3 Mark Question - Maths STD 8 Questions - Vidyadip

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26 questions · self-marked practice — reveal the answer and mark yourself.

Question 13 Marks

If the cost of 93 m of a certain kind of plastic sheet is Rs 1395, then what would it cost to buy 105 m of such plastic sheet?

Answer

Length of plastic sheet (in metre)	93	105
Cost(in Rs)	1395	x

Let the cost of the plastic sheet per metre be Rs x.
If more sheets are bought, the cost will also be more.
Therefore, it is a direct variation.
We get:
93 : 105 = 1395 : x
$\Rightarrow\frac{93}{105}=\frac{1395}{\text{x}}$
Applying cross multiplication, we get:
$\text{x}=\frac{105\times1395}{93}$
=1575
Thus, the required cost will be Rs 1, 575.

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Question 23 Marks

In 15 days, the earth picks up $1.2 \times 10^8 kg$ of dust from the atmosphere. In how many days it will pick up $4.8 \times 10^8 kg$ of dust?

Answer

Let $x$ be the number of days taken by the earth to pick up $4.8 \times 10^8 kg$ of dust.
Since the amount of dust picked up by the earth and the number of days are in direct variation, we have:
$\Rightarrow \frac{15}{x}=\frac{1.2 \times 10^8}{4.8 \times 10^8}$
$\Rightarrow x=15 \times \frac{4.8}{1.2}$
$\Rightarrow x=60$
Thus, the required number of days will be 60 .

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Question 33 Marks

The amount of extension in an elastic string varies directly as the weight hung on it. If a weight of 150gm produces an extension of 2.9cm, then what weight would produce an extension of 17.4cm?

Answer

Let x gm be the weight that would produce an extension of 17.4cm.

weight (in gm)	150	x
Length (in cm)	2.9	17.4

Since the amount of extension in an elastic string and the weight hung on it are in direct variation, we have:
$\frac{150}{\text{x}}=\frac{2.9}{17.4}$
$\Rightarrow 17.4\times150 =2.9\times\text{x}$
$\Rightarrow\text{x}=\frac{17.4\times150}{2.9}$
$=\frac{2610}{2.9}$
$= 900$
Thus, the required weight will be 900gm.

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Question 43 Marks

18 men can reap a field in 35 days. For reaping the same field in 15 days, how many men are required?

Answer

Let x be the number of cows that can graze the field in 10 days.

Number of days	35	15
Number of men	18	x

Since the number of days and the number of men required ro reap the field are in inverse variation, we have:
$35\times18 =15\times\text{x}$
$\Rightarrow\text{x}=\frac{35\times18}{15}$
$=42$
Thus, the required number of men is 42.

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Question 53 Marks

The cost of 97 metre of cloth is Rs 242.50. What length of this can be purchased for Rs 302.50?

Answer

Let x metre be the length of the cloth that can be purchased for Rs 302.50.

Length (in m)	97	x
Cost (in Rs.)	242.50	302.50

Since the pile of the cardboards and its thickness are in direct variation, we have:
$\frac{97}{\text{x}}=\frac{242.50}{302.50}$
$\Rightarrow 97\times302.50 =\text{x}\times242.50$
$\Rightarrow\text{x}=\frac{97\times302.50}{242.50}$
$=\frac{29342.50}{242.50}$
$= 121$
Thus, the required length will be 121 metre.

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Question 63 Marks

A person has money to buy 25 cycles worth Rs 500 each. How many cycles he will be able to buy if each cycle is costing Rs 125 more?

Answer

Let x be the number of cycles bought if each cycle costs Rs 125 more.

Cost of a cycle (in Rs.)	500	625
Number of cycles	25	x

It is in inverse variation. Therefore, we get:
$500\times25 =625\times\text{x}$
$\Rightarrow\text{x}=\frac{500\times25}{625}$
$=20$
$\therefore$ The required number of cycles is 20.

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Question 73 Marks

The amount of extension in an elastic spring varies directly with the weight hung on it. If a weight of 250gm produces an extension of 3.5cm, find the extension produced by the weight of 700gm.

Answer

Let x cm be the extension produced by the weight of 700gm.

weight (in gm)	250	700
Length (in cm)	3.5	x

Since the amount of extension in an elastic spring varies and the weight hung on it is in direct variation, we have:
$\frac{250}{700}=\frac{3.5}{\text{x}}$
$\Rightarrow \text{x}\times250 =3.5\times700$
$\Rightarrow\text{x}=\frac{3.5\times700}{250}$
$=\frac{2450}{250}$
$= 9.8$
Thus, the required extension will be 9.8cm.

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Question 83 Marks

If the thickness of a pile of 12 cardboards is 35 mm, find the thickness of a pile of 294 cardboards.

Answer

Let x cm be the thickness of a pile of 294 cardboards.

Thickness (in cm)	3.5	x
Cardboard	12	294

Since the pile of the cardboards and its thickness are in direct variation, we have:
$\frac{3.5}{\text{x}}=\frac{12}{204}$
$\Rightarrow 3.5\times294 =\text{x}\times12$
$\Rightarrow\text{x}=\frac{3.5\times294}{12}$
$=\frac{1029}{12}$
$= 85.75\text{cm}$
Thus, the thickness of 294 cardboards will be 85.75cm (or 857.5mm).

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Question 93 Marks

55 cows can graze a field in 16 days. How many cows will graze the same field in 10 days?

Answer

Let x be the number of cows that can graze the field in 10 days.

Number of days	16	10
Number of cows	55	x

Since the number of cows and the number of days taken by them to graze the field are in inverse variation, we have:
$16\times55 =10\times\text{x}$
$\Rightarrow\text{x}=\frac{16\times55}{10}$
$=88$
$\therefore$ The required number of cows is 88.

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Question 103 Marks

A worker is paid Rs 210 for 6 days work. If his total income of the month is Rs 875, for how many days did he work?

Answer

Let x be the number of days for which the worker is paid Rs.875.

Income (in Rs.)	210	875
Number of days	6	x

Since the income of the worker and the number of working days are in direct variation, we have:
$\frac{210}{875}=\frac{6}{\text{x}}$
$\Rightarrow 210\times\text{x} =875\times6$
$\Rightarrow\text{x}=\frac{875\times6}{210}$
$=\frac{5250}{210}$
$= 25$
Thus, the required number of days is 25.

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Question 113 Marks

If x and y vary inversely as:
x = 5 when y = 15, find x when y = 12

Answer

Since x and y vary inversely, we have:xy = k
For x = 5 and y = 15, we have:
5 × 15 = k
⇒ k = 75
For y = 12, we have:
12x = 75
$\Rightarrow\text{x}=\frac{75}{12}$
$=\frac{25}{4}$
$\therefore\text{x}=\frac{25}{4}$

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Question 123 Marks

Explain the concept of direct variation.

Answer

When two variables are connected to each other in such a way that if we increase the value of one variable, the value of other variable also increases and vice−versa. Similarly, if we decrease the value of one variable, the value of other variable also decreases and vice−versa.Therefore, if the ratio between two variables remains constant, it is said to be in direct variation.

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Question 133 Marks

In 10 days, the earth picks up $2.6 \times 10^8$ pounds of dust from the atmosphere. How much dust will it pick up in 45 days?

Answer

Let the amount of dust picked up by the earth in 45 days be x pounds.
Since the amount of dust picked up by the earth and the number of days are in direct variation, we have:
Ratio of the dust picked up by the earth in pounds = ratio of the number of days taken.
$\Rightarrow\frac{10}{45}=\frac{2.6\times10^{8}}{\text{x}}$
$\Rightarrow\text{x}=10=45\times2.6\times10^{8}$
$\Rightarrow\text{x}=\frac{45\times2.6\times10^{8}}{10}$
$=\frac{117\times10^{8}}{10}$
$=11.7\times10^{8}$
Thus, $11.7\times10^{8}$ pounds of dust will be picked up by the earth in 45 days.

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Question 143 Marks

Complite the following tables given that x varies directly as y.

x	3	5	7	9
y	...	20	28	...

Answer

Here, x and y vary directly.
$\therefore\text{x}=\text{ky}$
x = 5 and y = 20
i.e., 5 = k × 20
$\Rightarrow\text{k} =\frac{5}{20}=\frac{1}{4}$
For x = 3 and $\text{k}=\frac{1}{4},$ we have:
$\Rightarrow3 =\frac{1}{4}\times\text{y}$
⇒ y = 4 × 3 = 12
For x = 9 and $\text{k}=\frac{1}{4},$ we have:
x = ky
$\Rightarrow 9 = \frac{1}{4}\times\text{y}$
⇒ y = 9 × 4 = 36

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Question 153 Marks

If 36 men can do a piece of work in 25 days, in how many days will 15 men do it?

Answer

Let x be the number of days in which 15 men can do a piece of work.

Number of men	36	15
Number of days	25	x

Since the number of men hired and the number of days taken to do a piece of work are in inverse variation, we have:
$36\times25 = \text{x}\times15$
$\Rightarrow\text{x}=\frac{36\times25}{15}$
$=\frac{900}{15}$
$=60$
Thus, the required number of days is 60.

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Question 163 Marks

A work force of 50 men with a contractor can finish a piece of work in 5 months. In how many months the same work can be completed by 125 men?

Answer

Let x be the number days required to complete a piece of work by 125 men.

Number of men	50	125
Months	5	x

Since the number of men engaged and the number of days taken to do a piece of work are in inverse variation, we have:
$50\times5 = 125\text{x}$
$\Rightarrow\text{x}=\frac{50\times5}{125}$
$=2$
Thus, the required number of months is 2.

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Question 173 Marks

In a hostel of 50 girls, there are food provisions for 40 days. If 30 more girls join the hostel, how long will these provisions last?

Answer

Let x be the number of days with food provisions for 80 (i.e., 50 + 30) girls.

Number of girls	50	80
Number of days	40	x

Since the number of girls and the number of days with food provisions are in inverse variation, we have:
$50\times40 = 80\text{x}$
$\Rightarrow\text{x}=\frac{50\times40}{80}$
$=\frac{2000}{80}$
$=25$
Thus, the required number of days is 25.

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Question 183 Marks

11 men can dig $6\frac{3}{4}$ metre long trench in one day. How many men should be employed for digging 27 metre long trench of the same type in one day?

Answer

Let x be the number of men required to dig a trench of 27 metre.

Number of men	11	x
Length (in m)	$\frac{27}{4}$	27

Since the length of the trench and the number of men are in direct variation, we have:
$\frac{11}{\text{x}}=\frac{\frac{27}{4}}{27}$
$\Rightarrow 11\times27 =\text{x}\times\frac{27}{4}$
$\Rightarrow\text{x}=\frac{11\times27\times4}{27}$
$= 44$
Thus, 44 men will be required to dig a trench of 27m.

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Question 193 Marks

Three spraying machines working together can finish painting a house in 60 minutes. How long will it take for 5 machines of the same capacity to do the same job?

Answer

Let the time taken by 5 spraying machines to finish a painting job be x minutes.

Number of Machines	3	5
Time (in minutes)	60	x

Since the number of spraying machines and the time taken by them to finish a painting job are in inverse variation, we have:
$3\times60 =5\times\text{x}$
$\Rightarrow800=5\text{x}$
$\Rightarrow\text{x}=\frac{180}{5}$
$=36$
Thus, the required time will be 36 minutes.

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Question 203 Marks

1200 men can finish a stock of food in 35 days. How many more men should join them so that the same stock may last for 25 days?

Answer

Number of men	1200	x
Days	35	25

Let x be the number of additional men required to finish the stock in 25 days.
Since the number of men and the time taken to finish a stock are in inverse variation, we have:
$1200\times35 = 25\text{x}$
$\Rightarrow\text{x}=\frac{1200\times35}{25}$
$=1680$
$\therefore$ Required number of men = 1680 - 1200 = 480
Thus, an additional 480 men should join the existing 1200 men to finish the stock in 25 days.

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Question 213 Marks

Complite the following tables given that x varies directly as y.

x	6	8	10	...	20
y	15	20	...	40	...

Answer

Here, x and y vary directly.
$\therefore\text{x}=\text{ky}$
x = 6 and y = 15
i.e., 6 = k × 15
$\Rightarrow\text{k} =\frac{6}{15}=0.4$
For x = 10 and k = 0.4, we have:
$\Rightarrow\text{y}=\frac{10}{0.4}=25$
For y = 40 and k = 0.4, we have:
x = 0.4 × 40 = 16
For x = 20 and k = 0.4, we have:
$\Rightarrow\text{y}=\frac{20}{0.4}=50$

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Question 223 Marks

If x and y vary inversely as:
x = 3 when y = 8, find y when x = 4

Answer

Since x and y vary inversely, we have:
xy = k
For x = 3 and y = 8, we have:
3 × 8 = k
⇒ k = 24
For x = 4, we have:
4y = 24
$\Rightarrow\text{y}=\frac{24}{4}$
$=6$
$\therefore\text{y}=6$

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Question 233 Marks

Complite the following tables given that x varies directly as y.

x	2.5	...	...	15
y	5	8	12	...

Answer

Here, x and y vary directly.
$\therefore\text{x}=\text{ky}$
x = 2.5 and y = 5
i.e., 2.5 = k × 5
$\Rightarrow\text{k} =\frac{2.5}{5}=0.5$
For y = 8 and k = 0.5, we have:
x = ky
⇒ x = 8 × 0.5 = 4
For y = 12 and k = 0.5, we have:
x = ky
⇒ x = 12 × 0.5 = 6
For x = 15 and k = 0.5, we have:
x = ky
⇒ 15 = 0.5 × y
$\Rightarrow\text{y}=\frac{15}{0.5}=30$

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Question 243 Marks

A woker is paid Rs. 200 for 8 days work. If he works for 20 days, how much will he get?

Answer

Let x be the number of days for which the worker is paid Rs. 875

Income (in Rs.)	200	x
Number of days	8	20

Since the income and the number of working days are in direct variation, we have:
$\frac{200}{\text{x}}=\frac{8}{20}$
$\Rightarrow 200\times20 =8\text{x}$
$\Rightarrow\text{x}=\frac{200\times20}{8}$
$=\frac{4000}{8}$
$= 500$
Thus, the worker will get Rs. 500 for working 20 days.

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Question 253 Marks

A group of 3 friends staying together, consume 54kg of wheat every month. Some more friends join this group and they find that the same amount of wheat lasts for 18 days. How many new members are there in this group now?

Answer

Let x be the number of new members in the group.

Number of members	3	x
Number of days	30	18

Since more members can finish the wheat in less number of days, it is a case of inverse variation.
Therefore, we get:
$3\times30 =\text{x}\times18$
$\Rightarrow90=18\text{x}$
$\Rightarrow\text{x}=\frac{90}{18}$
Thus, the number of new members in the group = 5 - 3 = 2.

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Question 263 Marks

A work-force of 420 men with a contractor can finish a certain piece of work in 9 months. How many extra men must he employ to complete the job in 7 months?

Answer

Let x be the extra number of men employed to complete the job in 7 months.

Number of men	420	x
Months	9	7

Since the number of men hired and the time required to finish the piece of work are in inverse variation, we have:
$420\times9 = 7\text{x}$
$\Rightarrow\text{x}=\frac{420\times9}{7}$
$=540$
Thus, the number of extra men required to complete the job in 7 months = 540 - 420 = 120

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