Question 13 Marks
Find the other three proportions involving terms the following:
$12 : 18 = 14 : 21$
Answer$12 : 18 = 14 : 21$
$\frac{12}{18}=\frac{14}{21}$
$\Rightarrow\frac{2}{3}=\frac{2}{3}$
$\frac{21}{18}=\frac{14}{12}$
$\Rightarrow\frac{21\div3}{18\div3}=\frac{14\div2}{12\div2}$
$\Rightarrow\frac{7}{6}=\frac{7}{6}$
$21 : 18 = 14 : 21$
View full question & answer→Question 23 Marks
Of the $72$ persons working in an office, $28$ are men and the remaining are women. Find the ratio of the number of:
$i.\ $Men to that of women.
$ii.\ $Men to the total number persons.
$iii.\ $Persons to that of women.
Answer
$i.\ $Total no. of workers $= 72$
Women $= 28$
Women $= 72 - 28$
$= 44$
$ii.\ $Men to that of women $= 28 : 44$
$=\frac{28\div4}{44\div4}$
$=\frac{7}{11}$ $[H. C. F. = 4]$
$iii.\ $Men to the total no. of persons $=\frac{28\div4}{72\div4}$
$=\frac{7}{18}$
Persons to women $=\frac{72}{44}$
$=\frac{72\div4}{44\div4}$
$=\frac{18}{11}$
View full question & answer→Question 33 Marks
A bullock-cart travels $24\ km$ in $3$ hours and a train travels $120\ km$ in $2$ hours. Find the ratio of their speeds.
AnswerBullock-cart travels $24\ km$ in $3$ hours
Train travels $120\ km$ in $2$ hours Bullock-cart travels $\frac{24\text{km}}{3}$ in one hour i.e $8\ km$
Train travels $\frac{120}{2}\text{km}$ in one hour i.e $60\ km$
$\therefore\ \text{Ratio}=\frac{8\div4}{60\div4}$
$=\frac{2}{15}$
View full question & answer→Question 43 Marks
The ratio of the sale of eggs on a Sunday to that of the whole week of a grocery shop was $2 : 9$. If the total sale of eggs in the same week was $Rs. 360$, find the sale of eggs on Sunday.
AnswerThe ratio of eggs on a sunday to that of whole week of a grocery shop was $2 : 9.$
$GIT$ total sale of eggs in the week was $Rs. 360$ Let sale of eggs on sunday be $'x'$
$\Rightarrow\frac{2}{9}=\frac{\text{x}}{360}$
$\Rightarrow\frac{2\times360}{9}=\text{x}$
$\Rightarrow\text{x}=\frac{720}{9}=80$
$\therefore$ Sale of eggs on sunday $Rs. 80.$
View full question & answer→Question 53 Marks
If in a proportion, the first, second and fourth terms are $32, 112$ and $217$ respectively, find the third term.
Answer$\frac{32}{112}=\frac{\text{x}}{217}$
By cross multiplication we get,
$\Rightarrow32(217)=\text{x}(112)$
$\Rightarrow\frac{32(217)}{16}=\frac{\text{x}(112)}{16}$
$\Rightarrow2(217)=7\text{x}$
$\Rightarrow\frac{2(217)}{7}=\frac{7\text{x}}{7}$
$\Rightarrow\text{x}=2(31)$
$\Rightarrow\text{x}=62$
View full question & answer→Question 63 Marks
Tropical fruit juice is mixed from mango, papaya and orange juice in the ratio $3 : 1 : 4$. Aarushi has $400\ ml$ of orange juice. If she uses it all:
$i.\ $How many mango and papaya will he need?
$ii.\ $How much tropical fruit juice will he make altogether?
AnswerLet the quantity of mango juice be $3x$, the quantity of papaya juice be $x$ and the quantity of orange juice be $4x$.
According to the question, $4\text{x}=400\text{ml}$
$\Rightarrow\text{x}=\frac{400}{4}\text{ml}$
$\Rightarrow\text{x}=100\text{ml}$
$i.\ $Quantity of mango juice $= 3 \times 100 = 300\ ml$ and Quantity of papaya juice $= 100\ ml.$
$ii.\ $Total quantity of tropical juice $= 400 + 300 + 100 = 800\ ml.$ View full question & answer→Question 73 Marks
Which ratio is larger in the following pairs? $15 : 16$ or $24 : 24$
AnswerWriting the given ratios as fractions,
we have $15:16=\frac{15}{16}\ \text{and}\ 24:25=\frac{24}{25}$
$L.C.M$ of $25$ and $16$ is $= 400$
Making the denominator of each fraction equal to $400,$
we have $\frac{15}{16}=\frac{15\times25}{16\times25}=\frac{375}{400}\ \text{and}\ \frac{24}{25}=\frac{24\times16}{25\times16}=\frac{384}{400}$
Clearly $384 > 375$
$\therefore\frac{384}{400}>\frac{375}{400}\Rightarrow\frac{24}{25}>\frac{15}{16}$
View full question & answer→Question 83 Marks
Which ratio is larger in the following pairs?
$3 : 4$ or $9 : 16$
AnswerWriting the given ratios as fractions, we have
$3:4=\frac{3}{4}\ \text{and}\ 9:16=\frac{9}{16}$
Now $L.C.M$ of $4$ and is is $16$
Making the denominator of each fraction equal to $16$, we have
$\frac{3}{4}=\frac{3\times4}{4\times4}=\frac{12}{16}\ \text{and}\ \frac{9}{16}=\frac{9}{13}$
Clearly $12 > 9$
$\therefore\frac{12}{16}>\frac{9}{16}\Rightarrow\frac{3}{4}>\frac{9}{16}$
View full question & answer→Question 93 Marks
$10$ boys can dig a pitch in $12$ hours. How long will $8$ boys take to do it?
Answer$10$ boys can dig a pitch in $12$ hours,
$8$ boys can dig pitch in $x$ hrs One boy can dig pitch in $= 12 \times 10 = 120$ hours
$8$ boys can dig pitch in $=\frac{120}{8}$ $= 15$ hours.
View full question & answer→Question 103 Marks
The mean proportional of $a$ and $b$ is $10$ and the value of a is four times the value of $b$. If $a > 0, b > 0,$ find the value of $a + b$?
AnswerIt is given that, $a : 10 : : 10 : b$
$\Rightarrow\frac{\text{a}}{10}=\frac{10}{\text{a}}$
$\Rightarrow\text{a}\times\text{b}=10\times10$
$\Rightarrow\text{a}\times\text{b}=100\ ....(1)$ and $a = 4b ....(2)$ From $(1)$ and $(2), (4b)b = 100$
$\Rightarrow\text{b}\times\text{b}=\frac{100}{4}$
$\Rightarrow\text{b}\times\text{b}=25$
$\Rightarrow\text{b}\times\text{b}=5\times5$
$\Rightarrow\text{b}=5$
$\Rightarrow\text{a}=4\times5\ [\text{From} (1)]$
$\Rightarrow\text{a}=20$
$\therefore\text{a}+\text{b}=20+5$
$=25$
$\text{Hence, a + b}=25.$
View full question & answer→Question 113 Marks
The ratio of the income to the expenditure of a family is $7 : 6$. Find the saving if the income is $Rs. 1400.$
AnswerThe ratio of the income to the expenditure of a family is $7 : 6$ Saving = ? Income $Rs. 1400$
Let expenditure $= x$
$\Rightarrow\frac{7}{6}=\frac{1400}{\text{x}}$
$\Rightarrow7\text{x}=6\times1400$
$\Rightarrow\text{x}=\frac{6\times1400}{7}$
$\Rightarrow\text{x}=6\times200$
$\Rightarrow\text{x}=1200$
Expenditure $= 1200$.
Savings $= 1400 - 1200 = 200.$
View full question & answer→Question 123 Marks
$40$ men can finish a piece of work in $26$ days. How many men will be needed to finish it in $16$ days?
Answer$26$ days are required to finish a piece of work by $40$ men.
$1$ day is required to finish a piece of work by $40 \times 26$ men.
$16$ days are required to finish a piece of work by $\frac{40\times26}{16}=65\ \text{men}.$
Hence, $65$ men will be needed to finish a piece of work in $16$ days.
View full question & answer→Question 133 Marks
Divide $Rs. 3450$ among $A, B$ and $C$ in the ratio $3 : 5 : 7.$
AnswerLet the value of $A$ be $3x,$ the value of $B$ be $5x$ and the value of $C$ be $7x.$
According to the question, $3x + 5x + 7x = 3450$
$\Rightarrow15\text{x}=3450$
$\Rightarrow\text{x}=\frac{3450}{15}$
$\Rightarrow\text{x}=230$
Hence, the value of $A$ is $3 \times 230 = Rs. 690$,
the value of $B$ is $5 \times 230 = Rs. 1150,$ and
the value of $C$ is $7 \times 230 = Rs. 1610.$
View full question & answer→Question 143 Marks
The cost of $17$ chairs is $Rs. 9605$. Find the number of chairs that can be purchased in $Rs. 56500.$
AnswerCost of $17$ chairs $= Rs. 9605$
Cost of one chair $=\frac{\text{Rs. 9605}}{17}=565$
Number of chairs purchased By $Rs. 56500$ $=\frac{56500}{\text{cost of one chair}}$
$=\frac{56500}{565}$ $=100\ \text{chairs.}$
View full question & answer→Question 153 Marks
Find the other three proportions involving terms the following: $45 : 30 = 24 : 16$
Answer$45 : 30 = 24 : 16$
$\frac{45}{30}=\frac{24}{16}$
$\Rightarrow\frac{3}{2}=\frac{3}{2}$
$\frac{30}{45}=\frac{16}{24}$
$\Rightarrow\frac{2}{3}=\frac{2}{3}$
$\frac{16}{30}=\frac{24}{45}$
$\Rightarrow\frac{16\div2}{30\div2}=\frac{24\div3}{45\div3}$
$\Rightarrow\frac{8}{15}=\frac{8}{15}$
View full question & answer→Question 163 Marks
The ratio of copper and zinc in an alloy is $5 : 3$. If the weight of copper in the alloy is $30.5g$, find the weight of zinc in it.
AnswerLet the weight of copper be $5x$ and the weight of zinc be $3x$.
According to the question, $5x = 30.5g$
$\Rightarrow\text{x}=\frac{30.5}{5}=6.1\text{g}$
Hence, the weight of zinc in the alloy is $3 \times 6.1 = 18.3g.$
View full question & answer→Question 173 Marks
Which ratio is larger in the following pairs?
$4 : 7$ or $5 : 8$
Answer$4:7=\frac{4}{7}\ \text{and}\ 5:8=\frac{5}{8}$
Now, $LCM$ of $7$ and $8$ is $56$
$\frac{4}{7}=\frac{4\times8}{7\times8}=\frac{32}{56}\ \text{and } 5:8=\frac{5\times7}{8\times7}=\frac{35}{56}$
Clearly $35 > 32$
$\therefore\frac{35}{56}>\frac{32}{56}\Rightarrow\frac{5}{8}>\frac{4}{7}$
View full question & answer→Question 183 Marks
$12$ men can reap a field in $25$ days. In how many days can $20$ men reap the same field?
Answer$12$ men can reap a field in $25$ days.
$1$ man can reap a field in $25 \times 12$ days.
$20$ men can reap a field in $\frac{25\times12}{20}=15\ \text{days.}$
Hence, $20$ men can reap the same field in $15$ days.
View full question & answer→Question 193 Marks
The length of a steel tape for measurements of buildings is $10m$ and its width is $2.4\ cm$. What is the ratio of its length to width?
AnswerLength of steel tape $= 10m$ Width $= 2.4\ cm$
Ratio of its length and width $=\frac{10\text{m}}{2.4\text{cm}}$
$=\frac{1000\text{cm}}{2.4\text{cm}}$
$=\frac{1250}{3}$ $[H.C.F = 0.8\ cm]$
View full question & answer→Question 203 Marks
A train runs $200\ km$ in $5$ hours. How many kilometres does it run in $7$ hours?
AnswerTrain runs $200\ km$ in $5$ hours,Train runs in one hour $=\frac{200}{5}\text{km}$
$= 40\ km$
$\therefore$ No of kms does it run in $7$ hours $= 7 \times 40\ km$
$= 280\ km$
View full question & answer→Question 213 Marks
The number of boys and girls in a school are $1168$ and $1095$ respectively. Express the ratio of the number of boys to that of the girls in the simplest form.
AnswerThe number of boys $= 1168$
The number of girls $= 1095$
The number of boys to the number of girls
$=\frac{1168}{1095}$
$= 16 : 15$ $[$Dividing by $73]$
View full question & answer→Question 223 Marks
Shikha and Aarushi are sisters. The ratio of their ages is $3 : 4$. Their parents give them $Rs. 1400$ to share in the ratio of their ages. How much does each of them recieve?
AnswerLet the age of Shikha be $3x$ and the age of Aarushi be $4x$.
According to the question,
$3x + 4x = 1400$
$\Rightarrow 7x = 1400$
$\Rightarrow\text{x}=\frac{1400}{7}=200$
Hence, Shikha recieves $3 \times 200 = 600$ rupees
And Aarushi recieves $4 \times 200 = 800$ rupees.
View full question & answer→Question 233 Marks
Show that the following numbers are in continued proportion:
$36, 90, 225$
Answer$36, 90, 225$
$\frac{36}{90}=\frac{90}{225}$
$\Rightarrow\frac{36\div6}{90\div6}=\frac{90\div5}{225\div5}$
$\Rightarrow\frac{6}{15}=\frac{18}{45}$
$\Rightarrow\frac{2}{5}=\frac{2}{5}$
$\therefore$ $36, 90, 225$ are in continued proportion.
View full question & answer→Question 243 Marks
Show that the following numbers are in continued proportion: $16, 84, 441$
Answer$16, 84, 441$
$\Rightarrow\frac{16}{84}=\frac{84}{441}$
$\Rightarrow\frac{16\div4}{84\div4}=\frac{84\div21}{441\div21}$
$\Rightarrow\frac{4}{21}=\frac{4}{21}$
$\therefore 16, 84, 441$ are in continued proportion.
View full question & answer→Question 253 Marks
The ratio of the length of a school ground to its width is $5 : 2$. Find its length if the width is $40$ metres.
AnswerThe ratio of school ground to its width is $5 : 2 GIT$
width $= 40$ meters
Let $'x'$ be the length of the ground
$\Rightarrow\frac{5}{2}=\frac{\text{x meters}}{40\ \text{meters}}$
$\Rightarrow5\times40\ \text{meters}=2\text{x meters}$
$\Rightarrow200\ \text{meters}=2\text{x meters}$
$\Rightarrow\text{x}=\frac{200}{2}$
$\Rightarrow\text{x}=100$
$\therefore$ Lenght of the ground $100$ meters.
View full question & answer→Question 263 Marks
A given quantity of rice is sufficient for $60$ persons for $3$ days. How many days would the rice last for $18$ persons?
AnswerA given quantity of rice is sufficient for $60$ persons for $3$ days.
A given quantity of rice is sufficient for $1$ person for $3 \times 60$ days.
A given quantity of rice is sufficient for $18$ persons for $\frac{3\times60}{18}=10\ \text{days.}$
Hence, the rice last for $10$ days for $18$ persons.
View full question & answer→Question 273 Marks
If $a : b = 2 : 5$, find the value of $\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}.$
AnswerIt is given that,
$\frac{\text{a}}{\text{b}}=\frac{2}{5}\ ....(1)$
Now,
$\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}=\frac{(3\text{a}+2\text{b})\div\text{b}}{(4\text{a}+\text{b})\div\text{b}}$
$=\frac{3(\text{a}\div\text{b})+2}{4(\text{a}\div\text{b})+1}$
$=\frac{3\big(\frac{\text{a}}{\text{b}}\big)+2}{4\big(\frac{\text{a}}{\text{b}}\big)+1}$
$=\frac{3\big(\frac{2}{5}\big)+2}{4\big(\frac{2}{5}\big)+1}\ [\text{From (1)}]$
$=\frac{\frac{6+2\times5}{5}}{\frac{8+1\times5}{5}}$
$=\frac{6+10}{8+5}$
$=\frac{16}{13}$
$\text{Hence},\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}=\frac{16}{13}.$
View full question & answer→Question 283 Marks
Three ferryloads are needed to carry $150$ people across a river. How many people will be carried on $4$ ferryloads?
AnswerThree ferryloads carry $= 150$
people One ferryload carry $=\frac{150}{3}$
people $= 50$ people
Number of peoples can be carried by $4$ ferryloads $= 50 \times 4 = 200$ people.
View full question & answer→Question 293 Marks
Fifteen post cards cost $Rs 2.25$. What will be the cost of $36$ post cards? How many postcards can we buy in $Rs. 45?$
AnswerCost of $15$ postcards $= Rs. 2.25$ Cost of $1$ postcard $= Rs. 2.2515$ Cost of $36$
postcards $= 2.2515 \times 36 = 22515 \times 100 \times 36 = Rs. 5.40$
Number of postcards purchased in $Rs. 1 = 152.25$
Number of postcards purchased in $Rs. 45 = 152.25 \times 45 = 15 \times 100225 \times 45 = 300$
View full question & answer→Question 303 Marks
Which ratio is larger in the following pairs? $9 : 20$ or $8 : 13$
Answer$9:20=\frac{9}{20}\ \text{and}\ 8:13=\frac{8}{13}$
Now, $LCM$ of $20$ and $13$ is $260$
$\frac{9}{20}=\frac{9\times13}{20\times13}=\frac{117}{260}\ \text{and } \frac{8}{13}=\frac{8\times20}{20\times13}=\frac{160}{260}$Clearly $160 > 117$
$\therefore\frac{160}{260}>\frac{117}{260}\Rightarrow\frac{8}{13}>\frac{9}{20}$
View full question & answer→Question 313 Marks
If $48$ boxes contain $6000$ pens, how many such boxes will be needed for $1875$ pens?
Answer$48$ boxes contain $6000$ pens.
Number of pens contained in $1$ box $=\frac{6000}{48}=125\ \text{pens}.$
Therefore, number of boxes needed to contain $1875$ pens $=\frac{1875}{125}=15\ \text{boxes}.$
Hence, $15$ boxes will be needed for $1875$ pens.
View full question & answer→Question 323 Marks
A man can work $8$ hours daily and finishes a work in $12$ days. If he works $6$ hrs daily, in how many days will the same work be finished?
AnswerDaily $8$ hours
$\rightarrow $ work finishes in $72$ days, $6$ hrs daily
$\rightarrow $ ? Daily one hour $= 12 \times 8 = 96$ days No of day will take $6$ hrs daily works $=\frac{96}{6} = 16$ days.
View full question & answer→Question 333 Marks
The ratio of story books in a library to other books is $1 : 7$. The total number of story books is $800$. Find the total number of books in the library.
AnswerThe ratio of story books in a library to other books is $1 : 7.$
Out of $(1 + 7) = 8$ books, $1$ book is a story book
Therefore, When number of story books is $1$, the total number of books $= 8$
When number of story books is $800$, the total number of books $= 8 \times 800 = 6400$
View full question & answer→Question 343 Marks
The price of $3$ meters of cloth is $Rs. 79.50$. Find the price of $15$ meters of suoh cloth.
AnswerThe price of $3$ meters of cloth $= Rs. 79.50$
Let the price of $15$ meters cloth be $x$
Then, $\frac{3}{15}=\frac{79.50}{\text{x}}$ By cross multiplication we get,
$\Rightarrow3\text{x}=15\times79.50$
$\Rightarrow\text{x}=\frac{5\times79.50}{3}$
$\Rightarrow\text{x}=5\times79.50$
$\Rightarrow\text{x}=\text{Rs. 397.50}$
View full question & answer→Question 353 Marks
An office opens at $9 a.m$. and closed at $5 p.m$. with a lunch interval of $30$ minutes. What is the ratio of lunch interval to the total period in office?
AnswerTotal period office $= 9$ a.m to $5 p.m = 8$ hours
Lunch interval $= 30$ min Ratio $=\frac{30}{8\times60\text{ min}}$
$=\frac{30}{480}$
$=\frac{30\div30}{480\div30}$
$=\frac{1}{16}$
View full question & answer→Question 363 Marks
The ratio of copper and zinc in an alloy is $9 : 7$. If the weight of zinc in the alloy is $9.8\ kg$, find the weight of copper in the alloy.
AnswerThe ratio of copper and zinc in an alloy is $9 : 7$
Weight of zinc in the alloy $= 9.8\ kg$
Let weight of copper in the alloy $= x$
$\Rightarrow\frac{9}{7}=\frac{\text{x}}{9.8}$
$\Rightarrow\frac{9\times9.8}{7}=\text{x}$
$\Rightarrow\text{x}=12.6\text{kg}.$
View full question & answer→