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Ratio & Proportion, Alligation or Mixture

মোট প্রশ্ন১,০৮৬এই পাতা১০০প্রতি পাতা১০০
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উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ৯ / ১১ · ৮০১–৯০০ / ১,০৮৬

৮০১.
Last year, the ratio between the salaries of A and B was 3 : 4. But the ratios of their individual salaries between last year and this year were 4 : 5 and 2 : 3 respectively. If the sum of their present salaries is Tk. 4160, then how much is the salary of A now?
  1. ক
    Tk.1200
  2. খ
    Tk.1400
  3. গ
    Tk.1600
  4. ঘ
    Tk.1800
  5. ঙ
    Tk.2000
ব্যাখ্যা
Question: Last year, the ratio between the salaries of A and B was 3 : 4. But the ratios of their individual salaries between last year and this year were 4 : 5 and 2 : 3 respectively. If the sum of their present salaries is Tk. 4160, then how much is the salary of A now?

Solution:
Let,
the salaries of A and B last year be Tk. 3x and Tk. 4x respectively.

Then,
A's present salary = Tk. (5/4) × 3x
= Tk. 15x/4

B's present salary = Tk.(3/2) × 4x
= Tk. 6x.

According to the question,
(15x/4) + 6x = 4160
⇒ 39x = 4160 × 4
⇒ x = (4160 × 4)/39

So, A's present salary = Tk. (15/4) × (4160 × 4)/39
= Tk.1600
৮০২.
A dishonest milkman profess to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is-
  1. ক
    4%
  2. খ
    6.25%
  3. গ
    20%
  4. ঘ
    25%
ব্যাখ্যা
Question: A dishonest milkman profess to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is-

Solution:
Let, the cost price of 1 liter of milk be = 100 Tk
So, the selling price of 1 liter mixture is also = 100 Tk

Here, in 100 Tk, SP gain = 25% 
So, cost price of the mixture = (100 × 100)/125 = 80 Tk

So, water in the mixture = 100 - 80 = Tk 20
৮০৩.
The ratio of the areas of two squares is 225 : 196, then the ratio of their perimeters-
  1. ক
    ক) 17 : 18
  2. খ
    খ) 15 : 17
  3. গ
    গ) 14 : 15
  4. ঘ
    ঘ) 15 : 14
ব্যাখ্যা
Let
the sides of the two squares be x and y respectively.
Then,
Reqd. ratio of areas 

x2/​y2 = 225​/196
x2/​y2 =152/142
(x/y)2 = (15/14)2
x/y = 15/14 
4x/4y =  15 × 4 /14 × 4 
4x/4y = 15/14
4x : 4y = 15 : 14
৮০৪.
In a class, the number of girls is 20% more than that of the boys. The strength of the class is 66. If 4 more girls are admitted to the class, the ratio of the number of boys to that of the girls is-
  1. ক
    3 : 5
  2. খ
    1 : 3
  3. গ
    3 : 4
  4. ঘ
    1 : 2
ব্যাখ্যা

Question: In a class, the number of girls is 20% more than that of the boys. The strength of the class is 66. If 4 more girls are admitted to the class, the ratio of the number of boys to that of the girls is-

Solution:
Let the number of boys be 5k.
Then the number of girls = 20% more than the boys = 5k × 1.2 = 6k

Total students = 66
5k + 6k = 66
⇒ 11k = 66
⇒ k = 66/11 
∴ k = 6

∴ Number of boys = 5 × 6 = 30
∴ Number of girls = 6 × 6 = 36

Now, when 4 more girls are admitted than New number of girls = 36 + 4 = 40

∴ Ratio of boys to girls = 30 : 40 = 3 : 4 (dividing both by 10)

So the ratio of the number of boys to girls is 3 : 4

৮০৫.
How many kgs of sugar costing Tk 9 per kg must be mixed with 27 kg of sugar costing Tk 7 per kg so that may be a gain of 10% by selling the mixture at Tk 9.24 per kg?
  1. ক
    ক) 63 kg
  2. খ
    খ) 58 kg
  3. গ
    গ) 72 kg
  4. ঘ
    ঘ) 49 kg
ব্যাখ্যা
Question: How many kgs of sugar costing Tk 9 per kg must be mixed with 27 kg of sugar costing Tk 7 per kg so that may be a gain of 10% by selling the mixture at Tk 9.24 per kg?

Solution:
Let, the sugar of Tk 9 per kg is = x kg

ATQ,
110% of (7 × 27 + 9 × x) = 9.24(27 + x)
⇒ (11/10) × (189 + 9x) = 249.48 + 9.24x
⇒ 2079 + 99x = 2494.8 + 92.4x
⇒ 6.6x = 415.8
⇒ x = 63
৮০৬.
In a college union, there are 48 students. The ratio of the number of boys to the number of girls is 5 : 3. How many number of girls to be added in the union, so that the number of boys to girls in 6 : 5 ?
  1. ক
    5
  2. খ
    7
  3. গ
    13
  4. ঘ
    10
ব্যাখ্যা
Question : In a college union, there are 48 students. The ratio of the number of boys to the number of girls is 5 : 3. How many number of girls to be added in the union, so that the number of boys to girls in 6 : 5 is ?

Solution :
Let,
The ratio of numbers of boys to the number of girls is = 5x : 3x

According to the question,
⇒ 5x+3x=48
⇒ 8x=48
∴ x = 6

So the number of boys = 5x
= 5 × 6
= 30

and the number of girls = 3x
= 3 × 6
= 18

Let,
The number of girls to be added in the union = y

So,
30/(18 + y) = 6 : 5
⇒ 6(18 + y) = 30 × 5
⇒ 6(18 + y) = 150
⇒ 18 + y = 150/6
⇒ 18 + y = 25
⇒ y = 25 - 18
∴ y = 7

∴ The number of girls to be added in the union = 7
৮০৭.
In a mixture 120 liters, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then estimate the quantity of water in liter to be further added in the mixture.
  1. ক
    60 liters
  2. খ
    80 liters
  3. গ
    120 liters
  4. ঘ
    160 liters
ব্যাখ্যা
Question: In a mixture 120 liters, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then estimate the quantity of water in liter to be further added in the mixture.

Solution:
Sum of the given ratio = 2 +1=3

Quantity of milk = 120 × (2/3) = 80 liters
Hence, quantity of water will be
= 120 - 80 = 40 liters

Assume,
x liters water needs to be added.
ATQ,
80/(x + 40) = 1/2
⇒ 160 = x + 40
∴ x = 120

∴ 120 liters water needs to be added.
৮০৮.
If 4(Nahid's Capital ) = 6(Safin's Capital ) = 10(Robin's Capital ), then out of the total profit of Tk. 4650 , Robin will receive-
  1. ক
    Tk. 900
  2. খ
    Tk. 1200
  3. গ
    Tk. 1450
  4. ঘ
    Tk. 1600
ব্যাখ্যা
Question: If 4(Nahid's Capital ) = 6(Safin's Capital ) = 10(Robin's Capital ), then out of the total profit of Tk. 4650 , Robin will receive-

Solution:
Let
Nahid's capital = a
Safin's capital = b
Robin's capital = c

ATQ,
4a = 6b = 10c
⇒ 2a = 3b = 5c
∴ b = 2a/3

∴ c = 2a/5

Nahid : Safin : Robin = a : 2a/3 : 2a/5
= 15 : 10 : 6 [Multiply by 15]

Robin's share = 4650 × (6/31)
= 150 × 6
= Tk. 900
৮০৯.
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 speed of the stream is:
  1. ক
    ক) 5 : 3
  2. খ
    খ) 3 : 1
  3. গ
    গ) 2 : 3
  4. ঘ
    ঘ) 2 : 1
ব্যাখ্যা
Question: 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 speed of the stream is:

Solution: 
Ratio of downstream time : upstream time = 1 : 2
 Ratio of downstream speed : upstream speed = 2 : 1

Let
Downstream speed = 2x and
Upstream speed = x
Speed of the boat in still water = (2x + x)/2 = 3x/2
Speed of stream = (2x - x)/2 = x/2
∴ Required ratio = 3x/2 : x/2 = 3 : 1
৮১০.
If the ratio of numbers is 3: 4 and their least common multiple is 60, then the numbers are-
  1. ক
    ক) 15, 20
  2. খ
    খ) 12,16
  3. গ
    গ) 9, 12
  4. ঘ
    ঘ) 18, 24
ব্যাখ্যা

Let, these two numbers be 3x and 4x then their LCM = 12x
Now, according to question,
12x =  60
Or, x = 5
Thus, the numbers are (3x = 3 × 5) = 15 and (4x = 4 × 5) = 20

৮১১.
What is the relationship in the ratio of 6 inches to 6 feet?
  1. ক
    1 : 7
  2. খ
    1 : 5
  3. গ
    1 : 10
  4. ঘ
    1 : 12
ব্যাখ্যা
Question: What is the relationship in the ratio of 6 inches to 6 feet?

Solution:
We know,
1 feet = 12 inches
So, 6 feet = 6 × 12
= 72 inches

Now,
∴ The required ratio = 6 : 72 = 1 : 12
৮১২.
A can contains a mixture of two liquids A and B in the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?
  1. ক
    ক) 10
  2. খ
    খ) 20
  3. গ
    গ) 21
  4. ঘ
    ঘ) 25
ব্যাখ্যা

Suppose the can initially contains 7x and 5x litres of mixtures A and B respectively
Quantity of A in mixture left = {7x - (7/12) × 9} = 7x - 21/4
Quantity of B in mixture left = {5x - (5/12) × 9} = 5x - 15/4

According to the question,
(7x - 21/4)/{(5x - 15/4) + 9} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 147 + 189
⇒ 112x = 336
⇒ x = 3.

So, they can contain 21 litres of A.

৮১৩.
A cement mixture is composed of 3 elements. By weight, 1/3 of the mixture is sand, 3/5 is water and the remaining 12 pounds of the mixture is gravel. What is the weight of the entire mixture in pounds?
  1. ক
    60
  2. খ
    80
  3. গ
    90
  4. ঘ
    180
  5. ঙ
    None of these
ব্যাখ্যা
প্রশ্ন: A cement mixture is composed of 3 elements. By weight, 1/3 of the mixture is sand, 3/5 is water and the remaining 12 pounds of the mixture is gravel. What is the weight of the entire mixture in pounds?

সমাধান:
বালু ও পানির পরিমাণ = (1/3) + (3/5) অংশ 
= (5 + 9)/15 অংশ 
= 14/15 অংশ 

অবশিষ্ট কাঁকরের পরিমাণ = 1 - (14/15) অংশ
= 1/15 অংশ

প্রশ্নমতে, 
1/15 = 12 পাউন্ড
∴ 1 বা সম্পূর্ণ অংশ = 12 × 15 পাউন্ড
= 180 পাউন্ড
অতএব, সম্পূর্ণ মিশ্রণের পরিমাণ 180 পাউন্ড।
৮১৪.
If Arif work alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task.
  1. ক
    2 : 1
  2. খ
    5 : 1
  3. গ
    7 : 5
  4. ঘ
    None of these
ব্যাখ্যা
Question: If Arif work alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task.

Solution:
ধরি
আরিফ ও বাবু কাজটি করে x ঘণ্টায়

আরিফ একা কাজটি  করে (x + 20) ঘণ্টায়
বাবু একা কাজটি  করে (x + 5) ঘণ্টায় 

প্রশ্নমতে
{1/(x + 20)} + {1/(x + 5)} = 1/x
⇒ 1/(x + 20) = (1/x) - {1/(x + 5)}
⇒ 1/(x + 20) = (x + 5 - x)/(x2 + 5x)
⇒ 1/(x + 20) =5/(x2 + 5x)
⇒ x2 + 5x = 5x + 100
⇒ x2 = 5x - 5x + 100
⇒ x2 = 100
⇒ x2 = 102
∴ x = 10 

আরিফ একা কাজটি  করে (10 + 20) ঘণ্টা = 30 ঘণ্টায়
বাবু একা কাজটি  করে (10 + 5) ঘণ্টা  = 15 ঘণ্টায়

আরিফ ও বাবুর কাজের সময়ের অনুপাত = 30 : 15
= 2 : 1
৮১৫.
How many kilogram of sugar costing Tk. 9 per kg must be mixed with 27 kg of sugar costing Tk. 7 per kg so that there may be a gain of 10% by selling the mixture at Tk. 9.24 per kg?
  1. ক
    36 kg
  2. খ
    42 kg
  3. গ
    54 kg
  4. ঘ
    63 kg
ব্যাখ্যা

Question: How many kilogram of sugar costing Tk. 9 per kg must be mixed with 27 kg of sugar costing Tk. 7 per kg so that there may be a gain of 10% by selling the mixture at Tk. 9.24 per kg?

Solution:
Let,
Quantity of sugar costing Tk. 9 per kg = x
S.P. of 1 kg of mixture = Tk. 9.24,
Gain 10%.
C.P. of 1 kg of mixture = Tk. (100/110) × 9.24 = Tk. 8.40

By the rule of allilation, we have:

⇒ 27 : x = (9 - 8.40) : (8.40 - 7) = 0.60 : 1.40 = 6 : 14 = 3 : 7
⇒ 27/x = 3/7
⇒ x/27 = 7/3
⇒ x = (7 × 27)/3
∴ x = 63

৮১৬.
In 2 kg mixture of copper and aluminum, 30% is copper. How much aluminum powder should be added to the mixture so that the quantity of copper becomes 20%?
  1. ক
    1200 gm
  2. খ
    1000 gm
  3. গ
    700 gm
  4. ঘ
    900 gm
  5. ঙ
    1500 gm
ব্যাখ্যা

Question: In 2 kg mixture of copper and aluminum, 30% is copper. How much aluminum powder should be added to the mixture so that the quantity of copper becomes 20%?

Solution:
According to the question,
Mixture of copper and aluminum = 2 Kg = (2 × 1000) = 2000 gm
30% of this mixture is copper,
= (30/100) × 2000 gm
= 600 gm copper
∴ In 2 kg mixture of copper and aluminum, aluminum is = (2000 - 600) = 1400 gm

Let x be the mass of aluminum added. The new total mass is (2000 + x).
 We want the 600 gm of copper to be 20% of this new total.
⇒ 600 = 0.20(2000 + x)
⇒ 600 = 400 + 0.2x
⇒ 0.2x = 200
∴ x = 1000 gm

৮১৭.
8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the water is 16 : 65. How much wine the cask hold originally?
  1. ক
    18 litres
  2. খ
    24 litres
  3. গ
    32 litres
  4. ঘ
    42 litres
ব্যাখ্যা
Question: 8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the water is 16 : 65. How much wine the cask hold originally?

Solution:
Let the quantity of the wine in the cask originally be x litres
Then,
quantity of wine left in cask after 4 operations = [x(1 - 8/x)4] litres
[x(1 - 8/x)4]/x = 16/81
⇒ [1 - 8/x]4 = (2/3)4
⇒ 1 - 8/x = 2/3
⇒ 8/x = 1/3
∴ x = 24
৮১৮.
How much water should be added to 40 liters of pure milk to gain an extra 20% profit when selling the mixture at the price of pure milk?
  1. ক
    6 liters
  2. খ
    8 liters
  3. গ
    10 liters
  4. ঘ
    12 liters 
ব্যাখ্যা

Question: How much water should be added to 40 liters of pure milk to gain an extra 20% profit when selling the mixture at the price of pure milk?

Solution:
Assume,
Price of pure milk per liter = 100 Taka
So, the price of 40 liters of pure milk = 40 × 100 = 4000 Taka

Let, the amount of water added = x liters
Total mixture = (40 + x) liters

Since the mixture is sold at the price of pure milk,
Selling price of (40 + x) liters = 100(40 + x) Taka

According to the question,
100(40 + x) = 4000 + 4000 of 20%
⇒ 4000 + 100x = 4000 + (4000 × 20/100)
⇒ 4000 + 100x = 4000 + 800
⇒ 100x = 800
⇒ x = 800/100
⇒ x = 8

∴ Required water: 8 liters.

৮১৯.
When 25% of the first number is added to the second number, the second number becomes 3/2 times the first number. What is the ratio of the first number to the second number?
  1. ক
    5 : 6
  2. খ
    4 : 5
  3. গ
    3 : 4
  4. ঘ
    1 : 3
ব্যাখ্যা

Question: When 25% of the first number is added to the second number, the second number becomes 3/2 times the first number. What is the ratio of the first number to the second number?

Solution:
Let the first number = x
and the second number = y.

According to the question,
y + 25% of x = (3/2)x
⇒ y + (25/100)x = (3/2)x
⇒ y + (1/4)x = (3/2)x
⇒ y = (3/2)x - (1/4)x
⇒ y = (6 - 1)x/4
⇒ y = 5x/4
∴ y = 5x/4

Therefore, x : y = 4 : 5 

৮২০.
A and B start a business with initial investment in the ratio 12 : 11 and their annual profits were in the ratio 4 : 1. If A invested the money for 11 months B invested the money for –
  1. ক
    ক) 3 months
  2. খ
    খ) 4 months
  3. গ
    গ) 5 months
  4. ঘ
    ঘ) 6 months
ব্যাখ্যা

Let, B invested the money for t months
Then the ratio of investment = (12×11 : 11× t) = 12 : t
So, 12/t = 4/1
⇒ t = 3 months

৮২১.
The present age of three persons are in the proportion of 4 : 7 : 9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is -
  1. ক
    28
  2. খ
    36
  3. গ
    45
  4. ঘ
    None of these
ব্যাখ্যা
Question: The present age of three persons are in the proportion of 4 : 7 : 9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is -

Solution: 
Let, present ages of three persons 4x, 7x, 9x 

Eight years ago, the sum of their ages was 56 years
At present, the sum of their ages is = 56 + 8 + 8 + 8 
= 56 + 24 years 
= 80 years 

4x + 7x + 9x = 80 
⇒ 20x = 80
⇒ x = 4 

The present age of the eldest person is = 4 × 9 = 36 years 

৮২২.
In 60 litres of mixture, milk and water are in 3 : 2. How much water to add to make it 1 : 1?
  1. ক
    6 litres
  2. খ
    9 litres
  3. গ
    12 litres
  4. ঘ
    15 litres
ব্যাখ্যা
Question: In 60 litres of mixture, milk and water are in 3 : 2. How much water to add to make it 1 : 1?

Solution:
Find the quantity of milk and water in the initial mixture:
→ Total parts = 3 + 2 = 5 parts
→ Milk = (3/5) × 60 = 36 L
→ Water = (2/5) × 60 = 24 L

Let x liters of water be added.
New amount of water = 24 + x
Milk remains the same = 36 L

According to the question:
We want the final ratio to be 1:1, i.e.
→ Milk = Water
⇒ 36 = 24 + x
⇒ x = 36 − 24 = 12 L
৮২৩.
The ratio of milk to water in a mixture is 5:3. When adding 4 liters of water, the ratio becomes 5:5.
What was the quantity of milk in the original mixture?
  1. ক
    10 liters
  2. খ
    15 liters
  3. গ
    20 liters
  4. ঘ
    25 liters
ব্যাখ্যা

Question: The ratio of milk to water in a mixture is 5:3. When adding 4 liters of water, the ratio becomes 5:5. What was the quantity of milk in the original mixture? 

Solution:
Let the initial quantity of 
Milk = 5x liters
Water = 3x liters

When 4 liters of water are added, the new quantity of water becomes = (3x+4) liters
The new ratio becomes 5:5, which simplifies to 1:1. This means the amount of milk and water are now equal. 
5x=3x+4
2x = 4 
∴ x = 2

So, the initial quantity of Milk = 5 × 2 = 10 liters

৮২৪.
A right triangle has sides in the ratio of 5:12:13. What is the measure of the smallest angle in the triangle, in degrees?
  1. ক
    ক) 13.34
  2. খ
    খ) 22.62
  3. গ
    গ) 34.14
  4. ঘ
    ঘ) 42.71
ব্যাখ্যা

We know that, sinθ = AB/AC
⇒ sinθ = 5/13
⇒ θ = sin-1(5/13)
∴ θ = 22.62°

৮২৫.
A jar is filled with liquid, 2 parts of which are water and 4 parts milk. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half milk?
  1. ক
    1/2
  2. খ
    1/4
  3. গ
    1/8
  4. ঘ
    None of the above
ব্যাখ্যা

Question: A jar is filled with liquid, 2 parts of which are water and 4 parts milk. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half milk?

Solution:
মনে করি,
পাত্রটিতে তরল পদার্থ আছে = 6 ইউনিট
এবং এই তরলের x ইউনিট পানি দ্বারা প্রতিস্থাপন করা হল।

∴ নতুন মিশ্রণে পানির পরিমাণ = (2 - (2x/6) + x) 
= 2 - (x/3) + x 
= (6 - x + 3x)/3
= (2x + 6)/3 ইউনিট 

এবং, নতুন মিশ্রণে দুধের পরিমাণ = 4 - (4x/6)
= 4 - (2x/3)
= (12 - 2x)/3 ইউনিট

প্রশ্নমতে,
(2x + 6)/3 = (12 - 2x)/3
⇒ 2x + 6 = 12 - 2x
⇒ 2x + 2x = 12 - 6
⇒ 4x = 6
⇒ x = 3/2 
 
∴ মিশ্রণের প্রতিস্থাপিত অংশের পরিমাণ = (3/2) × (1/6)
= 1/4 

৮২৬.
The ratio of orange and water in a juice mixture is 2 : 3, then the percentage of water in mixture is-
  1. ক
    35%
  2. খ
    20%
  3. গ
    30%
  4. ঘ
    60%
ব্যাখ্যা

Question: The ratio of orange and water in a mixture is 2 : 3, then the percentage of water in mixture is-

Solution: 
The ratio of orange and water in a mixture is 2 : 3
Total = 5

∴ percentage of water = (3/5) × 100%
= 60%

৮২৭.
A jar contains black and white marbles. If there are ten marbles in the jar, then which of the following could not be the ratio of black to white marbles?
  1. ক
    1 : 4
  2. খ
    1 : 10
  3. গ
    7 : 3
  4. ঘ
    9 : 1
ব্যাখ্যা
Question: A jar contains black and white marbles. If there are ten marbles in the jar, then which of the following could not be the ratio of black to white marbles?

Solution:
Since the number of black and white marbles are whole numbers,
So the sum of the terms of the ratio must be a factor of 10.
ক) 1:4 = 1+4 = 5 a factor of 10.
খ) 1:10 = 1+10 = 11 not a factor of 10.
গ) 7:3 = 7+3 =10 a factor of 10.
ঘ) 9:1 = 9+1 = 10 a fac tor of 10.

Here we can see that the sum of only the second option is not a factor of 10
৮২৮.
In a race of 200 m, A can beat B by 31 m and C by 18 m. In a race of 350 m, C will beat B by -
  1. ক
    15m
  2. খ
    17m
  3. গ
    22m
  4. ঘ
    25m
ব্যাখ্যা
Question: In a race of 200 m, A can beat B by 31 m and C by 18 m. In a race of 350 m, C will beat B by -

Solution: 
A : B = 200 : 169
A : C = 200 : 182

B/C = (B/A) × (A/C)
= 169/182

So, in a 350 race B will pass = (169/182) × 350
= 325m 

Hence, C will beat B by (350 - 325) or, 25 metres
৮২৯.
If two times A is equal to three times of B and also equal to four times of C, then A : B : C is -
  1. ক
    2 : 3 : 4
  2. খ
    4 : 3 : 2
  3. গ
    3 : 4 : 6
  4. ঘ
    6 : 4 : 3
ব্যাখ্যা

Question: If two times A is equal to three times of B and also equal to four times of C, then A : B : C is -

Solution:
2A = 3B
Or, B = 2A/3
and 2A = 4C
Or, C = A/2

Hence, A : B : C = A : 2A/3 : A/2
= 1 : 2/3 : 1/2
= 6 : 4 : 3

৮৩০.
There are two containers containing milk and water in the ratio 2:3 and 4:1 respectively. Equal quantities from both the containers are mixed together. What will be the ratio of milk to water in the resultant solution?
  1. ক
    ক) 2 : 1
  2. খ
    খ) 5 : 3
  3. গ
    গ) 3 : 2
  4. ঘ
    ঘ) 6 : 5
ব্যাখ্যা
ধরি, প্রত্যেক পাত্রে তরলের পরিমাণ x litres
১ম পাত্রে দুধ ও পানির অনুপাত = 2 : 3
অতএব,
১ম পাত্রে দুধের পরিমাণ =  2x/5
২য় পাত্রে পানির পরিমাণ = 3x/5

 ২য় পাত্রে দুধ ও পানির অনুপাত = 4 : 1
অতএব,
১ম পাত্রে দুধের পরিমাণ =  4x/5
২য় পাত্রে পানির পরিমাণ = x/5

মোট দুধের পরিমাণ = 2x/5 + 4x/5 = 6x/5
মোট পানির পরমাণ = 3x/5 + x/5 = 4x/5

দুধ ও পানির অনুপাত  = 6x/5 : 4x/5 = 6 : 4 = 3 : 2
৮৩১.
One-fifth of Rahim’s investment in Mutual Funds is equal to one-third of his investment in Gold. If his total investment is Tk. 80,000. How much did he invest in Mutual Funds?
  1. ক
    Tk. 42,000
  2. খ
    Tk. 55,000
  3. গ
    Tk. 30,000
  4. ঘ
    Tk. 50,000
ব্যাখ্যা
Question: One-fifth of Rahim’s investment in Mutual Funds is equal to one-third of his investment in Gold. If his total investment is Tk. 80,000. How much did he invest in Mutual Funds?

Solution:
Let,
m = Investment in Mutual Funds
g = Investment in Gold

Given that,
(1​/5)m = (1/3​)g
g = (3/5)m ...... (1)

And
⇒ m + g = 80,000
⇒ m +  (3/5)m = 80000
⇒ (5m + 3m)/5 = 80000
⇒ 8m/5 = 80000
⇒ m = (80000 × 5)/8
∴ m = 50000

So Rahim invested Tk. 50,000 in Mutual Funds.
৮৩২.
When 30% of one number is subtracted from another number, the second number reduces to its 80%. What is the ratio of the first to the second number? 
  1. ক
    3 : 2
  2. খ
    2 : 3
  3. গ
    2 : 5
  4. ঘ
    4 : 7
ব্যাখ্যা
Question: When 30% of one number is subtracted from another number, the second number reduces to its 80%. What is the ratio of the first to the second number? 

Solution:
Let,
the first and second numbers be x and y,

ATQ,
y - 30% of x = 80% of y
⇒ y - (30x)/100 = (80y)/100
⇒ y - (30x)/100 = 4y/5
⇒ y - (3x)/10 = 4y/5
⇒ (10y - 3x)/10 = 4y/5
⇒ (10y - 3x)/2 = 4y
⇒ 10y - 3x = 8y
⇒ - 3x = 8y - 10y
⇒ 3x = 2y
⇒ x/y = 2/3
∴ x : y = 2 : 3
৮৩৩.
The ratio of milk and water in a mixture is 4 : 3. If we add 2 litres of water, the ratio of milk and water becomes 8 : 7. What is the quantity of the final mixture?
  1. ক
    ক) 18 litres
  2. খ
    খ) 30 litres
  3. গ
    গ) 24 litres
  4. ঘ
    ঘ) 28 litres
ব্যাখ্যা
Ratio of milk and water in a mixture is = 4x : 3x

According to the question,

4x/(3x + 2) = 8/7
⇒ 4x × 7 = 8 (3x + 2)
⇒ 28x = 24x + 16
⇒ 28x – 24x = 16
⇒ 4x = 16
⇒ x = 16/4
⇒ x = 4

Quantity of mixture in the last = (4x + 3x) = 7x = 7 × 4 = 28 litres
Quantity of mixture in the last = 28 + 2 = 30 litres

৮৩৪.
Some money is divided amongst three workers A, B and C such that 5 times A's share is equal to 12 times B's share which is equal to 6 times C's share . The ratio between the shares of A,B and C is? 
  1. ক
    ক) 5 : 10 : 12
  2. খ
    খ) 10 : 12 : 5
  3. গ
    গ) 12 : 5 : 10
  4. ঘ
    ঘ) 5 : 12 : 10
ব্যাখ্যা
ধরি,
5A = 12B = 6C = x  
5A= x        12B= x             6C = x  
A = x/5         B= x/12          C = x/6
A : B : C  = x/5 :  x/12 : x/6
              = 12 : 5 : 10
৮৩৫.
The sides of a triangle are in the ratio 4 : 5 : 6. The smallest angle is-
  1. ক
    ক) 48°
  2. খ
    খ) 60°
  3. গ
    গ) 72°
  4. ঘ
    ঘ) 55°
ব্যাখ্যা
Question: The sides of a triangle are in the ratio 4 : 5 : 6. The smallest angle is-

Solution: 
we know that, the sum of the angles of a triangle is 180°

Hence, 
The smallest angle is (180° × 4/15) = 48°
৮৩৬.
What is the mean proportional of √5 and √125?
  1. ক
    ক) 5
  2. খ
    খ) 5√5
  3. গ
    গ) 25
  4. ঘ
    ঘ) 25√5
ব্যাখ্যা
Question: What is the mean proportional of √5 and √125?

Solution: 
Mean proportional = √(√5 × √125)
= √(√625)
= √25
= 5
৮৩৭.
A box contains 56 in the form of coins of one tk, 50 paise and 25 paise. The number of 50 paise coins is double the number of 25 paise coins and four times the number of one tk coins. How many 50 paise coins are there in the box?
  1. ক
    46
  2. খ
    64
  3. গ
    72
  4. ঘ
    58
ব্যাখ্যা
Question: A box contains 56 in the form of coins of one tk, 50 paise and 25 paise. The number of 50 paise coins is double the number of 25 paise coins and four times the number of one tk coins. How many 50 paise coins are there in the box?

Solution:
Number of 1-tk coins = x
Number of 50 paise coins = 4x
Number of 25 paise coins = 2x

Ratio of their values = x : (4x/2) : (2x/4) = 2 : 4 : 1
Value of 50-paise coins = (4/7) × 56 = tk 32
Their number = 32 × 2 = 64

ATQ,
(x )(1) + (4x)(1/2) + 2x(1/4) = 56
⇒ x + 2x + (2/x) = 56
⇒ x = 56 × (2/7)
∴ x = 16

No. of 50p coins = 4 × 16 = 64.
৮৩৮.
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?
  1. ক
    5 liters   
  2. খ
    8 liters   
  3. গ
    10 liters   
  4. ঘ
    12 liters   
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:         
Amount of water = 150 × 20/100
= 30 

Amount of wine = 120 liter

ATQ, 
(30 + x)/(150 + x) = 25/100
⇒ (30 + x)/(150 + x) = 1/4
⇒ 120 + 4x = 150 + x
⇒ 4x - x = 150 - 120
⇒ 3x = 30
⇒ x = 30/3 = 10 liters                    
৮৩৯.
A jar contains white, red and green marbles in the ratios 2 : 3 : 5. Six more green marbles are added to the jars and the ratio becomes 2 : 3 : 7. How many white marbles are there in the jar?
  1. ক
    ক) 6
  2. খ
    খ) 7
  3. গ
    গ) 8
  4. ঘ
    ঘ) 9
ব্যাখ্যা
সাদা , লাল এবং সবুজ মার্বেলের অনুপাত 2 : 3 : 5
সাদা মার্বেল আছে = 2x
সবুজ মার্বেল আছে = 5x

এখানে 
2x : (5x + 6) = 2 : 7
2x/(5x + 6) = 2/7 
x/(5x + 6) = 1/7 
7x = 5x + 6
7x - 5x = 6
2x = 6
x = 3
সাদা মার্বেল আছে = 2 × 3 = 6
৮৪০.
In what ratio should milk costing Tk 40 per litre be mixed with milk costing Tk 60 per litre to get a mixture worth Tk 50 per litre?
  1. ক
    1 : 1
  2. খ
    2 : 3
  3. গ
    1 : 2
  4. ঘ
    1 : 3
ব্যাখ্যা
Question: In what ratio should milk costing Tk 40 per litre be mixed with milk costing Tk 60 per litre to get a mixture worth Tk 50 per litre?

Solution:
Use the allegation method:
Cheaper = 40, Costlier = 60, Mean = 50
=> Ratio = (60 – 50):(50 – 40) = 10:10 = 1:1

Alternative:
Let Tk 40 per liter is consists of x liters,
Tk 60 per liter is conists of y liters.
total milk = (x+y) liters
and price of mixture is 50 tk per liter

Now,
→ 40x + 60y = 50(x + y)
→ 40x + 60y = 50x + 50y
→ 10y = 10x
→ x : y = 1 : 1
৮৪১.
Alloy A contains 40% gold and 60% silver. Alloy B contains 35% gold and 40% silver and 25% copper. Alloys A and B are mixed in the ratio of 1 : 4. What is the ratio of gold and silver in the newly formed alloy?
  1. ক
    ক) 11 : 9
  2. খ
    খ) 20 : 30
  3. গ
    গ) 9 : 11
  4. ঘ
    ঘ) 25 : 35
  5. ঙ
    ঙ) 80 : 20
ব্যাখ্যা

A:: - G : S = 40 : 60
B:: - G : S : C = 35 : 40 : 25
New, G : S = {(1×40) + (4×35)} : {(40×4) + (1×60)}
= 180 : 220
= 18 : 22
= 9 : 11 [Answer. ]

৮৪২.
A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?
  1. ক
    20
  2. খ
    24
  3. গ
    16
  4. ঘ
    22
ব্যাখ্যা
Question: A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?

Solution: 
Let,
The capacity of the container in gallon is x gallons 

ATQ, 
(x/2) - 8 = x/10 
⇒ (x - 16)/2 = x/10 
⇒ 10(x - 16) = 2x 
⇒ 10x - 160 = 2x
⇒ 10x - 2x = 160 
⇒  8x = 160 
∴ x = 160/8 = 20 gallons
৮৪৩.
The ratio of expenditure and savings is 3 : 2 . If the income increases by 15% and the savings increase by 6%, then by how much percent should his expenditure increases?
  1. ক
    20%
  2. খ
    21%
  3. গ
    23%
  4. ঘ
    25%
ব্যাখ্যা
Question: The ratio of expenditure and savings is 3 : 2 . If the income increases by 15% and the savings increase by 6%, then by how much percent should his expenditure increases?

Solution: 
Let, expenditure 300 and savings be 200
Income = 300 + 200 
= 500 

New income = 500 + 500 × 15/100 
= 500 + 75
= 575 

New savings = 200 + 200 × 6/100
= 212 

 New expenditure = 575 - 212
= 363

% increase = {(363 - 300)/300} × 100%
= 21%
৮৪৪.
Two men earn a yearly salary in the ratio 10 : 13. If their spending is in the ratio of 4 : 5 and the man spending lesser of the two saves Tk. 6000 while the other one saves Tk. 8000, then find the salary of the person who is higher paid.
  1. ক
    Tk. 10000
  2. খ
    Tk. 13000
  3. গ
    Tk. 15000
  4. ঘ
    Tk. 17000
ব্যাখ্যা
Question: Two men earn a yearly salary in the ratio 10 : 13. If their spending is in the ratio of 4 : 5 and the man spending lesser of the two saves Tk. 6000 while the other one saves Tk. 8000, then find the salary of the person who is higher paid.

Solution: 
Let, their salary 10x, 13x taka

expenditure 4y, 5y taka

10x - 4y = 6000
⇒ 50x - 20y = 30000 

13x - 5y = 8000 
⇒ 52x - 20y = 32000

52x - 20y - 50x + 20y = 32000 - 30000 
⇒ 2x = 2000 
∴ x = 1000 

the salary of the person who is higher paid is 13000 taka.
৮৪৫.
The milk and water in a mixture are in the ratio 7 : 5. When 15 litres of water are added to it, the ratio of milk and water in the new mixture becomes 7 : 8. The total quantity of water in the new mixture is-
  1. ক
    35 litres
  2. খ
    40 litres
  3. গ
    45 litres
  4. ঘ
    32 litres
  5. ঙ
    None of these
ব্যাখ্যা
Question: The milk and water in a mixture are in the ratio 7 : 5. When 15 litres of water are added to it, the ratio of milk and water in the new mixture becomes 7 : 8. The total quantity of water in the new mixture is-

Solution:
Let the quantity of milk in the mixture = 7x litres and that of water = 5x litres.

ATQ,
⇒ 7x/5x+15 = 7/8
⇒ 56x = 35x + 105
⇒ 56x - 35x = 105
⇒ 21x = 105
⇒ x = 105/21
∴ x = 5

∴ Required quantity of water = (5x + 15) litres
= (5 × 5) + 15 = 40 litres
৮৪৬.
The salaries of P, Q, and R are in the ratio 5 : 8 : 6 and their expenses are in the ratio 4 : 7 : 5. If P saves 1/3rd of his salary then the savings of P, Q, and R are in the ratio of-
  1. ক
    11 : 15 : 17
  2. খ
    7 : 15 : 12
  3. গ
    13 : 19 : 9
  4. ঘ
    10 : 13 : 11
ব্যাখ্যা
Question: The salaries of P, Q, and R are in the ratio 5 : 8 : 6 and their expenses are in the ratio 4 : 7 : 5. If P saves 1/3rd of his salary then the savings of P, Q, and R are in the ratio of-

Solution:
Let the salaries of P, Q and R are 5x, 8x, and 6x respectively
and expenses of P, Q and R are 4y, 7y and 5y respectively
ATQ,
5x - 4y = 5x × 1/3
⇒ 15x - 12y = 5x
⇒ 10x = 12y
⇒ x : y = 12 : 10

∴ The ratio of savings of P, Q, and R
= (5x - 4y) : (8x - 7y) : (6x - 5y)
= (5 × 12 - 4 × 10) : (8 × 12 - 7 × 10) : (6 × 12 - 5 × 10)
= (60 - 40) : (96 - 70) : (72 - 50)
= 20 : 26 : 22
= 10 : 13 : 11
Therefore, the savings of P, Q, and R are in the ratio 10 : 13 : 11
৮৪৭.
Nirmal and Kapil started a business investing Tk 9000 and Tk 12000 respectively. After 6 months, Kapil withdrew half of his investment. If after a year, the total profit was Tk 4600, what was Kapil’s share initially?
  1. ক
    ক) Tk 2300
  2. খ
    খ) Tk 2400
  3. গ
    গ) Tk 2500
  4. ঘ
    ঘ) None of above
ব্যাখ্যা

Nirmal:Kapil = 9000×12:(12000×6 + 6000×6)= 1:1
Kapil's share = Tk [4600 ×(1/2)) = Tk. 2300

৮৪৮.
If A : B : C = 2 : 3 : 4, then A/B : B/C : C/A = ?
  1. ক
    7 : 10 : 24
  2. খ
    8 : 9 : 24
  3. গ
    11 : 14 : 23
  4. ঘ
    8 : 13 : 28
  5. ঙ
    None
ব্যাখ্যা
Question: If A : B : C = 2 : 3 : 4, then A/B : B/C : C/A = ?

Solution:
A/B : B/C : C/A
= 2/3 : 3/4 : 4/2
= (2/3) × 12 : (3/4) × 12 : (4/2) × 12
= 8 : 9 : 24
৮৪৯.
After a 25% discount, the price of a refrigerator is Tk. 18,000. What was the price before the discount? 
  1. ক
    Tk. 21,000
  2. খ
    Tk. 24,000
  3. গ
    Tk. 25,000
  4. ঘ
    Tk. 30,000
  5. ঙ
    None
ব্যাখ্যা

Question: After a 25% discount, the price of a refrigerator is Tk. 18,000. What was the price before the discount?

Solution:
In 25% discount,
Discount price = 75 when original price = 100

∴ Discount price 1 = 100/75 of original price

∴ Discount price 18,000 = (18,000 × 100)/75
= 24,000

∴ Original price = Tk. 24,000

৮৫০.
A woman spends a part of her monthly income and saves the rest. The ratio of her expenditure to her savings is 7 : 3. If her monthly income is Tk. 25,000, what is the amount of her monthly savings?
  1. ক
    Tk. 7000
  2. খ
    Tk. 7500
  3. গ
    Tk. 8000
  4. ঘ
    Tk. 8200
ব্যাখ্যা
Question: A woman spends a part of her monthly income and saves the rest. The ratio of her expenditure to her savings is 7 : 3. If her monthly income is Tk. 25,000, what is the amount of her monthly savings?

Solution:
Given,
Expenditure : Savings = 7 : 3

∴ Sum of the terms of ratio = (7 + 3) = 10

Given,
Total monthly salary = Tk. 25000

∴ Monthly savings = Tk.{(3/10) × 25000}
= Tk. 7500
৮৫১.
If 2/3 of X = 75% of Y = 0.4 of Z, then find the ratio X : Y : Z.
  1. ক
    9 : 8 : 15
  2. খ
    8 : 10 : 17
  3. গ
    16 : 7 : 21
  4. ঘ
    9 : 13 : 17
ব্যাখ্যা

Question: If 2/3 of X = 75% of Y = 0.4 of Z, then find the ratio X : Y : Z.

Solution:
According to the question,
(2/3) × X = 75% of Y
⇒ 2X/3 = 75Y/100
⇒ 2X/3 = 3Y/4
⇒ X/Y = 9/8

Now,
75% of Y = 0.4 of Z
⇒(75Y/100) = (2Z/5)
⇒ (3Y/4) = (2Z/5)
⇒ 15Y = 8Z
⇒ Y/Z = 8/15

X : Y : Z = (9/8) : 1 : (15/8)
Multiply all terms by 8:
X = 9 × 1 = 9
Y = 1 × 8 = 8
Z = (15/8) × 8 = 15

Therefore,
X : Y : Z = 9 : 8 : 15

৮৫২.
In what ratio salt costing 13 Tk/kg mixed with another salt costing 7 Tk/ kg to get a mixture costing 9 Tk/kg?
  1. ক
    3 : 2
  2. খ
    1 : 3
  3. গ
    1 : 2
  4. ঘ
    1 : 4
ব্যাখ্যা
Question: In what ratio salt costing 13 Tk/kg mixed with another salt costing  7 Tk/ kg to get a mixture costing  9 Tk/kg?

Solution: 
ধরি,
13 টাকা কেজি লবণের পরিমাণ x কেজি এবং 7 টাকা কেজি লবণের পরিমাণ y কেজি। 

(13x + 7y)/(x + y) = 9
⇒ (13x + 7y) = 9 (x + y)
⇒ 13x + 7y = 9x + 9y 
⇒ 13x - 9x = 9y - 7y
⇒ 4x = 2y 
∴ x / y = 2/4 = 1/2
x : y = 1 : 2
৮৫৩.
The ratio of income in two consecutive years is 2 : 3 respectively. The ratio of their expenditure is 5 : 9. Income of second-year is Tk. 45000 and Expenditure of first-year is Tk. 25000. Find the Savings in both years together.
  1. ক
    5000
  2. খ
    7000
  3. গ
    6075
  4. ঘ
    8025
ব্যাখ্যা
Question: The ratio of income in two consecutive years is 2 : 3 respectively. The ratio of their expenditure is 5 : 9. Income of second-year is Tk. 45000 and Expenditure of first-year is Tk. 25000. Find the Savings in both years together.

Solution:
Let the First-year income = 2x
And, Second-year income = 3x
But ATQ, the second-year income = 45000
So, x = 45000/3 = 15000
Then, first-year income = 2 × 15000 = 30000

Similarly,
Let the first-year expenditure = 5y
And, second-year expenditure = 9y
But ATQ, the first-year expenditure = 25000
So, y = 25000/5 = 5000
Second-year expenditure = 9y = 9 × 5000 = 45000

Income = expenditure + savings
Or, savings = income - expenditure

∴ first-year savings = 30000 - 25000 = 5000
Similarly,
Second-year savings = 45000 - 45000 = 0

Now, total saving in two years = first-year savings + second-year savings
= 5000 + 0 = 5000.
৮৫৪.
The vessels containing water and milk in the ratio of 1 : 2 and 2 : 5 are mixed is 1 : 4 then what will be the resulting ratio?
  1. ক
    ক) 31 : 25
  2. খ
    খ) 22 : 31
  3. গ
    গ) 47 : 31
  4. ঘ
    ঘ) 31 : 74
ব্যাখ্যা
Question: The vessels containing water and milk in the ratio of 1 : 2 and 2 : 5 are mixed is 1 : 4 then what will be the resulting ratio?

Solution:
We can create the following ratios:
Vessel 1: water : milk = x : 2x
Vessel 2: water : milk = 2y : 5y

We can create the equation:
(x + 2x)/(2y + 5y) = 1/4
3x/7y = 1/4
12x = 7y

We see that we can let x = 7 and y = 12.
So we have:
Vessel 1: water = 7 and milk = 14
Vessel 2: water = 24 and milk = 60

Therefore, in the final mixture, the ratio of water to milk is:
(7 + 24)/(14 + 60) = 31/74
৮৫৫.
A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is:
  1. ক
    ক) 10%
  2. খ
    খ) 15%
  3. গ
    গ) 20%
  4. ঘ
    ঘ) 27%
ব্যাখ্যা
Let the cost of 1 litre of milk be Tk.100.
He gains 25% profit if he sells the same milk at Tk.125.

Now he has to sell 1.25 litres of milk at Tk.125 (as the price of milk should be proportional to the quantity).
To accommodate for the additional 250 ml of milk, he uses water.

So, 250 ml of water is used in 1.25 litres of the mixture.
∴ Percentage of water in mixture = (.25/1.25) × 100 = 20%
 
Alternative Method:
Let C.P. of 1 litre milk be Tk.1
Then, S.P. of 1 litre of mixture = Tk.1, Gain = 25%
C.P. of 1 litre mixture = Tk. (100/125×1) = 4/5

By the rule of allegation, we have:
 
∴ Ratio of milk to water = 4/5 : 1/5 = 4 : 1
Hence, percentage of water in the mixture = (1/5×100)% = 20%
 
৮৫৬.
A student took five papers in an examination, where full marks were the same on each paper. Her marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all these papers together, the candidate obtained 60% of the total marks. Then the number of papers in which she got more than 50% marks is -
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 4
  4. ঘ
    ঘ) 5
ব্যাখ্যা

Let,
the full marks for each paper be x.
Let the marks obtained in the five papers be 6y, 7y, 8y, 9y and 10y respectively.
Then, (6y + 7y + 8y + 9y + 10y)/5x = 60/100
⇒ 40y/5x = 3/5
⇒ 40y = 3x
⇒ x = (40/3)y
50% of x = (50/100) × (40/3)y
= 20y/3
= 6(2/3)y.
Clearly, the student got more than 50% marks in each of the last 4 papers.

৮৫৭.
Fahim's collection contains US, Bangladeshi and British stamps. If the ration of US to Bangladeshi stamps is 5 : 2 and the ration of Bangladeshi of British stamps is 5 : 1, what is the ration of US to British stamps?
  1. ক
    ক) 5 : 1
  2. খ
    খ) 10 : 5
  3. গ
    গ) 15 : 2
  4. ঘ
    ঘ) 20 : 2
  5. ঙ
    ঙ) 25 : 2
ব্যাখ্যা
Question: Fahim's collection contains US, Bangladeshi and British stamps. If the ration of US to Bangladeshi stamps is 5 : 2 and the ration of Bangladeshi of British stamps is 5 : 1, what is the ration of US to British stamps?

Solution:
US : Bangladesh = 5 : 2  = 25 : 10
Bangladeshi : British = 5 : 1 = 10 : 2

US : British = 25 : 2
৮৫৮.
7 litres mixture of milk and water contains 30% water. 3.5 litres of milk is added to this mixture. What is the percentage of water in the new mixture?
  1. ক
    10
  2. খ
    15
  3. গ
    20
  4. ঘ
    25
ব্যাখ্যা
Question: 7 litres mixture of milk and water contains 30% water. 3.5 litres of milk is added to this mixture. What is the percentage of water in the new mixture?

Solution:
Quantity of water in the mixture = 30% of 7 litres = 2.1 litre
Quantity of milk in the mixture = 7 - 2.1 = 4.9 litres

New quantity of milk in the mixture after adding 3.5 litres of milk = 4.9 + 3.5 = 8.4 litres
Quantity of New mixture = 8.4 + 2.1  = 10.5 litres

∴ Percentage of water in new mixture = (2.1/10.5) × 100 % = 20%
৮৫৯.
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?
  1. ক
    ক) 10 liters
  2. খ
    খ) 20 liters
  3. গ
    গ) 30 liters
  4. ঘ
    ঘ) 40 liters
  5. ঙ
    ঙ) 50 liters
ব্যাখ্যা

Number of liters of water in 125 liters of the mixture = 20% of 150 = 1/5 of 150 = 30 liters
Let us assume that other ‘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

৮৬০.
Seats for Chemistry, Bangla and English in a school are in the ratio 5 : 8 : 10. There is a proposal to increase these seats by 20%, 30% and 40% respectively. What will be the ratio of increased seats?
  1. ক
    15 : 27 : 35
  2. খ
    25 : 26 : 35
  3. গ
    15 : 26 : 35
  4. ঘ
    15 : 26 : 37
ব্যাখ্যা

Question: Seats for Chemistry, Bangla and English in a school are in the ratio 5 : 8 : 10. There is a proposal to increase these seats by 20%, 30% and 40% respectively. What will be the ratio of increased seats?

Solution:

Originally, let the number of seats for Chemistry, Bangla and English be 5x, 8x and 10x respectively

Number of increased seats are (120% of 5x), (130% of 8x) and (140% of 10x)
= {(120/100) × 5x}, {(130/100) × 7x} and {(140/100) × 8x}
= 6x, 52x/5 and 14x

∴ The required ratio = 6x : 52x/5 : 14x
= 30 : 52 : 70
=15 : 26 : 35

৮৬১.
A container holds 4 quarts of chemicals and 4 quarts of water. How many quarts of water must be added to the container to create a mixture that is 3 parts chemical to 5 parts water by volume?
  1. ক
    ক) 4/3
  2. খ
    খ) 5/3
  3. গ
    গ) 7/3
  4. ঘ
    ঘ) 8/3
ব্যাখ্যা

Let x be the amount of water to be added.
New total amount of water = 4 + x
Total amount of chemical = 4
∴ New total = 4 + 4 + x = 8 + x
Final ratio required (for water) = 5/(5 + 3) = 5/8
Thus, (4 + x)/(8 + x) = 5/8
Or, 32 + 8x = 40 + 5x
Or, 3x = 8
Or, x = 8/3

৮৬২.
Rahim weighs 74.5 kg. If he reduces his weight in the ratio 5 : 3, find his reduced weight in kg.
  1. ক
    44.70 kg
  2. খ
    55.20 kg
  3. গ
    37.25 kg
  4. ঘ
    48.40 kg
ব্যাখ্যা

Question: Rahim weighs 74.5 kg. If he reduces his weight in the ratio 5 : 3, find his reduced weight in kg.

Solution:
মনেকরি,
রহিমের পূর্বের ওজন = 5x
রহিমের বর্তমান ওজন = 3x

প্রশ্নমতে,
⇒ 5x = 74.5
⇒ x = 74.5/5
∴ x = 14.9

∴ ওজন কমে যাওয়ার পর হবে = 3 × 14.9
= 44.70 kg

৮৬৩.
A cricket team has won 40 games out of 60 played. It has 32 more games to play. How many of these must the team win to make it record 70% win for the season?
  1. ক
    20
  2. খ
    25
  3. গ
    23
  4. ঘ
    32
ব্যাখ্যা
Question: A cricket team has won 40 games out of 60 played. It has 32 more games to play. How many of these must the team win to make it record 70% win for the season?

Solution: 
মনেকরি,
অবশিষ্ট খেলারগুলোর মধ্যে x টিতে জিততে হবে। 

প্রশ্নমতে, 
40 + x = (60 + 32) এর 70%
⇒ 40 + x = 92 এর 70/100
⇒ 40 + x = 64.4 
⇒ x = 64.4 - 40 
⇒ x = 24.4  ≈ 25
৮৬৪.
100 kg of solution A is mixed with 80 kg of solution B. If solution A has tin and copper in the ratio 1 : 4 and solution B has lead and tin in the ratio 3 : 2, then what is the amount of tin in the new solution?
  1. ক
    ক) 44 kg 
  2. খ
    খ) 24 kg 
  3. গ
    গ) 52 kg 
  4. ঘ
    ঘ) 70 kg 
ব্যাখ্যা
A তে টিন আছে = (100 এর 1/5)
                         = 20 kg 
A তে কপার আছে = (100 এর 4/5)
                             = 80 kg 
B তে লেড আছে = (80 এর 3/5) = 48kg 
B তে টিন আছে =(60 এর 2/5) = 32kg 

নতুন মিশ্রণে টিন আছে = (20 + 32)kg = 52 kg
৮৬৫.
An employer pays 3 workers A, B, and C a total of TK. 48,000 a week. A is paid 150% of the amount B is paid and 60% of the amount C is paid. How much does A make a week?
  1. ক
    12400 tk
  2. খ
    13200 tk
  3. গ
    14400 tk
  4. ঘ
    15600 tk
ব্যাখ্যা
Question: An employer pays 3 workers A, B, and C a total of TK. 48,000 a week. A is paid 150% of the amount B is paid and 60% of the amount C is paid. How much does A make a week?

Solution:
A = 159% of B
⇒ A = 150B/100
⇒ A/B = 150/100
∴ A : B = 3 : 2

A = 60% of C
⇒ A = 60C/100
⇒ A/C = 60/100
∴ A : C = 3 : 5

∴ A : B : C = 3 : 2 : 5

A makes the week = (3/10) × 48000
= 14400 tk
৮৬৬.
In a partnership, three friends invested Tk. 2700, Tk. 8100, and Tk. 7200 respectively. After one year, the third partner received Tk. 3600 as his share of the profit. How much total profit did the business make?
  1. ক
    Tk. 7200
  2. খ
    Tk. 9000
  3. গ
    Tk. 12000 
  4. ঘ
    Tk. 14400 
ব্যাখ্যা

Question: In a partnership, three friends invested Tk. 2700, Tk. 8100, and Tk. 7200 respectively. After one year, the third partner received Tk. 3600 as his share of the profit. How much total profit did the business make?

Solution:
Let the total profit be x

Here,
ratio of investment,
= first partner : second partner : third partner 
= 2700 : 8100 : 7200
= 3 : 9 : 8

Then, third partner's share = 8x/20 = 2x/5

According to the question,
2x/5 = 3600
⇒ x = (3600 × 5)/2 
⇒ x = 9000

So, The total profit is Tk. 9000

৮৬৭.
X and Y started a partnership business investing some amount in the ratio of 4 : 7. Z joined after 8 months with an amount equal to that of X. In what proportion should the profit at the end of one year be distributed among X, Y, and Z? 
  1. ক
    12 : 21 : 5
  2. খ
    12 : 11 : 4
  3. গ
    10 : 21 : 4
  4. ঘ
    12 : 21 : 4
  5. ঙ
    None
ব্যাখ্যা

Question: X and Y started a partnership business investing some amount in the ratio of 4 : 7. Z joined after 8 months with an amount equal to that of X. In what proportion should the profit at the end of one year be distributed among X, Y, and Z?

Solution:
Let the initial investments of X and Y be 4b and 7b.

X : Y : Z = (4b × 12) : (7b × 12) : (4b × 4)
= 48 : 84 : 16
= 12 : 21 : 4

∴ Required proportion - 12 : 21 : 4

৮৬৮.
Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?
  1. ক
    2 : 3 : 4
  2. খ
    6 : 7 : 8
  3. গ
    6 : 8 : 9
  4. ঘ
    2 : 4 : 3
  5. ঙ
    None of these
ব্যাখ্যা
Question: Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?

Solution:
Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
⇒ (140/100) × 5x , (150/100) × 7x and (175/100) × 8x
⇒ 7x, 21x/2 and 14x.

The required ratio = 7x : 21x/2 : 14x
= 14x : 21x : 28x
= 2 : 3 : 4.
৮৬৯.
An iron rod that weights 30 kg is cut into two pieces so that one of these pieces weights 16 kg and 40 m long. If the weight of each piece is proportional to its length, how long is the other one is -
  1. ক
    30
  2. খ
    45
  3. গ
    35
  4. ঘ
    42
ব্যাখ্যা
Question: An iron rod that weights 30 kg is cut into two pieces so that one of these pieces weights 16 kg and 40 m long. If the weight of each piece is proportional to its length, how long is the other one is -

Solution:
Total weight = 30 kg,
1st piece weight = 16 kg and Length = 40 m

So, 2nd piece weight = (30 - 16) = 14 kg
Let the Length of the 2nd piece = x m

Now,
16 : 14 = 40 : x
⇒ 16/14 = 40/x
⇒ x = (40 × 14)/16
∴ x = 35
৮৭০.
If x : y = 5 : 2, then (8x + 9y) : (8x + 2y) is- 
  1. ক
    20 : 22
  2. খ
    29 : 20
  3. গ
    29 : 22
  4. ঘ
    29 : 33
ব্যাখ্যা
Question: If x : y = 5 : 2, then (8x + 9y) : (8x + 2y) is- 

Solution: 
দেয়া আছে,
x : y = 5 : 2
⇒ x/y = 5/2
⇒ 2x = 5y
⇒ 8x = 20y [4 দিয়ে গুণ করে]
∴ 8x = 20y ........ (i)

এখন, (8x + 9y) : (8x + 2y)
= (20y + 9y) : (20y + 2y)
= 29y : 22y
= 29 : 22
৮৭১.
A 60-liter mixture contains milk and water in the ratio 5:1. How many liters of water must be added to the mixture so that the new ratio of milk to water becomes 2:1?
  1. ক
    12 liters
  2. খ
    15 liters
  3. গ
    18 liters
  4. ঘ
    20 liters
ব্যাখ্যা

Question: A 60-liter mixture contains milk and water in the ratio 5:1. How many liters of water must be added to the mixture so that the new ratio of milk to water becomes 2:1?

Solution:
Milk in the mixture = 60 × (5/6) = 50 liters
Water in the mixture = 60 - 50 = 10 liters

Let x liters of water be added.
Then the new ratio should be 2 : 1.

So,
50/(10 + x) = 2/1
⇒ 50 = 2(10 + x)
⇒ 50 = 20 + 2x
⇒ 2x = 30
⇒ x = 15

∴ 15 liters of water should be added.

৮৭২.
The number of students in 3 classes is in the ratio 2 : 3 : 4. If 12 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was?
  1. ক
    162
  2. খ
    148
  3. গ
    208
  4. ঘ
    320
  5. ঙ
    410
ব্যাখ্যা
Question: The number of students in 3 classes is in the ratio 2 : 3 : 4. If 12 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was-

Solution:
Let the number of students in the classes be 2x, 3x and 4x respectively;
∴ Total students = 2x + 3x + 4x = 9x

ATQ,
⇒ (2x + 12)/(3x + 12) = 8/11
⇒ 24x + 96 = 22x + 132
⇒ 24x - 22x = 132 - 96
⇒ 2x = 36
∴ x = 18

Hence, Original number of students,
9x = 9 × 18 = 162
৮৭৩.
A grocer mixes two types of pulses, one costing 15 taka per kg and the other 20 taka per kg. If the mixture is sold at 16.50 Taka per kg, what is the ratio of the two types of pulses in the mixture?
  1. ক
    3 : 7
  2. খ
    5 : 3
  3. গ
    7 : 3
  4. ঘ
    7 : 9
ব্যাখ্যা
Question: A grocer mixes two types of pulses, one costing 15 taka per kg and the other 20 taka per kg. If the mixture is sold at 16.50 Taka per kg, what is the ratio of the two types of pulses in the mixture? 

Solution:
ধরি,
প্রথম প্রকার ডালের পরিমাণ = x কেজি
দ্বিতীয় প্রকার ডালের পরিমাণ = y

প্রথম প্রকার ডালের x কেজির মূল্য = 15x টাকা 
দ্বিতীয় প্রকার ডালের y কেজির মূল্য = 20y টাকা 

প্রশ্নমতে,
15x + 20y = 16.50(x + y)
⇒ 15x + 20y = 16.50x + 16.50y
⇒ 20y - 16.50y = 16.50x - 15x
⇒ 3.50y = 1.50x
⇒ x/y = 3.50/1.50 
⇒ x/y = 7 : 3

শর্টকাট: 

∴ অনুপাত = 3.50 : 1.50
= 7 : 3
৮৭৪.
The incomes of A and B are in the ratio 3 : 2 and their expenditure are in the ratio 5 : 3. If each saves Tk 1000, then, A's income can be-
  1. ক
    ক) 4000 Tk
  2. খ
    খ) 6000 Tk
  3. গ
    গ) 3000 Tk
  4. ঘ
    ঘ) 8000 Tk
ব্যাখ্যা
Question: The incomes of A and B are in the ratio 3 : 2 and their expenditure are in the ratio 5 : 3. If each saves Tk 1000, then, A's income can be-

Solution:
Let the income of A and B be 3x and 2x respectively.
Also, their expenditure is 5y and 3y.

According to the question,
3x - 5y = 1000 ------- (i)
2x - 3y = 1000 ---------- (ii)

Now, {(i) × 3} - {(ii) × 5}
⇒ 9x - 15y - 10x + 15y = 3000 - 5000
⇒ - x = -2000
⇒ x = 2000

Then, income of A = 3x = 3 × 2000 = Tk 6000
৮৭৫.
P, Q and R start a business. P invests 3 times as much as Q invests 2/3rd as much as R invests. Find the ratio of capitals of P, Q and R ? 
  1. ক
    ক) 3 : 2 : 6
  2. খ
    খ) 6 : 2 : 3
  3. গ
    গ) 2 : 6 : 3
  4. ঘ
    ঘ) 5 : 2 : 3
ব্যাখ্যা
ধরি 
R বিনিয়োগ করে = x টাকা 
Q বিনিয়োগ করে = 2x/3 টাকা 
P বিনিয়োগ করে = 3× (2x/3) টাকা
                          = 2x টাকা 
P, Q এবং R এর বিনিয়োগের অনুপাত = 2x : (2x/3) : x
                                                          = 2 : (2/3) : 1
                                                           = 6 : 2 : 3
৮৭৬.
Ratio of milk and water in 630 litre of mixture is 4 : 3. 140 litre mixture taken out. Find the quantity of milk now?
  1. ক
    190 liters
  2. খ
    400 liters
  3. গ
    220 liters
  4. ঘ
    320 liters
  5. ঙ
    280 liters
ব্যাখ্যা
Question: Ratio of milk and water in 630 litre of mixture is 4 : 3. 140 litre mixture taken out. Find the quantity of milk now?

Solution:
Given that,
The ratio of milk to water in 630 liters of mixture is 4 : 3
140 liters of the mixture is taken out.

Now,
The total parts of the mixture = 4 + 3 = 7 parts.
Quantity of milk in the mixture = (4/7) × 630 = 360 liters of milk.
Quantity of water in the mixture = (3/7) × 630 = 270 liters of water.

When 140 liters of the mixture is taken out, the ratio of milk and water in the 140 liters will also be 4 : 3.
Milk taken out = (4/7) × 140 = 80 liters.

So, the remaining milk = 360 - 80 = 280 liters.

∴ The quantity of milk now is 280 liters.

৮৭৭.
73.04 : 72.03 is equal to-
  1. ক
    ক) 7 : 17
  2. খ
    খ) 7 : 5
  3. গ
    গ) 3 : 1
  4. ঘ
    ঘ) 7 : 1
ব্যাখ্যা
Question: 73.04 : 72.03 is equal to-

Solution: 
73.04 : 72.03
= 72.03 × 7 : 72.03
= 7 : 1
৮৭৮.
A, B, C have the total money of Tk. 1400. B have half of A and C have half of B. the amount of C is
  1. ক
    ক) 200
  2. খ
    খ) 400
  3. গ
    গ) 800
  4. ঘ
    ঘ) 300
ব্যাখ্যা
question: A, B, C have the total money of Tk. 1400. B have half of A and C have half of B. The amount of C is 

Solution: 
Let A have X

then, 
B have X/2 and
C have X/4

so,
X + (X/2) + (X/4) = 1400
7X/4 = 1400
X = 800

hence, 
the amount of C is (800/4) or, 200
৮৭৯.
If the income of A is 10% more than that of B and the income of B is 20% less than that of C, then the income of A, B and C respectively are in the ratio
  1. ক
    ক) 22 : 18 : 25
  2. খ
    খ) 22 : 20 : 25
  3. গ
    গ) 10 : 9 : 7
  4. ঘ
    ঘ) 11 : 10 : 8
ব্যাখ্যা

ধরি, C এর ইনকাম 100 টাকা
তাহলে B এর ইনকাম (100 - 100 এর 20%) = 80 টাকা
এবং A এর ইনকাম (80 + 80 এর 10 %) = 88 টাকা।
∴ A : B : C = 88 : 80 : 100 = 22 : 20 : 25

৮৮০.
The ratio of copper to gold in an ornament weighing 16 grams is 1 : 3. How much more gold must be added to it to make the ratio 1 : 4?
  1. ক
    ক) 3 grams
  2. খ
    খ) 4 grams
  3. গ
    গ) 5 grams
  4. ঘ
    ঘ) 8 grams
ব্যাখ্যা
Question: The ratio of copper to gold in an ornament weighing 16 grams is 1 : 3. How much more gold must be added to it to make the ratio 1 : 4?

Solution:
The ratio of copper to gold in an ornament weighing 16 grams is 3 : 1
gold weight = (16 × 3)/4 = 12 grams
copper weight = 16 - 12 = 4 grams

let, x grams gold to be added

So,
4/(12 + x) = 1/4
⇒ 12 + x = 16
∴ x = 4

So, 4 grams gold to be added
৮৮১.
The difference between two positive numbers is 15 and the ratio between them is 7 : 4. Find the sum of the two numbers.
  1. ক
    ক) 40
  2. খ
    খ) 50
  3. গ
    গ) 55
  4. ঘ
    ঘ) 60
ব্যাখ্যা
Question: The difference between two positive numbers is 15 and the ratio between them is 7 : 4. Find the sum of the two numbers.

Solution:
Let the two positive numbers be 7x and 4x respectively

According to the question,
7x - 4x = 15
⇒ 3x = 15
⇒ x = 5

Then numbers are 7 × 5 = 35 and 4 × 5 = 20
So, sum of numbers = 35 + 20 = 55
৮৮২.
If a + b : b + c : c + a = 6 : 7 : 8 and a + b + c = 14, Then the value of c is-
  1. ক
    ক) 8
  2. খ
    খ) 6
  3. গ
    গ) 4
  4. ঘ
    ঘ) 2
ব্যাখ্যা
Question: If a + b : b + c : c + a = 6 : 7 : 8 and a + b + c = 14, Then the value of c is- 

Solution: 
 a + b : b + c : c + a = 6 : 7 : 8

Let
a + b = 6k 
b + c = 7k
c + a = 8k

Now,
a + b + b + c + c + a = 6k + 7k + 8k
⇒ 2a + 2b + 2c = 6k + 7k + 8k
⇒ 2(a + b + c) = 21k
⇒ 2 × 14 = 21k
⇒ 21k = 28
⇒ k = 28/21
⇒ k =4/3

Again,
a + b = 6k
= 6 × (4/3)
= 8

∴ c = ( a + b + c) - (a + b)
= 14 - 8
= 6
৮৮৩.
In a mixture, the ratio of milk and water is 6 : 5. When 22 liter mixture is replaced by water, the ratio become 9 : 13. Find the quantity of water after replacement.
  1. ক
    ক) 40 liters
  2. খ
    খ) 48 liters
  3. গ
    গ) 57 liters
  4. ঘ
    ঘ) 52 liters
ব্যাখ্যা
Question: In a mixture, the ratio of milk and water is 6 : 5. When 22 liter mixture is replaced by water, the ratio become 9 : 13. Find the quantity of water after replacement.

Solution: 
Let milk = 6X, water = 5X
so, 
total mixture = 6X + 5X = 11X

When 22 liter of mixture is replaced with water, then remaining milk is = 6X - (22 × 6X/11X) = 6X - 12

remaining water = 5X - (22 × 5X/11X) + 22  = 5X +  12

ATQ,
6X - 12 : 5X + 12 = 9 : 13
13(6X - 12) = 9(5X + 12)
78X - 156 = 45X + 108
78X - 45X = 156 + 108
33X = 164
X = 8

∴ water after replacement = 5X + 12 = 40 + 12 = 52 liters 
৮৮৪.
A box has coins worth 50, 25, and 10 paisa in a 1 : 2 : 5 ratio. If all the coins add up to 36 Taka, how much of that is from the 50 paisa coins alone?
  1. ক
    Tk. 11
  2. খ
    Tk. 12
  3. গ
    Tk. 33
  4. ঘ
    Tk. 24
ব্যাখ্যা
Question: A box has coins worth 50, 25, and 10 paisa in a 1 : 2 : 5 ratio. If all the coins add up to 36 Taka, how much of that is from the 50 paisa coins alone?

Solution:
৮৮৫.
If the ratios A : B = 1 : 2, B : C = 4 : 3, hold true, and their total sum is 630, determine C.
  1. ক
    270
  2. খ
    140
  3. গ
    210
  4. ঘ
    70
ব্যাখ্যা

Question: If the ratios A : B = 1 : 2, B : C = 4 : 3, hold true, and their total sum is 630, determine C.

Solution: 
Given,
A : B = 1 : 2 = 2 : 4
B : C = 4 : 3

∴ A : B : C = 2 : 4 : 3

∴ Value of C = 630 × 3/9
= 210

৮৮৬.
A vessel is full of mixture of spirit and water in which there is 20% of spirit. 5 litres are drawn off and the vessel is filled up with water. If the spirit is now 12%, find the total quantity in the vessel (in ltrs).
  1. ক
    72
  2. খ
    12.5
  3. গ
    60
  4. ঘ
    7.2
ব্যাখ্যা
Question: A vessel is full of mixture of spirit and water in which there is 20% of spirit. 5 litres are drawn off and the vessel is filled up with water. If the spirit is now 12%, find the total quantity in the vessel (in ltrs).

Solution:
Let the total mixture be x liters.
It contains 20% spirit i.e. 0.2 of the total mixture.
When 5 liters are drawn off, quantity of spirit removed = 0.2(5) = 1 liters.
Remaining spirit = 0.2x - 1.
Also remaining spirit = 0.12x.

∴ 0.2x - 1 = 0.12x
⇒ 0.08x = 1
⇒ x = 1/0.08
⇒ x = 12.5
৮৮৭.
If A : B = 3 : 2, B : C = 5 : 4, C : D = 3 : 7 then A : B : C : D =?
  1. ক
    ক) 45 : 30 : 56 : 24
  2. খ
    খ) 30 : 45 : 56 : 24
  3. গ
    গ) 45 : 30 : 24 : 56
  4. ঘ
    ঘ) 30 : 56 : 45 : 24
ব্যাখ্যা
According to the question,
A : B = (3 : 2) × 5 = 15 : 10
B : C = (5 : 4) × 2 = 10 : 8
A : B : C = 15 : 10 : 8

Again,
A : B : C = (15 : 10 : 8) × 3 = 45 : 30 : 24
C : D = (3 : 7) × 8 = 24 : 56

A : B : C : D = 45 : 30 : 24 : 56

∴ The value of A : B : C : D is 45 : 30 : 24 : 56.
৮৮৮.
Two men, A and B, run a 4 km race on a circular course of 1/4 km. If their speeds are in the ratio of 5:4. How often does the winner pass the other?
  1. ক
    ক) Thrice
  2. খ
    খ) Four times
  3. গ
    গ) Once
  4. ঘ
    ঘ) Twice
ব্যাখ্যা

Number of rounds to be completed = 4 / 1/4 = 16 times
Since the ratio of speeds of A and B is 5 : 4
⇒ When A covers 5 rounds, B will complete 4 rounds and they will meet.
⇒ A will pass B after 5th, 10th & 15th round.
∴ The winner passes the other thrice.

৮৮৯.
The ratio of two numbers is 7:4. If 8 is added to both the numbers ratio becomes 13:8. What is the smaller number?
  1. ক
    ক) 52
  2. খ
    খ) 56
  3. গ
    গ) 38
  4. ঘ
    ঘ) 40
ব্যাখ্যা

Let the numbers be 7x and 4x
ATQ,
(7x + 8) : (4x + 8) = 13 : 8
Or, 56x + 64 = 52x + 104
Or, 4x = 40

The smaller number is 40

৮৯০.
Three numbers A, B and C are in the ratio of 12 : 15 : 25. If sum of these numbers is 312, ratio between the difference of B and A and the difference of C and B is –
  1. ক
    ক) 3 : 7
  2. খ
    খ) 10 : 3
  3. গ
    গ) 3 : 10
  4. ঘ
    ঘ) 7 : 3
ব্যাখ্যা

Let the number A, B and C be 12a , 15a and 25a respectively
Then from the question 12a + 15a + 25a = 312
⇒ 52a = 312
⇒ a = 312/52
⇒ a = 6
Required ratio = {(15 × 6) - (12 × 6)}/{(25 × 6) - (15 × 6)}
= 3/10
= 3 : 10.

৮৯১.
Two containers contain milk and water in the ratios 5:2 and 9:5. What ratio should the mixtures be combined in to achieve a final ratio of 2:1 milk to water?
  1. ক
    1 : 3
  2. খ
    3 : 2
  3. গ
    3 : 7
  4. ঘ
    5 : 8
  5. ঙ
    None of the above
ব্যাখ্যা
Question: Two containers contain milk and water in the ratios 5:2 and 9:5. What ratio should the mixtures be combined in to achieve a final ratio of 2:1 milk to water?

Solution:
Let,
P unit of the first mixture is added to Q unit of the second mixture.

So, in the P unit of the first mixture,
Amount of milk present = (5/7) × P = 5P/7
Amount of water present = (2/7) × P = 2P/7

In the Q unit of the second mixture,
Amount of milk present = (9/14) × Q = 9Q/14
Amount of water present = (5/14) × Q = 5Q/14

ATQ,
{(5P/7) + (9Q/14)}/{(2P/7) + (5Q/14)} = 2/1
⇒ {(10P + 9Q)/14}/{(4P + 5Q)/14} = 2
⇒ 10P + 9Q = 8P + 10Q
⇒ 2P = Q

∴ P : Q = 1 : 2
৮৯২.
The perimeter of a rectangle is 64 cm. If the ratio of the lengths of two adjacent sides is 7 : 9, find the lengths of these sides.
  1. ক
    24 cm, 28 cm
  2. খ
    14 cm, 18 cm
  3. গ
    7 cm, 9 cm
  4. ঘ
    None of these
ব্যাখ্যা
Question: The perimeter of a rectangle is 64 cm. If the ratio of the lengths of two adjacent sides is 7 : 9, find the lengths of these sides.

Solution:
Perimeter of a rectangle = 2(Length + Breadth)
Also Length :  Breadth = 9 : 7
Let actual values are 9x and 7x.

Hence,
2(9x + 7x) = 64
⇒ 16x = 32
∴ x = 2

∴ sides will be of 14 cm and 18 cm.
৮৯৩.
A bucket contains a mixture of two liquids A & B in the proportion 5 : 3. If 16 litres of the mixture is replaced by 16 litres of liquid B, then the ratio of the two liquids becomes 3 : 5. How much of the liquid B was there in the bucket?
  1. ক
    8 litres
  2. খ
    15 litres
  3. গ
    18 litres
  4. ঘ
    6 litres
ব্যাখ্যা
Question: A bucket contains a mixture of two liquids A & B in the proportion 5 : 3. If 16 litres of the mixture is replaced by 16 litres of liquid B, then the ratio of the two liquids becomes 3 : 5. How much of the liquid B was there in the bucket?

Solution:
Let, bucket contains 5a and 3a of liquids A and B respectively.
When 16 litres of mixture are drawn off, quantity of A in mixture left:
5a - (5/8) ×16 = 5a - 10
Similarly quantity of B in mixture left,
(3a - 6) + 16 = 3a + 10

Now the ratio becomes,
(5a - 10)/(3a + 10) = 3/5
⇒ 25a - 50 = 9a + 30
⇒ 16a = 80
∴ a = 5

So, quantity of liquid B initially = (3 × 5) = 15 litres
৮৯৪.
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, C receives:
  1. ক
    ক) 7200 Tk
  2. খ
    খ) 8200 Tk
  3. গ
    গ) 8400 Tk
  4. ঘ
    ঘ) 6200 Tk
ব্যাখ্যা
Let C = x.
Then, B = x + 5000
A = x + 5000 + 4000
    = x + 9000.

So,
x + x + 5000 + x + 9000 = 50000
3x = 36000
 x = 12000

A : B : C = 21000 : 17000 : 12000
              = 21 : 17 : 12

C's share = 35000 × 12/50 = Tk. 8400
৮৯৫.
A mixture contains wine and water in the ratio 3 : 2 and another mixture contains them in the ratio 4 : 5 . How many litres of the latter mixture must be mixed with 3 litres of the former mixture so that the resultant mixture may contain equal quantities of wine and water?
  1. ক
    4.5
  2. খ
    5.4
  3. গ
    3.75
  4. ঘ
    5.67
  5. ঙ
    6.00
ব্যাখ্যা
Question: A mixture contains wine and water in the ratio 3 : 2 and another mixture contains them in the ratio 4 : 5 . How many litres of the latter mixture must be mixed with 3 litres of the former mixture so that the resultant mixture may contain equal quantities of wine and water?

Solution:
The former mixture contains wine and water in a ratio of 3 : 2
3 litres of the mixture contains 1.8 litres of wine and 1.2 litres of water to maintain a 3 : 2 ratio.

Let
there be 4x litres of wine and 5x litres of water.
After mixing, Quantity of wine = Quantity of water
⇒1.8 + 4x = 1.2 + 5x
⇒ x = 0.6 litres
⇒9x = 9 × 0.6 = 5.4 litres
৮৯৬.
Helal and Tonmoy share some sweets in a ratio of 7 : 5. Helal has 12 more sweets than Tonmoy. How many sweets were there altogether?
  1. ক
    30
  2. খ
    42
  3. গ
    72
  4. ঘ
    None of these
ব্যাখ্যা
প্রশ্ন: Helal and Tonmoy share some sweets in a ratio of 7 : 5. Helal has 12 more sweets than Tonmoy. How many sweets were there altogether?

সমাধান: 
Let,
Helal has 7x sweets 
Tonmoy has 5x sweets 
∴ Total sweets 7x + 5x = 12x

ATQ,
7x - 5x = 12
⇒ 2x = 12
∴ x = 6

∴ There were 12 × 6 = 72 sweets altogether.
৮৯৭.
The ratio between two numbers is 7 : 9. If 25% of the smaller number is added to 30% of the larger number, the result is 178. What is the greater number?
  1. ক
    180
  2. খ
    270
  3. গ
    360
  4. ঘ
    450
ব্যাখ্যা

Question: The ratio between two numbers is 7 : 9. If 25% of the smaller number is added to 30% of the larger number, the result is 178. What is the greater number?

Solution:
Let the numbers be 7x and 9x.
Then, 
25% of the smaller = 0.25 × 7x = 1.75x
30% of the larger = 0.30 × 9x = 2.7x

According to the question,
1.75x + 2.7x = 178
⇒ 4.45x = 178
⇒ x = 178/4.45
∴ x = 40

∴ The greater number is (9 × 40) = 360

৮৯৮.
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- 
  1. ক
    1 : 5
  2. খ
    1 : 6
  3. গ
    1 : 4
  4. ঘ
    1 : 7
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: 
let, new mixture is 12 liters

old mixture = 12 × (1/4) = 3 liters 
sugar syrup = 9 liters 

new sugar syrup = 9 + (3/2) = 10.5 

ratio of lemon juice and sugar syrup = 1.5 : 10.5 
= 15 : 105 
= 1 : 7 
৮৯৯.
Consider three mixtures- the first having water and liquid A in the ratio 1 : 2, the second having water and liquid B in the ratio 1 : 3, and the third having water and liquid C in the ratio 1 : 4. These three mixtures of A, B, and C, respectively, are further mixed in the proportion 4 : 3 : 2. Then the resulting mixture has
  1. ক
    The same amount of water and liquid B
  2. খ
    The same amount of liquids B and C
  3. গ
    More water than liquid B
  4. ঘ
    More water than liquid A
ব্যাখ্যা
Question: Consider three mixtures- the first having water and liquid A in the ratio 1 : 2, the second having water and liquid B in the ratio 1 : 3, and the third having water and liquid C in the ratio 1 : 4. These three mixtures of A, B, and C, respectively, are further mixed in the proportion 4 : 3 : 2. Then the resulting mixture has

Solution:
Mixtures A, B, & C are mixed in the ratio 4 : 3 : 2
Let the volumes = 4k, 3k, 2k respectively

Assume k = 60 [LCM(3, 4 & 5) = 60]
Volume of Liquid A = 8k/3 = 160
Volume of Liquid B = 9k/4 = 135
Volume of Liquid C = 8k/5 = 96
Volume of water = (4k/3) + (3k/4) + (2k/5)
= 80 + 45 + 24
= 149

There is more water than Liquid B
৯০০.
8 liters Cold drink was added to a 20 liters mixture of water and alcohol such that the ratio of alcohol to that of cold drink and water is 1 : 3. Find the amount of alcohol in the solution.
  1. ক
    6 liters
  2. খ
    7 liters
  3. গ
    14 liters
  4. ঘ
    13 liters
ব্যাখ্যা
Question: 8 liters Cold drink was added to a 20 liters mixture of water and alcohol such that the ratio of alcohol to that of cold drink and water is 1 : 3. Find the amount of alcohol in the solution.

Solution:
Let,
Quantity of alcohol A
Quantity of water W
Quantity of cold drinks C = 8 liters

We have a 20 liters mixture of alcohol and water.
⇒ A + W = 20
∴ W = 20 - A ................(1)

ATQ,
A : (W + C) = 1 : 3
⇒ A/(W + 8) = 1/3
⇒ 3A = W + 8
⇒ W = 3A - 8  ................(2)

From (1) and (2)
3A - 8 = 20 - A
⇒ 4A = 28
⇒ A = 7

∴  7 liters of alcohol is present in the mixture.