Question 13 Marks
Find the sum:
$3\frac{1}{3}+4\frac{1}{4}+6\frac{1}{6}$
AnswerWe have,
$L.C.M$. of $3, 4$ and $6 = (2 \times 2 \times 3) = 12$
$\begin{array}{c|c}3&3,4,6\\\hline2&1,4,2\\\hline2&1,2,1\\\hline&1,1,1\end{array}$
Therefore,
$3\frac{1}{3}+4\frac{1}{4}+6\frac{1}{6}$
$=\frac{10}{3}+\frac{17}{4}+\frac{37}{6}$
$=\frac{(40+51+74)}{12}$
$\Big(\frac{12}{3}=4,4\times10=40\Big)$
$\Big(\frac{12}{4}=3,3\times17=51\Big)$
and $\Big(\frac{12}{6}=2,2\times37=74\Big)$
$=\frac{165}{12}$
$=\frac{55}{4}$
$=13\frac{3}{4}$
View full question & answer→Question 23 Marks
Convert $\frac{1}{4},\frac{5}{8},\frac{7}{12}$ and $\frac{13}{24}$ into like fractions.
AnswerThe given fractions are $\frac{1}{4},\frac{5}{8},\frac{7}{12}$ and $\frac{13}{24}$
$L.C.M.$ of $4, 8, 12$ and $24 = (4 \times 2 \times 3) = 24$
So, we convert the given fractions into equivalent fractions with $24$ as the denominator.
(But, one of the fractions already has $24$ as its denominator.
So, there is no need to convert it into an equivalent fraction.)
Thus, we have: $\frac{1}{4}=\frac{1\times6}{4\times6}=\frac{6}{24}$,
$\frac{5}{8}=\frac{5\times3}{8\times3}=\frac{15}{24}$,
$\frac{7}{12}=\frac{7\times2}{12\times2}=\frac{14}{24}$
Hence, the required like fractions are $\frac{6}{24},\frac{15}{24},\frac{14}{24}$ and $\frac{13}{24}$.
View full question & answer→Question 33 Marks
Find the sum:
$3\frac{2}{3}+1\frac{5}{6}+2$
AnswerWe have,
$L.C.M.$ of $3$ and $6 = (2 \times 3) = 6$
$\begin{array}{c|c}3&3,6\\\hline2&1,2\\\hline&1,1\end{array}$
Therefore,
$3\frac{2}{3}+1\frac{5}{6}+2$
$=\frac{11}{3}+\frac{11}{6}+\frac{2}{1}$
$=\frac{(22+11+12)}{6}$
$\Big(\frac{6}{3}=2,2\times11=22\Big)$
$\Big(\frac{6}{6}=1,1\times11=11\Big)$
and $\Big(\frac{6}{1}=6,6\times2=12\Big)$
$=\frac{45}{6}$
$=\frac{15}{2}$
$=7\frac{1}{2}$
View full question & answer→Question 43 Marks
Lalita read $30$ pages of a book containing $100$ pages while Sarita read $\frac{2}{5}$ of the book. Who read more?
AnswerLalita read $30$ pages of a book having $100$ pages Sarita read $\frac{2}{5}$ of the same book $\frac{2}{5}$ of $100$ pages = $\frac{2}{5}\times100$
$=\frac{200}{5}=40 = 40$ pages
Hence, Sarita read more pages than Lalita as $40$ is greater than $30.$
View full question & answer→Question 53 Marks
In a school $20$ students out of $25$ passed in $VI A$, while $24$ out of $30$ passed in $VI B$. Which section gave better result?
AnswerFraction of students who passed in $ VI A =\frac{20}{25}$
$=\frac{20\div5}{25\div5}=\frac{4}{5}$
Fraction of students who passed in $VI B =\frac{24}{30}$
$=\frac{24\div6}{30\div6}=\frac{4}{5}$
In both the sections, the fraction of students who passed is the same, so both the sections have the same result.
View full question & answer→Question 63 Marks
Reduce the following fractions into its simplest form: $\frac{150}{60}$
AnswerHere, numerator $= 150$ and denominator $= 60$ Factors of $150$ are $1, 2, 3, 5, 6, 10, 15, 25, 30, 75$ and $150$
Factors of $60$ are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30$ and $60$
Common factors of $150$ and $60$ are $1, 2, 3, 5, 6, 10, 15$ and $30 H.C.F$. of $150$ and $60$ is $30.$
$\therefore\frac{150}{60}=\frac{150\div30}{60\div30}=\frac{5}{2}$
Hence, the simplest form of $\frac{150}{60}$ is $\frac{5}{2}$.
View full question & answer→Question 73 Marks
Show that the following fractions is in the simplest form: $\frac{8}{11}$
View full question & answer→Question 83 Marks
Subtract the sum of $3\frac{5}{9}$ and $3\frac{1}{3}$ from the sum of $5\frac{5}{6}$ and $4\frac{1}{9}$.
Answer$\Big(5\frac{5}{6}+4\frac{1}{9}\Big)-\Big(3\frac{5}{9}+3\frac{1}{3}\Big)$$\Big(\frac{35}{6}+\frac{37}{9}\Big)-\Big(\frac{32}{9}+\frac{10}{3}\Big)$
$\begin{array}{c|c}2&6,9,3\\\hline3&3,9,3\\\hline3&1,3,1\\\hline&1,1,1\end{array}$
$L.C.M$. of $3, 6$ and $9 = (2 \times 2 \times 3) = 18$
$=\frac{[105+74]-[64+60]}{18}$
$\Big(\frac{18}{6}=3,3\times35=105\Big)$
and $\Big(\frac{18}{9}=2,2\times37=74\Big)$
$\Big(\frac{18}{9}=2,2\times32=64\Big)$
and $\Big(\frac{18}{3}=6,6\times10=60\Big)$
$=\frac{[179]-[124]}{18}$
$=\frac{55}{18}$
$=3\frac{1}{18}$
View full question & answer→Question 93 Marks
Find the sum: $3\frac{1}{8}+1\frac{5}{12}$
Answer$L.C.M.$ of $8$ and $12 = (2 × 2 × 2 × 3) = 34$
$\begin{array}{c|c}2&8,12\\\hline2&4,6\\\hline2&2,3\\\hline3&1,3\\\hline&1,1\end{array} $
Therefore, $3\frac{1}{8}+1\frac{5}{12}$
$=\frac{25}{8}+\frac{17}{12}$
$=\frac{(75+34)}{24}$
$\Big(\frac{24}{8}=3,3\times25=75\Big)$ and $\Big(\frac{24}{12}=2,2\times17=34\Big)$
$=\frac{109}{24}$
$=4\frac{13}{24}$
View full question & answer→Question 103 Marks
The weight of an empty gas cylinder is $16\frac{4}{5}\text{kg}$ and it contains $14\frac{2}{3}\text{kg}$ of gas. What is the weight of the cylinder filled with gas?
AnswerWeight of the cylinder filled with gas = Weight of the empty cylinder $+$ Weight of the gas inside the cylinder
Thus, we have: $(L.C.M .of 5$ and $3 = (5 \times 3) = 15$
$\Big(16\frac{4}{5}+14\frac{2}{3}\Big)\text{kg}$
$=\Big(\frac{84}{5}+\frac{44}{3}\Big)\text{kg}$
$=\frac{(252+220)}{15}\text{kg}$
$=\frac{472}{15}\text{kg}$
$=31\frac{7}{15}\text{kg}$
Hence, the weight of the cylinder filled with gas is $31\frac{7}{15}\text{kg}$.
View full question & answer→Question 113 Marks
Simplify:$\frac{5}{8}+\frac{3}{4}-\frac{7}{12}$
AnswerWe have:$\frac{5}{8}+\frac{3}{4}-\frac{7}{12}$
$\begin{array}{c|c}2&4,8,12\\\hline2&2,4,6\\\hline2&1,2,3\\\hline3&1,1,3\\\hline&1,1,1\end{array}$
$L.C.M.$ of $4, 8$ and $12 = (2 \times 2 \times 2 \times 3) = 24$
$=\frac{(15+18-14)}{24}$
$\Big(\frac{24}{8}=3,3\times5=15\Big)$
$\Big(\frac{24}{4}=6,6\times3=18\Big)$ and $\Big(\frac{24}{12}=2,2\times7=14\Big)$
$=\frac{(33-14)}{24}$
$=\frac{19}{24}$
View full question & answer→Question 123 Marks
Mrs Soni bought $7\frac{1}{2}$ litres of milk. Out of this milk, $5\frac{3}{4}$ litres was consumed. How much milk is left with her?
AnswerAmount of milk left with Mrs. Soni = Total amount of milk bought by her - Amount of milk consumed
$\therefore$ Amount of milk left with Mrs. Soni $=7\frac{1}{2}-5\frac{3}{4}$
$=\frac{15}{2}-\frac{23}{4}$
$L.C.M.$ of $2$ and $4 = (2 \times 2) = 4$
$=\frac{30-23}{4}$
$\Big(\frac{4}{2}=2,2\times15=30\Big)$ and $\Big(\frac{4}{4}=1,1\times23=23\Big)$
$=\frac{7}{4}$
$=1\frac{3}{4}\ \text{litres}$
Therefore, Milk left with Mrs. Soni $=1\frac{3}{4}\ \text{litres}$
View full question & answer→Question 133 Marks
What should be added to $9\frac{2}{3}$ to get $19?$
AnswerLet x be added to $9\frac{2}{3}$ to get $19$
Therefore, $9\frac{2}{3}+\text{x}=19$ Thus,
we have: $\text{x}=19-9\frac{2}{3}$ $=\frac{19}{1}-\frac{29}{3}$
$L.C.M.$ of $1$ and $3$ is 3$=\frac{(57-29)}{3}$
$\Big(\frac{3}{1}=3,3\times19=57\Big)$
and $\Big(\frac{3}{3}=1,1\times29=29\Big)$
$=\frac{28}{3}$
$=9\frac{1}{3}$
View full question & answer→Question 143 Marks
Show that the following fractions is in the simplest form: $\frac{9}{14}$
AnswerHere, numerator $= 9$ and denominator $= 14$
Factors of $9$ are $1, 3$ and $9$
Factors of $14$ are $1, 2, 7$ and $14$
Common factor of $9$ and $14$ is $1$
Thus, $H.C.F.$ of $9$ and $14$ is $1$
Hence, $\frac{9}{14}$ is the simplest form.
View full question & answer→Question 153 Marks
Sohini bought $4\frac{1}{2}\text{m}$ of cloth for her kurta and $2\frac{2}{3}\text{m}$ of cloth for her pyjamas. Ho much cloth did she purchase in all?
AnswerTotal cloth purchased by Sohini = Cloth for kurta + Cloth for pyjamas
Thus, we have:
$\Big(4\frac{1}{2}+2\frac{2}{3}\Big)\ \text{m}$
$=\Big(\frac{9}{2}+\frac{8}{3}\Big)\ \text{m}$
$(L.C.M.$ of $2$ and $3 = (2 \times 3) = 6)$
$=\Big(\frac{(27+16)}{6}\Big)\ \text{m}$
$\Big(\frac{16}{2}=3,3\times9=27\Big)$
and $\Big(\frac{6}{3}=2,2\times8=16\Big)$
$=\frac{43}{6}\ \text{m}$
$=7\frac{1}{6}\ \text{m}$
$\therefore$ Total length of cloth purchased $=7\frac{1}{6}\ \text{m}$
View full question & answer→Question 163 Marks
Simplify:$5\frac{3}{4}-4\frac{5}{12}+3\frac{1}{6}$
AnswerWe have:$5\frac{3}{4}-4\frac{5}{12}+3\frac{1}{6}$
$\frac{23}{4}-\frac{53}{12}+\frac{19}{6}$
$\begin{array}{c|c}2&4,12,6\\\hline2&2,6,3\\\hline3&1,2,3\\\hline2&1,2,1\\\hline&1,1,1\end{array}$
$L.C.M$. of $4, 12$ and $6 = (2 \times 2 \times 3) = 12$
$=\frac{(69-53+38)}{12}$
$\Big(\frac{12}{4}=3,3\times23=69\Big)$
$\Big(\frac{12}{12}=1,1\times53=53\Big)$ and
$\Big(\frac{12}{6}=2,2\times19=38\Big)$
$=\frac{(107-53)}{12}$
$=\frac{54}{12}$
$=\frac{9}{2}$
$=4\frac{1}{2}$
View full question & answer→Question 173 Marks
Neelam has $25$ pencils. She gives $\frac{4}{5}$ of them to Meena. How many pencils does Meena get? How many pencils are left with Neelam?
AnswerNeelam gives $\frac{4}{5}$ of $25$
pencils to Meena $\Big(\frac{4}{5}\times25\Big) = 20$
Pencils Thus, Meena gets $20$ pencils.
Therefore, Number of pencils left with Neelam $= 25 - 20 = 5$ pencils
Thus, $5$ pencils are left with Neelam
View full question & answer→Question 183 Marks
Rohit bought a pencil for Rs. $3\frac{2}{5}$ and an eraser for Rs. $2\frac{7}{10}$. What is the total cost of both the articles?
AnswerTotal cost of both articles $=$ Cost of pencil $+$ Cost of eraser
Thus, we have: $\text{Rs.}\ 3\frac{2}{5}+\text{Rs.}\ 2\frac{7}{10}$
$=\frac{17}{5}+\frac{27}{10}$
$=\frac{(34+27)}{10}$
$(L.C.M.$ of $5$ and $10 = (5 \times 2) = 10)$
$=\frac{61}{10}$
$=\text{Rs.}\ 6\frac{1}{10}$
Hence, the total cost of both the articles is $\text{Rs.}\ 6\frac{1}{10}$.
View full question & answer→Question 193 Marks
Show that the following fractions is in the simplest form: $\frac{8}{15}$
AnswerHere, numerator $= 8$ and denominator $= 15$
Factors of 8 are $1, 2, 4$ and $8$ Factors of $15$ are $1, 3, 5$ and $15$
Common factor of $8$ and $15$ is $1$
Thus, $H.C.F$. of $8$ and $15$ is $1$
Hence, $\frac{8}{15}$ is the simplest form.
View full question & answer→Question 203 Marks
Reduce the following fractions into its simplest form:
$\frac{48}{60}$
AnswerHere, numerator $= 84$ and denominator $= 98$
Factors of $84$ are $1, 2, 3, 4, 6, 7, 12, 14, 21, 42$ and $84$
Factors of $98$ are $1, 2, 7, 14, 49$ and $98$
Common factors of $84$ and $98$ are $1, 2, 7$ and $14$
$H.C.F.$ of $84$ and $98$ is $14$
$\therefore\frac{48}{60}=\frac{48\div12}{60\div12}=\frac{4}{5}$
Hence, the simplest form of $\frac{48}{60}$ is $\frac{4}{5}$.
View full question & answer→Question 213 Marks
What should be added to $6\frac{7}{15}$ to get $8\frac{1}{5}$?
AnswerLet $x$ be added to $6\frac{7}{15}$ to get $8\frac{1}{5}$
Therefore, $6\frac{7}{15}+\text{x}=8\frac{1}{5}$
Thus, we have: $\text{x}=8\frac{1}{5}-6\frac{7}{15}$
$=\frac{41}{5}-\frac{97}{15}$ $L.C.M$. of $5$ and $15 = (5 \times 3) = 15 =\frac{(123-97)}{15}$
$\Big(\frac{15}{5}=3,3\times41=123\Big)$
and $\Big(\frac{15}{15}=1,1\times97=97\Big)$
$=\frac{26}{15}$
$=1\frac{11}{15}$
View full question & answer→Question 223 Marks
Simplify:$3+1\frac{1}{5}+\frac{2}{3}-\frac{7}{15}$
AnswerWe have:$3+1\frac{1}{5}+\frac{2}{3}-\frac{7}{15}$
$=\frac{3}{1}+\frac{6}{5}+\frac{2}{3}-\frac{7}{15}$
$\begin{array}{c|c}5&5,3,15\\\hline3&1,3,3\ \\\hline&1,1,1\ \end{array}$
$L.C.M$. of $5, 3$ and $15 = (5 \times 3) = 15$
$=\frac{(45+18+10-7)}{15}$
$\Big(\frac{15}{1}=15,15\times3=45\Big)$
$\Big(\frac{15}{5}=3,3\times6=18\Big)$
$\Big(\frac{15}{3}=5,5\times2=10\Big)$ and
$\Big(\frac{15}{15}=1,1\times7=7\Big)$
$=\frac{(73-7)}{15}$
$=\frac{66}{15}$
$=\frac{22}{5}$
$=4\frac{2}{5}$
View full question & answer→Question 233 Marks
Show that the following fractions is in the simplest form:
$\frac{21}{10}$
AnswerHere, numerator $= 21$ and denominator $= 10$
Factors of $21$ are $1, 3, 7$ and $21$
Factors of $10$ are $1, 2, 5$ and $10$
Common factor of $21$ and $10$ is $1$
Thus, $H.C.F.$ of $21$ and $10$ is $1.$
Hence, $\frac{21}{10}$ is the simplest form.
View full question & answer→Question 243 Marks
Rafiq exercised for $\frac{2}{3}$ hour, while Rohit exercise for $\frac{3}{4}$ hour. Who exercised for a longer time?
AnswerTo know who exercised for a longer time,
we have to compare $\frac{2}{3}$ hour with $\frac{3}{4}$
hour On cross multiplying: $4 \times 2 = 8$ and $3 \times 3 = 9$
Clearly, $8 < 9$
$\therefore\frac{2}{3}$
hour $< \frac{3}{4}$ hour
Hence, Rohit exercised for a longer time.
View full question & answer→Question 253 Marks
Convert $\frac{3}{5},\frac{7}{10},\frac{8}{15}$ and $\frac{11}{30}$ into like fractions.
AnswerThe given fractions are $\frac{3}{5},\frac{7}{10},\frac{8}{15}$ and $\frac{11}{30}$
$\begin{array}{c|c}5&5,10,15,30\\\hline2&1,2,3,6\ \ \ \ \ \ \\\hline3&1,1,3,3\ \ \ \ \ \\\hline&1,1,1,1\ \ \ \ \ \end{array}$
$L.C.M$. of $5, 10, 15$ and $30 = (5 \times 2 \times 3) = 30$
So, we convert the given fractions into equivalent fractions with $30$ as the denominator.
(But, one of the fractions already has $30$ as its denominator.
So, there is no need to convert it into an equivalent fraction.)
Thus, we have: $\frac{3}{5}=\frac{3\times6}{5\times6}=\frac{18}{30}$,
$\frac{7}{10}=\frac{7\times3}{10\times3}=\frac{21}{30}$,
$\frac{8}{15}=\frac{8\times2}{15\times2}=\frac{16}{30}$
Hence, the required like fractions are $\frac{18}{30},\frac{21}{30},\frac{16}{30}$ and $\frac{11}{30}$.
View full question & answer→Question 263 Marks
Find the sum: $2\frac{1}{3}+1\frac{1}{4}+2\frac{5}{6}+3\frac{7}{12}$
AnswerWe have, $L.C.M$. of $3, 4, 6$ and $12 = (2 \times 2 \times 3) = 12$
$\begin{array}{c|c}2&3,4,6,12\\\hline2&3,2,3,6\ \ \\\hline3&3,1,3,3\ \ \\\hline&1,1,1,1\ \ \end{array}$
Therefore, $2\frac{1}{3}+1\frac{1}{4}+2\frac{5}{6}+3\frac{7}{12}$
$=\frac{7}{3}+\frac{5}{4}+\frac{17}{6}+\frac{43}{12}$
$=\frac{(28+15+34+43)}{12}$
$\Big(\frac{12}{3}=4,4\times7=28\Big)$
$\Big(\frac{12}{4}=3,3\times5=15\Big)$
$\Big(\frac{12}{6}=2,2\times17=34\Big)$ and
$\Big(\frac{12}{12}=1,1\times43=43\Big)$
$=\frac{120}{12}$
$=10$
View full question & answer→Question 273 Marks
Simplify:$2+5\frac{7}{10}-3\frac{14}{15}$
AnswerWe have:$2+5\frac{7}{10}-3\frac{14}{15}$
$=\frac{2}{1}+\frac{57}{10}-\frac{59}{15}$
$\begin{array}{c|c}5&4,10,15\\\hline2&1,2,3\ \ \\\hline3&1,1,3\ \ \\\hline&1,1,1\ \ \end{array}$
$L.C.M.$ of $10$ and $15 = (2 \times 5 \times 3) = 30$
$=\frac{(69+171-118)}{30}$
$\Big(\frac{30}{1}=30,30\times2=60\Big)$
$\Big(\frac{30}{10}=3,3\times57=171\Big)$ and $\Big(\frac{30}{15}=2,2\times59=118\Big)$
$=\frac{(231-118)}{30}$
$=\frac{113}{30}$
$=3\frac{23}{30}$
View full question & answer→Question 283 Marks
Reduce the following fractions into its simplest form: $\frac{84}{98}$
AnswerHere, numerator $= 84$ and
denominator $= 98$
Factors of $84$ are $1, 2, 3, 4, 6, 7, 12, 14, 21, 42$ and $84$
Factors of $98$ are $1, 2, 7, 14, 49$ and $98$
Common factors of $84$ and $98$ are $1, 2, 7$ and $14 H.C.F$. of $84$ and $98$ is $14$
$\therefore\frac{84}{98}=\frac{84\div14}{98\div14}=\frac{6}{7}$
Hence, the simplest form of $\frac{84}{98}$ is $\frac{6}{7}$.
View full question & answer→Question 293 Marks
Simplify:
$\frac{5}{6}-\frac{4}{9}+\frac{2}{3}$
AnswerWe have, $\frac{5}{6}-\frac{4}{9}+\frac{2}{3}$
$\begin{array}{c|c}3&3,6,9\\\hline3&1,2,3\\\hline2&1,2,1\\\hline&1,1,1\end{array}$
$L.C.M.$ of $3, 6$ and $9 = (2 \times 3 \times 3) = 18$
$=\frac{(15+8+12)}{18}$
$\Big(\frac{18}{6}=3,3\times5=15\Big)$
$\Big(\frac{18}{9}=2,2\times4=8\Big)$ and
$\Big(\frac{18}{3}=6,6\times2=12\Big)$
$=\frac{(27-8)}{18}$
$=\frac{19}{18}$
$=1\frac{1}{18}$
View full question & answer→Question 303 Marks
Find the difference:
$2\frac{7}{9}-1\frac{8}{15}$
Answer$\begin{array}{c|c}3&9,15\\\hline3&3,5\ \ \\\hline5&1,5\ \ \\\hline&1,1\ \ \end{array}$
$L.C.M$ of $9$ and $15 = (3 \times 3 \times 5) = 45$
$2\frac{7}{9}-1\frac{8}{15}$
$=\frac{25}{9}-\frac{23}{15}$
$=\frac{(125-69)}{45}$
$=\frac{56}{45}$
$=1\frac{11}{45}$
$\Big(\frac{45}{9}=5,5\times25=125\Big)$
and $\Big(\frac{45}{15}=3,3\times23=69\Big)$
View full question & answer→Question 313 Marks
Simplify:$8\frac{5}{6}-3\frac{3}{8}+2\frac{7}{12}$
AnswerWe have:$8\frac{5}{6}-3\frac{3}{8}+2\frac{7}{12}$
$=\frac{53}{6}-\frac{27}{8}+\frac{31}{12}$
$\begin{array}{c|c}2&6,8,12\\\hline2&3,4,6\ \\\hline3&3,2,3\ \\\hline2&1,2,1\ \\\hline&1,1,1\ \end{array}$
$L.C.M.$ of $6, 8$ and $12 = (2 \times 2 \times 2 \times 3) = 24$
$=\frac{(212-81+62)}{24}$
$\Big(\frac{24}{6}=4,4\times53=212\Big)$
$\Big(\frac{24}{8}=23,3\times7=81\Big)$ and
$\Big(\frac{24}{12}=2,2\times31=62\Big)$$=\frac{(274-81)}{24}$
$=\frac{193}{24}$
$=8\frac{1}{24}$
View full question & answer→Question 323 Marks
Show that the following fractions is in the simplest form: $\frac{25}{36}$
AnswerHere, numerator $= 25$ and denominator $= 36$
Factors of $25$ are $1, 5$ and $25$
Factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18$ and $36$
Common factor of $25$ and $36$ is $1$ Thus,
$H.C.F$. of $25$ and $36$ is $1$
Hence, $\frac{25}{36}$ is the simplest form.
View full question & answer→Question 333 Marks
Find the sum: $2+\frac{3}{4}+1\frac{5}{8}+3\frac{7}{16}$
AnswerWe have, $L.C.M$. of $4, 8$ and $16 = (2 \times 2 \times 2 \times 2) = 16$
$\begin{array}{c|c}2&4,8,16\\\hline2&2,4,8\\\hline2&1,2,4\\\hline2&1,1,2\\\hline&1,1,1\end{array}$
Therefore, $2+\frac{3}{4}+1\frac{5}{8}+3\frac{7}{16}$
$=\frac{2}{1}+\frac{3}{4}+\frac{13}{8}+\frac{55}{16}$
$=\frac{(32+12+26+55)}{16}$
$\Big(\frac{16}{1}=16,6\times2=32\Big)$
$\Big(\frac{16}{4}=4,4\times3=12\Big)$
$\Big(\frac{16}{8}=2,2\times13=26\Big)$ and
$\Big(\frac{16}{16}=1,1\times55=55\Big)$
$=\frac{125}{16}$
$=7\frac{13}{16}$
View full question & answer→Question 343 Marks
Compare the fractions given below: $\frac{7}{8},\frac{9}{10}$
Answer$L.C.M$. of $8$ and $10 = (2 \times 5 \times 2 \times 2) = 40$
Now, we convert $\frac{7}{8}$ and $\frac{9}{10}$
into equivalent fractions having $40$ as the denominator.
$\therefore\frac{7}{8}=\frac{7\times5}{8\times5}=\frac{35}{40}$ and
$\frac{9}{10}=\frac{9\times4}{10\times4}=\frac{36}{40}$
Clearly, $\frac{35}{40}<\frac{36}{40}$
$\therefore\frac{7}{8}<\frac{9}{10}$
View full question & answer→Question 353 Marks
Simplify:$8-3\frac{1}{2}-2\frac{1}{4}$
AnswerWe have:$8-3\frac{1}{2}-2\frac{1}{4}$
$=\frac{8}{1}-\frac{7}{2}-\frac{9}{4}$
$\begin{array}{c|c}2&1,2,4\\\hline2&1,1,2\\\hline&1,1,1\end{array}$
$L.C.M.$ of $1, 2$ and $4 = (2 \times 2) = 4$
$=\frac{(32-14-9)}{4}$
$\Big(\frac{4}{1}=4,4\times8=32\Big)$
$\Big(\frac{4}{2}=2,2\times7=14\Big)$ and
$\Big(\frac{4}{4}=1,1\times9=9\Big)$
$=\frac{(32-23)}{34}$
$=\frac{9}{4}$
$=2\frac{1}{4}$
View full question & answer→Question 363 Marks
In one day, a rickshaw puller earned $\text{Rs.}\ 137\frac{1}{2}$. Out of this money, he spent $\text{Rs.}\ 56\frac{3}{4}$ on food. How much money is left with him?
AnswerMoney left with the rickshaw puller $=$ Money earned by him in a day $-$ Money spent by him on food
$=\text{Rs.}\ \Big(137\frac{1}{2}-56\frac{3}{4}\Big)$
$L.C.M.$ of $2$ and $4 = (2 \times 2) = 4$
$=\text{Rs.}\ \Big(\frac{275}{2}-\frac{227}{4}\Big)$
$\Big(\frac{4}{2}=2,2\times275=550\Big)$ and $\Big(\frac{4}{4}=1,1\times227=227\Big)$
$=\text{Rs.}\ \Big(\frac{550-227}{4}\Big)$
$=\text{Rs.}\ \Big(\frac{323}{4}\Big)$
$=\text{Rs.}\ 80\frac{3}{4}$ Hence, $\text{Rs.}\ 80\frac{3}{4}$ is left with the rickshaw puller.
View full question & answer→Question 373 Marks
Find the sum:
$\frac{2}{3}+3\frac{1}{6}+4\frac{2}{9}+2\frac{5}{18}$
AnswerWe have,
$L.C.M.$ of $3, 6, 9$ and $18 = (3 \times 3 \times 2) = 18$
$\begin{array}{c|c}3&3,6,9,18\\\hline3&1,2,3,6\ \ \\\hline2&1,2,1,2\ \ \\\hline&1,1,1,1\ \end{array}$
Therefore,
$\frac{2}{3}+3\frac{1}{6}+4\frac{2}{9}+2\frac{5}{18}$
$=\frac{2}{3}+\frac{19}{6}+\frac{38}{9}+\frac{41}{18}$
$=\frac{(12+57+76+41)}{18}$
$\Big(\frac{18}{3}=6,6\times2=12\Big)$
$\Big(\frac{18}{6}=3,3\times19=57\Big)$
$\Big(\frac{18}{9}=2,2\times38=76\Big)$
and $\Big(\frac{18}{18}=1,1\times41=41\Big)$
$=\frac{186}{18}$
$=\frac{31}{3}$
$=10\frac{1}{3}$
View full question & answer→Question 383 Marks
Find the difference: $\frac{5}{6}-\frac{4}{9}$
Answer$\begin{array}{c|c}3&6,9\\\hline2&2,3\\\hline3&1,3\\\hline&1,1\end{array}$
$L.C.M$ of $6$ and $9 = (3 \times 2 \times 3) = 18$
Now, we have:
$\frac{5}{6}=\frac{5\times3}{6\times3}=\frac{15}{18}$
$\frac{4}{9}=\frac{4\times2}{9\times2}=\frac{8}{18}$
Therefore,
$\frac{5}{6}-\frac{4}{9}$
$=\frac{15}{18}-\frac{8}{18}$
$=\frac{(15-8)}{18}$
$=\frac{7}{18}$
View full question & answer→Question 393 Marks
Convert the fractions $\frac{1}{2},\frac{ 2}{3}, \frac{4}{9}$ and $\frac{5}{6}$ into like fractions.
AnswerThe given fractions are $\frac{1}{2},\frac{ 2}{3}, \frac{4}{9},\frac{5}{6}$
$L.C.M$. of $2, 3, 9$ and $6 = (2 ⨯ 3 ⨯ 3) = 18$
Now, we have: $\frac{1}{2}=\frac{1\times9}{2\times9}=\frac{9}{18}$
$\frac{2}{3}=\frac{2\times6}{3\times6}=\frac{12}{18}$
$\frac{4}{9}=\frac{4\times2}{9\times2}=\frac{8}{18}$
$\frac{5}{6}=\frac{5\times3}{6\times3}=\frac{15}{18}$
Hence, $\frac{9}{18},\frac{12}{18}, \frac{8}{18}$ and $\frac{15}{18}$ are like fractions.
View full question & answer→Question 403 Marks
Find the sum:
$3+1\frac{4}{15}+1\frac{3}{20}$
AnswerWe have,
$L.C.M$. of $15$ and $20 = (2 \times 2 \times 3 \times 5) = 60$
$\begin{array}{c|c}5&15,20\\\hline3&3,4\ \ \ \ \\\hline2&1,4\ \ \ \ \\\hline2&1,2\ \ \ \ \ \\\hline&1,1\ \ \ \ \ \end{array} $
Therefore,
$3+1\frac{4}{15}+1\frac{3}{20}$
$=\frac{3}{1}+\frac{19}{15}+\frac{23}{20}$
$=\frac{(180+76+69)}{60}$
$\Big(\frac{60}{1}=60,60\times3=180\Big)$
$\Big(\frac{60}{15}=4,4\times19=76\Big)$
and $\Big(\frac{60}{20}=3,3\times23=69\Big)$
$=\frac{325}{60}$
$=\frac{65}{12}$
$=5\frac{5}{12}$
View full question & answer→Question 413 Marks
Simplify:$6\frac{1}{6}-5\frac{1}{5}+3\frac{1}{3}$
AnswerWe have:$6\frac{1}{6}-5\frac{1}{5}+3\frac{1}{3}$
$=\frac{37}{6}-\frac{26}{5}+\frac{10}{3}$
$\begin{array}{c|c}2&6,5,3\\\hline3&3,5,3\\\hline5&1,5,1\\\hline&1,1,1\end{array}$
$L.C.M$. of $6, 5$ and $3 = (2 \times 5 \times 3) = 30$
$=\frac{(185-156+100)}{30}$
$\Big(\frac{30}{6}=5,5\times37=185\Big)$
$\Big(\frac{30}{5}=6,6\times26=156\Big)$ and
$\Big(\frac{30}{3}=10,10\times10=100\Big)$
$=\frac{(285-156)}{30}$
$=\frac{129}{30}$
$=\frac{43}{10}$
$=4\frac{3}{10}$
View full question & answer→Question 423 Marks
Find the sum: $2\frac{7}{10}+3\frac{8}{15}$
AnswerWe have, $L.C.M$. of $10$ and $15 = (2 \times 3 \times 5) = 30$
$\begin{array}{c|c}5&10,15\\\hline2&2,3\\\hline3&1,3\\\hline&1,1\end{array} $
Therefore, $2\frac{7}{10}+3\frac{8}{15}$
$=\frac{27}{10}+\frac{53}{15}$
$=\frac{(81+106)}{30}$
$\Big(\frac{30}{10}=3,3\times27=81\Big)$ and
$\Big(\frac{30}{15}=2,2\times53=106\Big)$
$=\frac{187}{30}$
$=6\frac{7}{30}$
View full question & answer→Question 433 Marks
Reduce $\frac{84}{98}$ to the simplest form.
AnswerThe factors of $84$ are $1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$
The factors of $98$ are $1, 2, 7, 14, 49, 98$
The common factors of $84$ and $98$ are $1, 2, 7, 14$
The $H.C.F.$ of $84$ and $98$ is $14$
Dividing both the numerator and the denominator by the
$H.C.F \frac{84}{98}=\frac{84\div14}{98\div14}=\frac{6}{7}$
View full question & answer→Question 443 Marks
Compare the fractions given below:
$\frac{4}{5},\frac{7}{10}$
Answer$L.C.M.$ of $5$ and $10 = (5 \times 2) = 10$
Now, we convert $\frac{4}{5}$ into equivalent fractions having $10$ as the denominator.
$\therefore\frac{4}{5}=\frac{4\times2}{5\times2}=\frac{8}{10}$
Clearly, $\frac{8}{10}<\frac{7}{10}$
$\therefore\frac{4}{5}<\frac{7}{10}$
View full question & answer→Question 453 Marks
Simplify:$2+\frac{11}{15}-\frac{5}{9}$
AnswerWe have:$2+\frac{11}{15}-\frac{5}{9}$
$\begin{array}{c|c}3&1,15,9\\\hline3&1,5,3\ \\\hline5&1,5,1\ \\\hline&1,1,1\ \end{array}$
$L.C.M.$ of $15$ and $9 = (3 \times 3 \times 5) = 45$
$=\frac{(90+33-25)}{45}$
$\Big(\frac{45}{1}=45,45\times2=90\Big)$
$\Big(\frac{45}{15}=3,3\times11=33\Big)$ and
$\Big(\frac{45}{9}=5,5\times5=25\Big)$
$=\frac{(98+8)}{45}$
$=\frac{98}{45}$
$=2\frac{8}{45}$
View full question & answer→Question 463 Marks
Compare the fractions given below: $\frac{4}{9},\frac{5}{6}$
Answer$L.C.M.$ of $9$ and $6 = (3 \times 3 \times 2) = 18$
Now, we convert $\frac{4}{9}$ and $\frac{5}{6}$ into equivalent fractions having $18$ as the denominator.
$\therefore\frac{4}{9}=\frac{4\times2}{9\times2}=\frac{8}{18}$ and
$\frac{5}{6}=\frac{5\times3}{6\times3}=\frac{15}{18}$
Clearly, $\frac{8}{18}<\frac{15}{18}$
$\therefore\frac{4}{9}<\frac{5}{6}$
View full question & answer→Question 473 Marks
Of $\frac{3}{4}$ and $\frac{5}{7}$, which is greater and by how much?
AnswerLet us compare $\frac{3}{4}$ and $\frac{5}{7}$ By
cross multiplying: $3 ⨯ 7 = 21$ and $4 ⨯ 5 = 20$
Clearly, $21 > 20$
$\therefore\frac{3}{4}>\frac{5}{7}$
Their difference: $\frac{3}{4}-\frac{5}{7}$
$L.C.M.$ of $4$ and $7 = (2 \times 2 \times 7) = 28$
$(28\div4=7,7\times3=21)$ and $(28\div7=4,4\times5=20)$
$=\frac{21-20}{28}$
$=\frac{1}{28}$
Hence, $\frac{3}{4}$ is greater than $\frac{5}{7}$ by $\frac{1}{28}$.
View full question & answer→Question 483 Marks
Find the equivalent fraction of $\frac{3}{5}$ having denominator $30.$
AnswerLet $\frac{3}{5}=\frac{}{30}$
$30 = 5 ⨯ 6$
So, we have to multiply the numerator by $6$ to get the equivalent fraction having denominator $30$
$\therefore\frac{3}{5}=\frac{3\times6}{5\times6}=\frac{18}{30}$
Thus, $\frac{18}{30}$ is the equivalent fraction of $\frac{3}{5}$.
View full question & answer→Question 493 Marks
Find the difference: $\frac{5}{8}-\frac{7}{12}$
Answer$\begin{array}{c|c}2&8,12\\\hline2&4,6\\\hline2&2,3\\\hline3&1,3\\\hline&1,1\end{array}$
$L.C.M$ of $2$ and $8 = (2 \times 2 \times 2 \times 3) = 24$
Now, we have:
$\frac{5}{8}=\frac{5\times3}{8\times3}=\frac{15}{24}$
$\frac{7}{12}=\frac{7\times2}{12\times2}=\frac{14}{24}$
Therefore,
$\frac{5}{8}-\frac{7}{12}$
$=\frac{15}{24}-\frac{14}{24}$
$=\frac{(15-14)}{4}$
$=\frac{1}{24}$
View full question & answer→Question 503 Marks
While coming back home from his school, Kishan covered $4\frac{3}{4}\text{km}$ by rickshaw and $1\frac{1}{2}\text{km}$ on foot. What is the distance of his house from the school?
AnswerDistance from Kishan's house to school $=$ Distance covered by him by rickshaw $+$ Distance covered by him on foot
Thus, we have:$\Big(4\frac{3}{4}+1\frac{1}{2}\Big)\text{km}$
$=\frac{19}{4}+\frac{3}{2}\text{km}$
$=\frac{(19+6)}{4}\text{km}$
$=\frac{25}{4}\text{km}$
$=6\frac{1}{4}\text{km}$
$(L.C.M $.of $2$ and $4 = (2 × 2) = 4$
$\begin{array}{c|c}2&2,4\\\hline2&1,2\\\hline&1,1\end{array}$
Hence, the distance from Kishan's house to school is $6\frac{1}{4}\text{km}$.
View full question & answer→Question 513 Marks
Of $\frac{3}{4}$ and $\frac{5}{7}$, which is greater and by how much?
AnswerLet us compare $\frac{3}{4}$ and $\frac{5}{7}$
$3 \times 7 = 21$ and $4 \times 5 = 20$
Clearly, 21 > 20 Therefore, $\frac{3}{4}>\frac{5}{7}$ Required difference:$=\frac{3}{4}-\frac{5}{7}$
$L.C.M.$ of $4$ and $7 = (2 \times 2 \times 7) = 28$
$=\frac{21-20}{28}$
$\Big(\frac{28}{4}=7,7\times3=21\Big)$
and $\Big(\frac{28}{7}=4,4\times5=20\Big)$
$=\frac{1}{28}$
Hence, $\frac{3}{4}$ is greater than $\frac{5}{7}$ by $\frac{1}{28}$.
View full question & answer→Question 523 Marks
A piece of wire, $2\frac{3}{4}$ metres long, broke into two pieces. One piece is $\frac{5}{8}$ metre long. How long is the other piece?
AnswerThe length of the other piece = (Length of the wire $-$ Length of one piece)
$=\Big(2\frac{3}{4}-\frac{5}{8}\Big)\ \text{m}$
$=\Big(\frac{11}{4}-\frac{5}{8}\Big)\ \text{m}$
$L.C.M.$ of $4$ and $8 = (2 \times 2 \times 2) = 8$
$=\Big(\frac{22-5}{8}\Big)\ \text{m}$
$\Big(\frac{8}{4}=2,2\times11=22\Big)$
and $\Big(\frac{8}{8}=1,1\times5=5\Big)$
$=\Big(\frac{17}{8}\Big)\ \text{m}$
$=2\frac{1}{8}\ \text{m}$
Hence, the other piece is $2\frac{1}{8}\ \text{m}$ long.
View full question & answer→Question 533 Marks
The weight of an empty gas cylinder is $16\frac{4}{5}\text{kg}$ and it contains $14\frac{2}{3}\text{kg}$ of gas. What is the weight of the cylinder filled with gas?
AnswerWeight of the cylinder filled with gas = Weight of the empty cylinder $+$ Weight of the gas inside the cylinder Thus, we have: $(L.C.M$ .of $5$ and $3 = (5 \times 3) = 15$
$\Big(16\frac{4}{5}+14\frac{2}{3}\Big)\text{kg}$
$=\Big(\frac{84}{5}+\frac{44}{3}\Big)\text{kg}$
$=\frac{(252+220)}{15}\text{kg}$
$=\frac{472}{15}\text{kg}$
$=31\frac{7}{15}\text{kg}$
Hence, the weight of the cylinder filled with gas is $31\frac{7}{15}\text{kg}$.
View full question & answer→Question 543 Marks
Find the sum: $2\frac{3}{4}+5\frac{5}{6}$
AnswerWe have, $L.C.M$. of $4$ and $6 = (2 \times 2 \times 3) = 12$
$\begin{array}{c|c}2&4,6\\\hline2&2,3\\\hline3&1,3\\\hline&1,1\end{array} $
Therefore, $2\frac{3}{4}+5\frac{5}{6}$
$=\frac{11}{4}+\frac{35}{6}$
$=\frac{(66+140)}{24}$
$\Big(\frac{24}{4}=6,6\times11=66\Big)$ and
$\Big(\frac{24}{6}=4,4\times35=140\Big)$
$=\frac{206}{24}$
$=\frac{103}{12}$
$=8\frac{7}{12}$
View full question & answer→Question 553 Marks
A film show lasted for $3\frac{1}{3}$ hours. Out of his time, $1\frac{3}{4}$ hours was spent on advertisements. What was the actual duration of the film?
AnswerActual duration of the film = Total duration of the show - Time spent on advertisements
$=\Big(3\frac{1}{3}-1\frac{3}{4}\Big)\ \text{hours}$
$=\Big(\frac{10}{3}-\frac{7}{4}\Big)\ \text{hours}$
$L.C.M$. of $3$ and $4 = (2 \times 2 \times 3) = 12$
$=\Big(\frac{40-21}{12}\Big)\ \text{hours}$
$\Big(\frac{12}{3}=4,4\times10=40\Big)$
and $\Big(\frac{12}{4}=3,3\times7=21\Big)$
$=\Big(\frac{19}{12}\Big)\ \text{hours}$
$=1\frac{7}{12}\ \text{hours}$
Thus, the actual duration of the film was $1\frac{7}{12}\ \text{hours}$.
View full question & answer→Question 563 Marks
Reduce the following fractions into its simplest form: $\frac{9}{15}$
AnswerHere, numerator $= 9$ and denominator $= 15$
Factors of $9$ are $1, 3$ and $9$
Factors of $15$ are $1, 3, 5$ and $15$
Common factors of $9$ and $15$ are $1$ and $3 H.C.F$. of $9$ and $15$ is $3$
$\therefore\frac{9}{15}=\frac{9\div3}{15\div3}=\frac{3}{5}$
Hence, the simplest form of $\frac{9}{15}$ is $\frac{3}{5}$.
View full question & answer→Question 573 Marks
Reduce the following fractions into its simplest form: $\frac{72}{90}$
AnswerHere, numerator $= 72$ and denominator $= 90$
Factors of $72$ are $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36$ and $72$
Factors of $90$ are $1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45$ and $90$
Common factors of $72$ and $90$ are $1, 2, 3, 6, 9$ and $18 H.C.F.$ of $72$ and $90$ is $18$
$\therefore\frac{72}{90}=\frac{72\div18}{90\div18}=\frac{4}{5}$
Hence, the simplest form of $\frac{72}{90}$ is $\frac{4}{5}$.
View full question & answer→Question 583 Marks
Arrange the following fractions in descending order: $\frac{3}{7},\frac{3}{11},\frac{3}{5},\frac{3}{13},\frac{3}{4},\frac{3}{17}$
AnswerThe given fractions are $\frac{3}{7},\frac{3}{11},\frac{3}{5},\frac{3}{13},\frac{3}{4},\frac{3}{17}$
As the fractions have the same numerator,
we can follow the rule for the comparison of such fractions.
This rule states that when two fractions have the same numerator,
the fraction having the smaller denominator is the greater one.
Clearly, $\frac{3}{4} >\frac{3}{5} >\frac{3}{7} >\frac{3}{11}>\frac{3}{13}>\frac{3}{17}$
Hence, the given fractions can be arranged in the descending order as follows: $\frac{3}{4},\frac{3}{5},\frac{3}{7} ,\frac{3}{11},\frac{3}{13},\frac{3}{17}$
View full question & answer→Question 593 Marks
Arrange the following fractions in descending order:
$\frac{1}{12},\frac{1}{23},\frac{1}{7},\frac{1}{9},\frac{1}{17},\frac{1}{50}$
AnswerThe given fractions are $\frac{1}{12},\frac{1}{23},\frac{1}{7},\frac{1}{9},\frac{1}{17},\frac{1}{50}$
As the fractions have the same numerator, we can follow the rule for the comparison of such fractions.
This rule states that when two fractions have the same numerator,
the fraction having the smaller denominator is the greater one.
Clearly, $\frac{1}{7} >\frac{1}{9} >\frac{1}{12} >\frac{1}{17}>\frac{1}{23}>\frac{1}{50}$
Hence, the given fractions can be arranged in the descending order as follows:
$\frac{1}{7},\frac{1}{9},\frac{1}{12},\frac{1}{17},\frac{1}{23},\frac{1}{50}$
View full question & answer→Question 603 Marks
Compare the fractions given below: $\frac{11}{12},\frac{13}{15}$
Answer$L.C.M.$ of $12$ and $15 = (2 \times 2 \times 3 \times 5) = 60$
Now, we convert $\frac{11}{12}$ and $\frac{13}{15}$ into equivalent fractions having $40$ as the denominator.
$\therefore\frac{11}{12}=\frac{11\times5}{12\times5}=\frac{55}{60}$ and
$\frac{13}{15}=\frac{13\times4}{15\times4}=\frac{52}{60}$
Clearly, $\frac{55}{60}>\frac{52}{60}$
$\therefore\frac{11}{12}>\frac{13}{15}$
View full question & answer→