all. If $8\frac{1}{9}\text{kg}$ of these be apples, $3\frac{1}{6}\text{kg}$ be oranges and the rest pears, what is the weight of the pears in the basket?
AnswerTotal weight of three types of fruits $=19\frac{1}{3}\text{kg}$
$=\frac{58}{3}\text{kg}$
Weight of apples $=8\frac{1}{9}\text{kg}=\frac{73}{9}\text{kg}$
Weight of oranges $=3\frac{1}{6}\text{kg}=\frac{19}{6}\text{kg}$
$\therefore$ Weight of apples and oranges
$=\frac{73}{9}+\frac{19}{6}$
$=\frac{146+57}{18}$
$=\frac{203}{18}\text{kg}$
$\therefore$ Weight of pears $=\frac{58}{3}-\frac{203}{18}$
$=\frac{348-203}{18}$
$=\frac{145}{18}$
$=8\frac{1}{18}\text{kg}$
View full question & answer→Question 303 Marks
Find the area of a rectangular park which is $36\frac{3}{5}\text{m}$ long and $16\frac{2}{3}\text{m}$board.
Answer Length of the rectangular park $=36\frac{3}{5}\text{m}$ $=\frac{183}{5}\text{m}$ and breadth $=16\frac{2}{3}\text{m}$ $=\frac{50}{3}\text{m}$ $\therefore$ Area = Length × Breadth $=\frac{183}{5}\times\frac{50}{3}$ $=\frac{183\times50}{5\times3}\ \text{km}^2$ $=\frac{9150}{15}\text{km}^2$ $=610\text{km}^2$
View full question & answer→Question 313 Marks
Find two rational numbers lying between $\frac{-1}{3}$ and $\frac{1}{2}$.
Answer Required number $=\frac{1}{2}\times\Big(\frac{-1}{3}+\frac{1}{2}\Big)$
$=\frac{1}{2}\times\Big(\frac{-2+3}{6}\Big)$
$=\frac{1}{2}\times\frac{1}{6}$
$=\frac{1}{12}$
$\frac{-1}{3}<\frac{1}{12}<\frac{1}{2}$
Required number between $\frac{-1}{3}$ and $\frac{1}{2}$:
$=\frac{1}{2}\times\Big(\frac{-1}{3}+\frac{1}{12}\Big)$
$=\frac{1}{2}\times\Big(\frac{1-4}{12}\Big)$
$=\frac{1}{2}\times\Big(\frac{-3}{12}\Big)$
$=\frac{-3}{24}$
$=\frac{-3\div3}{24\div3}$
$=\frac{-1}{8}$
Thus, $\frac{1}{12}$ and $\frac{-1}{8}$ are two rational numbers between $\frac{-1}{3}$ and $\frac{1}{2}$.
View full question & answer→Question 323 Marks
The product of two fractions is $9\frac{3}{5}$. If one of the fractions is $9\frac{3}{7}$, find the other.
Answer Product of two fractions $=9\frac{3}{5}$ $=\frac{48}{5}$ One fraction $=9\frac{3}{7}=\frac{66}{7}$ $\therefore$ Second fraction $=\frac{48}{5}\div\frac{66}{7}$
$=\frac{48}{5}\times\frac{7}{66}$ $=\frac{48\times7}{5\times66}$ $=\frac{336}{330}$ $=\frac{336\div6}{330\div6}$ $=\frac{56}{55}=1\frac{1}{55}$ View full question & answer→Question 333 Marks
Which of the two rational numbers is greater in the given pair$?$
$\frac{9}{-13}\ \text{or}\ \frac{7}{-12}$
Answer$\frac{9}{-13}\ \text{or}\ \frac{7}{-12}$
Making their denominators positive $\frac{9}{-13}\ \text{=}\ \frac{-9}{13}$ and $\frac{7}{-12}\ \text{=}\ \frac{-7}{12}$
Now, $LCM$ of $13$ and $12 = 156$
$\therefore\frac{9}{-13}=\frac{-9\times12}{13\times12}=\frac{-108}{156}$
$\frac{-7}{12}=\frac{-7\times13}{12\times13}=\frac{-91}{156}$
Clearly $\frac{-91}{156}$ is greaterOr $\frac{-7}{12}\ \text{or}\ \frac{7}{-12}$ is greater.
View full question & answer→Question 343 Marks
A cord of length $71\frac{1}{2}\text{m}$ has been cut into $26$ pieces of equal length. What is the length of each piece$?$
AnswerTotal length of each piece of cord $=71\frac{1}{2}\text{m}$
No. of pieces $= 26$
$\therefore$ length of each piece $=71\frac{1}{2}\div26\text{m}$
$=\frac{143}{2}\div26$
$=\frac{143}{2}\times\frac{1}{26}\text{m}$
$=\frac{143}{52}\text{m}$
$=\frac{143\div13}{52\div13}\text{m}$
$=\frac{11}{4}\text{m}$
$=2\frac{3}{4}\text{m}$
View full question & answer→Question 353 Marks
Represent the following numbers on the number line.
$-2\frac{5}{6}$
Answer$-2\frac{5}{6}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O$, take $OA, AB, BC$ on the left of $D$ representing integers $-1, -2, -3$ respectively,
Divide $C$ into $6$ equal parts and take $5$ parts so that $\text{BP}=\frac{5}{6}$
Then,
$\text{OP}=-\Big(2+\frac{5}{6}\Big)=-2\frac{5}{6}$ as shown on the number line given below,
View full question & answer→Question 363 Marks
At a cricket test match $\frac{2}{7}$ of the spectators were in a covered place while $15000$ were in open. Find the total number of spectators.
AnswerLet total number of spectators $= 1$
No. of spectators in covered place $=\frac{2}{7}$ of $1-\frac{2}{7}$
Balance $=1-\frac{2}{7}$
$=\frac{7-2}{7}$
$=\frac{5}{7}$
$\therefore\frac{5}{7}$ of total spectators $= 15000$
$\therefore$ Total number of spectators
$=15000\div\frac{5}{7}$
$=15000\times\frac{7}{5}$
$=\frac{105000}{5}$
$=21000$
View full question & answer→Question 373 Marks
Which of the two rational numbers is greater in the given pair? $\frac{4}{-5}\ \text{or}\ \frac{-7}{10}$
Answer$\frac{4}{-5}\ \text{or}\ \frac{-7}{10}$
Making denominators of $\frac{4}{-5}$ positive
$\frac{4}{-5}\ \text{=}\ \frac{-4}{5}$
Now, $LCM$ of $5$ and $10 = 10$
$\therefore\frac{-4}{5}=\frac{-4\times2}{5\times2}=\frac{-8}{10}$
$\frac{-7}{10}=\frac{-7}{10}$
Clearly $\frac{-7}{10}$ is greater.
View full question & answer→Question 383 Marks
Which of the two rational numbers is greater in the given pair? $\frac{-1}{2}\ \text{or}\ -1$
Answer$\frac{-1}{2}\ \text{or}\ -1$
Here, denominator are different.
$\therefore$ Making denominator same $LCM$ of $2, 1 = 2$
$\therefore\frac{-1}{2}=\frac{-1}{2}$
$-1=\frac{-1\times2}{1\times2}=\frac{-2}{2}$ and clearly $\frac{-1}{2}$ is greater.
View full question & answer→Question 393 Marks
Represent the following numbers on the number line. $-3\frac{1}{7}$
Answer$-3\frac{1}{7}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O,$ take $OA, AB$ to the left of $O,$ is representing integers $-1, -2, -3, -4$
Divide $CD$ into $7$ equal parts and take $1$ parts at $P.$
Then,
$\text{CP}=\frac{1}{7}$
$\therefore\text{OP}=\text{OC}+\text{CP}$
$-\Big(3+\frac{1}{7}\Big)=-3\frac{1}{7}$ as shown on the number line given below,
View full question & answer→Question 403 Marks
The product of two rational numbers is $\frac{-16}{9}$. If one of the numbers is $\frac{-4}{3}$, find the other.
AnswerProduct of two rational number $=\frac{-16}{9}$
One number $=\frac{-4}{3}$
Let $x$ be the second number
Then,
$\text{x}+\frac{-4}{3}=\frac{-16}{3}$
$\text{x}=\frac{-16}{9}\div\frac{4}{3}$
$\text{x}=\frac{-16}{9}\times\frac{3}{-4}$
$=\frac{-48}{-36}=\frac{48}{36}$
$=\frac{48\div12}{36\div12}$
$=\frac{4}{3}$
$\therefore$ Second number $=\frac{4}{3}$
View full question & answer→Question 413 Marks
Verify the following:
$\frac{3}{7}\times\Big(\frac{5}{6}+\frac{12}{13}\Big)=\Big(\frac{3}{7}\times\frac{5}{6}\Big)+\Big(\frac{3}{7}\times\frac{12}{13}\Big)$
Answer $\text{L.H.S.}=\frac{3}{7}\times\Big(\frac{5}{6}+\frac{12}{13}\Big)$
$=\frac{3}{7}\times\Big(\frac{65+72}{78}\Big)$
$=\frac{3}{7}\times\frac{137}{78}$
$=\frac{411}{546}$
$\text{R.H.S.}=\Big(\frac{3}{7}\times\frac{5}{6}\Big)+\Big(\frac{3}{7}\times\frac{12}{13}\Big)$
$=\frac{3\times5}{7\times6}+\frac{3\times12}{7\times13}$
$=\frac{15}{42}+\frac{36}{91}$
$=\frac{195+216}{546}$
$=\frac{411}{546}$
$\text{L.H.S.}=\text{R.H.S.}$
$\therefore\frac{3}{7}\times\Big(\frac{5}{6}+\frac{12}{13}\Big)=\Big(\frac{3}{7}\times\frac{5}{6}\Big)+\Big(\frac{3}{7}\times\frac{12}{13}\Big)$
View full question & answer→Question 423 Marks
Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.
AnswerSum of $\frac{65}{12}$ and $\frac{8}{3}$
$=\frac{65}{12}+\frac{8}{3}$
$=\frac{65+32}{12}$
$=\frac{97}{12}$
Difference of $\frac{65}{12}$ and $\frac{8}{3}$
$=\frac{65}{12}-\frac{8}{3}$
$=\frac{65-32}{12}$
$=\frac{33}{12}$
$\therefore\frac{97}{12}\div\frac{33}{12}$
$=\frac{97}{12}\times\frac{12}{33}$
$=\frac{97}{33}$
View full question & answer→Question 433 Marks
The sum of two rational numbers is $-4.$ If one of them is $\frac{-11}{5}$, find the other.
AnswerLet the other number be $x$ Thus, we have: $\text{x}+\frac{-11}{5}=-4$
$\Rightarrow\text{x}-\frac{11}{5}=-4$
$\Rightarrow\text{x}=-4+\Big(\text{Additive inverse of }\frac{-11}{5}\Big)$
$\Rightarrow\text{x}=-4+\frac{11}{5}$
$\Rightarrow\text{x}=\frac{-4}{1}+\frac{11}{5}$
$\Rightarrow\text{x}=\frac{(-4\times5)+(11\times1)}{5}$
$\Rightarrow\text{x}=\frac{-20+11}{5}$
$\Rightarrow\text{x}=\frac{-9}{5}$
View full question & answer→Question 443 Marks
Using the rearrangement property find the sum: $\frac{4}{3}+\frac{3}{5}+\frac{-2}{3}+\frac{-11}{5}$
Answer$\frac{4}{3}+\frac{3}{5}+\frac{-2}{3}+\frac{-11}{5}$ $=\Big(\frac{4}{3}+\frac{-2}{3}\Big)+\Big(\frac{3}{5}+\frac{-11}{5}\Big)$ $=\frac{4+(-2)}{3}+\frac{3-11}{5}$ $=\frac{2}{3}+\Big(\frac{-8}{5}\Big)$ $=\frac{10+(-24)}{15}$ $=\frac{10-24}{15}$ $=\frac{-14}{15}$
View full question & answer→Question 453 Marks
Which of the two rational numbers is greater in the given pair? $\frac{-4}{3}\ \text{or}\ \frac{-8}{7}$
Answer$\frac{-4}{3}\ \text{or}\ \frac{-8}{7}$ Here, denominator are not same.
$LCM$ of $3$ and $7 = 21$
$\therefore\frac{-4}{3}=\frac{-4\times7}{3\times7}=\frac{-28}{21}$
$=\frac{-8}{7}=\frac{-8\times3}{7\times3}=\frac{-24}{21}$
Clearly $\frac{-24}{21}$ is greater.Or $\frac{-8}{7}$ is greater.
View full question & answer→Question 463 Marks
Verify whether the given statement is true or false: $\frac{-8}{9}\div\frac{-4}{3}=\frac{-4}{3}\div\frac{-8}{9}$
AnswerFalse.
Solution:
$\text{L.H.S.}=\frac{-8}{9}\div\frac{-4}{3}$
$=\frac{-8}{9}\times\frac{3}{-4}$
$=\frac{-8\times3}{9\times(-4)}$
$=\frac{-24}{-36}=\frac{24}{36}$
$=\frac{24\div12}{36\div12}$
$=\frac{2}{3}$
$\text{R.H.S.}=\frac{-4}{3}\div\frac{-8}{9}$
$=\frac{-4}{3}\times\frac{9}{-8}$
$=\frac{-36}{-24}=\frac{36}{24}$
$=\frac{36\div12}{24\div12}$
$=\frac{3}{2}$
$\text{L.H.S}\neq\text{R.H.S}$
$\therefore$ It is false.
View full question & answer→Question 473 Marks
A drum full of rice weighs $40\frac{1}{6}\text{kg}$. If the empty drum weighs $13\frac{3}{4}\text{kg}$, find the weight of rice in the drum.
AnswerTotal weight of rice and drum
$=40\frac{1}{6}\text{kg}$
$=13\frac{3}{4}\text{kg}$
Weight of empty drum $=13\frac{3}{4}=\frac{55}{4}\text{kg}$
$\therefore$ Weight of rice $=\Big(\frac{241}{6}-\frac{55}{4}\Big)\text{kg}$
$=\frac{482-165}{12}$
$=\frac{317}{12}\text{kg}$
$=26\frac{5}{12}\text{kg}$
View full question & answer→Question 483 Marks
Verify the following: $\Big(\frac{-8}{3}+\frac{-13}{12}\Big)\times\frac{5}{6}=\Big(\frac{-8}{3}\times\frac{5}{6}\Big)+\Big(\frac{-13}{12}\times\frac{5}{6}\Big)$
Answer$\text{L.H.S.}=\Big(\frac{-8}{3}+\frac{-13}{12}\Big)\times\frac{5}{6}$ $=\Big(\frac{-32-13}{12}\Big)\times\frac{5}{6}$ $=\frac{-45}{12}\times\frac{5}{6}$ $=\frac{-225}{72}$ $\text{R.H.S.}=\Big(\frac{-8}{3}\times\frac{5}{6}\Big)+\Big(\frac{-13}{12}\times\frac{5}{6}\Big)$ $=\frac{-8\times5}{3\times6}+\frac{-13\times5}{12\times6}$ $=\frac{-40}{18}+\frac{-65}{72}$ $=\frac{-160+(-65)}{72}$ $=\frac{-160-65}{72}$ $=\frac{-225}{72}$ $\text{L.H.S.}=\text{R.H.S.}$ $\therefore\Big(\frac{-8}{3}+\frac{-13}{12}\Big)\times\frac{5}{6}=\Big(\frac{-8}{3}\times\frac{5}{6}\Big)+\Big(\frac{-13}{12}\times\frac{5}{6}\Big)$
View full question & answer→Question 493 Marks
Find two rational number between $-3$ and $-2.$
AnswerFirst rational number between $-3$ and $-2$
$=\frac{1}{2}[-3+(-2)]$
$=\frac{1}{2}(-3-2)$
$=\frac{1}{2}(-5)$
$=\frac{-5}{2}$
$\therefore-3<\frac{-5}{2}<-2$
Second rational number between $-3$ and $\frac{-5}{2}$
$=\frac{1}{2}\Big[-3+\Big(\frac{-5}{2}\Big)\Big]$
$=\frac{1}{2}\Big[-3-\frac{5}{2}\Big]$
$=\frac{1}{2}\Big[\frac{-6-5}{2}\Big]$
$=\frac{1}{2}\times\frac{-11}{2}$
$=\frac{-11}{4}$
View full question & answer→Question 503 Marks
Represent the following numbers on the number line. $-4\frac{3}{5}$
Answer$-4\frac{3}{5}$
Draw a line and take a point $O$ on it.
Let it represent $0.$
Now, From $O$, take $OA, AB, BC, CD, DE$ on the left of $O$, is representing integers $-1, -2, -3, -4, -5$ respectively.
Divide $DE$ into $5$ equal parts and take $3$ equal parts so that $\text{DP}=\frac{3}{5}$
Then $\therefore\text{OP}=-\Big(4+\frac{3}{5}\Big)=-4\frac{3}{5}$ as shown on the number line given below
View full question & answer→