Question 12 Marks
Insert $3$ equivalent rational numbers between.
$\frac{-1}{2}$ and $\frac{1}{5}$
AnswerGiven, rational numbers are $-\frac{1}{2}$ and $\frac{1}{5}.$
For common denomiator, $LCM$ of $2$ and $5 = 10$
$\therefore\frac{-1\times5}{2\times5}=\frac{-5}{10}$ and
$\frac{1\times2}{5\times2}=\frac{2}{10}$
Hence, three equivalent rational numbers between
$\frac{-5}{10}$ and $\frac{2}{10}$ are
$\frac{-3}{10},\frac{-6}{20},\frac{-9}{30}.$
View full question & answer→Question 22 Marks
Simplify:
$1\div\Big(-\frac{1}{2}\Big)$
Answer Given, $1\div\Big(-\frac{1}{2}\Big)$
The reciprocal of $\Big(-\frac{1}{2}\Big)$ is $\frac{2}{-1}$
So, $1\div\Big(-\frac{1}{2}\Big)\frac{1}{1}\times\frac{2}{-1}$
$=\frac{1\times2}{1\times (-1)}$
$=\frac{2}{-1}=-1$
View full question & answer→Question 32 Marks
Write a rational number in which the numerator is less than $‘-7 \times 11′$ and the denominator is greater than $’12+ 4’.$
AnswerLet, $-7 \times 11 = p = -77$ and $12 ÷ 4 = q = 16$
Rational number $=\frac{\text{P}}{\text{q}}=\frac{-77}{16}$
Hence, it has more than one answer like $\frac{-78}{17},\frac{-79}{18},\frac{-80}{19}.$
View full question & answer→Question 42 Marks
Express the following rational numbers in its standard form:
$\frac{14}{-49}$
AnswerGiven rational number is $\frac{14}{-49}$
For standard form of given rational number,
$=\frac{14\div7}{-49\div7}$
$[\therefore HCF$ of $14$ and $49 = 7]$
$=\frac{2}{-7}=\frac{-2}{7}$
Hence, the standard form of $\frac{14}{-49}$ is $\frac{-2}{7}.$
View full question & answer→Question 52 Marks
Write the following as rational numbers in their standard forms.
$35\%$
AnswerGiven, $35\%=\frac{35}{100}$
$\begin{array}{c|c} 7 & 35 \\ \hline 5 & 5\\ \hline&1 \end{array}$
$\begin{array}{c|c} 2 & 100 \\ \hline 2 & 50\\ \hline5&25 \\ \hline5&5 \end{array}$
By using prime factorisation, we get
$35 = 7 \times 5$ and $100 = 2 \times 2 \times 5 \times 5$
$\therefore HCF$ of $35$ and $100 = 5$
On dividing numerator and denominator by their $HCF,$ we get
$\frac{35+\div5}{100\div5}=\frac{2}{-7}$
View full question & answer→MCQ 62 Marks
Find the odd one out of the following and give reason.
- ✓
$\frac{4}{3}\times\frac{2}{4}$
- B
$\frac{-3}{2}\times\frac{-2}{3}$
- C
$2\times\frac{1}{2}$
- D
$\frac{-1}{3}\times\frac{3}{1}$
AnswerCorrect option: A. $\frac{4}{3}\times\frac{2}{4}$
$a.$ Given, $\frac{4}{3}\times\frac{2}{4}$
$\therefore$ Product of rational numbers
$=\frac{\text{Product of numerators}}{\text{Product of denominators}}$
$=\frac{4\times3}{3\times4}$
$=\frac{12}{12}$
$=1$
$b.$ Similarly,
$\frac{-3}{1}\times\frac{-2}{3}=\frac{(-3)\times(-2)}{2\times3}=\frac{6}{6}=1$
$c. \frac{2}{1}\times\frac{1}{2}=\frac{2\times1}{1\times2}=\frac{2}{2}=1$
$d. -\frac{1}{3}\times\frac{3}{1}=\frac{(-1)\times3}{3\times1}=-1$
Since, the value of options $(a), (b), (c)$ are $1$ and option $(d)$ is $-1$
Hence, option $(d)$ is odd out.
View full question & answer→Question 72 Marks
Write the next three rational numbers to complete the pattern: $\frac{4}{-5},\frac{8}{-10},\frac{12}{-15},\frac{16}{-20},$ ___, ____, ____.
AnswerGiven rational number is $\frac{4}{-5}.$So, the equivaient rational numbers are
$\frac{4\times5}{-5\times5}=\frac{20}{-25},\frac{4\times6}{-5\times6}=\frac{24}{-30}$
and $\frac{4\times7}{-5\times7}=\frac{28}{-35}$
Hence, three equivalent rational numbers are,
$\frac{20}{-25},\frac{24}{30},\frac{28}{-35}.$
View full question & answer→Question 82 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ find $x - (y + z)$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$ Here, $x - (y + z)$
$=\frac{-4}{9}+\Big(\frac{5}{12}+\frac{7}{18}\Big)$
$=\frac{-4}{9}+\Big(\frac{5\times3+7\times2}{36}\Big)$
$=\frac{-4}{9}-\frac{29}{36}$
$=\frac{-4\times4-29}{36}=\frac{-16-29}{36}$
$=\frac{-45}{36}=\frac{-5}{4}$
View full question & answer→Question 92 Marks
Simplify: $\frac{3}{7}\div\Big(\frac{21}{-55}\Big)$
AnswerGiven, $\frac{3}{7}\div\Big(\frac{21}{-55}\Big)$ The reciprocal of $\Big(\frac{21}{-55}\Big)$ is $\frac{-55}{21}$ So, $\frac{3}{7}\div\Big(\frac{21}{-55}\Big)$ $=\frac{3}{7}\times\frac{(-55)}{21}$ $\frac{(-55)\times3}{7\times21}=\frac{-55}{49}$
View full question & answer→Question 102 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ find$(x – y) + z$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
We have,$(x - y) + z$
$=\Big(\frac{-4}{9}-\frac{5}{12}\Big )+\frac{7}{18}$
$=\frac{-4\times4-5\times3}{36}+\frac{7}{18}$
$=\frac{-16-15}{36}+\frac{7}{18}=\Big(\frac{-31}{36}+\frac{7}{18}\Big )$
$=\frac{-31+7\times2}{36}=\frac{-31+14}{36}$
$=\frac{-17}{36}$
View full question & answer→Question 112 Marks
Find the reciprocal of the following:
$\frac{20}{51}\times\frac{4}{91}$
Answer Given, $\frac{20}{51}\times\frac{4}{91}$ $=\frac{20\times4}{51\times91}$ $\Big[\therefore$ Product of ation nimbers $=\frac{\text {Product of numerators}}{\text {Product of denominators}}\Big]$
$=\frac{80}{4641}$ Hence, the reciprocal of $\frac{80}{4641}$ is $\frac{4641}{80}.$ View full question & answer→Question 122 Marks
$\frac{-22}{11}$ and $\frac{-21}{11}$
AnswerGiven, $\frac{-22}{11}$ and $\frac{-21}{11}$
$\therefore$ Product of rational numbers
$=\frac{\text{Product of numerators}}{\text{Product of denominators}}$
$=\frac{(-22)\times(-21)}{11\times11}=\frac{462}{121}$$=\frac{462+11}{121+11} [$divingnumerator and denominator by $11]$
$=\frac{42}{11}$
View full question & answer→Question 132 Marks
Simplify: $1\div\Big(-\frac{1}{2}\Big)$
AnswerGiven, $1\div\Big(-\frac{1}{2}\Big)$ The reciprocal of $\Big(-\frac{1}{2}\Big)$ is $\frac{2}{-1}$ So, $1\div\Big(-\frac{1}{2}\Big)\frac{1}{1}\times\frac{2}{-1}$ $=\frac{1\times2}{1\times (-1)}$ $=\frac{2}{-1}=-1$
View full question & answer→Question 142 Marks
List four rational numbers between $\frac{5}{7}$ and $\frac{7}{8.}$
AnswerGiven rational numbers $\frac{5}{7}.$ and $\frac{7}{8}.$
For making the same denominators: $LCM$ of $7$ and $8 = 56.$
e.i. $\frac{5\times8}{7\times8}=\frac{40}{56}$ and $\frac{7\times7}{8\times7}=\frac{49}{56}$
So, the four rational numbers between $\frac{40}{56}$ and $\frac{49}{56}$ are,
$\frac{42}{56},\frac{44}{56},\frac{46}{56},\frac{48}{56}.$
View full question & answer→Question 152 Marks
What should be subtracted from $\frac{-2}{3}$ to obtain the nearest integer?
AnswerGiven rational number is $\frac{-2}{3}.$
We knoe that,nearest natural number of $\frac{-2}{3}$ is $-1.$
Let $x$ be subtracted to $\frac{-2}{3}$ to obtain $-1.$
Then, $\frac{-2}{3}-\text{x}=-1$
$\Rightarrow\text{x}=\frac{-2}{3}+1=\frac{1}{3}$
So, we subtract $\frac{1}{3}$ from $\frac{-2}{3}$ to get the neartes integer.
View full question & answer→Question 162 Marks
Find the reciprocal of the following: $\frac{3}{13}\div\frac{-4}{65}$
AnswerGiven, $\frac{3}{13}\div\frac{-4}{65}$ The recipocal of $\frac{-4}{65}$ is $\frac{65}{-4}.$ $\therefore\frac{3}{13}\div\frac{-4}{65}=\frac{3}{13}\times\frac{65}{-4}$$=\frac{65\times3}{13\times(-4)}=\frac{15}{-4}$
Hence, the reciprocal of $\frac{15}{-4}$ is $\frac{-4}{15}.$
View full question & answer→Question 172 Marks
What should be multiplied with $\frac{-5}{8}$ to obtain the nearest integer$?$
AnswerLet number be $x.$
We know that, neartest integer of $-\frac{5}{8}$ is $-1$
According to the question, $\frac{-5}{8}\times\text{x}=-1$
$\Rightarrow\text{x}=-1\times\frac{8}{-5}=\frac{8}{5}$
Hence, the required number is $\frac{8}{5}.$
View full question & answer→Question 182 Marks
if $\frac{-5}{7}=\frac{\text{x}}{28}$ find the value of $x.$
AnswerGiven, $\frac{-5}{7}=\frac{\text{x}}{28}$
$\Rightarrow7\times\text{x}=-5\times28 [$by cross-multiplication$]$
$\Rightarrow\text{x}=-\frac{5\times28}{7}=-5\times4$
$\Rightarrow=-20$
Hence, the value of $x$ is $-20.$
View full question & answer→Question 192 Marks
Find the reciprocal of the following:
$\Big(\frac{1}{2}\times\frac{1}{4}\Big)\div\Big(\frac{1}{2}\times6\Big) $
Answer Given, $\Big(\frac{1}{2}\times\frac{1}{4}\Big)\div\Big(\frac{1}{2}\times6\Big) $
$=\frac{1\times1}{2\times4}\div\frac{1\div6}{2\div1}=\frac{1}{8}\div\frac{6}{2}$
$=\frac{1\times1}{8\times1}\div\frac{6\times4}{2\times4}$ $\Big[\therefore$ Product of ation nimbers $=\frac{\text {Product of numerators}}{\text {Product of denominators}}\Big]$
$=\frac{1}{8}\div\frac{24}{8}=\frac{1\div24}{8}$
$=\frac{25}{8}$
Hence, the reciprocal of $\frac{25}{8}$ is $\frac{8}{25}.$
View full question & answer→Question 202 Marks
$‘o’$ and $‘b’$ are two different numbers taken from the numbers $1 - 50.$ What is the largest value that $\frac{\text{a}-\text{b}}{\text{a}+\text{b}}$ can have? What is the largest $\frac{\text{a}+\text{b}}{\text{a}-\text{b}}$ can have$?$
AnswerGiven, $a$ and $b$ are two different numbers between $1$ to $50.$
Let $a = 50$ and $b = 1$
$\therefore\frac{\text{a}-\text{b}}{\text{a}+\text{b}}=\frac{50-1}{50+1}=\frac{49}{51},$
Which is the largest value. Similary.
Let $a = 50$ and $b = 49$
$\therefore\frac{\text{a}+\text{b}}{\text{a}-\text{b}}=\frac{50+49}{50-49}=\frac{99}{1}=99,$
Which is the largest value.
View full question & answer→Question 212 Marks
What should be divided by $\frac{-1}{2}$ to obtain the greatest negative integer$?$
AnswerLet the number be $x.$
We know that, greatest negative integer of is $-1$ According to the question,
$\frac{1}{2}\div\text{x}=-1$
$\Rightarrow\frac{1}{2}\times\frac{1}{\text{x}}=-1$
$\Rightarrow\frac{1}{\text{x}}=-1\times\frac{2}{1}$
$\frac{1}{\text{x}}=\frac{-2}{1}$
$\Rightarrow\text{x}=\frac{-1}{2}$
Hence, the required number is $\frac{-1}{2}.$
View full question & answer→Question 222 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ findThe sum of reciprocals of $x$ and $y.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Reciprocal of $x$ and $y$ is $\frac{1}{\text{x}}$ and $\frac{1}{\text{y}}.$
$\therefore$ Sum of reciproccals $=\frac{1}{\text{x}}+\frac{1}{\text{y}}=\frac{1}{\frac{-4}{9}}+\frac{1}{\frac{5}{12}}$
$=\frac{-9}{4}+\frac{12}{5}=\frac{-45+48}{20}$
$=\frac{3}{20}.$
View full question & answer→Question 232 Marks
If $12$ shirts of equal size can be prepared from $27m$ cloth, what is length of cloth required for each shirt$?$
AnswerGiven, total size of avilable cloth $= 27m$
Since, $12$ shrits and required for each shirt $=\frac{\text{Total avilable cloth}}{\text{Number of shirts}}$
$=\frac{27}{12}=\frac{9}{4}$
$=2.25\text{m}$
Hence, $2.25m$ cloth required for each shirt.
View full question & answer→Question 242 Marks
Express the following rational numbers in its standard form: $\frac{-15}{35}$
AnswerGiven rational number is $\frac{-15}{35}.$
For standard form of given rational number, $=\frac{-15\div5}{35\div5}$
$[\therefore HCF$ of $15$ and $35 = 5]$
$=\frac{-3}{7}$
Hence, the standard form of $\frac{-15}{35}$ is $\frac{-3}{7}.$
View full question & answer→Question 252 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ find $(x ÷ y) × z.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
We have,$(x ÷ y) × z$
$=\Big(\frac{-4}{9}\div\frac{5}{12}\Big)\times\frac{7}{18}$
$=\Big(\frac{-4}{9}\times\frac{12}{5}\Big)\times\frac{7}{18}$
$=\frac{-4\times12\times7}{9\times5\times18}$
$=\frac{-56}{135}$
View full question & answer→Question 262 Marks
Find a rational number exactly halfway between. $\frac{1}{15}$ and $\frac{1}{12}$
AnswerWe know that, a rational number, which is haifway between two rational number i.e.
$a$ and $b =\frac{\text{a}+\text{b}}{2}.$ Given rational numbers are $\frac{1}{15}$ and $\frac{1}{12}.$
Here, $\text{a} =\frac{1}{15}$ and $\text{b}=\frac{1}{12}$
$\therefore\frac{\text{a}+\text{b}}{2}=\frac{\frac{1}{15}+\frac{1}{12}}{2}=\frac{\frac{1\times4}{15\times4}+\frac{1\times5}{12\times5}}{2}$
$=\frac{\frac{4}{60}+\frac{5}{60}}{2}=\frac{\frac{4+5}{60}}{2}=\frac{9}{60\times2}=\frac{9}{120}$
$=\frac{3}{40}$
Hecne, the exaclty halfway betweenm $\frac{1}{15}$ and $\frac{1}{12}$ is $\frac{3}{40}.$
View full question & answer→Question 272 Marks
Solve: $\frac{29}{4}-\frac{30}{7}$
AnswerGiven, $\frac{29}{4}-\frac{30}{7}=\frac{29\times7}{4\times7}-\frac{30\times4}{7\times4}$
$[\therefore LCM $ of $4$ and $4$ and $7$ is $28,$ so convert each of the given fraction to equivalent fractions with denominator $28]$
$=\frac{203}{28}-\frac{120}{28}$ $=\frac{203-120}{28}=\frac{83}{28}$
View full question & answer→Question 282 Marks
Express the following rational numbers in its standard form: $\frac{-12}{-30}$
AnswerGiven rational number is $\frac{-12}{-30}$
For standard form of given rational number, $=\frac{-12\div6}{-30\div6}$
$[\therefore HCF$ of $12$ and $30 = 6]$
$=\frac{-2}{-5}=\frac{2}{5}$
Hence, the standard form of $\frac{-12}{-30}$ is $\frac{2}{5}.$
View full question & answer→Question 292 Marks
Find a rational number exactly halfway between. $\frac{1}{6}$ and $\frac{1}{9}$
AnswerWe know that, a rational number, which is haifway between two rational number i.e. $a$ and $b$
$=\frac{\text{a}+\text{b}}{2}.$
Given rational numbers are $\frac{1}{6}$ and $\frac{1}{9}.$
Hence, $\text{a} =\frac{1}{6}$ and $\text{b}=\frac{1}{9}$
$\therefore\frac{\text{a}+\text{b}}{2}=\frac{\frac{1}{6}+\frac{1}{9}}{2}=\frac{\frac{1\times3}{6\times3}+\frac{1\times2}{9\times2}}{2}$
$=\frac{\frac{3}{18}+\frac{2}{18}}{2}$
$=\frac{\frac{3+2}{18}}{2}=\frac{\frac{5}{18}}{2}=\frac{5}{18\times2}=\frac{5}{36}$
Hecne, the exaclty halfway betweenm $\frac{1}{6}$ and $\frac{1}{9}$ is $\frac{5}{36.}$
View full question & answer→Question 302 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ findThe rational number which when added to $x$ gives $y.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Let we add a to $x$ to get $y.$
$\therefore\text{A}+\text{x}=\text{y}$
$\Rightarrow\text{A}+ \Big(\frac{-4}{9}\Big)=\frac{5}{12}$
$\Rightarrow\text{A}=\frac{5}{12}-\Big(-\frac{4}{9}\Big)$
$=\frac{5}{12}+\frac{4}{9}=\frac{5\times3+4\times4}{36}$
$=\frac{15+16}{36}=\frac{31}{36}$
View full question & answer→Question 312 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ findThe rational number which subtracted from $y$ gives $z.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Let we add $A$ to $y$ to get $z.$
$\therefore\text{A}-\text{x}=\text{y}$
$\Rightarrow\frac{5}{12}-\text{A}=\frac{7}{18}$
$\Rightarrow-\text{A}=\frac{7}{18}-\frac{5}{12}$
$=\frac{7\times2-5\times3}{36}$
$=\frac{14-15}{36}=\frac{-1}{36}$
$\Rightarrow\text{A}=\frac{1}{36}$
View full question & answer→Question 322 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ find $x + (y + z)$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Here, $x + (y + z)$
$=\frac{-4}{9}+\Big(\frac{5}{12}+\frac{7}{18}\Big)$
$=\frac{-4}{9}+\frac{15}{14}=\frac{-4}{9}\times\frac{14}{15}=\frac{-56}{135}$
View full question & answer→Question 332 Marks
Write the next three rational numbers to complete the pattern: $\frac{-8}{7},\frac{16}{14},\frac{-24}{21},\frac{-32}{28},$ ___, ____, ____.
AnswerGiven rational number is $\frac{-8}{7}.$So, the equivaient rational numbers are
$\frac{8\times5}{7\times5}=\frac{40}{35},\frac{8\times6}{7\times6}=\frac{48}{42}$
and $\frac{-8\times7}{7\times7}=\frac{-56}{49}$
Hence, three equivalent rational numbers are,
$\frac{40}{35},\frac{48}{42},\frac{-56}{49}.$
View full question & answer→Question 342 Marks
Find the reciprocal of the following: $\Big(-5\times\frac{12}{15}\Big)-\Big(-3\times\frac{2}{9}\Big)$
AnswerGiven, $\Big(-5\times\frac{12}{15}\Big)-\Big(-3\times\frac{2}{9}\Big)$$=\Big(-\frac{12}{3}\Big)\Big(-\frac{2}{3}\Big)$
$=-\frac{12}{3}+\frac{2}{3}=\frac{-12+2}{3}=-\frac{10}{3}$
Hence, the reciprocal of $\frac{10}{3}$ is $\frac{3}{10}.$
View full question & answer→Question 352 Marks
Find a rational number exactly halfway between. $\frac{5}{-13}$ and $\frac{-7}{9}$
AnswerWe know that, a rational number, which is haifway between two rational number
i.e. $a$ and $b =\frac{\text{a}+\text{b}}{2}.$
Given rational numbers are $\frac{5}{-13}$ and $\frac{-7}{9}.$
Hence, $\text{a} =-\frac{5}{13}$ and $\text{b}=-\frac{7}{9}$
$\therefore\frac{\text{a}+\text{b}}{2}=\frac{\frac{-5}{13}+\big(-\frac{7}{9}\big)}{2}=\frac{\frac{-5}{13}-\frac{7}{9}}{2}$
$=\frac{\frac{-5\times9}{13\times9}-\frac{7\times13}{9\times13}}{2}$
$=\frac{\frac{-45}{117}-\frac{91}{117}}{2}=\frac{\frac{-45-49}{117}}{2}$
$=\frac{-136}{117\times2}=\frac{-136}{234}$
Hecne, the exaclty halfway betweenm $\frac{5}{-13}$ and $\frac{-7}{9}$ is $-\frac{136}{234}.$
View full question & answer→Question 362 Marks
Give three rational numbers equivalent to: $\frac{7}{11}$
AnswerGiven rational number is $\frac{7}{11}.$So, the equivaient rational numbers are
$\frac{7\times2}{11\times2}=\frac{14}{22},\frac{7\times3}{11\times3}=\frac{21}{33}$
and $\frac{7\times4}{11\times4}=\frac{28}{44}$
Hence, three equivalent rational numbers are,
$\frac{14}{22},\frac{21}{23}$ and $\frac{28}{44}.$
View full question & answer→Question 372 Marks
What’s the Error? Chhaya simplified a rational number is this manner $\frac{-25}{-30}=\frac{-5}{-6}$ What error did the student make?
AnswerIf anegative $(-)$ sign comes in both numerator and denominator, then it will be cancelled.
So, the resulting fraction will be posotive.
$\therefore\frac{-25}{-30}=\frac{25}{30}=\frac{5}{6}$
Here, chhaya divied numerator by $5$ but denominator by $-5.$
View full question & answer→Question 382 Marks
$150$ students are studying English, Maths or both. $62\%$ of the students are studying English and $68\%$ are studying Maths. How many students are studying both$?$
AnswerGiven, total students in the class studying English, Maths or both $= 150$
Students studying English $= 62\%$ of $150 =\frac{68}{100}\times150=93$
Students studying Maths $= 68\%$ of $150 =\frac{68}{100}\times150=102$
Total students studing both $=$ Students studying English $+$ students studing Maths $-$ Students studying English,
Maths or both $= 93 + 102 - 150 = 45$
View full question & answer→Question 392 Marks
Find the product of: $\frac{-4}{5}$ and $\frac{-5}{12}$
AnswerGiven, $\frac{-4}{5}$ and $\frac{-5}{12}$
$\therefore$ Prduct of rational numbers, $=\frac{\text{Product of numerators}}{\text{Product of denominators}}$
$=\frac{(-4)\times(-5)}{5\times12}=\frac{20}{60}$
$[$diving numerator and denominator by $20]$
$=\frac{20\div20}{60\div20}$
$=\frac{1}{3}$
View full question & answer→Question 402 Marks
A body floats $\frac{2}{9}$ of its volume above the surface. What is the ratio of the body submerged volume to its exposed volume? Rewrite it as a rational number.
AnswerGiven, volume of body exposed $=\frac{2}{9}$
$\therefore$ Volume of body submerged $1\ -$ volume of body exposed
$=1-\frac{2}{9}=\frac{9-2}{9}=\frac{7}{9}$
$\therefore$ Required ratio
$=\frac{7}{9}:\frac{2}{9}=\frac{7}{9}\div\frac{2}{9}=\frac{7}{9}\times\frac{9}{2}=\frac{7}{2}=7:2$
In rationl number $=\frac{7}{2}$
View full question & answer→Question 412 Marks
Find a rational number exactly halfway between. $\frac{5}{-13}$ and $\frac{-7}{9}$
AnswerWe know that, a rational number, which is haifway between two rational number i.e.
$a$ and $b =\frac{\text{a}+\text{b}}{2}.$
Given rational numbers are $\frac{-1}{3}$ and $\frac{1}{3}.$
Hence, $\text{a} =-\frac{1}{3}$ and $\text{b}=\frac{1}{3}$
$\therefore\frac{\text{a}+\text{b}}{2}=\frac{\frac{1}{3}+\frac{1}{3}}{2}=\frac{0}{2}=0$
Hecne, the exaclty halfway betweenm $-\frac{1}{3}$ and $\frac{1}{3}$ is $0\ ($zero$).$
View full question & answer→Question 422 Marks
Give three rational numbers equivalent to: $\frac{-3}{4}$
AnswerGiven rational number is $\frac{-3}{4}.$So, the equivaient rational numbers are
$\frac{3\times2}{4\times2}=\frac{-6}{8},\frac{-3\times3}{4\times3}=\frac{-9}{12}$
and $\frac{-3\times4}{4\times4}=\frac{-12}{16}$
Hence, three equivalent rational numbers are,
$\frac{-6}{8},\frac{-9}{12}$ and $\frac{-12}{16}.$
View full question & answer→Question 432 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$
findThe reciprocal of $x + y.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Here, $x + y =\frac{-4}{9}+\frac{5}{12}=\frac{-4\times4+5\times3}{36}$
$\Rightarrow\text{x}+\text{y}=\frac{1}{\frac{-1}{36}}=-\frac{-1}{36}$
$\therefore$ Reciprocal of $\text{x}+\text{y}=\frac{1}{\frac{-1}{36}}=-36$
View full question & answer→Question 442 Marks
Reduce the following rational numbers in its lowest form: $\frac{-60}{72}$
Answer$\frac{-60}{72}$ can be written as $=\frac{-60+12}{72+12}$
$[$dividing numerator and denominator by $HCF$ of $60$ and $72$ i.e. $12]$
$=\frac{-60\times\frac{1}{12}}{72\times\frac{1}{12}}$
$\Big[\because\text{Reciprocal of }12=\frac{1}{12}\Big]$
$=\frac{-5}{6},$ which is the lowest form.
View full question & answer→Question 452 Marks
From a rope $68m$ long, pieces of equal size are cut. If length of one piece is $4\frac{1}{4}\text{m},$ find the number of such pieces.
AnswerGiven, length of the rope $= 68m$ and
length of small piece $=4\frac{1}{4}\text{m}=\frac{(4\times4)+1}{4}\text{m}=\frac{17}{4}\text{m}$
$\therefore$ Number of pieces $=\frac{\text{Total length of rope}}{\text{Length of small piece}}=\frac{68}{\frac{17}{4}}$
$=\frac{68}{1}\times\frac{4}{17}$
$=4\times4=16$
Hence, the number of pieces is $16.$
View full question & answer→Question 462 Marks
What should be added to $\frac{-1}{2}$ to obtain the nearest natural number$?$
AnswerWe know that, nearest number of $\frac{-1}{2}$ is $1.$
Let $x$ be added to $-\frac{1}{2}$ to obtain $1.$
Then, $-\frac{1}{2}+\text{x}=1$
$\Rightarrow \text{x}=1+\frac{1}{2}=\frac{2+1}{2}$
$\Rightarrow\text{x}=\frac{3}{2}$
Hence, $\frac{3}{2}$ should be added to $\frac{-1}{2}$ to obtain nearest natural number.
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Write the following as rational numbers in their standard forms. $115 ÷ 207$
AnswerGiven, $115 ÷ 207$
$=\frac{115}{207}$
$\begin{array}{c|c} 5 & 1158 \\ \hline 23 & 23\\ \hline&1 \end{array}$
$\begin{array}{c|c} 3 & 207 \\ \hline 3 & 69\\ \hline23&23 \\ \hline&1 \end{array}$
By using prime factorisation, we get, $115 = 5 \times 23$ and $207 = 3 \times 23 \times 3$
$\therefore HCF$ of $115$ and $207 = 23$
On dividing numerator and denominator by their
$HCF,$ we get $\frac{115\div23}{207\div23}=\frac{5}{9}$
View full question & answer→Question 482 Marks
Taking $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18},$ findThe rational number which when multiplied by $y$ to get $x.$
AnswerGiven, $\text{x}=\frac{-4}{9},\text{y}=\frac{5}{12}$ and $\text{z}=\frac{7}{18}$
Suppose, if $A$ is multiplied by $y,$
then we get $x$ i.e. $\text{A}\times\text{y}=\text{x}$
$\Rightarrow\text{A}\times\frac{5}{12}=\frac{-4}{9}$
$\Rightarrow\text{A}=\frac{-4}{9}-\frac{12}{5}=\frac{-48}{45}$
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Which is greater in the following? $\frac{3}{4},\frac{7}{8}$
AnswerGien rational numbers are $\frac{3}{4}$ and $\frac{7}{8}.$ Here, $\frac{3}{4}=\frac{3\times2}{4\times2}=\frac{6}{8}$ and $\frac{7}{8}=\frac{7\times1}{8\times1}=\frac{7}{8}$ $\therefore 7>6$ [Since, the denominators of bhot rational numbers are sane] So, $\frac{7}{8}>\frac{3}{4}$ Hence, the greater numbers is $\frac{7}{8}.$
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Solve: $\frac{5}{ 13}-\frac{-8}{26}$
AnswerGiven, $\frac{5}{13}-\Big(\frac{-8}{27}\Big)$
$=\frac{5}{13}+\frac{8}{26}=\frac{5\times2}{13\times2}+\frac{8\times1}{26\times1}$
$[\therefore LCM$ of $4$ and $4$ and $7$ is $28,$ so convert each of the given fraction to equivalent fractions with denominator $28]$
$=\frac{10}{26}+\frac{8}{26}$
$[$dividing numerator and denominatore by $2]$
$=\frac{10+8}{26}=\frac{18}{26}$
$=\frac{18+2}{26+2}=\frac{9}{13}$
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