3 Mark Question

Question 13 Marks

Find the other three proportions involving terms the following:
12 : 18 = 14 : 21

Answer

12 : 18 = 14 : 21
$\frac{12}{18}=\frac{14}{21}$
$\Rightarrow\frac{2}{3}=\frac{2}{3}$
$\frac{21}{18}=\frac{14}{12}$
$\Rightarrow\frac{21\div3}{18\div3}=\frac{14\div2}{12\div2}$
$\Rightarrow\frac{7}{6}=\frac{7}{6}$
$21 : 18 = 14 : 21$

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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:

Men to that of women.
Men to the total number persons.
Persons to that of women.

Answer

Total no. of workers = 72

Women = 28

Women = 72 - 28

= 44

Men to that of women = 28 : 44

$=\frac{28\div4}{44\div4}$

$=\frac{7}{11}$ [H. C. F. = 4]

Men to the total no. of persons $=\frac{28\div4}{72\div4}$

$=\frac{7}{18}$

Persons to women $=\frac{72}{44}$

$=\frac{72\div4}{44\div4}$

$=\frac{18}{11}$

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Question 33 Marks

A bullock-cart travels 24km in 3 hours and a train travels 120km in 2 hours. Find the ratio of their speeds.

Answer

Bullock-cart travels 24km in 3 hours
Train travels 120km in 2 hours
Bullock-cart travels $\frac{24\text{km}}{3}$ in one hour
i.e 8km
Train travels $\frac{120}{2}\text{km}$ in one hour
i.e 60km
$\therefore\ \text{Ratio}=\frac{8\div4}{60\div4}$
$=\frac{2}{15}$

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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.

Answer

The 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.

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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,
$\Rightarrow32(217)=\text{x}(112)$
$\Rightarrow\frac{32(217)}{16}=\frac{\text{x}(112)}{16}$
$\Rightarrow2(217)=7\text{x}$
$\Rightarrow\frac{2(217)}{7}=\frac{7\text{x}}{7}$
$\Rightarrow\text{x}=2(31)$
$\Rightarrow\text{x}=62$

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Question 63 Marks

Tropical fruit juice is mixed from mango, papaya and orange juice in the ratio 3 : 1 : 4. Aarushi has 400ml of orange juice. If she uses it all:

How many mango and papaya will he need?
How much tropical fruit juice will he make altogether?

Answer

Let 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,
$4\text{x}=400\text{ml}$
$\Rightarrow\text{x}=\frac{400}{4}\text{ml}$
$\Rightarrow\text{x}=100\text{ml}$

Quantity of mango juice = 3 × 100 = 300ml and Quantity of papaya juice = 100ml.
Total quantity of tropical juice = 400 + 300 + 100 = 800ml.

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Question 73 Marks

Which ratio is larger in the following pairs?
15 : 16 or 24 : 24

Answer

Writing 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
$\therefore\frac{384}{400}>\frac{375}{400}\Rightarrow\frac{24}{25}>\frac{15}{16}$

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Question 83 Marks

Which ratio is larger in the following pairs?
3 : 4 or 9 : 16

Answer

Writing 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
$\therefore\frac{12}{16}>\frac{9}{16}\Rightarrow\frac{3}{4}>\frac{9}{16}$

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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 × 10 = 120 hours
8 boys can dig pitch in $=\frac{120}{8}$
= 15 hours.

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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?

Answer

It 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
$\Rightarrow\text{b}\times\text{b}=\frac{100}{4}$
$\Rightarrow\text{b}\times\text{b}=25$
$\Rightarrow\text{b}\times\text{b}=5\times5$
$\Rightarrow\text{b}=5$
$\Rightarrow\text{a}=4\times5\ [\text{From} (1)]$
$\Rightarrow\text{a}=20$
$\therefore\text{a}+\text{b}=20+5$
$=25$
$\text{Hence, a + b}=25.$

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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.

Answer

The 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.

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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 × 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.

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Question 133 Marks

Divide Rs. 3450 among A, B and C in the ratio 3 : 5 : 7.

Answer

Let 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 × 230 = Rs. 690, the value of B is 5 × 230 = Rs. 1150, and the value of C is 7 × 230 = Rs. 1610.

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Question 143 Marks

The cost of 17 chairs is Rs. 9605. Find the number of chairs that can be purchased in Rs. 56500.

Answer

Cost of 17 chairs = Rs. 9605
Cost of one chair $=\frac{\text{Rs. 9605}}{17}=565$
Number of chairs purchased
By Rs. 56500 $=\frac{56500}{\text{cost of one chair}}$
$=\frac{56500}{565}$
$=100\ \text{chairs.}$

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Question 153 Marks

Find the other three proportions involving terms the following:
45 : 30 = 24 : 16

Answer

45 : 30 = 24 : 16
$\frac{45}{30}=\frac{24}{16}$
$\Rightarrow\frac{3}{2}=\frac{3}{2}$
$\frac{30}{45}=\frac{16}{24}$
$\Rightarrow\frac{2}{3}=\frac{2}{3}$
$\frac{16}{30}=\frac{24}{45}$
$\Rightarrow\frac{16\div2}{30\div2}=\frac{24\div3}{45\div3}$
$\Rightarrow\frac{8}{15}=\frac{8}{15}$

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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.

Answer

Let 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 × 6.1 = 18.3g.

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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
$\therefore\frac{35}{56}>\frac{32}{56}\Rightarrow\frac{5}{8}>\frac{4}{7}$

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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 × 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.

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Question 193 Marks

The length of a steel tape for measurements of buildings is 10m and its width is 2.4cm. What is the ratio of its length to width?

Answer

Length of steel tape = 10m
Width = 2.4cm
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 = 0.8cm]

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Question 203 Marks

A train runs 200km in 5 hours. How many kilometres does it run in 7 hours?

Answer

Train runs 200km in 5 hours,Train runs in one hour $=\frac{200}{5}\text{km}$
= 40km
$\therefore$ No of kms does it run in 7 hours = 7 × 40km
= 280km

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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.

Answer

The 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]

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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?

Answer

Let the age of Shikha be 3x and the age of Aarushi be 4x.
According to the question,
3x + 4x = 1400
⇒ 7x = 1400
$\Rightarrow\text{x}=\frac{1400}{7}=200$
Hence, Shikha recieves 3 × 200 = 600 rupees
And Aarushi recieves 4 × 200 = 800 rupees.

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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.

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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.

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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.

Answer

The 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.

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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?

Answer

A given quantity of rice is sufficient for 60 persons for 3 days.
A given quantity of rice is sufficient for 1 person for 3 × 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.

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Question 273 Marks

If a : b = 2 : 5, find the value of $\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}.$

Answer

It is given that,
$\frac{\text{a}}{\text{b}}=\frac{2}{5}\ ....(1)$
Now,
$\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}=\frac{(3\text{a}+2\text{b})\div\text{b}}{(4\text{a}+\text{b})\div\text{b}}$
$=\frac{3(\text{a}\div\text{b})+2}{4(\text{a}\div\text{b})+1}$
$=\frac{3\big(\frac{\text{a}}{\text{b}}\big)+2}{4\big(\frac{\text{a}}{\text{b}}\big)+1}$
$=\frac{3\big(\frac{2}{5}\big)+2}{4\big(\frac{2}{5}\big)+1}\ [\text{From (1)}]$
$=\frac{\frac{6+2\times5}{5}}{\frac{8+1\times5}{5}}$
$=\frac{6+10}{8+5}$
$=\frac{16}{13}$
$\text{Hence},\frac{3\text{a}+2\text{b}}{4\text{a}+\text{b}}=\frac{16}{13}.$

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Question 283 Marks

Three ferryloads are needed to carry 150 people across a river. How many people will be carried on 4 ferryloads?

Answer

Three ferryloads carry = 150 people
One ferryload carry $=\frac{150}{3}$ people
= 50 people
Number of peoples can be carried by 4 ferryloads = 50 × 4
= 200 people.

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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?

Answer

Cost of 15 postcards = Rs. 2.25
Cost of 1 postcard = Rs. 2.2515
Cost of 36 postcards = 2.2515 × 36
= 22515 × 100 × 36 = Rs. 5.40
Number of postcards purchased in Rs. 1 = 152.25
Number of postcards purchased in Rs. 45
= 152.25 × 45 = 15 × 100225 × 45 = 300

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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
$\therefore\frac{160}{260}>\frac{117}{260}\Rightarrow\frac{8}{13}>\frac{9}{20}$

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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.

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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?

Answer

Daily 8 hours → work finishes in 72 days,
6 hrs daily → ?
Daily one hour = 12 × 8 = 96 days
No of day will take 6 hrs daily works $=\frac{96}{6}$
= 16 days.

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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.

Answer

The 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 × 800
= 6400

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Question 343 Marks

The price of 3 meters of cloth is Rs. 79.50. Find the price of 15 meters of suoh cloth.

Answer

The 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,
$\Rightarrow3\text{x}=15\times79.50$
$\Rightarrow\text{x}=\frac{5\times79.50}{3}$
$\Rightarrow\text{x}=5\times79.50$
$\Rightarrow\text{x}=\text{Rs. 397.50}$

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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?

Answer

Total period office = 9 a.m to 5 p.m
= 8 hours
Lunch interval = 30 min
Ratio $=\frac{30}{8\times60\text{ min}}$
$=\frac{30}{480}$
$=\frac{30\div30}{480\div30}$
$=\frac{1}{16}$

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Question 363 Marks

The ratio of copper and zinc in an alloy is 9 : 7. If the weight of zinc in the alloy is 9.8kg, find the weight of copper in the alloy.

Answer

The ratio of copper and zinc in an alloy is 9 : 7
Weight of zinc in the alloy = 9.8kg
Let weight of copper in the alloy = x
$\Rightarrow\frac{9}{7}=\frac{\text{x}}{9.8}$
$\Rightarrow\frac{9\times9.8}{7}=\text{x}$
$\Rightarrow\text{x}=12.6\text{kg}.$

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