Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ৮ / ১১ · ৭০১–৮০০ / ১,০৮৬

৭০১.

In a 300 m race, A beats B by 30 m and C by 40 m. In a race of 540 m, B will beat C by -

ক
ক) 12 m
খ
খ) 16 m
গ
গ) 20 m
ঘ
ঘ) None of these

সঠিক উত্তর: গ

গ) 20 m

উত্তর

সঠিক উত্তর: গ

গ) 20 m

গ

ব্যাখ্যা

Question: In a 300 m race, A beats B by 30 m and C by 40 m. In a race of 540 m, B will beat C by -

Solution:
Given that,
A : B = 300 : 270
and A : C = 300 : 260

A/B = 300/270
and A/C = 300/260

∴ B/C = (B/A) × (A/C)
⇒(270/300) × (300/260) = 270/260
⇒ (270 × 2)/(260 × 2) = 540/520
⇒ B/C = 540/520

∴ B : C = 540 : 520

∴ In a 360 m race, B beats C by (540 - 520) m = 20 m

৭০২.

Given that the milk and water ratio in a 45-liter mixture is 4 : 1, how much additional water is required to achieve a 3 : 2 ratio?

ক
9 litres
খ
10 litres
গ
12 litres
ঘ
15 litres

সঠিক উত্তর: ঘ

15 litres

উত্তর

সঠিক উত্তর: ঘ

15 litres

ঘ

ব্যাখ্যা

Question: Given that the milk and water ratio in a 45-liter mixture is 4 : 1, how much additional water is required to achieve a 3 : 2 ratio?
(৪৫ লিটার মিশ্রণের মধ্যে দুধ ও পানির অনুপাত ৪:১, তাহলে ৩:২ অনুপাত করার জন্য কতটা পানি আরও যোগ করতে হবে?)

Solution:
৪৫ লিটার মিশ্রণে দুধের পরিমাণ ৪৫ × (৪/৫) লিটার = ৩৬ লিটার
অতএব, মিশ্রণে পানির পরিমাণ = ৪৫ - ৩৬ লিটার = ৯ লিটার

ধরা যাক,
মিশ্রণে x লিটার পানি যোগ করা হয়েছে।
তাহলে,
৩৬ / (৯ + x) = ৩ / ২
⇒ ৭২ = ২৭ + ৩x
⇒ ৩x = ৪৫

অতএব, x = ১৫

৭০৩.

A 150-liter mixture contains 60% milk and the rest water. How many liters of water should be evaporated to make the mixture 75% milk?

ক
25 liters
খ
30 liters
গ
40 liters
ঘ
45 liters
ঙ
50 liters

সঠিক উত্তর: খ

30 liters

উত্তর

সঠিক উত্তর: খ

30 liters

খ

ব্যাখ্যা

Question: A 150-liter mixture contains 60% milk and the rest water. How many liters of water should be evaporated to make the mixture 75% milk?

Solution:
There is 60% milk in 150 liters of mixture,
That means, (60/100) × 150 = 90 liters of milk.
∴ The amount of water = 150 - 90 = 60 liters.

Let,
x liters of water need to be evaporated.
Then the new volume of the solution will be (150 - x) liters.

ATQ,
90/(150 - x) = 75/100
⇒ 90/(150 - x) = 3/4
⇒ 3(150 - x) = 360
⇒ 450 - 3x = 360
⇒ 3x = 450 - 360
⇒ 3x = 90
⇒ x = 90/3
∴ x = 30

∴ 30 liters of water need to be evaporated so that the milk concentration becomes 75%.

৭০৪.

A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 10%. The percentage of water in the mixture is:

ক
10.25%
খ
9.09%
গ
7.45%
ঘ
8.56%

সঠিক উত্তর: খ

9.09%

উত্তর

সঠিক উত্তর: খ

9.09%

খ

ব্যাখ্যা

Question: A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 10%. The percentage of water in the mixture is-

Solution:
Let,
the cost price = 100
selling price = 110

So the amount of milk = 100/110
= 10/11

∴ Amount of water = (1 - 10/11) × 100%
= 9.09%

৭০৫.

Class A has boys to girls in the ratio 2 : 3, Class B has girls to boys in the ratio 5 : 3. If the number of students in Class A is at least twice as many as the number of students in Class B, what is the minimum percentage of boys when both classes are considered together?

ক
39.17%
খ
40%
গ
33.33%
ঘ
37.5%

সঠিক উত্তর: ক

39.17%

উত্তর

সঠিক উত্তর: ক

39.17%

ক

ব্যাখ্যা

Question: Class A has boys to girls in the ratio 2 : 3, Class B has girls to boys in the ratio 5 : 3. If the number of students in Class A is at least twice as many as the number of students in Class B, what is the minimum percentage of boys when both classes are considered together?

Solution:
let, in class B there are 40 students
girls = 40 × (5/8) = 25
boys = 15

the number of students in Class A is at least twice as many as the number of students in Class B
class A students = 80
minimum number of boys = 80 × 2/5 = 32 and girls = 48

The minimum percentage of boys when both classes are considered together is = {(15 + 32)/(40 + 80)} × 100%
= (47/120) × 100%
= 39.17%

৭০৬.

If 4 kg of metal of which 1/3 is silver and the rest is copper, be mixed with 3 kg of metal, of which 1/4 is silver and the rest is copper, then what will be the ratio of silver to copper in the mixture?

ক
ক) 25 : 41
খ
খ) 17 : 35
গ
গ) 25 : 59
ঘ
ঘ) 23 : 63

সঠিক উত্তর: গ

গ) 25 : 59

উত্তর

সঠিক উত্তর: গ

গ) 25 : 59

গ

ব্যাখ্যা

Question: If 4 kg of metal of which 1/3 is silver and the rest is copper, be mixed with 3 kg of metal, of which 1/4 is silver and the rest is copper, then what will be the ratio of silver to copper in the mixture?

Solution:
1st metal,
Quantity of silver = 4 × (1/3) = 4/3 kg
∴ Quantity of copper = 4 - (4/3) = 8/3 kg

2nd metal,
Quantity of silver = 3 × (1/4) = 3/4 kg
∴ Quantity of copper = 3 - 3/4 = 9/4 kg

In the mixture, the required ratio = (4/3 + 3/4/) : (8/3 + 9/4)
= 25/12 : 59/12
= 25 : 59

৭০৭.

The ratio of Books and Calculators in a shop is 5 : 2 respectively. The average number of Books and Calculators is 644. What is the number of Calculators in the shop?

ক
ক) 920 Pieces
খ
খ) 820 Pieces
গ
গ) 364 Pieces
ঘ
ঘ) 368 Pieces

সঠিক উত্তর: ঘ

ঘ) 368 Pieces

উত্তর

সঠিক উত্তর: ঘ

ঘ) 368 Pieces

ঘ

ব্যাখ্যা

Question: The ratio of Books and Calculator in a shop is 5 : 2 respectively. The average number of Books and Calculator is 644. What is the number of Calculators in the shop?

Solution:
Let,
There are books in the shop = 5x
There are calculators in the shop = 2x

ATQ,
(5x + 2x)/2 = 644
⇒ 7x/2 = 644
⇒ 7x = 1288
∴ x = 184

There are calculators in the shop = (2 × 184) = 368 Pieces.

৭০৮.

The salaries of X and Y are in the ratio 3 : 7. If the salary of each is increased by 4,800, then the new ratio becomes 5 : 9. What is X's salary?

ক
8750 Tk
খ
7900 Tk
গ
7200 Tk
ঘ
6850 Tk

সঠিক উত্তর: গ

7200 Tk

উত্তর

সঠিক উত্তর: গ

7200 Tk

গ

ব্যাখ্যা

Question: The salaries of X and Y are in the ratio 3 : 7. If the salary of each is increased by 4,800, then the new ratio becomes 5 : 9. What is X's salary?

Solution:
Let,
The salaries of X and Y are 3x and 7x respectively.

After increasing each salary by 4800, the new salaries become: 3x + 4800 and 7x + 4800

ATQ,
(3x + 4800) : (7x + 4800) = 5 : 9
⇒ (3x + 4800)/(7x + 4800) = 5/9
⇒ 9(3x + 4800) = 5(7x + 4800)
⇒ 27x + 43200 = 35x + 24000
⇒ 43200 - 24000 = 35x - 27x
⇒ 19200 = 8x
∴ x = 2400

So the salary of "X" = 3 × 2400
= 7200 Tk

৭০৯.

There are 174 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

ক
72
খ
60
গ
84
ঘ
66

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: There are 174 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

Solution:
Total students = 174.
Ratio of students in 1st and 2nd standards = 2 : 3 = (2 × 4) : (3 × 4) = 8 : 12

Ratio of students in 2nd and 3rd standards = 4 : 3 = (4 × 3) : (3 × 3) = 12 : 9
Hence combined ratio i.e. 1st : 2nd: 3rd is = 8 : 12 : 9.

∴ Number of students in 2nd standard = (174 × 12)/29 = 72

৭১০.

A and B are the two alloys of copper and brass prepared by mixing metals in the proportion of 7 : 2 and 7 : 11 respectively. If equal quantities of two alloys are melted to form a third alloy called C, then the proportion of copper and brass in C will be

ক
3 : 5
খ
7 : 5
গ
5 : 2
ঘ
7 : 9

সঠিক উত্তর: খ

7 : 5

উত্তর

সঠিক উত্তর: খ

7 : 5

খ

ব্যাখ্যা

Question: A and B are the two alloys of copper and brass prepared by mixing metals in the proportion of 7 : 2 and 7 : 11 respectively. If equal quantities of two alloys are melted to form a third alloy called C, then the proportion of copper and brass in C will be

Solution:
Let, A is 18 kg
copper = 18 × (7/9)
= 14 kg
brass = 4 kg

Let, B is 36 kg
copper = 36 × (7/18)
= 14 kg
brass = 22 kg

After mixing, total amount = 18 + 36 = 54 kg

From part A, copper = (54/2) × (7/9) = 21 kg
brass = 27 - 21 = 6 kg

From part B, copper = (54/2) × (7/18) = 10.5 kg
brass = 27 - 10.5 = 16.5 kg

Ratio = (21 + 10.5) : (6 + 16.5)
= 31.5 : 22.5
= 315 : 225
= 63 : 45
= 7 : 5

৭১১.

The ratio of cups of flour : cups of water in a pizza dough recipe is 9 : 4. A pizza restaurant makes a large quantity of dough, using 36 cups of flour. How much water should they use?

ক
16 cups
খ
13 cups
গ
11 cups
ঘ
81 cups
ঙ
None of these

সঠিক উত্তর: ক

16 cups

উত্তর

সঠিক উত্তর: ক

16 cups

ক

ব্যাখ্যা

Question: The ratio of cups of flour : cups of water in a pizza dough recipe is 9 : 4. A pizza restaurant makes a large quantity of dough, using 36 cups of flour. How much water should they use?

Solution:
Let,
cups of flour 9x
cups of water 4x

∴ 9x = 36
⇒ x = 4

∴ cups of water 4x = 4 × 4 = 16 cups

৭১২.

There are 145 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

ক
40
খ
45
গ
60
ঘ
65

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: There are 145 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

Solution:
Total students = 145.
Ratio of students in 1st and 2nd standards = 2 : 3 = (2 × 4) : (3 × 4) = 8 : 12

Ratio of students in 2nd and 3rd standards = 4 : 3 = (4 × 3) : (3 × 3) = 12 : 9
Hence combined ratio i.e. 1st : 2nd: 3rd is = 8 : 12 : 9.

∴ Number of students in 2nd standard = (145 × 12)/29 = 60

৭১৩.

If a carton containing a dozen mirrors is dropped. which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

ক
ক) 2 : 1
খ
খ) 3 : 1
গ
গ) 3 : 2
ঘ
ঘ) 7 : 5
ঙ
ঙ) None of these

সঠিক উত্তর: গ

গ) 3 : 2

উত্তর

সঠিক উত্তর: গ

গ) 3 : 2

গ

ব্যাখ্যা

অনুপাতগুলোর যোগফল ১২ এর উৎপাদক হতে হবে। এখানে 3:2 এর যোগফল 12 এর কোনো উৎপাদক নয়। সুতরাং, সঠিক উত্তর হবে 3:2।

৭১৪.

What is the ratio of 4 inches to 7 feet?

ক
1 : 12
খ
1 : 18
গ
1 : 21
ঘ
1 : 25
ঙ
None of the above

সঠিক উত্তর: গ

1 : 21

উত্তর

সঠিক উত্তর: গ

1 : 21

গ

ব্যাখ্যা

Question: What is the ratio of 4 inches to 7 feet?

Solution:
We know,
1 feet = 12 inches
So, 7 feet = 7 × 12
= 84 inches

Now,
4 inches : 7 feet = 4 : 84 = 1 : 21

৭১৫.

Two numbers are such that the ratio between them is 4 : 7. If each is increased by 4, the ratio becomes 3 : 5. The sum of two numbers is-

ক
ক) 56
খ
খ) 32
গ
গ) 88
ঘ
ঘ) 66

সঠিক উত্তর: গ

গ) 88

উত্তর

সঠিক উত্তর: গ

গ) 88

গ

ব্যাখ্যা

Question: Two numbers are such that the ratio between them is 4 : 7. If each is increased by 4, the ratio becomes 3 : 5. The sum of two numbers is-

Solution:
Let the numbers be 4x and 7x
Then,
⇒(4x + 4)/(7x + 4) = 3/5
⇒5(4x + 4) = 3(7x+4)
⇒20x + 20 = 21x + 12
⇒21x - 20x = 20 - 12
x = 8

∴ The sum of two numbers = 4x + 7x
= 11x
= 11 × 8
= 88

৭১৬.

A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:

ক
ক) 2 : 1
খ
খ) 3 : 2
গ
গ) 4 : 3
ঘ
ঘ) 3 : 1

সঠিক উত্তর: ঘ

ঘ) 3 : 1

উত্তর

সঠিক উত্তর: ঘ

ঘ) 3 : 1

ঘ

ব্যাখ্যা

Let man's rate upstream be x kmph
Then, his rate downstream = 2x kmph
∴ (speed in still water) : (Speed of stream)
=(2x+x)/2 : (2x−x)/2
= 3x/2 : x/2 = 3 : 1

৭১৭.

Motin bought two varieties of rice, costing Tk. 50 kg and Tk. 60 kg each, and mixed them in some ratio. Then he sold the mixture at Tk. 70 kg making a profit of 20%, what was the ratio of the mixture?

ক
2 : 3
খ
1 : 10
গ
1 : 5
ঘ
2 : 7
ঙ
None

সঠিক উত্তর: গ

1 : 5

উত্তর

সঠিক উত্তর: গ

1 : 5

গ

ব্যাখ্যা

Question: Motin bought two varieties of rice, costing Tk. 50 kg and Tk. 60 kg each, and mixed them in some ratio. Then he sold the mixture at Tk. 70 kg making a profit of 20%, what was the ratio of the mixture?

Solution:
মনে করি,
প্রথম পদের চাল কিনলো x কেজি
দ্বিতীয় পদের চাল কিনলো y কেজি

মোট খরচ = (50x + 60y) টাকা
মোট চালের পরিমাণ = (x + y) কেজি
মোট বিক্রয়মূল্য = 70 (x + y) = 70x + 70y

লাভ = 70x + 70y - 50x - 60y
= 20x + 10y টাকা

প্রশ্নমতে,
20x + 10y = (50x + 60y) 20%
বা, 20x + 10y = (50x + 60y) 20/100
বা, 10(2x + y) = 10(5x + 6y) 1/5
বা, 2x + y = (5x + 6y) 1/5
বা, 10x + 5y = 5x + 6y
বা, 5x = y
বা, x/y = 1/5
∴ x : y = 1 : 5

৭১৮.

How much sugar at Tk. 95 a kg should be added to 17 kg of tea at Tk. 200 a kg so that the mixture be worth Tk. 130 a kg.?

ক
11 kg
খ
17 kg
গ
21 kg
ঘ
34 kg

সঠিক উত্তর: ঘ

34 kg

উত্তর

সঠিক উত্তর: ঘ

34 kg

ঘ

ব্যাখ্যা

Question: How much sugar at Tk. 95 a kg should be added to 17 kg of tea at Tk. 200 a kg so that the mixture be worth Tk. 130 a kg.?

Solution:
Ratio in which tea and sugar should be mixed
= 200 - 130 : 130 - 95 = 70 : 35 = 10 : 5 = 2 : 1
Let x be quantity at 95/kg.

∴ 2 : 1 = x : 17
⇒ 2/1 = x/17
⇒ x = 34
hence x = 34 kg.

৭১৯.

A mixture of 200 liters of wine and water contains 25% water. How much more water should be added so that water becomes 40% of the new mixture?

ক
50 liters
খ
40 liters
গ
30 liters
ঘ
45 liters

সঠিক উত্তর: ক

50 liters

উত্তর

সঠিক উত্তর: ক

50 liters

ক

ব্যাখ্যা

Question: A mixture of 200 liters of wine and water contains 25% water. How much more water should be added so that water becomes 40% of the new mixture?

Solution:
Number of liters of water in 200 liters of the mixture = 25% of 200 = 1/4 of 200 = 50 liters
Let us Assume that another 'P' liters of water are added to the mixture to make water 40% of the new mixture.
So, the total amount of water becomes (50 + P) and the total volume of the mixture becomes (200 + P)
Thus, (50 + P) = 40% of (200 + P)
⇒ 50 + P = (40/100) × (200 + P)
⇒ 5000 + 100P = 8000 + 40P
⇒ 60P = 3000
∴ P = 50 liters

৭২০.

A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture.

ক
ক) 10
খ
খ) 12
গ
গ) 15
ঘ
ঘ) 18

সঠিক উত্তর: ক

ক) 10

উত্তর

সঠিক উত্তর: ক

ক) 10

ক

ব্যাখ্যা

Let the quantity of alcohol and water be 4x litres and 3x litres respectively
4x/(3x + 5) = 4/5
⇒ 20x = 4(3x + 5)
⇒ 8x = 20
⇒ x = 2.5
Quantity of alcohol = (4 x 2.5) litres = 10 litres.

৭২১.

800 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

ক
ক) 120 gm
খ
খ) 220 gm
গ
গ) 320 gm
ঘ
ঘ) 420 gm

সঠিক উত্তর: গ

গ) 320 gm

উত্তর

সঠিক উত্তর: গ

গ) 320 gm

গ

ব্যাখ্যা

Question: 800 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
amount of sugar = 800 × 30/100
= 240 grams

let, x gm sugar to be added
ATQ,
(240 + x)/(800 + x) = 50%
⇒ (240 + x)/(800 + x) = = 1/2
⇒ 480 + 2x = 800 + x
⇒ 2x - x = 800 - 480
∴ x = 320 gm

৭২২.

If A : B = 2 : 3 and B : C = 4 : 5 then A : C is -

ক
ক) 6 : 15
খ
খ) 8 : 20
গ
গ) 8 : 15
ঘ
ঘ) 8 : 12

সঠিক উত্তর: গ

গ) 8 : 15

উত্তর

সঠিক উত্তর: গ

গ) 8 : 15

গ

ব্যাখ্যা

Question: If A : B = 2 : 3 and B : C = 4 : 5 then A : C is -

Solution:
A : B = 2 : 3
= (2 × 4 ) : (3 × 4)
= 8 : 12

B : C = 4 : 5
= (4 × 3) : (5 × 3)
= 12 : 15

∴ A : B : C = 8 : 12 : 15
∴ A : C = 8 : 15

৭২৩.

If 3 : 7 :: 12 : x, then x is equal to:

ক
22
খ
23
গ
28
ঘ
27
ঙ
None

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If 3 : 7 :: 12 : x, then x is equal to:

Solution:
3 : 7 :: 12 : x
⇒ 3/7 = 12/x
⇒ 3x = 84
⇒ x = 28

∴ x = 28

৭২৪.

Two numbers are in the ratio 2 : 5. If 16 is added to both the numbers, their ratio becomes 1 : 2. What is the bigger number?

ক
ক) 92
খ
খ) 80
গ
গ) 56
ঘ
ঘ) 78

সঠিক উত্তর: খ

খ) 80

উত্তর

সঠিক উত্তর: খ

খ) 80

খ

ব্যাখ্যা

Let the numbers be 2x, 5x
ATQ,
(2x + 16) / (5x + 16) = 1/2
Or, 4x + 32 = 5x + 16
Or, x = 16
∴ The numbers: 2x = 2 × 16 = 32
And, 5x = 5 × 16 = 80

৭২৫.

If a : b = 2 : 3 and a : c = 4 : 7, then b : c =?

ক
6 : 5
খ
3 : 7
গ
6 : 7
ঘ
6 : 1

সঠিক উত্তর: গ

6 : 7

উত্তর

সঠিক উত্তর: গ

6 : 7

গ

ব্যাখ্যা

প্রশ্ন: If a : b = 2 : 3 and a : c = 4 : 7, then b : c =?

সমাধান:
a : b = 2 : 3
⇒ a/b = 2/3

a : c = 4 : 7
⇒ a/c = 4/7

(a/b) / (a/c) = (2/3)/(4/7)
⇒ c/b = 7/6
∴ b/c = 6/7
∴ b : c = 6 : 7

৭২৬.

In what ratio, water must be mixed with fruit juice costing Tk. 24 per litre so that the juice would be worth of Tk. 20 per litre?

ক
1 : 4
খ
1 : 5
গ
1 : 6
ঘ
2 : 5
ঙ
None of these

সঠিক উত্তর: খ

1 : 5

উত্তর

সঠিক উত্তর: খ

1 : 5

খ

ব্যাখ্যা

Question: In what ratio, water must be mixed with fruit juice costing Tk. 24 per litre so that the juice would be worth of Tk. 20 per litre?

Solution:
Cost of 1 litre of water = Tk. 0 = cheaper quantity.
Cost of 1 litre of juice = Tk. 24 = dearer quantity.
And, the mean price = m = Tk. 20

Therefore, (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c) = 4 : 20 = 1 : 5
Hence, the required answer is 1 : 5.

৭২৭.

The ratio of male students to female students in a class is 13 to 19. If there are 224 people in the class, including one teacher, one administrator and thirty evaluations. How many people in the class are male students?

ক
ক) 78
খ
খ) 80
গ
গ) 91
ঘ
ঘ) 114

সঠিক উত্তর: ক

ক) 78

উত্তর

সঠিক উত্তর: ক

ক) 78

ক

ব্যাখ্যা

দেওয়া আছে,
মোট লোক সংখ্যা = 224
1 জন শিক্ষক, 1 জন প্রশাসক ও 30 জন মূল্যায়নকারী বাদ দিলে অবশিষ্ট লোক সংখ্যা
= 224 - (1 + 1 + 30)
= 224 - 32
= 192 জন
এই 192 জনের মধ্যে ছাত্র : ছাত্রী = 13:19
∴ ছাত্র সংখ্যা = 13/(13+19) × 192
= (13/32) × 192
= 78 জন

৭২৮.

How many kgs of sugar costing Tk 12 per kg must be mixed with 35 kg of sugar costing Tk 8 per kg so that may be a gain of 15% by selling the mixture at Tk 12.65 per kg?

ক
100 kg
খ
85 kg
গ
90 kg
ঘ
105 kg

সঠিক উত্তর: ঘ

105 kg

উত্তর

সঠিক উত্তর: ঘ

105 kg

ঘ

ব্যাখ্যা

Question: How many kgs of sugar costing Tk 12 per kg must be mixed with 35 kg of sugar costing Tk 8 per kg so that may be a gain of 15% by selling the mixture at Tk 12.65 per kg?

Solution:
Let, the sugar of Tk 12 per kg is = x kg

ATQ,
115% of (8 × 35 + 12 × x) = 12.65(35 + x)
⇒ (115/100) × (280 + 12x) = 442.75 + 12.65x
⇒ 322 + 13.8x = 442.75 + 12.65x
⇒ 1.15x = 120.75
⇒ x = 105

৭২৯.

The ratio of the salary of A, B and C is 7 : 5 : 3. If C gets Tk. 222 less than what B gets, then what is the salary of A?

ক
ক) Tk. 770
খ
খ) Tk. 777
গ
গ) Tk. 780
ঘ
ঘ) Tk. 788

সঠিক উত্তর: খ

খ) Tk. 777

উত্তর

সঠিক উত্তর: খ

খ) Tk. 777

খ

ব্যাখ্যা

Question: The ratio of the salary of A, B and C is 7 : 5 : 3. If C gets Tk. 222 less than what B gets, then what is the salary of A?

Solution:
Let,
A gets 7x
B gets 5x
C gets 3x

ATQ,
3x = 5x - 222
⇒ 5x - 3x = 222
⇒ 2x = 222
∴ x = 111

∴ The salary of A is 7 × 111 = 777 Tk.

৭৩০.

A, B and C enter into a partnership with capitals in the ratio 5:6:8. At the end of the business term, they received the profit in the ratio 5:3:12. Find the ratio of time for which they contributed their capitals?

ক
ক) 2 : 1 : 3
খ
খ) 1 : 2 : 3
গ
গ) 2 : 3 : 1
ঘ
ঘ) 3 : 2 : 1

সঠিক উত্তর: ক

ক) 2 : 1 : 3

উত্তর

সঠিক উত্তর: ক

ক) 2 : 1 : 3

ক

ব্যাখ্যা

Profit= Time×Capital invested
Time= Profit/ Capital invested
Required ratio of time
=5/5:3/6:12/8
=1:1/2:3/2
=2:1:3

৭৩১.

If 5 men and 2 boys working together can do four times as much work per hour as a man and a boy together, Working capacities of a man and a boy are in the ratio:

ক
ক) 3 : 1
খ
খ) 2 : 1
গ
গ) 4 : 1
ঘ
ঘ) 6 : 1

সঠিক উত্তর: খ

খ) 2 : 1

উত্তর

সঠিক উত্তর: খ

খ) 2 : 1

খ

ব্যাখ্যা

Qustion: If 5 men and 2 boys working together can do four times as much work per hour as a man and a boy together, Working capacities of a man and a boy are in the ratio:

Solution:
Let
1 man 1 day work = x
1 boy 1 day work = y

Now
5x + 2y = 4(x + y)
5x + 2y = 4x + 4y
5x - 4x = 4y - 2y
x = 2y
x/y = 2/1
x : y = 2 : 1

৭৩২.

Let x : y = a/b : - b/a, If (x - y) = (a/b + b/a) then x is equal to -

ক
ক) (a - b)/a
খ
খ) (a + b)/a
গ
গ) (a + b)/b
ঘ
ঘ) None of these

সঠিক উত্তর: ঘ

ঘ) None of these

উত্তর

সঠিক উত্তর: ঘ

ঘ) None of these

ঘ

ব্যাখ্যা

Question: Let x : y = a/b : - b/a, If (x - y) = (a/b + b/a) then x is equal to -

Solution:
Given,
⇒ x/y =(a/b)/(-b/a)
⇒ x/y = - a²/b²
⇒ y =(−b²/a²)x

Now,
x - y = a/b + b/a
⇒ x + (b²/a²)x = (a²+ b²)/ab
⇒ x(a²+ b²)/a² = (a²+b²)/ab
⇒ x = a²/ab
⇒ x = a/b

৭৩৩.

In what ratio a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?

ক
1 : 2
খ
1 : 3
গ
1 : 5
ঘ
1 : 4

সঠিক উত্তর: খ

1 : 3

উত্তর

সঠিক উত্তর: খ

1 : 3

খ

ব্যাখ্যা

Question: In what ratio a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?

Solution:
Let, X contains 30% alcohol strength and
Y contains 50% alcohol strength

According to the question,
(30% of X) + (50% of Y) = 45% of (X + Y)
⇒ 30X + 50Y = 45X + 45Y
⇒ 15X = 5Y
⇒ X : Y = 1 : 3

৭৩৪.

Alcohol and water in two vessels A and B are in the ratio 5 : 3 and 5 : 4 respectively. In what ratio, the liquid of both the vessels be mixed to obtain a new mixture in vessel C in the ratio 7 : 5?

ক
ক) 2 : 3
খ
খ) 3 : 2
গ
গ) 3 : 5
ঘ
ঘ) 2 : 5
ঙ
ঙ) 2 : 1

সঠিক উত্তর: ক

ক) 2 : 3

উত্তর

সঠিক উত্তর: ক

ক) 2 : 3

ক

ব্যাখ্যা

According to the question,
Alcohol : Water -
Vessel A - 4 : 3
Vessel B - 2 : 3
Now using alligation,

৭৩৫.

A man spent 1/2 of his money and then lost 1/4 of the reminder. He was left with Tk. 3,600. How much did he start with?

ক
ক) 7,200
খ
খ) 8,800
গ
গ) 9,600
ঘ
ঘ) 10,400

সঠিক উত্তর: গ

গ) 9,600

উত্তর

সঠিক উত্তর: গ

গ) 9,600

গ

ব্যাখ্যা

Question: A man spent 1/2 of his money and then lost 1/4 of the reminder. He was left with Tk. 3,600. How much did he start with?

Solution:
ধরি
মোট ছিল = x টাকা

সে খরচ করলো = x/2 টাকা
অবশিষ্ট থাকে = x - (x/2)
= (2x - x)/2
= x/2

হারিয়ে ফেলে = x/2 এর 1/4 = x/8

অবশিষ্ট থাকে = (x/2) - (x/8)
= (4x - x)/8
= 3x/8

প্রশ্নমতে,
3x/8 = 3600
3x = 3600 × 8
x = (3600 × 8)/3
x = 9600 টাকা

৭৩৬.

A solution contains 200 liters with 25% salt. How many liters of water must be evaporated so that the concentration of salt in the solution increases to 50%?

ক
100 liters
খ
120 liters
গ
115 liters
ঘ
200 liters
ঙ
None

সঠিক উত্তর: ক

100 liters

উত্তর

সঠিক উত্তর: ক

100 liters

ক

ব্যাখ্যা

Question: A solution contains 200 liters with 25% salt. How many liters of water must be evaporated so that the concentration of salt in the solution increases to 50%?

Solution:
200 লিটারের 25% লবণ রয়েছে, অর্থাৎ,
(25/100) × 200 = 50 লিটার লবণ

∴ পানির পরিমাণ = 200 - 50 = 150 লিটার

ধরি,
x লিটার পানি বাষ্পীভূত করতে হবে।
তাহলে নতুন দ্রবণের পরিমাণ হবে, 200 - x লিটার

প্রশ্নমতে,
50/(200 - x) = 50/100
⇒ 100 × 50 = 50(200 - x)
⇒ 5000 = 10000 - 50x
⇒ 50x = 10000 - 5000
⇒ 50x = 5000
⇒ x = 5000/50
⇒ x = 100

∴ 100 লিটার পানি বাষ্পীভূত করতে হবে যাতে লবণের পরিমাণ ৫০% হয়।

৭৩৭.

A shopkeeper mixed low-quality vegetable oil costing Tk. 40 per litre with sunflower refined oil costing Tk. 80 per litre in a ratio of 2 ∶ 3 respectively. If he sold the mixture at Tk. 100 per litre, find his profit percentage.

ক
42.75%
খ
47.5%
গ
51.5%
ঘ
56.25%

সঠিক উত্তর: ঘ

56.25%

উত্তর

সঠিক উত্তর: ঘ

56.25%

ঘ

ব্যাখ্যা

Solution: A shopkeeper mixed low-quality vegetable oil costing Tk. 40 per litre with sunflower refined oil costing Tk. 80 per litre in a ratio of 2 ∶ 3 respectively. If he sold the mixture at Tk. 100 per litre, find his profit percentage.

Solution:
Let,
Quantity of low-quality vegetable oil in mixure 2x litre.
∴ The costing price of low-quality vegetable oil Tk. (40 × 2x) = Tk. 80x

Quantity of sunflower refined oil in mixure 3x litre.
∴ The costing price of sunflower refined oil Tk. (80 × 3x) = Tk. 240x

∴ Total costing price = Tk. (80x + 240x) = Tk. 320x

Total selling price = Tk. (100 × 5x) = Tk. 500x

∴ Profit = Tk. (500 - 320) = Tk. 180

∴ Profit percentage = (180/320) × 100%
= 56.25%

৭৩৮.

In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

ক
3 : 1
খ
1 : 3
গ
2 : 3
ঘ
3 : 2
ঙ
None of the above

সঠিক উত্তর: ঘ

3 : 2

উত্তর

সঠিক উত্তর: ঘ

3 : 2

ঘ

ব্যাখ্যা

Question: In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

Solution:
Quantity of Tk. 60 tea is x kg.
Quantity of Tk. 65 tea is y kg

S.P. of 1 kg of the mixture = Tk. 68.20,
Gain = 10%.
C.P of 1 kg of the mixture = Tk. (100/110 × 68.20) = Tk. 62

ATQ,
60x + 65y = (x + y)62
⇒ 60x + 65y = 62x + 62y
⇒ 62x - 60x = 65y - 62y
⇒ 2x = 3y

∴ x/y = 3/2

৭৩৯.

A certain basketball team that has played 2/3 of its games has a record of 17 wins and 3 losses. What is the greatest number of the remaining games that the team can lose and still win at least 3/4 of all of its games?

ক
6
খ
5
গ
4
ঘ
3

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: A certain basketball team that has played 2/3 of its games has a record of 17 wins and 3 losses. What is the greatest number of the remaining games that the team can lose and still win at least 3/4 of all of its games?

Solution:
Total match x
∴ (2/3) × x = 20
∴ x = 30

Total match = 30
Played = 20
remaining = 10

total win = 3/4 of 30 = 22.5 ≅ 23
total losses = 30 - 23 = 7 games
so max losses from the remaining 10 games = 7 - 3 = 4

৭৪০.

A and B started a partnership business investing some amount in the ratio of 3:5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C?

ক
ক) 3 : 5 : 2
খ
খ) 3 : 5 : 5
গ
গ) 6 : 10 : 5
ঘ
ঘ) Data inadequate

সঠিক উত্তর: গ

গ) 6 : 10 : 5

উত্তর

সঠিক উত্তর: গ

গ) 6 : 10 : 5

গ

ব্যাখ্যা

Let initial investment of A is 3x and B is 5x, then C investment is also 5x, but most important to note in this question is the time duration of investment
Like, A invested for 12 months, B invested for 12 months and C invested for 6 months.
A : B : C = (3x x 12) : (5x x 12) : (5x x 6)
= 36 : 60 : 30
= 6 : 10 : 5.

৭৪১.

Coffee A normally costs 100 taka per pound. It is mixed with coffee B, which normally costs 70 taka per pound, to form a mixture which costs 88 taka per pound. If there are 10 pounds of the mix, how many pounds of coffee A are used in the mix?

ক
4
খ
5
গ
6
ঘ
7
ঙ
None of these

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Amount of A in the mixture = 10 × (3/5)
= 6

৭৪২.

If (2x + 3y) : (3x + 5y) = 18 : 29, then what is the value of x : y?

ক
ক) 2 : 3
খ
খ) 3 : 4
গ
গ) 4 : 7
ঘ
ঘ) 9 : 13

সঠিক উত্তর: খ

খ) 3 : 4

উত্তর

সঠিক উত্তর: খ

খ) 3 : 4

খ

ব্যাখ্যা

Question: If (2x + 3y) : (3x + 5y) = 18 : 29, then what is the value of x : y?

Solution:
(2x + 3y) : (3x + 5y) = 18 : 29
⇒ (2x + 3y)/(3x + 5y) = 18/29
⇒ 58x + 87y = 54x +90y
⇒ 4x = 3y
⇒ x/y = 3/4
⇒ x : y = 3 : 4

৭৪৩.

A vessel of capacity 90 litres is fully filled with pure milk. Nine litres of milk is removed from the vessel and replaced with water. Nine litres of the solution thus formed is removed and replaced with water. Find the quantity of pure milk in the final milk solution?

ক
ক) 72
খ
খ) 72.9
গ
গ) 73.8
ঘ
ঘ) 74.7

সঠিক উত্তর: খ

খ) 72.9

উত্তর

সঠিক উত্তর: খ

খ) 72.9

খ

ব্যাখ্যা

Let the initial quantity of milk in vessel be T litres.
Let us say y litres of the mixture is taken out and replaced by water for n times, alternatively.
Quantity of milk finally in the vessel is then given by [(T - y)/T]n × T
For the given problem, T = 90, y = 9 and n = 2.
Hence, quantity of milk finally in the vessel
= [(90 - 9)/90]2 (90) = 72.9 litres.

৭৪৪.

Between two consecutive years my incomes are in the ratio of 2 : 3 and expenses in the ratio 5 : 9. If my income in the second year is Tk. 45000 and my expenses in the first year is Tk. 25000 my total savings for the two years is -

ক
ক) Nil
খ
খ) Tk. 5000
গ
গ) Tk. 10000
ঘ
ঘ) Tk. 15000

সঠিক উত্তর: খ

খ) Tk. 5000

উত্তর

সঠিক উত্তর: খ

খ) Tk. 5000

খ

ব্যাখ্যা

Let,
income in the first year be Tk. x and expenses in the second year be Tk. y
Then, x/45000 = 2/3 and 25000/y = 5/9
⇒ x = (2 × 45000)/3
= 30000
and y (25000 × 9)/5
= 45000.
∴ Total savings for 2 years
= Tk. [(30000 - 25000) + (45000 - 45000)]
= Tk. 5000.

৭৪৫.

In your wallet, there are Tk 1000, Tk 500, and Tk 100 notes in the ratio 3 : 5 : 2, amounting to Tk 22,800. Find the number of each note respectively.

ক
12, 20, 10
খ
22, 10, 17
গ
12, 20, 8
ঘ
10, 18, 27

সঠিক উত্তর: গ

12, 20, 8

উত্তর

সঠিক উত্তর: গ

12, 20, 8

গ

ব্যাখ্যা

Question: In your wallet, there are Tk 1000, Tk 500, and Tk 100 notes in the ratio 3 : 5 : 2, amounting to Tk 22,800. Find the number of each note respectively.

Solution:
Let,
The number of Tk 1000 notes is 3x
The number of Tk 500 notes is 5x
The number of Tk 100 notes is 2x

ATQ,
1000 × 3x + 500 × 5x + 100 × 2x = 22800
⇒ 3000x + 2500x + 200x = 22800
⇒ 5700x = 22800
⇒ x = 22800 / 5700
⇒ x = 4

Number of Tk 1000 note = 3x = 3 × 4 = 12
Number of Tk 500 note = 5x = 5 × 4 = 20
Number of Tk 100 note = 2x = 2 × 4 = 8

Therefore, the number of Tk 1000, Tk 500, and Tk 100 notes are respectively 12, 20, and 8.

৭৪৬.

If A : B = 3 : 4, C : B = 5 : 4, C : D = 10 : 9, then A : B : C : D = ?

ক
6 : 8 : 12 : 9
খ
6 : 8 : 10 : 9
গ
11 : 8 : 10 : 9
ঘ
6 : 8 : 10 : 15

সঠিক উত্তর: খ

6 : 8 : 10 : 9

উত্তর

সঠিক উত্তর: খ

6 : 8 : 10 : 9

খ

ব্যাখ্যা

A : B = 3 : 4 = 6 : 8
B : C = 4 : 5 = 8 : 10
C : D = 10 : 9
A : B : C : D = 6 : 8 : 10 : 9

৭৪৭.

A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

ক
Tk. 8400
খ
Tk. 12500
গ
Tk. 14700
ঘ
Tk. 15000

সঠিক উত্তর: গ

Tk. 14700

উত্তর

সঠিক উত্তর: গ

Tk. 14700

গ

ব্যাখ্যা

Question: A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

Solution:
let, C subscribes x taka
B subscribes x + 5000 taka
A subscribes x + 5000 + 4000
= x + 9000 taka

x + x + 5000 + x + 9000 = 50000
⇒ 3x + 14000 = 50000
⇒ 3x = 36000
⇒ x = 12000 taka

A receives = {(x + 9000)/50000} × 35000
= (21000/50000) × 35000
= 14700 taka

৭৪৮.

In a mixture of 35 liters, milk and water are in the ratio of 3:4. How much milk would be added to the mixture to make the ratio equal?

ক
7 liters
খ
4 liters
গ
6 liters
ঘ
5 liters

সঠিক উত্তর: ঘ

5 liters

উত্তর

সঠিক উত্তর: ঘ

5 liters

ঘ

ব্যাখ্যা

Question: In a mixture of 35 liters, milk and water are in the ratio of 3:4. How much milk would be added to the mixture to make the ratio equal?

Solution:
amount of milk
= (3/7)*35
= 15 litres
amount of water
= (4/7)*35
= 20 litres

the amount of milk to be added is
= (20-15)
= 5 litres.

৭৪৯.

A mixture of 30litre of sprit and water contains 20% water. How much water must be added to the mixture to raise the percentage of water to 25%

ক
2 litre
খ
3 litre
গ
8 litre
ঘ
None

সঠিক উত্তর: ক

2 litre

উত্তর

সঠিক উত্তর: ক

2 litre

ক

ব্যাখ্যা

Question: A mixture of 30 litre of sprit and water contains 20% water. How much water must be added to the mixture to raise the percentage of water to 25%

Solution:
20 লিটার মিশ্রণে পানি আছে = (30 এর 20/100) = 6 লিটার

∴ স্পিরিট আছে = 30 - 6 = 24 লিটার

এখন নতুন মিশ্রণে পানি থাকবে 25 ভাগ এবং স্পিরিট থাকবে 75 ভাগ ।

প্রশ্নমতে,
⇒ 24/(6 + x) = 75/25
⇒ 24/(6 + x) = 3
⇒ 18 + 3x = 24
⇒ 3x = 24 - 18
⇒ 3x = 6
⇒ x = 6/3
∴ x = 2

∴ ২ লিটার পানি মিশ্রণে যোগ করতে হবে।

৭৫০.

Three numbers are in the ratio 1 : 2 : 3 and their H.C.F 14. The smallest number is-

ক
ক) 14
খ
খ) 28
গ
গ) 42
ঘ
ঘ) 84

সঠিক উত্তর: ক

ক) 14

উত্তর

সঠিক উত্তর: ক

ক) 14

ক

ব্যাখ্যা

Given that
The ratio of three numbers 1 : 2 : 3
Let the numbers be x, 2x and 3x.
The HCF in x, 2x and 3x is x

Hence,
x = 14;

The smallest number is 14

৭৫১.

A container contains 40 litres of milk. From this container, 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

ক
ক) 26.34 litres
খ
খ) 27.36 litres
গ
গ) 28 litres
ঘ
ঘ) 29.16 litres

সঠিক উত্তর: ঘ

ঘ) 29.16 litres

উত্তর

সঠিক উত্তর: ঘ

ঘ) 29.16 litres

ঘ

ব্যাখ্যা

Amount of milk left after 3 operations
= [40 {1 - (4/40)}³] litres
= (40 × 9/10 × 9/10 × 9/10) litres
= 29.16 litres.

৭৫২.

If Tk. 945 is allocated into three portions according to the ratio (2/3) : (3/4) : (5/6), what is the amount of the first portion?

ক
174 Tk.
খ
374 Tk.
গ
224 Tk.
ঘ
274 Tk.
ঙ
None

সঠিক উত্তর: ঙ

None

উত্তর

সঠিক উত্তর: ঙ

None

ঙ

ব্যাখ্যা

Question: If Tk. 945 is allocated into three portions according to the ratio (2/3) : (3/4) : (5/6), what is the amount of the first portion?

Solution:
The given ratio = 2/3 : 3/4 : 5/6

Take LCM of denominators 3, 4, 6 = 12

∴ The ratio = 8 : 9 : 10 (Multipy thr ratio with 12)

Sum of parts = 8 + 9 + 10 = 27

The first portion = 945 × (8/27) = 280 Tk

∴ First portion = 280 Tk.

৭৫৩.

The number of boys and girls in a school is in the proportion 7 : 5. When 20 more girls are added and 14 boys leave, the ratio becomes 6 : 5. Determine the total boys.

ক
266
খ
250
গ
310
ঘ
180

সঠিক উত্তর: ক

266

উত্তর

সঠিক উত্তর: ক

266

ক

ব্যাখ্যা

Question: The number of boys and girls in a school is in the proportion 7 : 5. When 20 more girls are added and 14 boys leave, the ratio becomes 6 : 5. Determine the total boys.

Solution:
Given that,
the ratio of the boys and girls is 7 : 5
Let the number of the boys and girls be ,
boys = 7x
girls = 5x

After changing the number of the students,
Number of boys = 7x - 14
Number of girls = 5x + 20

Now the ratio becomes,
⇒ (7x - 14)/(5x + 20) = 6/5
⇒ 35x - 70 = 30x + 120
⇒ 35x - 30x = 120 + 70
⇒ 5x = 190
∴ x = 38

So the number of the boys = 7x = 7 × 38 = 266

৭৫৪.

In a zoo, there are Rabbits and Pigeons. If heads are counted, there are 200 and if legs are counted, there are 580. Find the ratio of number of Rabbits and Pigeons?

ক
ক) 8 : 7
খ
খ) 9 : 11
গ
গ) 7 : 9
ঘ
ঘ) 10 : 9

সঠিক উত্তর: খ

খ) 9 : 11

উত্তর

সঠিক উত্তর: খ

খ) 9 : 11

খ

ব্যাখ্যা

Question: In a zoo, there are Rabbits and Pigeons. If heads are counted, there are 200 and if legs are counted, there are 580. Find the ratio of number of Rabbits and Pigeons?

Solution:
ধরি,
খরগোশ আছে X টি
তাহলে, কবুতর আছে (200 - X) টি

প্রশ্নমতে,
4X + 2(200 - X) = 580
4X + 400 - 2X = 580
2X = 180
X = 90টি

∴ কবুতর আছে = (200 - 90) বা, 110 টি

Rabbits : Pigeons = 90 : 110 = 9 : 11

৭৫৫.

A merchant has 1000 kg of sugar part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The Quantity sold at 18% profit is-

ক
400 kg
খ
560 kg
গ
600 kg
ঘ
640 kg

সঠিক উত্তর: গ

600 kg

উত্তর

সঠিক উত্তর: গ

600 kg

গ

ব্যাখ্যা

Question: A merchant has 1000 kg of sugar part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The Quantity sold at 18% profit is-

Solution:
By the rule of alligation:
Profit of first part Profit of second part

So, ratio of 1st and 2nd parts = 4 : 6 = 2 : 3.
Therefore, Quantity of 2nd kind = (3/5) × 1000 kg = 600 kg.

৭৫৬.

The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio of 7 : 9. If each saves BDT 50 per month, find their monthly income.

ক
ক) 400 and 2100
খ
খ) 400 and 500
গ
গ) 500 and 400
ঘ
ঘ) None of these

সঠিক উত্তর: খ

খ) 400 and 500

উত্তর

সঠিক উত্তর: খ

খ) 400 and 500

খ

ব্যাখ্যা

Question: The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio of 7 : 9. If each saves BDT 50 per month, find their monthly income.

Solution:
Let the monthly income of one person be 4x and that of the other be 5x
Let the monthly expenses of one person be 7y and that of other be 9y

According to the question,
4x - 7y= 50...................(1)
5x - 9y= 50....................(2)

(1) × 9 - (2) × 7
36x - 63y - 35x + 63y = 450 - 350
x = 100

Monthly income of one person
= 4 ×100 = 400

Monthly income of the other person
= 5 × 100 = 500

৭৫৭.

The angles in a triangle are in the ratio 3 : 4 : 5. Work out the size of each angle.

ক
30°, 40° and 50°
খ
22.5°, 30° and 37.5°
গ
60°, 60° and 60°
ঘ
45°, 60° and 75°
ঙ
None of these

সঠিক উত্তর: ঘ

45°, 60° and 75°

উত্তর

সঠিক উত্তর: ঘ

45°, 60° and 75°

ঘ

ব্যাখ্যা

Question: The angles in a triangle are in the ratio 3 : 4 : 5. Work out the size of each angle.

Solution:
The angles in a triangle add up to 180°.
Therefore 180° is the whole and we need to divide 180° in the ratio 3 : 4 : 5.

The total number of shares is 3 + 4 + 5 = 12
Each share is worth 180 ÷ 12 = 15°

3 shares is 3 × 15 = 45°
4 shares is 4 × 15 = 60°
5 shares is 5 × 15 = 75°

৭৫৮.

In your wallet, there are Tk 50, Tk 20 and Tk 10. notes in the ratio 5 : 9 : 4, amounting to Tk. 1880. Find the number of each note respectively.

ক
20, 36 and 16
খ
15, 27 and 12
গ
10, 18 and 8
ঘ
25, 45 and 20
ঙ
None

সঠিক উত্তর: ক

20, 36 and 16

উত্তর

সঠিক উত্তর: ক

20, 36 and 16

ক

ব্যাখ্যা

Question: In your wallet, there are Tk 50, Tk 20 and Tk 10. notes in the ratio 5:9:4, amounting to Tk 1880. Find the number of each note respectively.

Solution:
In your wallet, there are Tk. 50, Tk. 20 and Tk. 10 notes in the ratio = 5 : 9 : 4

Let,
Number of Tk. 50 note is 5x
Number of Tk. 20 note is 9x
Number of Tk. 10 note is 4x

ATQ,
(50 × 5x) + (20 × 9x) + (10 × 4x) = 1880
⇒ 250x + 180x + 40x = 1880
⇒ 470x = 1880
⇒ x = 1880/470
∴ x = 4

∴ Number of Tk. 50 note is 5x = 5 × 4 = 20
∴ Number of Tk. 20 note is 9x = 9 × 4 = 36
∴ Number of Tk. 10 note is 4x = 4 × 4 = 16

৭৫৯.

Father divides his property equally between his two sons and 150000 Tk left. If the remaining part is 7.5% of actual property than every brother owns Tk. -

ক
ক) 1850000 Tk
খ
খ) 925000 Tk
গ
গ) 750000 Tk
ঘ
ঘ) 2000000 Tk

সঠিক উত্তর: খ

খ) 925000 Tk

উত্তর

সঠিক উত্তর: খ

খ) 925000 Tk

খ

ব্যাখ্যা

Question: Father divides his property equally between his two sons and 150000 Tk left. If the remaining part is 7.5% of actual property than every brother owns Tk. -

Solution:
৭.৫% = ১৫০০০০ টাকা
∴ সম্পূর্ণ সম্পত্তি = ২০০০০০০ টাকা
২ পুত্র পেল = (২০০০০০০ -১৫০০০০) = ১৮৫০০০০

∴ প্রত্যেক পুত্র পেল = ১৮৫০০০০/২ = ৯২৫০০০ টাকা

৭৬০.

B invested three-fourths of what C invested. A invested 20% more than B. What is C's investment, if total investment is Tk. 3816

ক
ক) Tk. 1300
খ
খ) Tk. 1440
গ
গ) Tk. 1650
ঘ
ঘ) Tk. 1800

সঠিক উত্তর: খ

খ) Tk. 1440

উত্তর

সঠিক উত্তর: খ

খ) Tk. 1440

খ

ব্যাখ্যা

Let investment of C be Tk. 100
So the investment of B = Three-fourths of C = 3/4 of Tk. 100 = Tk. 75
Investment of A = 20% more than B = 20% more than Tk. 75 = Tk. 90
Ratio of investment of A, B, and C = 90 : 75 : 100 = 18 : 15 : 20
Investment of C = 20/(18 + 15 + 20) × 3816
= Tk. 1440.

৭৬১.

How much water should be added to 50 kg of pure milk to gain 12% profit when selling the mixture at the price of pure milk?

ক
4 liters
খ
5 liters
গ
6 liters
ঘ
10 liters

সঠিক উত্তর: গ

6 liters

উত্তর

সঠিক উত্তর: গ

6 liters

গ

ব্যাখ্যা

Question: How much water should be added to 50 liters of pure milk to gain extra 12% profit when selling the mixture at the price of pure milk?

Solution:
Let’s assume,
Price of pure milk per liter = 100 Taka
So, the price of 50 liters of pure milk = 100 × 50 = 5000 Taka

Now assume,
Water added to the milk = x liters
Then the total quantity of the milk-water mixture = (50 + x) liters

Since the mixture is sold at the price of pure milk,
The selling price of (50 + x) liters = 100(50 + x) Taka

According to the question,
100(50 + x) = 5000 + 5000 এর 12%
⇒ 5000 + 100x = 5000 + {5000 × (12/100)}
⇒ 5000 + 100x = 5000 + 600
⇒ 5000 - 5000 + 100x = 600
⇒ 100x = 600
⇒ x = 600/100
⇒ x = 6

∴ Amount of water to be added = 6 liters

Shortcut:
Amount of water to be added = (Profit%/100) × Quantity of pure milk
= (12/100) × 50
= 6 liters

৭৬২.

The ratio of the number of boys to that of girls in a group becomes 2 : 1 when 15 girls leave. But afterwards when 45 boys also leave, the ratio becomes 1 : 5 . Originally the number of girls in the group was

ক
ক) 30
খ
খ) 35
গ
গ) 40
ঘ
ঘ) 45

সঠিক উত্তর: গ

গ) 40

উত্তর

সঠিক উত্তর: গ

গ) 40

গ

ব্যাখ্যা

Let the number of boys and girls be x and y respectively.
Now
x/(y - 15) = 2/1
2y - 30 = x
x - 2y = - 30
x = 2y - 30...........(1)

Again
(x - 45)/(y - 15) = 1/5
5(x - 45) = y - 15
5x - 225 = y - 15
5x - y = 225 - 15
5(2y - 30) - y = 210
10y - 150 - y = 210
10y - y = 210 + 150
9y = 360
y = 360/9
y = 40

৭৬৩.

A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

ক
50 liters
খ
25 liters
গ
15 liters
ঘ
10 liters

সঠিক উত্তর: ঘ

10 liters

উত্তর

সঠিক উত্তর: ঘ

10 liters

ঘ

ব্যাখ্যা

Question: A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

Solution:
Number of liters of water in 125 liters of the mixture = 20% of 150 = 1/5 of 150 = 30 liters

Let us Assume that another 'P' liters of water are added to the mixture to make water 25% of the new mixture.
So, the total amount of water becomes (30 + P) and the total volume of the mixture becomes (150 + P)

Thus, (30 + P) = 25% of (150 + P)
Solving, we get P = 10 liters

৭৬৪.

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is-

ক
3 : 7
খ
1 : 5
গ
1 : 7
ঘ
1 : 3

সঠিক উত্তর: গ

1 : 7

উত্তর

সঠিক উত্তর: গ

1 : 7

গ

ব্যাখ্যা

Question: A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is-

Solution:
ধরি,
পুরাতন মিশ্রণে লেমন জুস এবং চিনির সিরাপ আছে সম পরিমাণ।

নতুন যে মিশ্রণ তৈরি হবে সেখানে পুরাতন মিশ্রণ থাকবে y/৪ ভাগ এবং বাকি ৩y/৪ ভাগ হবে নতুন চিনির সিরাপ।
এখানে,
যে y/৪ ভাগ পুরাতন মিশ্রণ থাকে তাতে লেবুর জুস থাকে y/৪ এর ১/২ অংশ = y/৮ এবং চিনির সিরাপ থাকে y/৪ এর ১/২ অংশ = y/৮

∴ নতুন মিশ্রণে লেবুর জুস থাকবে y/৮
∴ নতুন মিশ্রণে চিনির সিরাপ থাকবে ৩y/৪ + y/৮ = (৬y + y)/৮ = ৭y/৮

∴ নতুন মিশ্রণে লেবুর জুস ও চিনির সিরাপের অনুপাত = y/৮ : ৭y/৮
= y : ৭y
= ১ : ৭

৭৬৫.

X, Y, and Z share Tk. 1680 in such a way that X has 3.5 times as much as Y, and Y has 2 times as much as Z. How much money does Z receive?

ক
140 Tk.
খ
320 Tk.
গ
560 Tk.
ঘ
980 Tk.
ঙ
None

সঠিক উত্তর: ঙ

None

উত্তর

সঠিক উত্তর: ঙ

None

ঙ

ব্যাখ্যা

Question: X, Y, and Z share Tk. 1680 in such a way that X has 3.5 times as much as Y, and Y has 2 times as much as Z. How much money does Z receive?

Solution:
Given,
X = 3.5Y
Y = 2Z

∴ X = 3.5 × 2Z = 7Z

So, the ratio of X, Y, Z = 7 : 2 : 1

∴ Z receive = [1680 × (1/10)] Tk.
= 168 Tk.

৭৬৬.

If A : B = 3 : 4, B : C = 5 : 6, and C : D = 7: 9 then A : D is equal to-

ক
72 : 35
খ
35 : 72
গ
25 : 72
ঘ
72 : 25

সঠিক উত্তর: খ

35 : 72

উত্তর

সঠিক উত্তর: খ

35 : 72

খ

ব্যাখ্যা

Question: If A : B = 3 : 4, B : C = 5 : 6, and C : D = 7: 9 then A : D is equal to-

Solution:
Given that,
A : B = 3 : 4, B : C = 5 : 6, and C : D = 7: 9
Then,
A/D = (A/B) × (B/C) × (C/D)
= (3/4) × (5/6) × (7/9)
= 35/72
= 35 : 72
∴ A : D = 35 : 72

৭৬৭.

Rice and wheat are in a mixture in the ratio 5 : 3. If 16 kg wheat is added to this mixture, the ratio of rice to wheat changes to 5 : 7. How much wheat is in new mixture?

ক
28 kg
খ
36 kg
গ
42 kg
ঘ
24 kg

সঠিক উত্তর: ক

28 kg

উত্তর

সঠিক উত্তর: ক

28 kg

ক

ব্যাখ্যা

Question: Rice and wheat are in a mixture in the ratio 5 : 3. If 16 kg wheat is added to this mixture, the ratio of rice to wheat changes to 5 : 7. How much wheat is in new mixture?

Solution:
Let the initial amount of rice = 5x kg
Initial amount of wheat = 3x kg

After adding 16 kg wheat,
Rice remains = 5x kg
And wheat becomes = 3x + 16 kg

New ratio of rice : wheat = 5 : 7
So,
⇒ 5x/(3x + 16) = 5/7
⇒ 7 × 5x = 5 × (3x + 16)
⇒ 35x = 15x + 80
⇒ 35x - 15x = 80
⇒ 20x = 80
⇒ x = 80/20
∴ x = 4

Now, wheat in the new mixture = 3x + 16
= 3 × 4 + 16
= 12 + 16
= 28 kg

So, 28 kg of wheat is in the new mixture.

৭৬৮.

A jar contains a mixture of oil and water in the ratio 25 : 5. 30 liters of the mixture was taken out and 15 liters of water was added to it. If water was 20% in the resultant mixture, what was the initial quantity of the mixture (in liters) in the jar?

ক
ক) 390 liters
খ
খ) 450 liters
গ
গ) 500 liters
ঘ
ঘ) 300 liters

সঠিক উত্তর: ক

ক) 390 liters

উত্তর

সঠিক উত্তর: ক

ক) 390 liters

ক

ব্যাখ্যা

Question: A jar contains a mixture of oil and water in the ratio 25 : 5. 30 liters of the mixture was taken out and 15 liters of water was added to it. If water was 20% in the resultant mixture, what was the initial quantity of the mixture (in liters) in the jar?

Solution:
ধরি
মিশ্রণে তেলের পরিমাণ = 25x লিটার
মিশ্রণে পানির পরিমাণ = 5x লিটার
মিশ্রণে মোট পরিমাণ = 25x + 5x = 30x

30 লিটার মিশ্রণে পানির পরিমাণ = 30 × 5/30 = 5 লিটার

প্রশ্নমতে,
5x - 5 + 15 = (30x - 30 + 15) × (20/100)
5x + 10 = (30x - 15) × (1/5)
5x + 10 = 3(2x - 1)
5x + 10 = 6x - 3
x = 13
মিশ্রণের মোট পরিমাণ = 30x = 30 x 13 = 390 লিটার

৭৬৯.

The sum of three number is 330. If the ratio of the first to the second is 3 : 2 and then that of the second to third is 5 : 4, what is the second number?

ক
10
খ
25
গ
50
ঘ
100
ঙ
150

সঠিক উত্তর: ঘ

100

উত্তর

সঠিক উত্তর: ঘ

100

ঘ

ব্যাখ্যা

Question: The sum of three number is 330. If the ratio of the first to the second is 3 : 2 and then that of the second to third is 5 : 4, what is the second number?

Solution:
Let the three numbers be a, b, c.

Here,
a : b = 3 : 2
⇒ a/b = 3/2
⇒ a = 3b/2

And,
b : c = 5 : 4
⇒ b/c = 5/4
⇒ c = 4b/5

Given,
Sum of the three numbers, a + b + c = 330
⇒ (3b/2) + b + (4b/5) = 330
⇒ (15b + 10b + 8b)/10 = 330
⇒ 33b = 3300
⇒ b = 3300/33
⇒ b = 100

∴ The second number is 100.

৭৭০.

The price of a variety of a commodity is Tk. 7 per kg and that of another is Tk. 12 per kg. Find the ratio in which two varieties should be mixed so that the price of the mixture is Tk. 10 per kg.

ক
3 : 4
খ
4 : 3
গ
2 : 3
ঘ
5 : 4

সঠিক উত্তর: গ

2 : 3

উত্তর

সঠিক উত্তর: গ

2 : 3

গ

ব্যাখ্যা

Question: The price of a variety of a commodity is Tk. 7 per kg and that of another is Tk. 12 per kg. Find the ratio in which two varieties should be mixed so that the price of the mixture is Tk. 10 per kg.

Solution:
Let the mixed amount of the first and second commodity is A and B

∴ A : B = (12-10) : (10-7)
A : B = 2 : 3

৭৭১.

25% of A's income is equal to 35% of B's income. The ratio of the incomes of A and B is-

ক
ক) 5 : 7
খ
খ) 7 : 5
গ
গ) 13 : 15
ঘ
ঘ) 15 : 13
ঙ
ঙ) 7 : 13

সঠিক উত্তর: খ

খ) 7 : 5

উত্তর

সঠিক উত্তর: খ

খ) 7 : 5

খ

ব্যাখ্যা

25% of A = 35% of B
⇒ (25/100)A = (35/100)B
⇒ A/4 = 7B/20
⇒ A/B = (7/20)×4 = 7/5
⇒ A:B = 7:5

৭৭২.

The ratio of the present ages of Riya and her mother is 3 : 7. The mother’s age at the time of Riya’s birth was 20 years. Find the mother’s present age.

ক
35 years
খ
30 years
গ
38 years
ঘ
None of these

সঠিক উত্তর: ক

35 years

উত্তর

সঠিক উত্তর: ক

35 years

ক

ব্যাখ্যা

Question: The ratio of the present ages of Riya and her mother is 3 : 7. The mother’s age at the time of Riya’s birth was 20 years. Find the mother’s present age.

Solution:
Present ratio is 7 : 3.
Let actual ages are 7x and 3x.

∴ 7x - 3x = 20
⇒ 4x = 20
∴ x = 5

Hence the mother’s present age 7 × 5 = 35 years

৭৭৩.

Mr. Rifat divides Tk. 1573 such that 4 times the 1st share, thrice the 2nd share and twice the third share amount to the same. Then the value of the 2nd share is-

ক
Tk. 484
খ
Tk. 242
গ
Tk. 363
ঘ
Tk. 726

সঠিক উত্তর: ক

Tk. 484

উত্তর

সঠিক উত্তর: ক

Tk. 484

ক

ব্যাখ্যা

Question: Mr. Rifat divides Tk. 1573 such that 4 times the 1st share, thrice the 2nd share and twice the third share amount to the same. Then the value of the 2nd share is-

Solution:
Total amount = Tk. 1573
Let,
The shares are A, B and C
Here,
4A = 3B = 2C.
Now,
4A = 3B
∴ B = 4A/3

4A = 2C
∴ C = 2A

∴ A : B : C = A : 4A/3 : 2A
= 1 : 4/3 : 2
= 3 : 4 : 6

The value of the 2nd share = (4/13) × 1573 = Tk. 484

৭৭৪.

What is the ratio of 3/4 to the product 4(3/4)?

ক
ক) 1/3
খ
খ) 1/4
গ
গ) 4/9
ঘ
ঘ) 9/4

সঠিক উত্তর: খ

খ) 1/4

উত্তর

সঠিক উত্তর: খ

খ) 1/4

খ

ব্যাখ্যা

Question: What is the ratio of 3/4 to the product 4(3/4)?

Solution:
ratio = (3/4)/{4(3/4)}
= (3/4)/{(4 × 3)/4}
= (3/4)/(3/1)
= (3/4) × (1/3)
= 1/4

৭৭৫.

Ratul and Sajib were classmates. Their earnings now are in the ratio 5:6. The ratio of their expenses is 7:9. Sajib saves Tk. 3,000 every month while Ratul saves Tk. 1000/- more than Somesh. Find the total earnings and expenses of each of them.

ক
ক) Ratul - 25000, 21000; Sajib - 30000, 27000
খ
খ) Ratul - 30000, 27000; Sajib - 36000, 32000
গ
গ) Ratul - 36000, 32000; Sajib - 30000, 27000
ঘ
ঘ) None of the above

সঠিক উত্তর: ক

ক) Ratul - 25000, 21000; Sajib - 30000, 27000

উত্তর

সঠিক উত্তর: ক

ক) Ratul - 25000, 21000; Sajib - 30000, 27000

ক

ব্যাখ্যা

Income ratio = Ratul : Sajib = 5 : 6 = 5/6
Common factor helps in finding actual values easily
So, take 'A' as a common factor.
Income of Ratul = 5A; Income of Sajib = 6A
(Expenses of Ratul)/(Expenses of Sajib) = (Ratul Income - Ratul Saving)/(Sajib Income -Sajib Saving) = 7/9
Since Ratul save, Tk. 1000/- more than Sajib, Ratul's savings = Tk. 4000/-
5A - 4000/6A - 3000 = 7/9
∴ 9(5A-4000) = 7(6A-3000)
∴ A = 5000
Income of Ratul = 5A = 25000 ; Income of Sajib = 6A = 30000
Spending of Ratul =25000 - 4000 = 21000
Spending of Sajib = 30000 - 3000 = 27000.

৭৭৬.

A, B and C play cricket. The ratio of A’s runs to B’s runs and B’s runs to C’s is 3 ∶ 2. They make altogether 342 runs. How many runs did A make?

ক
ক) 108
খ
খ) 72
গ
গ) 162
ঘ
ঘ) 99

সঠিক উত্তর: গ

গ) 162

উত্তর

সঠিক উত্তর: গ

গ) 162

গ

ব্যাখ্যা

A + B + C = 342

A : B = 3 : 2 = 9 : 6
B : C = 3 : 2 = 6 : 4
A : B : C = 9 : 6 : 4

∴ Runs made by A = 9/19 × 342 = 162

৭৭৭.

If 40% of a number is equal to two - third of another number, what is the ratio of first number to the second number?

ক
2 : 5
খ
5 : 4
গ
5 : 3
ঘ
7 : 3

সঠিক উত্তর: গ

5 : 3

উত্তর

সঠিক উত্তর: গ

5 : 3

গ

ব্যাখ্যা

Question: If 40% of a number is equal to two - third of another number, what is the ratio of first number to the second number?

Solution:
Suppose, 40% of A = 2B/3
⇒ 2A/5 = 2B/3
⇒ A/B = 5/3
⇒ A : B = 5 : 3

৭৭৮.

If (7x - 3y) : (x - 3y) = 5 : 11, find the value of x/y.

ক
1/2
খ
2/3
গ
3/4
ঘ
1/4

সঠিক উত্তর: ঘ

1/4

উত্তর

সঠিক উত্তর: ঘ

1/4

ঘ

ব্যাখ্যা

Question: If (7x - 3y) : (x - 3y) = 5 : 11, find the value of x/y.

Solution:
Given that,
(7x - 3y) : (x - 3y) = 5 : 11
⇒ (7x - 3y)/(x - 3y) = 5/11
⇒ 77x - 33y = 5x - 15y
⇒ 77x - 5x = 33y - 15y
⇒ 72x = 18y
⇒ x/y = 18/72
∴ x/y = 1/4

৭৭৯.

1050 tk are divided among P, Q and R. The share of Q is 2/5 of the combined share of P and R. Q gets-

ক
ক) 750 tk
খ
খ) 300 tk
গ
গ) 350 tk
ঘ
ঘ) 900 tk

সঠিক উত্তর: খ

খ) 300 tk

উত্তর

সঠিক উত্তর: খ

খ) 300 tk

খ

ব্যাখ্যা

Question: 1050 tk are divided among P, Q and R. The share of Q is 2/5 of the combined share of P and R. Q gets-

Solution:
let, combined share of P and R is x
share of Q is 2x/5

x + 2x/5 = 1050
⇒ (5x + 2x)/5 = 1050
⇒ 7x/5 = 1050
⇒ x = 1050 × 5/7
∴ x = 750 tk

Q gets - 750 × 2/5
= 300 tk

৭৮০.

If y : x = 1 : 5 and 2x + y = 22, then what is the value of y?

ক
0
খ
1
গ
2
ঘ
5
ঙ
10

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If y : x = 1 : 5 and 2x + y = 22, then what is the value of y?

Solution:
Given,
y : x = 1 : 5
⇒ y/x = 1/5
∴ y = x/5 ..........(1)

and,
2x + y = 22
⇒ 2x + (x/5) = 22
⇒ (10x + x)/5 = 22
⇒ 11x = 22 × 5
⇒ 11x = 110
⇒ x = 110/11
∴ x = 10

From equation (1),
y = 10/5 = 2

৭৮১.

If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then the ratio of the distances traveled by them in one hour is

ক
5 : 7
খ
5 : 8
গ
3 : 8
ঘ
None of these

সঠিক উত্তর: খ

5 : 8

উত্তর

সঠিক উত্তর: খ

5 : 8

খ

ব্যাখ্যা

Question: If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then the ratio of the distances traveled by them in one hour is

Solution:
Distance traveled by bus in one hour= 160 / 4
= 40

Distance traveled by bus in one hour= 320 / 5
= 64

Ratio = 40 : 64
= 5 : 8

৭৮২.

A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

ক
12 liters
খ
16 liters
গ
11 liters
ঘ
10 liters

সঠিক উত্তর: খ

16 liters

উত্তর

সঠিক উত্তর: খ

16 liters

খ

ব্যাখ্যা

Question: A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

Solution:
ধরি, মিশ্রণের প্রাথমিক পরিমাণ = 5x লিটার

A এর পরিমাণ = 4x লিটার
B এর পরিমাণ = x লিটার

∴ 10 লিটার মিশ্রণ তুলে নেওয়ার পর,
A এর পরিমাণ = 4x - (4/5) × 10 = 4x - 8 লিটার
B এর পরিমাণ = x - (1/5) × 10 = x - 2 লিটার

আবার,
B তে 10 লিটার যোগ করার পর,
B এর পরিমাণ = x - 2 + 10 = x + 8 লিটার

∴ প্রদত্ত অনুপাত,
⇒ (4x - 8)/(x + 8) = 2/3
⇒ 12x - 24 = 2x + 16
⇒ 10x = 16 + 24
⇒ x = 40/10
⇒ x = 4

∴ A এর পরিমাণ = 4 × 4 = 16 লিটার

৭৮৩.

A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

ক
80 liters
খ
98 liters
গ
92liters
ঘ
96 liters
ঙ
None of these

সঠিক উত্তর: ঘ

96 liters

উত্তর

সঠিক উত্তর: ঘ

96 liters

ঘ

ব্যাখ্যা

Question: A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

Solution:
Let,
Original Quantity of P = 5x
Original Quantity of Q = 3x

The quantity of P and Q in 16 liters of the mixture,
Quantity of P = (16 × 5x)/8x = 10 liters
Quantity of Q = (16 × 3x)/8x = 6 liters

Now,
5 liters of P poured in so the Quantity of P will be = 5x - 10 + 5 liters
= 5x - 5 liters

ATQ,
(5x - 5)/(3x - 6) = 11/6
⇒ 6(5x - 5) = 11(3x - 6)
⇒ 30x - 30 = 33x - 66
⇒ 3x = 36
∴ x = 12

So, total mixture originally = 8x
= (8 × 12) = 96 liters

৭৮৪.

A company blends two varieties of tea from two different tea gardens, one variety costing Tk. 30 per kg and the other Tk. 20 per kg in the ratio 4 : 3. He sells the blended tea at Tk. 27 per kg. Find the profit or loss percent.

ক
ক) 5%
খ
খ) 7%
গ
গ) 8%
ঘ
ঘ) 10%

সঠিক উত্তর: ক

ক) 5%

উত্তর

সঠিক উত্তর: ক

ক) 5%

ক

ব্যাখ্যা

Question: A company blends two varieties of tea from two different tea gardens, one variety costing Tk. 30 per kg and the other Tk. 20 per kg in the ratio 4 : 3. He sells the blended tea at Tk. 27 per kg. Find the profit or loss percent.

Solution:
Let 4 kg of the first variety be mixed with 3 kg of second variety.
Then total cost price of 7 kg of tea = (30 × 4) + (20 × 3) = Tk. 180
Selling price of 7 kg of tea = (27 × 7) = Tk. 189

∴ Profit = Tk. (189 - 180) = Tk. 9

∴ Percentage of profit = (9 × 100)/180 = 5%

৭৮৫.

Given that an office opens at 9 a.m. and closes at 5:30 p.m., with a 17-minute lunch break, what is the proportion of the lunch break to the total workday?

ক
1 : 15
খ
1 : 30
গ
1 : 20
ঘ
1 : 25
ঙ
None of the above

সঠিক উত্তর: খ

1 : 30

উত্তর

সঠিক উত্তর: খ

1 : 30

খ

ব্যাখ্যা

Question: Given that an office opens at 9 a.m. and closes at 5:30 p.m., with a 17-minute lunch break, what is the proportion of the lunch break to the total workday?

Solution:
The ratio of lunch breaks to the total period in the office
= 17/{(8 × 60) + 30}
= 17/510
= 1/30
= 1 : 30

৭৮৬.

Tom, Dick and Harry went for lunch to a restaurant. Tom had $100 with him, Dick had $60 and Harry had $40. They got a bill for $104 and decided to give a tip of $16. They further decided to share the total expenses in the ratio of the amounts of money each carried. The amount of money which Tom paid more than what Harry paid is-

ক
ক) 120
খ
খ) 200
গ
গ) 60
ঘ
ঘ) 24
ঙ
ঙ) 36

সঠিক উত্তর: ঙ

ঙ) 36

উত্তর

সঠিক উত্তর: ঙ

ঙ) 36

ঙ

ব্যাখ্যা

Question: Tom, Dick and Harry went for lunch to a restaurant. Tom had $100 with him, Dick had $60 and Harry had $40. They got a bill for $104 and decided to give a tip of $16. They further decided to share the total expenses in the ratio of the amounts of money each carried. The amount of money which Tom paid more than what Harry paid is-

Solution:
Tom : Dick : Harry = 100 : 60 : 40
= 10 : 6 : 4
= 5 : 3 : 2

Total bill = 104 + 16
= 120

Tom paid = 120 × (5/10) = 60
Harry paid = 120 × (2/10) = 24

Tom paid ( 60 - 24) = 36 more than Harry paid.

৭৮৭.

In a class, the number of girls is 5/6 of the number of boys. The total students of the class is 121. If 5 more girls join the class, what will be the new ratio of girls to boys?

ক
11 : 10
খ
5 : 6
গ
7 : 5
ঘ
10 : 11

সঠিক উত্তর: ঘ

10 : 11

উত্তর

সঠিক উত্তর: ঘ

10 : 11

ঘ

ব্যাখ্যা

Question: In a class, the number of girls is 5/6 of the number of boys. The total students of the class is 121. If 5 more girls join the class, what will be the new ratio of girls to boys?

Solution:
Let the number of boys be x
Then, number of girls = 5/6 of x = 5x/6

ATC,
x + (5x/6) = 121
⇒ (6x + 5x)/6 = 121
⇒ 6x + 5x = 6 × 121
⇒ 11x = 6 × 121
⇒ x = (6 × 121)/11
∴ x = 66

So, number of boys = x = 66
number of girls = 5/6 of x = 5/6 of 66 = 55

Now, 5 more girls join,
∴ New number of girls = 55 + 5 = 60

∴ ratio (girls : boys) = 60/66
= 10/11
= 10 : 11

৭৮৮.

The ratio of A's and B's salaries is 2 : 3 respectively. When A's salary is increased by 10%, it becomes Tk 13200. What is the salary of B?

ক
ক) 15000 Tk
খ
খ) 16000 Tk
গ
গ) 18000 Tk
ঘ
ঘ) 21000 Tk

সঠিক উত্তর: গ

গ) 18000 Tk

উত্তর

সঠিক উত্তর: গ

গ) 18000 Tk

গ

ব্যাখ্যা

Question: The ratio of A's and B's salaries is 2 : 3 respectively. When A's salary is increased by 10%, it becomes Tk 13200. What is the salary of B?

Solution:
Let the salaries of A and B be Tk 2x and 3x respectively.

Then,
110% of 2x = 13200
⇒ (110/100) × 2x = 13200
⇒ x = (13200 × 100)/(110 × 2)
⇒ x = 6000

So, salary of B is = 3 × 6000 = 18000 Tk

৭৮৯.

A mixture contains copper, zinc, and tin in the ratio 2 : 3 : 5. If the total weight of the mixture is 50 grams, how many grams of tin are there in the mixture?

ক
20 grams
খ
15 grams
গ
10 grams
ঘ
25 grams

সঠিক উত্তর: ঘ

25 grams

উত্তর

সঠিক উত্তর: ঘ

25 grams

ঘ

ব্যাখ্যা

Question: A mixture contains copper, zinc, and tin in the ratio 2 : 3 : 5. If the total weight of the mixture is 50 grams, how many grams of tin are there in the mixture?

Solution:
Given,
Copper : Zinc : Tin = 2 : 3 : 5

Total parts = 2 + 3 + 5 = 10

Weight of tin = (5/10) × 50 = 25 grams

৭৯০.

A mixture of 20kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?

ক
4
খ
8
গ
12
ঘ
16
ঙ
None of the above

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: A mixture of 20kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?

Solution:
In 1st mixture, water = 20 of 10%
= 20 of 10/100
= 2 kg

So, Spirit = 20 - 2
= 18 kg

Let,
The volume of water has to be mixed x litre

ATQ,
18/(2 + x) = 75/25
⇒ 18/(2 + x) = 3/1
⇒ 3(2 + x) = 18
⇒ 6 + 3x = 18
⇒ 3x = 18 - 6
⇒ 3x =12
⇒ x = 4

৭৯১.

একটি ব্যবসায়ে A, B এবং C তিনজন অংশীদার। তারা সবাই মিলে মোট ১৪০০০ টাকা বিনিয়োগ করলো। বছর শেষে A ৩৩৭.৫০ টাকা, B ১১২৫ টাকা এবং C ৬৩৭.৫ টাকা মুনাফা পেলো। B এবং A এর বিনিয়োগের পার্থক্য কত?

ক
৪০৫৬ টাকা
খ
৪৮৯০ টাকা
গ
৫০৩৬ টাকা
ঘ
৫২৫০ টাকা
ঙ
৬৪২৮ টাকা

সঠিক উত্তর: ঘ

৫২৫০ টাকা

উত্তর

সঠিক উত্তর: ঘ

৫২৫০ টাকা

ঘ

ব্যাখ্যা

প্রশ্ন: একটি ব্যবসায়ে A, B এবং C তিনজন অংশীদার। তারা সবাই মিলে মোট ১৪০০০ টাকা বিনিয়োগ করলো। বছর শেষে A ৩৩৭.৫০ টাকা, B ১১২৫ টাকা এবং C ৬৩৭.৫ টাকা মুনাফা পেলো। B এবং A এর বিনিয়োগের পার্থক্য কত?

সমাধান:
A, B এবং C এর বিনিয়োগের অনুপাত = A, B এবং C এর লাভের অনুপাত
= ৩৩৭.৫০ : ১১২৫ : ৬৩৭.৫
= ৯ : ৩০ : ১৭

∴ যোগফল = (৯ + ৩০ + ১৭) = ৫৬

এখন,
A এর বিনিয়োগ = {১৪০০০ × (৯/৫৬)} = ২২৫০ টাকা
B এর বিনিয়োগ = {১৪০০০ × (৩০/৫৬)} = ৭৫০০ টাকা

∴ A এবং B এর বিনিয়োগের পার্থক্য = (৭৫০০ - ২২৫০) টাকা
= ৫২৫০ টাকা

৭৯২.

Tk. 180 is to be divided among 66 persons (men and women). The ratio of the total amount of money received by men and women is 5 : 4. But the ratio of the money received by each man and woman is 3 : 2. The number of men is:

ক
22
খ
26
গ
30
ঘ
36

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Tk. 180 is to be divided among 66 persons (men and women). The ratio of the total amount of money received by men and women is 5 : 4. But the ratio of the money received by each man and woman is 3 : 2. The number of men is:

Solution:
Let, amount received by men = 5x.
and amount received by women = 4x

ATQ,
5x + 4x = 180
⇒ 9x = 180
⇒ x = 20

Amount received by men = TK.100
Amount received by women = TK.80

Suppose, the number of men be y and that of women be = (66 - y).
∴ (100/y)/{80/(66 - y)} = 3/2
⇒ (100/y) × {(66 - y)/80} = 3/2
⇒ {5(66 - y)}/4y = 3/2
⇒ 660 - 10y = 12y
⇒ 22y = 660
⇒ y = 30

৭৯৩.

Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

ক
35%
খ
45%
গ
50%
ঘ
60%

সঠিক উত্তর: গ

50%

উত্তর

সঠিক উত্তর: গ

50%

গ

ব্যাখ্যা

Question: Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

Solution:
ধরি, মিশ্রণে Detergent A এর অংশ x%
∴ Detergent B এর অংশ হবে (100 - x%)

এখানে,
ডিটারজেন্ট A থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (x% এর 30%)।
ডিটারজেন্ট B থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (100 - x)% এর 50%।
চূড়ান্ত মিশ্রণে ব্লিচের মোট পরিমাণ হলো (100% এর 40%)।

প্রশ্নমতে,
0.30x + 0.50 × (100 - x) = 0.40 × 100
⇒ 0.30x + 50 - 0.50x = 40
⇒ - 0.20x = - 10
⇒ x = - 10/(- 0.20)
∴ x = 50

অর্থাৎ, মিশ্রণের 50% হবে Detergent A।

৭৯৪.

The ratio of men : women working in a company is 3 : 5. What proportion of the employees are women?

ক
3/5
খ
3/8
গ
5/8
ঘ
5/3
ঙ
None of these

সঠিক উত্তর: গ

5/8

উত্তর

সঠিক উত্তর: গ

5/8

গ

ব্যাখ্যা

Question: The ratio of men : women working in a company is 3 : 5. What proportion of the employees are women?

Solution:
In this company, the ratio of men : women is 3 : 5
so for every 3 men there are 5 women.
This means that for every 8 employees, 5 of them are women.
Therefore 5/8 of the employees are women.

৭৯৫.

Two alloys contain zinc and copper in the ratio of 2 : 1 and 4 : 1.In what ratio the two alloys should be added together to get a new alloy having zinc and copper in the ratio of 3 : 1?

ক
ক) 3 : 5
খ
খ) 5 : 9
গ
গ) 7 : 5
ঘ
ঘ) None of these

সঠিক উত্তর: ক

ক) 3 : 5

উত্তর

সঠিক উত্তর: ক

ক) 3 : 5

ক

ব্যাখ্যা

Zinc in first allow = 2/3 units;
Zinc in second alloy = 4/5 units.
copper in first alloy = 1/3 units;
copper in second alloy = 1/5 units.
Let the first and second alloys be mixed in the ratio 1 : y.
Then, {(2/3) + (4y/5)}/{(1/3 + (y/5)) = 3/1
⇒ 10 + 12y = 3 (5 + 3y)
⇒ 10 + 12y = 15 + 9y
⇒ 3y = 5
⇒ y = 5/3.
∴ Required ratio = 1 : 5/3
= 3:5

৭৯৬.

P and O invested in a business. The profit earned was divided in the ratio 2: 3. If P invested Tk. 40,000 the amount invested by Q is

ক
ক) Tk. 40,000
খ
খ) Tk. 50,000
গ
গ) Tk. 70,000
ঘ
ঘ) Tk. 60,000

সঠিক উত্তর: ঘ

ঘ) Tk. 60,000

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 60,000

ঘ

ব্যাখ্যা

ধরি, P বিনিয়োগ করেছিল 2x টাকা এবং Q বিনিয়োগ করেছিল 3x টাকা
প্রশ্নমতে, 2x = 40,000
∴ x = 20,000
অতএব, Q বিনিয়োগ করেছিল = 3x = 3 × 20,000
= 60,000 টাকা।

৭৯৭.

Two numbers P and Q are such that the sum of 10% of P and 15% of Q is three-fourths of the sum of 20% of P and 18% of Q. Find the ratio of P : Q.

ক
10 : 3
খ
3 : 7
গ
5 : 12
ঘ
3 : 10

সঠিক উত্তর: ঘ

3 : 10

উত্তর

সঠিক উত্তর: ঘ

3 : 10

ঘ

ব্যাখ্যা

Question: Two numbers P and Q are such that the sum of 10% of P and 15% of Q is three-fourths of the sum of 20% of P and 18% of Q. Find the ratio of P : Q.

Solution:
10% of P + 15% of Q = 3/4 × (20% of P + 18% of Q)
⇒ 10P/100 + 15Q/100 = 3/4 × (20P/100 + 18Q/100)
⇒ P/10 + 3Q/20 = 3/4 × (P/5 + 9Q/50)
⇒ 2P/20 + 3Q/20 = 3/4 × (10P + 9Q)/50
⇒ (2P + 3Q)/20 = (30P + 27Q)/200
⇒ 10(2P + 3Q) = 30P + 27Q
⇒ 20P + 30Q = 30P + 27Q
⇒ 30Q - 27Q = 30P - 20P
⇒ 3Q = 10P
⇒ P/Q = 3/10
∴ P : Q = 3 : 10

৭৯৮.

If 2 : 5 :: 6 : x, then x is equal to:

ক
25
খ
15
গ
20
ঘ
30
ঙ
None of the above

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If 2 : 5 :: 6 : x, then x is equal to:

Solution:
2 : 5 :: 6 : x
⇒ 2/5 = 6/x
⇒ 30 = 2x
∴ x = 15

৭৯৯.

In a mixture of milk and water, the proportion of water by weight was 75%. If in the 60 gm mixture, 15 gm of water was added, what would be the percentage of water?

ক
70%
খ
75%
গ
80%
ঘ
90%

সঠিক উত্তর: গ

80%

উত্তর

সঠিক উত্তর: গ

80%

গ

ব্যাখ্যা

Question: In a mixture of milk and water, the proportion of water by weight was 75%. If in the 60 gm mixture, 15 gm of water was added, what would be the percentage of water?

Solution:
Weight of water in 60 gm mixture = 75% of 60 gm
= {(75/100) × 60} gm
= 45 gm.

Weight of water in 75 gms mixture = (45 + 15) gm
= 60 gm.

∴ Required percentage = {(60/75) × 100}%
= 80%.

৮০০.

3 litre of water is added to 11 litres of a solution containing 42% of alcohol in the water. The percentage of alcohol in the new mixture is -

ক
20%
খ
33%
গ
30%
ঘ
25%

সঠিক উত্তর: খ

33%

উত্তর

সঠিক উত্তর: খ

33%

খ

ব্যাখ্যা

We have an 11-liter solution containing 42% of alcohol in the water.
=> quantity of alcohol in the solution = (11 × 42)/100
Now 3 liter of water is added to the solution.
=> Total quantity of the new solution = 11 + 3 = 14
Percentage of alcohol in the new solution = {(11 × 42)/100}/14 × 100
= (11 × 3)/100
= 33%