Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ৩ / ১১ · ২০১–৩০০ / ১,০৮৬

২০১.

A solution containing 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

ক
13.12%
খ
18.18%
গ
19.19%
ঘ
28.13%

সঠিক উত্তর: খ

18.18%

উত্তর

সঠিক উত্তর: খ

18.18%

খ

ব্যাখ্যা

Question: A solution containing 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

Solution:
let, solution is 100 unit
amount of sugar = 100 × 10%
= 100 × 10/100
= 10 unit

by doubling, amount of sweet = 20 unit
solutin = 100 + 10 = 110 unit

percent of sugar = 20 × 100%/110
= 18.18%

২০২.

Asim, Raju and Rokon agree to pay their total electricity bill in the proportion 3 : 4 : 5. Asim pays the first day's bill of Tk. 50, Raju pays the second day's bill of Tk. 55 and Rokon pays the third day's bill of Tk. 75. How much should Asim pay/get to settle the accounts?

ক
15
খ
17
গ
12
ঘ
5

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Total bill paid by Asim, Raju and Rokon = ( 50 + 55 +75 ) = Tk. 180

Let the amount paid by Asim, Raju, and Rokon be Tk. 3x, 4x and 5x respectively.
Therefore, (3x + 4x + 5x ) = 180
12x = 180
x = 15

Therefore, the amount to be paid by,
Asim = Tk. 45
Raju = Tk. 60
Rokon = Tk. 75

But actually as given in the question, Asim pays Tk. 50, Raju pays Tk. 55 and Rokon pays Tk. 80.
Hence, Asim pays Tk. 5 more and Raju 5 Tk less than the actual amount to be paid.
Hence Raju needs to pay Tk. 5 to Asim to settle the amount.

২০৩.

Two numbers are in the raton 3 : 4. If the difference of their squares is 63, then what is the smaller number?

ক
ক) 6
খ
খ) 9
গ
গ) 12
ঘ
ঘ) 15

সঠিক উত্তর: খ

খ) 9

উত্তর

সঠিক উত্তর: খ

খ) 9

খ

ব্যাখ্যা

Question: Two numbers are in the raton 3 : 4. If the difference of their squares is 63, then what is the smaller number?

Solution:
Let the numbers be 3x and 4x.
Then, (4x)² - (3x)² = 63
⇒ 16x² - 9x² = 63
⇒ 7x² = 63
⇒ x² = 9
∴ x = 3

∴ The smaller number (3 × 3) = 9

২০৪.

The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the numbers of teachers were to increase by 5, the ratio of students would then be 25 to 1. What is the present number of teachers?

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 15

সঠিক উত্তর: ঘ

ঘ) 15

উত্তর

সঠিক উত্তর: ঘ

ঘ) 15

ঘ

ব্যাখ্যা

Question: The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the numbers of teachers were to increase by 5, the ratio of students would then be 25 to 1. What is the present number of teachers?

Solution:
Let the number of student and teacher be represented by S and T respectively.
S : T = 30 : 1
S/T = 30/1
S = 30T.............(1)

If number of students increases by 50 and number of teachers increases by 5
50 + S : T + 5 = 25 : 1
⇒ (50 + S)/(T + 5) = 25/1
⇒ S + 50 = 25(T + 5)
⇒ S + 50 = 25T + 125
⇒ 30T + 50 = 25T + 125 [From (1)]
⇒ 30T - 25T = 125 - 50
⇒ 5T = 75
⇒ T= 75/5 =15

∴ No of teachers=T=15

২০৫.

An 80L solution of alcohol and water has 45% alcohol in it. If you want the mixture to be 75% alcohol, how much alcohol would you add to it?

ক
30 litres
খ
75 litres
গ
96 litres
ঘ
110 litres

সঠিক উত্তর: গ

96 litres

উত্তর

সঠিক উত্তর: গ

96 litres

গ

ব্যাখ্যা

Currently alcohol quantity = (45/100) × 80 = 36 litres.
Let A be alcohol added.

So,
36 + A = (75/100) × (80 + A)
⇒ 36 + A = (3/4) × (80 + A)
⇒ 144 + 4A = 240 + 3A
⇒ A = 240 - 144 = 96

∴ A = 96 Litres = This is the additional quantity of alcohol to be added.

২০৬.

A jar contains white, red and green marbles in the ratios 2 : 3 : 5. Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7. How many green marbles are there in the jar?

ক
ক) 15
খ
খ) 18
গ
গ) 21
ঘ
ঘ) 24

সঠিক উত্তর: গ

গ) 21

উত্তর

সঠিক উত্তর: গ

গ) 21

গ

ব্যাখ্যা

Question: A jar contains white, red and green marbles in the ratios 2 : 3 : 5. Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7. How many green marbles are there in the jar?

Solution:
A jar contains white, red and green marbles in the ratios 2 : 3 : 5
let there are 2x white, 3x red and 5x green marbles.

Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7
let there are 2y white, 3y red and 7y green marbles.
2x = 2y
⇒ x = y

7y - 6 = 5x
⇒ 7x - 6 = 5x
⇒ 7x - 5x = 6
∴ x = 6/2
= 3

green marbels = 7 × 3
= 21

২০৭.

The ratio of the area of a square to that of the square drawn on its diagonal, is-

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

সঠিক উত্তর: খ

খ) 1 : 2

উত্তর

সঠিক উত্তর: খ

খ) 1 : 2

খ

ব্যাখ্যা

Let the side of a square be a.
Area of square = a²
Length of the diagonal = √(a² + a²) =a√2
Area of a square formed on diagonal of first square is = (a√2)²=2a²

Now
a²/2a²=1/2
= 1 : 2

২০৮.

A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Tk. 6500 for 6 months, B, Tk. 8400 for 5 months and C, Tk. 10,000 for 3 months. B wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Tk. 7400. Calculate the share of C in the profit.

ক
Tk. 1680
খ
Tk. 1900
গ
Tk. 2470
ঘ
Tk. 2660
ঙ
None

সঠিক উত্তর: খ

Tk. 1900

উত্তর

সঠিক উত্তর: খ

Tk. 1900

খ

ব্যাখ্যা

Question: A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Tk. 6500 for 6 months, B, Tk. 8400 for 5 months and C, Tk. 10,000 for 3 months. B wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Tk. 7400. Calculate the share of C in the profit.

Solution:

For managing, B received = 5% of Tk. 7400
= (5/100) × 7400
= Tk. 370

Balance = Tk. (7400 - 370)
= Tk. 7030

∴ Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3)

= 39000 : 42000 : 30000

= 13 : 14 : 10

Sum of the ratio = (13 + 14 + 10) = 37

C's share = [7030 × (10/37)]
= Tk. 1900

২০৯.

Two containers have mixtures of milk and water, respectively, in the ratios 3 ∶ 2 and 6 ∶ 5. In what ratio should the contents be mixed so that the ratio of milk to water in the final mixture is 4 ∶ 3?

ক
ক) 9 ∶ 14
খ
খ) 10 ∶ 11
গ
গ) 6 ∶ 13
ঘ
ঘ) 5 ∶ 8

সঠিক উত্তর: খ

খ) 10 ∶ 11

উত্তর

সঠিক উত্তর: খ

খ) 10 ∶ 11

খ

ব্যাখ্যা

Two containers have mixtures of milk and water, respectively, in the ratios 3 ∶ 2 and 6 ∶ 5.
The ratio of milk to water in the final mixture is 4 : 3.

Let P unit of the first mixture is added to Q unit of the second mixture.
So, in P unit of first mixture,

Amount of milk present = 3/5 × P = 3P/5
Amount of water present = 2/5 × P = 2P/5

So, in Q unit of second mixture,
Amount of milk present = 6/11 × Q = 6Q/11
Amount of water present = 5/11 × Q = 5Q/11

According to the question,
(3P/5 + 6Q/11) ÷ (2P/5 + 5Q/11) = 4 ÷ 3
⇒ {(33P + 30Q)/55} ÷ {(22P + 25Q)/55} = 4 ÷ 3
⇒ 99P + 90Q = 88P + 100Q
⇒ 11P = 10Q
⇒ P : Q = 10 : 11

∴ In 10 : 11 the contents should be mixed so that the ratio of milk to water in the final mixture is 4 ∶ 3.

২১০.

Two numbers are in the ratio 5 : 4 and their difference is 10. Find the largest number.

ক
72
খ
40
গ
70
ঘ
50

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Two numbers are in the ratio 5 : 4 and their difference is 10. Find the largest number.

Solution:
Let the two numbers be,
5x and 4x

According to the problem,
5x - 4x = 10
⇒ x = 10

So, the numbers are,
5x = 5 × 10 = 50 and 4x = 4 × 10 = 40

∴ The largest number = 50

২১১.

If x : y = 2 : 3 and y : z = 4 : 5, then (x + y) : (y + z) is equal to -

ক
ক) 5 : 9
খ
খ) 20 : 27
গ
গ) 10 : 27
ঘ
ঘ) 15 : 23

সঠিক উত্তর: খ

খ) 20 : 27

উত্তর

সঠিক উত্তর: খ

খ) 20 : 27

খ

ব্যাখ্যা

Given that
x : y = 2 : 3 = 8 : 12
y : z = 4 : 5 = 12 : 15

x : y : z = 8 : 12 : 15
Let
x = 8a , y = 12a , z = 15a

x + y = 8a + 12a = 20a
y + z =12a + 15a = 27a

(x + y) : (y + z) = 20a : 27a
= 20 : 27

২১২.

The ratio of copper and zinc in an alloy is 5 : 2. If 12 kg of zinc is added, the ratio becomes 5 : 4. What is the initial weight of the copper?

ক
24 kg
খ
36 kg
গ
30 kg
ঘ
40 kg

সঠিক উত্তর: গ

30 kg

উত্তর

সঠিক উত্তর: গ

30 kg

গ

ব্যাখ্যা

Question: The ratio of copper and zinc in an alloy is 5 : 2. If 12 kg of zinc is added, the ratio becomes 5 : 4. What is the initial weight of the copper?

Solution:
Given that,
The initial ratio of copper to zinc is 5 : 2.
Let the initial quantity of copper = 5x kg
and initial quantity of zinc = 2x kg

When 12 kg of zinc is added when,
Copper remains = 5x kg
And zinc becomes = (2x + 12) kg

Now the new ratio becomes 5 : 4, so we get,
⇒ 5x/(2x + 12) = 5/4
⇒ 4 × 5x = 5 × (2x + 12)
⇒ 20x = 10x + 60
⇒ 20x - 10x = 60
⇒ 10x = 60
∴ x = 6
∴ Initial quantity of copper = 5x = 5 × 6 = 30 kg

So the initial weight of the copper is 30 kg.

২১৩.

A jar contains white, red and green marbles in the ratios 2 : 3 : 5. Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7. How many green marbles are there in the jar?

ক
15
খ
18
গ
21
ঘ
24

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: A jar contains white, red and green marbles in the ratios 2 : 3 : 5. Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7. How many green marbles are there in the jar?

Solution:
A jar contains white, red and green marbles in the ratios 2 : 3 : 5
let there are 2x white, 3x red and 5x green marbles.

Six more green marbles are added to the jars and then the ratio becomes 2 : 3 : 7
let there are 2y white, 3y red and 7y green marbles.
2x = 2y
⇒ x = y

7y - 6 = 5x
⇒ 7x - 6 = 5x
⇒ 7x - 5x = 6
∴ x = 6/2
= 3

green marbels = 7 × 3
= 21

২১৪.

An iron rod that weights 24 kg is cut into pieces so that one of these pieces weighs 16 kg and is 34m long. If the weight of each piece is proportional to its length, how long is the other piece?

ক
15 m
খ
21 m
গ
17 m
ঘ
12 m

সঠিক উত্তর: গ

17 m

উত্তর

সঠিক উত্তর: গ

17 m

গ

ব্যাখ্যা

Question: An iron rod that weights 24 kg is cut into pieces so that one of these pieces weighs 16 kg and is 34m long. If the weight of each piece is proportional to its length, how long is the other piece?

Solution:
Given that,
Total weight of rod = 24 kg
Cut into two pieces, one weighs 16 kg and is 34 m long
Weight ∝ Length
Let the length of the other piece be L₂.
Its weight is W₂ = 24 - 16 = 8 kg.

Since weight is proportional to length,

W₁/L₁ = W₂/L₂
⇒ 16/34 = 8/L₂
⇒ 16 × L₂ = 8 × 34
⇒ L₂ = (8 × 34)/16
∴ L₂ = 17 m

So the other piece is 17 meters long.

২১৫.

Two friends Roman and Shimul invest in a grocery shop. Shimul invests Tk. 25000/- while Roman invests Tk. 35000. Third friend Salim, joins them with the condition that all of them must get equal share of profit. To do so, he gives Tk. 400000 to Roman and Shimul to share between themselves. Find the ratio in which Roman and Shimul should share the money given by Salim?

ক
ক) 5:7
খ
খ) 7:5
গ
গ) 7:13
ঘ
ঘ) 13:7

সঠিক উত্তর: গ

গ) 7:13

উত্তর

সঠিক উত্তর: গ

গ) 7:13

গ

ব্যাখ্যা

The total value of investment of Roman and Shimul after 12 months is =
Tk. 35000 x 12 months : Tk. 25000 x 12 months = 420000 : 300000

Profits need to be the same so investment share must be the same too.

Now 400000 given by Salim needs to be shared by Roman and Shimul so that their investment value becomes the same.
∴ If Tk. X must be given to Roman,
Then,
420000 + X = 300000 + (400000-X)
∴ X = Tk. 140000 = Roman should get this much
Required ratio = Share of Roman: Share of Shimul
= 140000 : (400000 - 140000)
= 7 : 13.

২১৬.

Two equal glasses are respectively two-third and three-fourth full of milk. They are then filled up with water and the contents are mixed in a tumbler. Ratio of milk and water in tumbler is

ক
3 : 19
খ
5 : 17
গ
9 : 13
ঘ
17 : 7

সঠিক উত্তর: ঘ

17 : 7

উত্তর

সঠিক উত্তর: ঘ

17 : 7

ঘ

ব্যাখ্যা

Question: Two equal glasses are respectively two-third and three-fourth full of milk. They are then filled up with water and the contents are mixed in a tumbler. Ratio of milk and water in tumbler is

Solution:
In 1st glass milk = 2/3
Water = (1 - 2/3)
= 1/3

In 2nd glass milk = 3/4
Water = (1 - 3/4)
= 1/4

Ratio of milk and water = (2/3 + 3/4) : (1/3 + 1/4)
= {(8 + 9)/12} : {(4 + 3)/12}
= (17/12) : (7/12)
= 17 : 7

২১৭.

A 60g silver-copper alloy contains 70% silver. How much additional silver is needed to raise the silver percentage to 85%?

ক
80g
খ
70g
গ
60g
ঘ
100g
ঙ
None

সঠিক উত্তর: গ

60g

উত্তর

সঠিক উত্তর: গ

60g

গ

ব্যাখ্যা

Question: A 60 g silver-copper alloy contains 70% silver. How much additional silver is needed to raise the silver percentage to 85%?

Solution:
Silver in alloy = 60 × 70% = 42 g
Copper in alloy = 60 × 30% = 18 g

Let the additional silver be x g.

Then, total weight after adding silver = 42 + x + 18 = 60 + x

ATQ,
(42 + x)/(60 + x) = 85/100
⇒ 100(42 + x) = 85(60 + x)
⇒ 4200 + 100x = 5100 + 85x
⇒ 100x - 85x = 5100 - 4200
⇒ 15x = 900
∴ x = 60 g

২১৮.

Reena and Shaloo are partners in a business, Reena invest Tk. 35,000 for 8 months and Shallo invest Tk. 42,000 for 10 months. Out of a profit of Tk. 31,570, Reena's share is:

ক
Tk. 9471
খ
Tk. 12,628
গ
Tk. 18,040
ঘ
Tk. 18,942

সঠিক উত্তর: খ

Tk. 12,628

উত্তর

সঠিক উত্তর: খ

Tk. 12,628

খ

ব্যাখ্যা

Question: Reena and Shaloo are partners in a business, Reena invest Tk. 35,000 for 8 months and Shallo invest Tk. 42,000 for 10 months. Out of a profit of Tk. 31,570, Reena's share is:

Solution:
Ratio of Reena and Shaloo shares = (35000 × 8) : (42000 × 10)
= 280000 : 420000
= 2 : 3.

Reena's share Tk. 31570 × (2/5)
= Tk. 12628

২১৯.

To fill a tank, 28 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to four-fifth of its present?

ক
35
খ
28
গ
25
ঘ
None of the above

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: To fill a tank, 28 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to four-fifth of its present?

Solution:

ধরি,
বালতির ধারণক্ষমতা x
ট্যাঙ্কের ধারণক্ষমতা 28x

আবার,
বালতির নতুন ধারণক্ষমতা (4x/5)

∴ প্রয়োজনীয় বালতির সংখ্যা = 28x/(4x/5)
= 28x × (5/4x)
= (7 × 5) টি
= 35 টি

২২০.

What is the ratio of one third of 7/15 and two third of the same number?

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

সঠিক উত্তর: খ

খ) 1 : 2

উত্তর

সঠিক উত্তর: খ

খ) 1 : 2

খ

ব্যাখ্যা

Question: What is the ratio of one third of 7/15 and two third of the same number?

Solution:
one third of 7/15 is {(1/3) × (7/15)} = 7/45
two third of 7/15 is {(2/3) × (7/15)} = 14/45

ratio = 7/45 : 14/45 = 1 : 2

২২১.

A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk is respectively -

ক
1 : 10
খ
1 : 5
গ
1 : 4
ঘ
2 : 5

সঠিক উত্তর: গ

1 : 4

উত্তর

সঠিক উত্তর: গ

1 : 4

গ

ব্যাখ্যা

Question: A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk is respectively -

Solution:
ধরি,
১ লিটার দুধের ক্রয়মূল্য ১০০ টাকা
২৫% লাভে মিশ্রনের বিক্রয়মূল্য = (১০০ + ২৫) = ১২৫ টাকা

১০০ টাকার দুধ = ১ লিটার
১২৫ টাকার দুধ = (১২৫/১০০) = ৫/৪ লিটার।

কিন্তু মিশ্রনে দুধের পরিমান ১ লিটার।
∴ পানির পরিমান = (৫/৪) - ১ = ১/৪ লিটার।

∴ পানি : দুধ = ১/৪ : ১
= ১ : ৪

২২২.

100 kg of solution A is mixed with 60 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?

ক
ক) 70 kg
খ
খ) 36 kg
গ
গ) 44 kg
ঘ
ঘ) 56 kg

সঠিক উত্তর: গ

গ) 44 kg

উত্তর

সঠিক উত্তর: গ

গ) 44 kg

গ

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 = (60 এর 3/5) = 36kg
B তে টিন আছে =(60 এর 2/5) = 24kg

নতুন মিশ্রণে টিন আছে = (20 + 24)kg = 44 kg

২২৩.

A seller bought 60 pens for 600 taka. How many pens does he need to sell for 600 taka to make a profit of 20%?

ক
30
খ
60
গ
40
ঘ
50

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A seller bought 60 pens for 600 taka. How many pens does he need to sell for 600 taka to make a profit of 20%?

Solution:
Cost price of 60 pens = Tk. 600
Cost price per pen = 600 ÷ 60
= Tk. 10

Since the seller wants 20% profit,
The selling price per pen will be = (10 × 120%) tk
= {10 × (120/100)} tk
= 12 tk

Now, the number of pens the seller can sell = 600/12
= 50

২২৪.

The number of students in 3 classes is in the ratio of 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 at the start was-

ক
162
খ
142
গ
152
ঘ
122

সঠিক উত্তর: ক

162

উত্তর

সঠিক উত্তর: ক

162

ক

ব্যাখ্যা

Question: The number of students in 3 classes is in the ratio of 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 at the start was-

Solution:
The number of students in 3 classes is in the ratio of 2 : 3 : 4.
Let the number of students in 3 classes be as follows 2x, 3x, 4x.
Total students = 9x.
According to the question, 2x + 12 : 3x + 12 : 4x + 12 = 8 : 11 : 14

Equating the ratio,
(2x + 12)/(3x + 12) = 8/11
→ 22x + 132 = 24x + 96
→ 2x = 36
→ x= 18

So, the total number of students in the three classes at the start was- 9 × 18 = 162.

২২৫.

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

ক
120 grams
খ
140 grams
গ
160 grams
ঘ
240 grams

সঠিক উত্তর: গ

160 grams

উত্তর

সঠিক উত্তর: গ

160 grams

গ

ব্যাখ্যা

Question: 400 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 = 400 × (30/100) = 120 grams
let,
x gm sugar to be added

According to the question,
(120 + x)/(400 + x) = 50%
⇒ (120 + x)/(400 + x) = 50/100
⇒ (120 + x)/(400 + x) = 1/2
⇒ 2 × (120 + x) = (400 + x)
⇒ 240 + 2x = 400 + x
⇒ 2x - x = 400 - 240
⇒ x = 160 grams

২২৬.

A, B, C started a business with their investments in the ratio 1:3:5. After 4 months, A invested the same amount as before and B as well as C withdrew half of their investments. The ratio of their profits at the end of the year is?

ক
ক) 4 : 3 : 5
খ
খ) 5 : 6 : 10
গ
গ) 6 : 5 : 10
ঘ
ঘ) 10 : 5 : 6

সঠিক উত্তর: খ

খ) 5 : 6 : 10

উত্তর

সঠিক উত্তর: খ

খ) 5 : 6 : 10

খ

ব্যাখ্যা

Let their initial investments be x, 3x and 5x respectively
Then,
=A:B:C
= (x×4+2x×8) : {3x×4+(3x/2)×8} : {5x×4+(5x/2)×8}
= 20x:24x:40x
= 5:6:10

২২৭.

After adding some water in a 50 litres milk-water mixture the ratio of milk and water changes to 6 : 4 to 5 : 5. What was the amount of extra water that was added?

ক
ক) 8 litres
খ
খ) 9 litres
গ
গ) 10 litres
ঘ
ঘ) 12 litres

সঠিক উত্তর: গ

গ) 10 litres

উত্তর

সঠিক উত্তর: গ

গ) 10 litres

গ

ব্যাখ্যা

Question: After adding some water in a 50 litres milk-water mixture the ratio of milk and water changes to 6 : 4 to 5 : 5. What was the amount of extra water that was added?

Solution:
initially the amount of milk was = 50 × 6/10 = 30 litres
so, the amount of water was = 20 litres

let, X litres of water is added to the mixture

ATQ,
30 : (20 + X) = 5 : 5
100 + 5X = 150
5X = 50
X = 10

hence, 10 litres of water were added to the mixture

২২৮.

There are deer and peacock in zoo. The total number of their heads is 80 and the total number of their legs is 200. How many peacocks are there?

ক
20
খ
30
গ
50
ঘ
60

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: There are deer and peacock in zoo. The total number of their heads is 80 and the total number of their legs is 200. How many peacocks are there?

Solution:
Let there are "X" deer and "Y" peacocks.

Total heads are 80.
∴ X + Y = 80 .................(1)

Total legs are 200. Deer has 4 legs and peacock has 2.
∴ 4X + 2Y = 200 ..................(2)

Multiplying Equation (1) by 4 and substracting with Equation (2), we get
4X + 4Y - 4X - 2Y = 320 - 200
⇒ 2Y = 120
∴ Y = 60

Putting this value of Y in Equation (1), we get
X + 60 = 80
∴ X = 20

∴ Total peacocks are Y = 60.
Hence, the correct answer is 60.

২২৯.

Rafi weighs 72 kg. If he reduces his weight in the ratio 6 : 5, find his new weight in kg.

ক
50 kg
খ
60 kg
গ
65 kg
ঘ
54 kg

সঠিক উত্তর: খ

60 kg

উত্তর

সঠিক উত্তর: খ

60 kg

খ

ব্যাখ্যা

Question: Rafi weighs 72 kg. If he reduces his weight in the ratio 6 : 5, find his new weight in kg.

Solution:
ধরি, রাফির পূর্বের ওজন = 6x
রাফির পরের ওজন = 5x

প্রশ্নমতে,
6x = 72
⇒ x = 72 / 6 = 12

∴ ওজন কমে যাওয়ার পর হবে = 5x = 5 × 12 = 60 kg

২৩০.

In what ratio water must be mixed with milk costing Tk. 48 per liter to get a mixture worth Tk. 32 per liter?

ক
2 : 3
খ
3 : 2
গ
3 : 4
ঘ
1 : 2

সঠিক উত্তর: ঘ

1 : 2

উত্তর

সঠিক উত্তর: ঘ

1 : 2

ঘ

ব্যাখ্যা

Question: In what ratio water must be mixed with milk costing Tk. 48 per liter to get a mixture worth Tk. 32 per liter?

Solution:
Cost Price of water Tk. 0
Cost Price of milk Tk. 48
Mean Price of mixure Tk. 32
Let,
Quantity of water be x
Quantity of milk be y

Using the formula,

⇒ x/y = (48 - 32)/(32 - 0)
⇒ x/y = 16/32
⇒ x/y = 1/2
∴ x : y = 1 : 2

২৩১.

A, B and C start a business. A invests 2 times as much as B invests 5/2 times as much as C invests. Find the ratio of capitals of A, B and C?

ক
ক) 5 : 10 : 2
খ
খ) 10 : 5 : 2
গ
গ) 10 : 5 : 3
ঘ
ঘ) 5 : 2 : 10

সঠিক উত্তর: খ

খ) 10 : 5 : 2

উত্তর

সঠিক উত্তর: খ

খ) 10 : 5 : 2

খ

ব্যাখ্যা

Question: A, B and C start a business. A invests 2 times as much as B invests 5/2 times as much as C invests. Find the ratio of capitals of A, B and C?

Solution:
ধরি,
C বিনিয়োগ করে = x টাকা
B বিনিয়োগ করে = 5x/2 টাকা
A বিনিয়োগ করে = 2 × (5x/2) টাকা
= 5x টাকা
A, B এবং C এর বিনিয়োগের অনুপাত = 5x : (5x/2) : x
= 5 : (5/2) : 1
= 10 : 5 : 2

২৩২.

The ratio of boys and girls in a club is 3 : 2. Which of the following could be the actual number of members?

ক
ক) 18
খ
খ) 16
গ
গ) 25
ঘ
ঘ) 24

সঠিক উত্তর: গ

গ) 25

উত্তর

সঠিক উত্তর: গ

গ) 25

গ

ব্যাখ্যা

Question: The ratio of boys and girls in a club is 3 : 2 Which of the following could be the actual number of members?

Solution:
Sum of the ratio = 2 + 3 = 5
So, the actual number must be divided by 5.
Here, only 25 is the only number that can be divided by 5.

২৩৩.

On a shelf, books with green cover and that with brown cover are in the ratio 2 : 3. If there are 18 books with green cover, then the number of books with brown cover is

ক
21
খ
24
গ
27
ঘ
30

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: On a shelf, books with green cover and that with brown cover are in the ratio 2 : 3. If there are 18 books with green cover, then the number of books with brown cover is

Solution:
Green cover books 2x
Brown cover books 3x

2x = 18
⇒ x = 18/2 = 9

Brown cover books = 3 × 9
= 27

২৩৪.

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?

ক
18 litres
খ
24 litres
গ
32 litres
ঘ
42 litres

সঠিক উত্তর: খ

24 litres

উত্তর

সঠিক উত্তর: খ

24 litres

খ

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 - 2/3
⇒ 8/x = 1/3
⇒ x/8 = 3
∴ x = 24

২৩৫.

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-

ক
2 : 5
খ
3 : 5
গ
4 : 5
ঘ
6 : 7

সঠিক উত্তর: গ

4 : 5

উত্তর

সঠিক উত্তর: গ

4 : 5

গ

ব্যাখ্যা

Question: Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-

Solution:
Let the third number be x.
Then,
first number = 120% of x = 120x/100 = 6x/5
Second number = 150% of x = 150x/100 = 3x/2

∴ Ratio of first two numbers = (6x/5 : 3x/2) = 12x : 15x = 4 : 5

২৩৬.

A mixture contains 1/5 of element A and 4/5 of element B. When 5 ml of A is added to the mixture, the proportion of B in the mixture changes to 1/5. What amount of A was originally present in the mixture before the addition was made?

ক
5/3 ml
খ
2/3 ml
গ
1/3 ml
ঘ
7/3 ml

সঠিক উত্তর: গ

1/3 ml

উত্তর

সঠিক উত্তর: গ

1/3 ml

গ

ব্যাখ্যা

Question: A mixture contains 1/5 of element A and 4/5 of element B. When 5 ml of A is added to the mixture, the proportion of B in the mixture changes to 1/5. What amount of A was originally present in the mixture before the addition was made?

Solution:
Let,
Mixture be x ml.
A = x/5 ml
B = 4x/5 ml

After adding 5 ml of A to mixture, amount of B remained same.
And the mixure be x + 5 ml.
New
B = (x + 5)/5

Now,
(4x)/5 = (x + 5)/5
⇒ 20x = 5x + 25
⇒ 15x = 25
⇒ x = 25/15
∴ x = 5/3

Original amount of A = (5/3)/5 ml
= 1/3 ml

২৩৭.

Of the people who responded to a market survey, 120 preferred Brand X and the rest preferred Brand Y. If the respondents indicated a preference for Brand X over Brand Y by ratio of 3 to 1, how many people responded to the survey?

ক
80
খ
160
গ
240
ঘ
360
ঙ
480

সঠিক উত্তর: খ

160

উত্তর

সঠিক উত্তর: খ

160

খ

ব্যাখ্যা

Question: Of the people who responded to a market survey, 120 preferred Brand X and the rest preferred Brand Y. If the respondents indicated a preference for Brand X over Brand Y by ratio of 3 to 1, how many people responded to the survey?

Solution:
Ratio = 3 : 1
Let, 3a respondents preferred Brand X and a preferred Brand Y

Since, no. of respondents who preferred Brand X = 120
∴ 3a =120
∴ a = 40

Hence Total no. of respondents = 120 + 40 = 160

২৩৮.

15 men, 18 women and 12 boys working together earned Tk. 2070. If the daily wages of a man, a woman and a boy are in the ratio of 4 : 3 : 2, the daily wage (in Tk) of 1 man, 2 women and 3 boys are -

ক
ক) 135
খ
খ) 180
গ
গ) 205
ঘ
ঘ) 240

সঠিক উত্তর: ঘ

ঘ) 240

উত্তর

সঠিক উত্তর: ঘ

ঘ) 240

ঘ

ব্যাখ্যা

Let,
the daily wage of a man, a woman and a boy be Tk. 4x, Tk. 3x and Tk. 2x respectively.
Then, 15 × 4x + 18 × 3x + 12 × 2x = 2070
⇒ 60x + 54x + 24x = 2070
⇒ 138x = 2070
⇒ x = 15
∴ daily wages of 1 man, 2 women and 3 boys
= Tk. (4x + 2 × 3x + 3 × 2x)
= Tk. (4x + 6x + 6x)
= Tk. 16x
= Tk. (16 × 15)
= Tk. 240.

২৩৯.

The salaries a of A, B and C are in the ratio of 1:2:3. The salary of B and C together is Tk. 6,000. By what percent is the salary of C more than that of A?

ক
ক) 100%
খ
খ) 150%
গ
গ) 200%
ঘ
ঘ) 250%

সঠিক উত্তর: গ

গ) 200%

উত্তর

সঠিক উত্তর: গ

গ) 200%

গ

ব্যাখ্যা

Let,
Salary of A = 1x
Salary of B = 2x
Salary of C = 3x

Salary of B + C = 5x = 6000 so x = 6000/5 = 1200
Salary of A = 1200, salary of C = 3600
Salary of C more than A = 3600 – 1200 = 2400
% salary of C more than A = 2400/1200 × 100 = 200%

২৪০.

A and B started a business jointly A's investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B received Tk. 4000 as profit, then their total profit is :

ক
ক) Tk. 22000
খ
খ) Tk. 28000
গ
গ) Tk. 32000
ঘ
ঘ) Tk. 36000

সঠিক উত্তর: খ

খ) Tk. 28000

উত্তর

সঠিক উত্তর: খ

খ) Tk. 28000

খ

ব্যাখ্যা

Suppose B invested Tk. x for y months.
Then, A invested Tk. 3x for 2y months.

So,
A : B = (3x × 2y) : (x × y)
= 6xy : xy
= 6 : 1.
B's profit : Total profit = 1 : 7.

Let the total profit be Tk. x
Then, 1/7 = 4000/x
⇒ x = Tk. 28000.

২৪১.

If a : b = 3 : 2, find ratio (4a + 5b) : (4a - 5b).

ক
5 : 1
খ
11 : 13
গ
11 : 1
ঘ
8 : 1

সঠিক উত্তর: গ

11 : 1

উত্তর

সঠিক উত্তর: গ

11 : 1

গ

ব্যাখ্যা

Question: If a : b = 3 : 2, find ratio (4a + 5b) : (4a - 5b).

Solution:
(4a + 5b) : (4a - 5b)
= b(4a/b + 5) : b (4a/b - 5)
= (4 × 3/2 + 5) : (4 × 3/2 - 5)
= (6 + 5) : (6 - 5)
= 11 : 1

২৪২.

In a mayoral election, Candidate X received 1/3 more votes than Candidate Y, and Candidate Y received 1/4 fewer votes than Candidate Z. If Candidate Z received 24,000 votes, how many votes did Candidate X receive?

ক
18,000
খ
22,000
গ
24,000
ঘ
26,000
ঙ
32,000

সঠিক উত্তর: গ

24,000

উত্তর

সঠিক উত্তর: গ

24,000

গ

ব্যাখ্যা

Question: In a mayoral election, Candidate X received 1/3 more votes than Candidate Y, and Candidate Y received 1/4 fewer votes than Candidate Z. If Candidate Z received 24,000 votes, how many votes did Candidate X receive?

Solution:
Candidate Y received 1/4 fewer votes than Candidate Z
In other words, Candidate Y received 3/4 of the votes that Candidate Z received
3/4 of 24,000 = (3/4)(24,000) = 18,000

Candidate X received 1/3 more votes than Candidate Y
1/3 of 18,000 = 6,000
Number of votes that Candidate X received = 18,000 + 6,000 = 24,000

২৪৩.

Two numbers are in the ratio 5 : 8. If 3 is subtracted to the first number, the ratio becomes 1 : 2. The numbers are = ?

ক
10 and 16
খ
12 and 30
গ
11 and 27
ঘ
15 and 24

সঠিক উত্তর: ঘ

15 and 24

উত্তর

সঠিক উত্তর: ঘ

15 and 24

ঘ

ব্যাখ্যা

Question : Two numbers are in the ratio 5 : 8. If 3 is subtracted to the first number, the ratio becomes 1 : 2. The numbers are = ?

Solution :
Let,
The numbers are = 5x and 8x

∴ According to question,
⇒ (5x - 3) : 8x = 1 : 2
⇒ (5x - 3)/8x = 1/2
⇒ 2(5x - 3) = 8x
⇒ 10x - 6 = 8x
⇒ 10x - 8x = 6
⇒ 2x = 6
∴ x = 3

So,The first numbers =5x
=5×3
=15
The second number = 8x
= 8 × 3
= 24

২৪৪.

A basketball team has a ratio of win to loss of 3 : 2. After winning 6 games in a row, the team's ratio of win to loss became 2 : 1. How many games had the team won before it played the last six games?

ক
10
খ
6
গ
12
ঘ
14
ঙ
18

সঠিক উত্তর: ঙ

উত্তর

সঠিক উত্তর: ঙ

ঙ

ব্যাখ্যা

Question: A basketball team has a ratio of win to loss of 3 : 2. After winning 6 games in a row, the team's ratio of win to loss became 2 : 1. How many games had the team won before it played the last six games?

Solution:
ধরি, দলটির জেতা খেলার সংখ্যা ছিল 3x
এবং হারা খেলার সংখ্যা ছিল 2x.
পরপর 6টি খেলা জেতার পর,
জেতা খেলার নতুন সংখ্যা = (3x + 6)
হারা খেলার সংখ্যা = 2x (যেহেতু কোনো খেলা হারেনি)

প্রশ্নমতে,
(3x + 6)/(2x) = 2/1
⇒ 3x + 6 = 2 × (2x)
⇒ 3x + 6 = 4x
⇒ 4x - 3x = 6
⇒ x = 6

∴ প্রথম অবস্থায় জেতা খেলার সংখ্যা ছিল 3x = 3 × 6 = 18
অতএব, শেষ 6টি খেলা খেলার আগে দলটি 18টি খেলায় জিতেছিল।

২৪৫.

Rahim's expenditures and savings are in the ratio of 3 : 2. His income increases by 10%. His expenditure also increases by 12%. How much percent does his savings increase?

ক
ক) 10%
খ
খ) 5%
গ
গ) 7%
ঘ
ঘ) 12%

সঠিক উত্তর: গ

গ) 7%

উত্তর

সঠিক উত্তর: গ

গ) 7%

গ

ব্যাখ্যা

Question: Rahim's expenditures and savings are in the ratio of 3 : 2. His income increases by 10%. His expenditure also increases by 12%. How much percent does his savings increase?

Solution:
Let Rahim's expenditures be 3x and his savings be 2x
So, his income = 3x + 2x = 5x

Increased income = 110% of 5x = 5.5x
Increased expenditures = 112% of 3x = 3.36x

New savings = 5.5x - 3.36x = 2.14x

Increased savings = 2.14x - 2x = 0.14x

So, increases in percentage = (0.14x/2x) × 100 = 7%

২৪৬.

A shopkeeper has two varieties of lentils priced at Taka 60 per kg and Taka 80 per kg. In what ratio should he mix them to get a mixture worth Taka 68 per kg?

ক
2 : 3
খ
3 : 2
গ
3 : 4
ঘ
4 : 3

সঠিক উত্তর: খ

3 : 2

উত্তর

সঠিক উত্তর: খ

3 : 2

খ

ব্যাখ্যা

Question: A shopkeeper has two varieties of lentils priced at Taka 60 per kg and Taka 80 per kg. In what ratio should he mix them to get a mixture worth Taka 68 per kg?

Solution:
Cheaper variety price of lentils, p = 60 Taka
Expensive variety price of lentils, q = 80 Taka
mixture price, r = 68 Taka

Ratio = (q - r)/(r - p)
=(80 - 68)/(68 - 60)
=12/8
= 3 : 2

২৪৭.

In a zoo, there are Lions and Pigeons. If heads are counted, there are 140 and if legs are counted, there are 440. How many pigeons are there?

ক
ক) 80
খ
খ) 60
গ
গ) 50
ঘ
ঘ) 90

সঠিক উত্তর: খ

খ) 60

উত্তর

সঠিক উত্তর: খ

খ) 60

খ

ব্যাখ্যা

Question: In a zoo, there are Lions and Pigeons. If heads are counted, there are 140 and if legs are counted, there are 440. How many pigeons are there?

Solution:
Let, the pigeons have x
So, Loins has = 140 - x

ATQ,
2x + 4(140 - x) =440
⇒ 2x + 560 - 4x = 440
⇒ 2x = 120
⇒ x = 60

২৪৮.

The sum of three numbers is 360. The ratio of the first number to the second number is 3 : 4 and between the second and third numbers, this ratio is 4 : 5, ﬁnd the second number.

ক
100
খ
120
গ
160
ঘ
200

সঠিক উত্তর: খ

120

উত্তর

সঠিক উত্তর: খ

120

খ

ব্যাখ্যা

Question: The sum of three numbers is 360. The ratio of the first number to the second number is 3 : 4 and between the second and third numbers, this ratio is 4 : 5, ﬁnd the second number.

Solution:
Given that,
The sum of three numbers is = 360
The ratio of first number to second number is = 3 : 4
The ratio of second number of third number is = 4 : 5

The ratio of first, second and third number = 3 : 4 : 5
Let, the numbers are = 3x, 4x , 5x

ATQ,
3x + 4x + 5x = 360
⇒ 12x = 360
⇒ x = 360/12
⇒ x = 30

∴ The second number is = 4 × 30
= 120

২৪৯.

The ratio of X : Y is 3 : 4, and the ratio of Y : Z is 8 : 9. If X = 18, what is the value of Z?

ক
24
খ
27
গ
30
ঘ
36

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The ratio of X : Y is 3 : 4, and the ratio of Y : Z is 8 : 9. If X = 18, what is the value of Z?

Solution:
দেওয়া আছে,
X : Y = 3 : 4
Y : Z = 8 : 9
এবং X = 18

এখন,
X : Y = 3 : 4
⇒ X/Y = 3/4
⇒ 3Y = 4X
⇒ Y = (4 × 18) / 3 ; [X = 18]
∴ Y = 24

আবার,
Y : Z = 8 : 9
⇒ Y/Z = 8/9
⇒ 8Z = 9Y
⇒ Z = (9 × 24) / 8
∴ Z = 27

সুতরাং, Z এর মান হলো 27

২৫০.

In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the number of girls becomes-

ক
110
খ
128
গ
132
ঘ
144

সঠিক উত্তর: গ

132

উত্তর

সঠিক উত্তর: গ

132

গ

ব্যাখ্যা

Question: In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the number of girls becomes-

Solution:
Boys : girls = 8 : 5
Let, the boys = 8x, girl = 5x

According to the 1st condition,
8x + 5x = 286
⇒ 13x = 286
⇒ x = 286/13
∴ x = 22
Boys = 8 × 22 = 176 and girls = 5 × 2 = 110

22 more girls get admitted then the number of girls become = (5x + 22)
= 110 + 22
= 132

২৫১.

In two identical glasses, one is half full of milk and the other is three-fifth full of milk. Both glasses are then filled with water and poured into a tumbler. What is the ratio of milk to water in the tumbler?

ক
7 : 13
খ
9 : 11
গ
11 : 9
ঘ
13 : 7

সঠিক উত্তর: গ

11 : 9

উত্তর

সঠিক উত্তর: গ

11 : 9

গ

ব্যাখ্যা

Question: In two identical glasses, one is half full of milk and the other is three-fifth full of milk. Both glasses are then filled with water and poured into a tumbler. What is the ratio of milk to water in the tumbler?

Solution:
In the 1st glass,
Milk = 1/2
Water = 1 − 1/2 = 1/2

In the 2nd glass,
Milk = 3/5
Water = 1 − 3/5 = 2/5

Now total milk in tumbler = (1/2 + 3/5)
= {(5 + 6)/10}
= 11/10

Total water in tumbler = (1/2 + 2/5)
= {(5 + 4)/10}
= 9/10

∴ Ratio of milk and water
= (11/10) : (9/10)
= 11 : 9

২৫২.

200kg of solution A is mixed with 80kg of solution B. If solution A has tin and copper in the ratio 3 : 5 and solution B has lead and tin in the ratio 3 : 1, then what is the amount of tin in the new solution?

ক
ক) 75 kg
খ
খ) 125 kg
গ
গ) 95 kg
ঘ
ঘ) 65 kg

সঠিক উত্তর: গ

গ) 95 kg

উত্তর

সঠিক উত্তর: গ

গ) 95 kg

গ

ব্যাখ্যা

Question: 200kg of solution A is mixed with 80kg of solution B. If solution A has tin and copper in the ratio 3 : 5 and solution B has lead and tin in the ratio 3 : 1, then what is the amount of tin in the new solution?

Solution:
A এর মিশ্রণে টিনের পরিমাণ = (200 এর 3/(3 + 5)} কেজি
= 75 কেজি

B এর মিশ্রণে টিনের পরিমাণ =(80 এর 1/(3 + 1)} কেজি
= 20 কেজি

A এবং B এর মিশ্রণে মোট টিনের পরিমাণ = (75 + 20) কেজি
= 95 কেজি

২৫৩.

In what ratio must a mixture of 25% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 40% alcohol strength?

ক
2 : 5
খ
2 : 3
গ
1 : 3
ঘ
1 : 5

সঠিক উত্তর: খ

2 : 3

উত্তর

সঠিক উত্তর: খ

2 : 3

খ

ব্যাখ্যা

Question: In what ratio must a mixture of 25% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 40% alcohol strength?

Solution:
Let, X contains 25% alcohol strength and
Y contains 50% alcohol strength

ATQ,
(25% of X) + (50% of Y) = 40% of (X + Y)
⇒ 25X + 50Y = 40X + 40Y
⇒ 15X = 10Y
∴ X : Y = 2 : 3

২৫৪.

A national issue was won by a vote of 9 to 6. What part of the total vote was against the issue?

ক
ক) 3/8
খ
খ) 2/5
গ
গ) 6/8
ঘ
ঘ) 5/2

সঠিক উত্তর: খ

খ) 2/5

উত্তর

সঠিক উত্তর: খ

খ) 2/5

খ

ব্যাখ্যা

Question: A national issue was won by a vote of 9 to 6. What part of the total vote was against the issue?

Solution:
পক্ষে ভোট ছিল ৯ টি, বিপক্ষে ভোট ছিল ৬ টি।

মোট ভোট = (৯ + ৬) = ১৫

∴ বিরোধী দল পেয়েছে = ৬/১৫
= ২/৫ অংশ ভোট

২৫৫.

A mixture contains two liquids 'A' and 'B' in the ratio 5 : 2. If 14 litres of mixture is withdrawn and replaced with 14 litres of 'B', then the ratio becomes 3 : 4. What was the initial quantity of A?

ক
8 litres
খ
22 litres
গ
10 litres
ঘ
25 litres

সঠিক উত্তর: ঘ

25 litres

উত্তর

সঠিক উত্তর: ঘ

25 litres

ঘ

ব্যাখ্যা

Question: A mixture contains two liquids 'A' and 'B' in the ratio 5 : 2. If 14 litres of mixture is withdrawn and replaced with 14 litres of 'B', then the ratio becomes 3 : 4. What was the initial quantity of A?

Solution:
ধরি, মিশ্রণের প্রাথমিক পরিমাণ = 7x লিটার

A এর পরিমাণ = 5x লিটার
B এর পরিমাণ = 2x লিটার

∴ 14 লিটার মিশ্রণ তুলে নেওয়ার পর,
A এর পরিমাণ = 5x - (5/7) × 14 = 5x - 10 লিটার
B এর পরিমাণ = 2x - (2/7) × 14 = 2x - 4 লিটার

আবার, B তে 14 লিটার যোগ করার পর,
B এর পরিমাণ = 2x - 4 + 14 = 2x + 10 লিটার

∴ প্রদত্ত অনুপাত,
⇒ (5x - 10)/(2x + 10) = 3/4
⇒ 4(5x - 10) = 3(2x + 10)
⇒ 20x - 40 = 6x + 30
⇒ 20x - 6x = 30 + 40
⇒ 14x = 70
⇒ x = 5

∴ A এর পরিমাণ = 5 × 5 = 25 লিটার

২৫৬.

A merchant has 1500 kg of wheat, part of which he sells at 10% profit and the rest at 20% profit. He gains 14% overall. The quantity sold at 20% profit is-

ক
450 kg
খ
600 kg
গ
750 kg
ঘ
900 kg

সঠিক উত্তর: খ

600 kg

উত্তর

সঠিক উত্তর: খ

600 kg

খ

ব্যাখ্যা

Question: A merchant has 1500kg of wheat, part of which he sells at 10% profit and the rest at 20% profit. He gains 14% overall. The quantity sold at 20% profit is-

Solution:
ধরি, 20% লাভে বিক্রি করা গমের পরিমাণ = x কেজি
∴ 10% লাভে বিক্রি করা গমের পরিমাণ = (1500 - x) কেজি

প্রশ্নমতে,
10% of (1500 - x) + 20% of x = 14% of 1500
⇒ 10(1500 - x)/100 + 20x/100 = (14 × 1500)/100
⇒ {10(1500 - x) + 20x}/100 = 210
⇒ 15000 - 10x + 20x = 21000
⇒ 10x = 6000
⇒ x = 600

সুতরাং, 20% লাভে বিক্রি করা গমের পরিমাণ হলো 600 কেজি।

২৫৭.

The difference of the ages of Raju and Manik is 21 years and the product of their age is 72 years. Find the ratio of the ages of Raju and Manik.?

ক
8:1
খ
6:5
গ
7:4
ঘ
2:3

সঠিক উত্তর: ক

8:1

উত্তর

সঠিক উত্তর: ক

8:1

ক

ব্যাখ্যা

Let a be the age of Raju and b be the age of manik.
According to the question,
a - b = 21 ...(1) and ab = 72
⇒ b = 72/a
Then (1) becomes, a - 72/a = 21
⇒ a² - 72 = 21a
⇒ a² - 21a - 72 = 0
⇒ a² - 24a + 3a -72 = 0
⇒ a(a - 24) + 3(a - 24) = 0
⇒ (a - 24)(a + 3) = 0
⇒ a = 24 or a = -3
Since age cannot be a negative number, the age of Raju will be 24 years
Therefore b = 3.
Hence the age of manik is 3 years
Then the required ratio = Raju/manik = 24/3 = 8/1
Hence the answer is 8:1

২৫৮.

Rice worth Tk. 126 per kg and Tk. 134 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 177 per kg, the price of the third variety per kg will be:

ক
ক) Tk 224
খ
খ) Tk 200
গ
গ) Tk 250
ঘ
ঘ) Tk 150
ঙ
ঙ) Tk 75

সঠিক উত্তর: ক

ক) Tk 224

উত্তর

সঠিক উত্তর: ক

ক) Tk 224

ক

ব্যাখ্যা

Let, the price of third variety = x
ATQ,
Or, 126×1 + 134×1 + x×2 = 177×4
Or, 260 + 2x = 708
Or, 2x = 448
Or, x = 224 [Answer.]

২৫৯.

A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

ক
1/3
খ
1/4
গ
1/5
ঘ
1/7

সঠিক উত্তর: গ

1/5

উত্তর

সঠিক উত্তর: গ

1/5

গ

ব্যাখ্যা

Question: A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

Solution:
Suppose the vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.

Quantity of water in new mixture = {3 - (3x/8) + x} litres

Quantity of syrup in new mixture = (5 - 5x/8) litres

ATQ,
{3 - (3x/8) + x} = (5 - 5x/8)
⇒ 5x + 24 = 40 - 5x
⇒ 10x = 16
∴ x = 8/5

So, part of the mixture replaced = (8/5) × (1/8) = 1/5

২৬০.

If 2 kg of metal, of which 1/3 is zinc and the rest is copper be mixed with 3 kg of metal of which 1/4 is zinc and the rest is copper, What is the ratio of zinc to copper in the mixture?

ক
23 : 39
খ
5 : 9
গ
17 : 43
ঘ
3 : 7

সঠিক উত্তর: গ

17 : 43

উত্তর

সঠিক উত্তর: গ

17 : 43

গ

ব্যাখ্যা

Question: If 2 kg of metal, of which 1/3 is zinc and the rest is copper be mixed with 3 kg of metal of which 1/4 is zinc and the rest is copper, What is the ratio of zinc to copper in the mixture?

Solution:
Zinc = {2 × (1/3)} + {3 × (1/4)} = 17/12
Copper = 5 - (17/12) = 43/12

∴ Zinc : Copper = (17/12) : (43/12)
= 17 : 43

২৬১.

The ratio of ages of Mary and Maria is 4 : 5. After 12 years their ratio becomes 5 : 6. What will be the age of Mary after 2 years?

ক
ক) 45 years
খ
খ) 48 years
গ
গ) 50 years
ঘ
ঘ) 56 years

সঠিক উত্তর: গ

গ) 50 years

উত্তর

সঠিক উত্তর: গ

গ) 50 years

গ

ব্যাখ্যা

Since the ages of Mary and Maria are in the ratio 4 : 5,
the actual ages can be taken as 4x years and 5x years respectively.
After 12 years, their ages will be 4x + 12 years and 5x + 12 years.
Therefore, as per the problem (4x + 12)/(5x + 12) = 5/6 => x = 12
Mary's age now is 4x years = 48 years.
So after 2 years Mary will be 50 years.

২৬২.

The ratio of girls to boys in a class is 8 : 7. Which of the following can't be the number of the total students-

ক
30
খ
45
গ
50
ঘ
60

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The ratio of girls to boys in a class is 8 : 7. Which of the following can't be the number of the total students-

Solution:
ছাত্রী ও ছাত্রের অনুপাত = ৮ : ৭

যোগফল = ৮ + ৭ = ১৫
অতএব, মোট শিক্ষার্থী সংখ্যা ১৫ দ্বারা নি:শেষে বিভাজ্য হতে হবে।

৫০, ১৫ দ্বারা নি:শেষে বিভাজ্য নয়।

২৬৩.

In a certain town, the ratio of the number of rented residences to the number of owned residences is 7 : 4. Those are the only two kinds of residences in this town. If there are 150 more rented than owned residences in the town?

ক
ক) 200
খ
খ) 350
গ
গ) 550
ঘ
ঘ) 700

সঠিক উত্তর: গ

গ) 550

উত্তর

সঠিক উত্তর: গ

গ) 550

গ

ব্যাখ্যা

ধরি,
বাড়িওয়ালার সংখ্যা x জন।
ভাড়াটের সংখ্যা (x + 150) জন।

প্রশ্নমতে,
(x + 150) : x = 7 : 4
(x + 150)/x = 7/4
7x = 4x + 600
3x = 600
x = 200

অতএব
বাড়িওয়ালার সংখ্যা 200 জন।
ভাড়াটের সংখ্যা = 200 + 150 = 350 জন।
মোট অধিবাসীর সংখ্যা = 200 + 350 = 550 জন।

২৬৪.

In what ratio must rice at Tk 10 per kg be mixed with rice at Tk 16 per kg so that the mixture be worth Tk 12 per kg?

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

সঠিক উত্তর: ঘ

ঘ) 2 : 1

উত্তর

সঠিক উত্তর: ঘ

ঘ) 2 : 1

ঘ

ব্যাখ্যা

Question: In what ratio must rice at Tk 10 per kg be mixed with rice at Tk 16 per kg so that the mixture be worth Tk 12 per kg?

Solution:
ধরি, প্রতি কেজি ১০ টাকায় বিক্রি করে x কেজি
প্রতি কেজি ১৬ টাকায় বিক্রি করে y কেজি

10x + 16y = 12(x + y)
⇒ 10x + 16y = 12x + 12y
⇒ 12x - 10x = 16y - 12y
⇒ 2x = 4y
∴ x/y = 2 = 2 : 1

২৬৫.

a : b = 3 : 4 and b : c = 9 : 7. What is a : b : c?

ক
ক) 3 : 13 : 7
খ
খ) 3 : 36 : 28
গ
গ) 27 : 16 : 28
ঘ
ঘ) 27 : 36 : 28

সঠিক উত্তর: ঘ

ঘ) 27 : 36 : 28

উত্তর

সঠিক উত্তর: ঘ

ঘ) 27 : 36 : 28

ঘ

ব্যাখ্যা

B is common to both the ratios.
Values of b = 4 and 9 (That means they are not the same)
Make the values of 'b' the same as follows -
Multiply 3 : 4 up and down with 9 as shown
∴ (3 × 9)/(4 × 9) = 27/36 = a/b
Multiply 9 : 7 up and down with 4 as shown
∴ (9 × 4)/(7 × 4) = 36/28 = b/c
Since values of b are the same = 36
a : b : c = 27 : 36 : 28

২৬৬.

The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries?

ক
13 : 33 : 60
খ
23 : 43 : 6
গ
23 : 33 : 60
ঘ
3 : 3 : 10

সঠিক উত্তর: গ

23 : 33 : 60

উত্তর

সঠিক উত্তর: গ

23 : 33 : 60

গ

ব্যাখ্যা

Let, A = 2y
B = 3y
C = 5y
New salary of A = 115/100 of 2y = 23y/10
New salary of B = 110/100 of 2y = 33y/10
New salary of C = 120/100 of 5y = 6Y
Required ratio = 23y/10 : 33y/10 : 6Y = 23 : 33 : 60

২৬৭.

In business, A and C invested amounts in the ratio 2:1, whereas the ratio between amounts invested by A and B was 3:2, If Tk 157300 was their profit, how much amount did B receive?

ক
ক) 48000
খ
খ) 47000
গ
গ) 47400
ঘ
ঘ) 48400

সঠিক উত্তর: ঘ

ঘ) 48400

উত্তর

সঠিক উত্তর: ঘ

ঘ) 48400

ঘ

ব্যাখ্যা

A:B = 3:2 = 6:4
=> A:C = 2:1 = 6:3
=> A:B:C = 6:4:3
B share = (4/13)×157300
= 48400

২৬৮.

In a mixture of 100 litres, milk and water are in the ratio 4:1. 40 litres of mixture drawn off. Find the ratio of water and milk in the remaining mixture?

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

সঠিক উত্তর: খ

খ) 1 : 4

উত্তর

সঠিক উত্তর: খ

খ) 1 : 4

খ

ব্যাখ্যা

Mixture = 100 litres

M : W = 4 : 1

Milk = 4/5 x100 = 80 litres

Water = 100 - 80 = 20 litres

In 40 litres of mixture, ratio will be same.

Milk = 4/5 x 40 = 32 litres
Water = 40 - 32 = 8 litres

Remaining Milk = 80 - 32 = 48 litres

Remaining Water = 20 - 8 = 12 litres

Water : Milk = 12 : 48 = 1 : 4

২৬৯.

By mixing two qualities of pulse in the ratio 2 : 3 and selling the mixture at the rate of Tk. 22 per kilogram, a shopkeeper makes a profit of 10%. If the cost of the smaller quantity be Tk. 14 per kg, the cos in Tk per kg of the larger quantity is :

ক
23
খ
24
গ
25
ঘ
26

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: By mixing two qualities of pulse in the ratio 2 : 3 and selling the mixture at the rate of Tk. 22 per kilogram, a shopkeeper makes a profit of 10%. If the cost of the smaller quantity be Tk. 14 per kg, the cos in Tk per kg of the larger quantity is :

Solution:
মনেকরি,
বেশি পরিমাণের ডাল এর মূল্য = x টাকা

দেওয়া আছে,
কম পরিমাণ : বেশি পরিমাণ = 2 : 3
অনুপাতদ্বয়ের যোগফল = 2 + 3 = 5

5 কেজি এর ক্রয়মূল্য = 14 × 2 + x × 3
= 28 + 3x

5 কেজি এর বিক্রয়মূল্য = 22 × 5 = 110

.: লাভ = 110 - (28 + 3x)
= 110 - 28 - 3x
= 82 - 3x

প্রশ্নমতে,
{(82 - 3x)/(28 + 3x)}× 100 % = 10%
(82 - 3x)/(28 + 3x) = 1/10
820 - 30x = 28 + 3x
3x + 30x = 820 - 28
33x = 792
x = 792/33
x = 24

২৭০.

A man distributes Tk.16500 among his daughter, wife and son in such a manner that the daughter's share and the wife's share are of 1 : 2 ratio, and the son gets half of the total amount. Find the daughter's share?

ক
3040 Tk
খ
3300 Tk
গ
2000 Tk
ঘ
2750 Tk

সঠিক উত্তর: ঘ

2750 Tk

উত্তর

সঠিক উত্তর: ঘ

2750 Tk

ঘ

ব্যাখ্যা

Question: A man distributes Tk.16500 among his daughter, wife and son in such a manner that the daughter's share and the wife's share are of 1 : 2 ratio, and the son gets half of the total amount. Find the daughter's share?

Solution:
The daughter's share and the wife's share are of 1 : 2 ratio
daughter's share = x
wife's share = 2x

As the son gets half of the total amount
∴ son's share = 16500/2
= 8250 Tk

→ Rest of the share = 3x
According to the question,
→ 3x= 8250
→ x= 2750

→ daughter's share = x = 2750 Tk

২৭১.

To gain 12% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 60 kg. of pure milk is-

ক
7 kg
খ
8.5 kg
গ
12 kg
ঘ
7.2 kg
ঙ
None of these

সঠিক উত্তর: ঘ

7.2 kg

উত্তর

সঠিক উত্তর: ঘ

7.2 kg

ঘ

ব্যাখ্যা

Question: To gain 12% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 60 kg. of pure milk is-

Solution:
দেওয়া আছে,
বিশুদ্ধ দুধের পরিমাণ = ৬০কেজি
বিক্রয়মূল্য বৃদ্ধি না করে অতিরিক্ত লাভ করতে হবে = ১২%
এখন,
বিক্রয়মূল্য বৃদ্ধি না করে আরো ১২% লাভ করতে হলে দুধের সাথে এর মোট পরিমাণের ১২% পানি যোগ করতে হবে।

∴ মিশ্রনে পানি পরিমাণ = ৬০ এর ১২% = ৬০ × (১২/১০০) = ৭.২ কেজি

সুতরাং, বিশুদ্ধ ৬০ কেজি দুধের সাথে ৭.২ কেজি পানি মিশ্রিত করলে বিক্রয়মূল্য বৃদ্ধি না করেও আরো ১২% লাভ করা যাবে।

২৭২.

How much water should be mixed with 100 kg of pure milk to achieve a 10% profit when selling the milk at its cost price?

ক
10 kg
খ
8 kg
গ
6 kg
ঘ
12 kg
ঙ
None of the above

সঠিক উত্তর: ক

10 kg

উত্তর

সঠিক উত্তর: ক

10 kg

ক

ব্যাখ্যা

Question: How much water should be mixed with 100 kg of pure milk to achieve a 10% profit when selling the milk at its cost price?

Solution:
Let the cost price of 100 kg pure milk be C.

If we add x kg of water, the total mixture will be (100 + x) kg.

The cost of this mixture remains C (since water is free).
When we sell this mixture at the pure milk's cost price, we sell (100 + x) kg at the price of 100 kg.

Our selling price will be C per 100 kg × (100 + x) kg = C × (100 + x)/100
For a 10% profit, this selling price must equal 1.1 × C

Setting up the equation:
C × (100 + x)/100 = 1.1 × C

Simplifying the equation,
⇒ (100 + x)/100 = 1.1
⇒ 100 + x = 110
⇒ x = 10

২৭৩.

A person has two acid solutions — one containing 40% acid and the other 60% acid. In what quantities should he mix each to obtain 10 liters of a 50% acid solution?

ক
1 liters
খ
2 liters
গ
3 liters
ঘ
5 liters

সঠিক উত্তর: ঘ

5 liters

উত্তর

সঠিক উত্তর: ঘ

5 liters

ঘ

ব্যাখ্যা

Question: A person has two acid solutions — one containing 40% acid and the other 60% acid. In what quantities should he mix each to obtain 10 liters of a 50% acid solution?

Solution:
ধরি,
প্রথম দ্রবণটি মেশাতে হবে = x লিটার
দ্বিতীয় দ্রবণটি মেশাতে হবে = (10 - x) লিটার

প্রশ্নমতে,
x লিটার 40% এসিডের দ্রবণ + (10 - x) লিটার 60% এসিডের দ্রবণ = 10 লিটার 50% এসিডের দ্রবণ
⇒ x × (40/100) + (10 - x) × (60/100) = 10 × (50/100)
⇒ (2x/5) + {3(10 - x)/5} = 5
⇒ (2x/5) + {(30 - 3x)/5} = 5
⇒ (2x + 30 - 3x)/5 = 5
⇒ (30 - x)/5 = 5
⇒ 30 - x = 25
⇒ x = 30 - 25
⇒ x = 5

40% এসিডের প্রথম দ্রবণটি মেশাতে হবে = 5 লিটার
60% এসিডের দ্বিতীয় দ্রবণটি মেশাতে হবে = (10 - 5) লিটার = 5 লিটার

অর্থাৎ 40% ও 60% এসিডের প্রতিটি দ্রবণ 5 লিটার করে মিশ্রিত করলে 50% এসিডের দ্রবণ পাওয়া যাবে।

২৭৪.

The circumference of the circle and the perimeter of the square is equal and the ratio between the diameter of the circle and the side of the square is 7 : 11. What is the area of the circle?

ক
ক) 154 cm²
খ
খ) 160 cm²
গ
গ) 132 cm²
ঘ
ঘ) Can’t be determined

সঠিক উত্তর: ঘ

ঘ) Can’t be determined

উত্তর

সঠিক উত্তর: ঘ

ঘ) Can’t be determined

ঘ

ব্যাখ্যা

Let, the side of the square = 11x
and, diameter of the circle = 7x
ATQ,
2π(7x/2) = 4×11x
Or, 7πx = 44x
x omits from both side.
So, the radius of the circle can't be determined from the given information.

২৭৫.

Two numbers are in the ratio of 21 : 26. If 8 is added in each, the new numbers are in ratio of 5 : 6. Find the ratio of numbers, if 6 is subtracted from each number?

ক
ক) 19 : 25
খ
খ) 18 : 23
গ
গ) 9 : 16
ঘ
ঘ) 6 : 7

সঠিক উত্তর: খ

খ) 18 : 23

উত্তর

সঠিক উত্তর: খ

খ) 18 : 23

খ

ব্যাখ্যা

Let the numbers be 21x and 26x.
⇒ (21x + 8)/(26x + 8) = 5/6
⇒ 6(21x + 8) = 5(26x + 8)
⇒ 126x + 48 = 130x + 40
⇒ x = 2.
So, the numbers will be 42 and 52.
If 6 is subtracted, then numbers will be 36 and 46.
Required ratio = 36 : 46.
i.e. 18 : 23.

২৭৬.

Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 153 per kg, the price of the third variety per kg will be:

ক
Tk. 169.50
খ
Tk. 170
গ
Tk. 175.50
ঘ
Tk. 180

সঠিক উত্তর: গ

Tk. 175.50

উত্তর

সঠিক উত্তর: গ

Tk. 175.50

গ

ব্যাখ্যা

Question: Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 153 per kg, the price of the third variety per kg will be:

Solution:
let, price of third variety x tk per kg

126y + 135 y + x × 2y = 153 (y + y + 2y)
⇒ 126y + 135y + 2xy = 153 × 4y
⇒ 126 + 135 + 2x = 612
⇒ 2x + 261 = 612
⇒ 2x = 351
∴ x = 175.5 tk

২৭৭.

a : b = 3 : 5 and c : b = 7 : 3 then what is the ratio of c : b : a?

ক
ক) 35 : 15 : 9
খ
খ) 9 : 15 : 35
গ
গ) 35 : 9 : 15
ঘ
ঘ) 9 : 35 : 15

সঠিক উত্তর: ক

ক) 35 : 15 : 9

উত্তর

সঠিক উত্তর: ক

ক) 35 : 15 : 9

ক

ব্যাখ্যা

Question: a : b = 3 : 5 and c : b = 7 : 3 then what is the ratio of c : b : a?

Solution:
here,
a : b = 3 : 5 = 9 : 15 [multiplying by 3]
c : b = 7 : 3 = 35 : 15 [multiplying by 5]

c : b : a = 35 : 15 : 9

২৭৮.

A's income is Tk. 140 more than B's income and C's income is Tk 80 more than D's. If the ratio of A's and C's income is 2 : 3 and the ratio of B's and D's income is 1 : 2, then the incomes of A, B, C and D are respectively

ক
ক) Tk. 60, Tk.120, Tk. 320 and Tk. 240
খ
খ) Tk.300, Tk.160, Tk. 600 and Tk. 520
গ
গ) Tk.400, Tk.260, Tk. 600 and Tk. 520
ঘ
ঘ) Tk.320, Tk.180, Tk. 480 and Tk. 360

সঠিক উত্তর: গ

গ) Tk.400, Tk.260, Tk. 600 and Tk. 520

উত্তর

সঠিক উত্তর: গ

গ) Tk.400, Tk.260, Tk. 600 and Tk. 520

গ

ব্যাখ্যা

A : C = 2 : 3 or 2x : 3x
B : D = 1 : 2 or y : 2y
According to question,
2x - 140 = y .... (i)
3x - 80 = 2y .... (ii)

multiply equation (i) by 2 and solved
4x - 280 - 3x + 80 = 2y - 2y
⇒ x - 200 = 0
⇒ x = 200

Now put value of x in equation (i)
(2 × 200) - 140 = y
⇒ 400 - 140 = y
⇒ y = 260

A's salary = 2x = Tk. 400
B's salary = y = Tk. 260
C's salary = 3x = Tk. 600
D's Salary = 2y = Tk. 520

২৭৯.

A fruit vendor claims to sell oranges at cost price, but he secretly adds lower quality oranges to his premium ones and gains 25%. Find the percentage of lower quality oranges in the mixture.

ক
18%
খ
20%
গ
22%
ঘ
None of the above

সঠিক উত্তর: খ

20%

উত্তর

সঠিক উত্তর: খ

20%

খ

ব্যাখ্যা

Question: A fruit vendor claims to sell oranges at cost price, but he secretly adds lower quality oranges to his premium ones and gains 25%. Find the percentage of lower quality oranges in the mixture.

Solution:
let,
cost price = 100
sell price = 125
∴ the amount of premium oranges = 100/125
= 4/5
∴ the amount of lower quality oranges = {1 - (4/5)} × 100%
= 20%

২৮০.

The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is-

ক
5 : 2
খ
3 : 2
গ
5 : 3
ঘ
5 : 1

সঠিক উত্তর: ঘ

5 : 1

উত্তর

সঠিক উত্তর: ঘ

5 : 1

ঘ

ব্যাখ্যা

Question: The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is-

Solution:
Let, the third proportional to 12 and 30 be x.

ATQ,
12 : 30 : : 30 : x
⇒ 12/30 = 30/x
⇒ 12x = 30 × 30
⇒ x = (30 × 30)/12
∴ x = 75

∴ Third proportional to 12 and 30 = 75.
Mean proportional between 9 and 25 = √(9 × 25) = 15
∴ Required ratio = 75/15 = 5 : 1 .

২৮১.

A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

ক
10 liters
খ
20 liters
গ
21 liters
ঘ
25 liters

সঠিক উত্তর: গ

21 liters

উত্তর

সঠিক উত্তর: গ

21 liters

গ

ব্যাখ্যা

Question: A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

Solution:
ধরি,
শুরুতে জারের মধ্যে তেলের পরিমাণ = 7x লিটার
পানির পরিমাণ = 5x লিটার
∴ মোট অংশ = 12x

9 লিটার মিশ্রণ ফেলে দিলে,
ফেলে দেওয়া মিশ্রণে তেলের পরিমাণ = 9 এর (7x/12x) = 21/4 লিটার
এবং পানির পরিমাণ = 9 এর (5x/12x) = 15/4 লিটার

বাকি মিশ্রণে,
তেলের পরিমাণ = 7x - (21/4) = (28x - 21)/4
পানির পরিমাণ = 5x - (15/4) = (20x - 15)/4

মিশ্রণে 9 লিটার পানি যোগ করা হলে পানির নতুন পরিমাণ = {(20x - 15)/4} + 9 = (20x - 15 + 36)/4 = (20x + 21)/4

প্রশ্নমতে,
{(28x - 21)/4}/{(20x + 21)/4} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 9(28x - 21) = 7(20x + 21)
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 189 + 147
⇒ 112x = 336
⇒ x = 336/112
⇒ x = 3

∴ শুরুতে মিশ্রণে তেলের পরিমাণ ছিলো = (7 × 3) লিটার = 21 লিটার

২৮২.

Out of three positive numbers, the ratio of the first and the second numbers is 3 : 4 that of the second and the third numbers is 5 : 6 if the product of the second and the third numbers is 4320. What is the sum of three numbers?

ক
ক) 177
খ
খ) 165
গ
গ) 185
ঘ
ঘ) 160

সঠিক উত্তর: ক

ক) 177

উত্তর

সঠিক উত্তর: ক

ক) 177

ক

ব্যাখ্যা

The ratio of 1st and 2nd numbers = 3 : 4
The ratio of 2nd and 3rd numbers = 5 : 6
Let,
the 2nd number = 5x, third number = 6x
Product of 2nd and 3rd numbers = 4320
5x × 6x = 4320
x²= 144
x = 12
2nd number = 60, 3rd number = 72,
1st number = (60/4) × 3 = 45
Sum of three numbers = 60 + 72 + 45 = 177

২৮৩.

If then 2A : B : 2C = ?

ক
ক) 2 : 3 : 4
খ
খ) 3 : 2 : 4
গ
গ) 6 : 3 : 8
ঘ
ঘ) 4 : 3 : 8

সঠিক উত্তর: ঘ

ঘ) 4 : 3 : 8

উত্তর

সঠিক উত্তর: ঘ

ঘ) 4 : 3 : 8

ঘ

ব্যাখ্যা

Question: If then 2A : B : 2C = ?

Solution:
দেওয়া আছে,
A/2 = B/3
A : B = 2 : 3
এবং,
B/3 = C/4
B : C = 3 : 4

এখন,
A : B = 2 : 3
B : C = 3 : 4
A : B : C = 2 : 3 : 4

∴ 2A : B : 2C = (2 × 2) : 3 : (4 × 2)
= 4 : 3 : 8

২৮৪.

If A and B are in the ratio 5 : 7 and B and C are in the ratio 14 : 15 then what is the ratio of A to C?

ক
3 : 7
খ
3 : 2
গ
2 : 3
ঘ
14 : 15

সঠিক উত্তর: গ

2 : 3

উত্তর

সঠিক উত্তর: গ

2 : 3

গ

ব্যাখ্যা

Question: If A and B are in the ratio 5 : 7 and B and C are in the ratio 14 : 15 then what is the ratio of A to C?

Solution:
Given that,
A : B = 5 : 7 and B : C = 14 : 15

Now,
(A/B) × (B/C) = (5/7) × (14/15)
⇒ A/C = (2/3)
∴ A : C = 2 : 3

২৮৫.

Three Rabbits A, B and C move in such a way that when A takes 7 steps, B takes 8 steps and C takes 9 steps. But 4 steps of A are equal to 5 steps of B and 6 steps of C. What is the ratio of their speeds?

ক
28 : 40 : 54
খ
42 : 40 : 36
গ
35 : 32 : 30
ঘ
30 : 32 : 35

সঠিক উত্তর: গ

35 : 32 : 30

উত্তর

সঠিক উত্তর: গ

35 : 32 : 30

গ

ব্যাখ্যা

A : B : C = 7 : 8 : 9
Size of step, 4A = 5B = 6C
the ratio of speeds = 7/4 : 8/5 : 9/6
= 35 : 32 : 30.

২৮৬.

In a basket the ratio of banana and apple is 3:2. If 5 bananas are removed from the basket then the ratio becomes 1:1. How many apples were there in the basket?

ক
ক) 5
খ
খ) 10
গ
গ) 15
ঘ
ঘ) 20
ঙ
ঙ) None

সঠিক উত্তর: খ

খ) 10

উত্তর

সঠিক উত্তর: খ

খ) 10

খ

ব্যাখ্যা

ধরি, ঝুড়িতে কলা ছিলো 3x টি এবং আপেল ছিলো 2x টি
প্রশ্নমতে , 3x-5 / 2x = 1/1
⇒ 3x - 5 = 2x
⇒ x = 5
∴ আপেল ছিল = 2x = 2 × 5 = 10 টি

২৮৭.

A sum of Tk. 6400 is divided among three workers in the ratio 3/5 : 2 : 5/3. The share of the second worker is-

ক
ক) Tk. 900
খ
খ) Tk. 2500
গ
গ) Tk. 3000
ঘ
ঘ) Tk. 3600

সঠিক উত্তর: গ

গ) Tk. 3000

উত্তর

সঠিক উত্তর: গ

গ) Tk. 3000

গ

ব্যাখ্যা

Question: A sum of Tk. 6400 is divided among three workers in the ratio 3/5 : 2 : 5/3. The share of the second worker is-

Solution:
ratio of share = 3/5 : 2 : 5/3
= 9 : 30 : 25
The share of the second worker is = 6400 × (30/64) = Tk. 3000

২৮৮.

The ratio of Pens ant Pencils in a shop is 3 : 2 respectively. The average number of Pens and Pencils is 170. What is the number of Pens in the shop?

ক
ক) 144
খ
খ) 154
গ
গ) 184
ঘ
ঘ) 204

সঠিক উত্তর: ঘ

ঘ) 204

উত্তর

সঠিক উত্তর: ঘ

ঘ) 204

ঘ

ব্যাখ্যা

Question: The ratio of Pens ant Pencils in a shop is 3 : 2 respectively. The average number of Pens and Pencils is 170. What is the number of Pens in the shop?

Solution:
ধরি,
দোকানে কলম আছে = 3x
দোকানে পেন্সিল আছে = 2x

প্রশ্নমতে,
(3x + 2x)/2 = 170
5x = 340
x = 340/5
x = 68

দোকানে কলম আছে = 3 × 68 = 204টি

২৮৯.

A vessel contains 40 litres of orange juice. 4 litres of orange juice in the vessel was replaced by water. This process was again repeated twice with the mixture. How much orange juice is there in the final mix?

ক
ক) 28 litre
খ
খ) 29.16 litre
গ
গ) 32 litre
ঘ
ঘ) 32.29 litre

সঠিক উত্তর: খ

খ) 29.16 litre

উত্তর

সঠিক উত্তর: খ

খ) 29.16 litre

খ

ব্যাখ্যা

First 4L orange juice was removed
So now in 40L mixture, there is 40 - 4 = 36 litres orange juice and 4 litres water
Now, remove 4L mixture. While doing this proportionate amount of juice and water gets removed.
Amount of juice removed = 4 × (Juice quantity/Mixture quantity)
= 4 × (36/40)
= 3.6 Litres.

Orange Juice remaining = 36-3.6 = 32.4 Litres
Again 4L mixture removed
Amount of juice removed = 4 × (Juice quantity/Mixture quantity)
= 4 × (32.4/40)
= 3.24 Litres.

Juice remaining = 32.4 - 3.24
= 29.16 Litres.

২৯০.

In your wallet, there are Tk 1000, Tk 500, and Tk 200 notes in the ratio 3:7:5. The total amount of money in the wallet is Tk 22,500. Find the number of each note.

ক
12, 28, 20
খ
10, 23, 17
গ
8, 19, 13
ঘ
9, 21, 15

সঠিক উত্তর: ঘ

9, 21, 15

উত্তর

সঠিক উত্তর: ঘ

9, 21, 15

ঘ

ব্যাখ্যা

Question: In your wallet, there are Tk 1000, Tk 500, and Tk 200 notes in the ratio 3 : 7 : 5. The total amount of money in the wallet is Tk 22,500. Find the number of each note respectively.

Solution:
Let,
The number of Tk 1000 notes = 3x
The number of Tk 500 notes = 7x
The number of Tk 200 notes = 5x

According to the question,
(1000 × 3x) + (500 × 7x) + (200 × 5x) = 22500
⇒ 3000x + 3500x + 1000x = 22500
⇒ 7500x = 22500
⇒ x = 22500/7500
∴ x = 3

∴ Number of Tk 1000 notes = 3 × 3 = 9
∴ Number of Tk 500 notes = 7 × 3 = 21
∴ Number of Tk 200 notes = 5 × 3 = 15

২৯১.

Moyna and Noyna respectively got 20% more and 10% less marks than Jolin in exam. What is the ratio of Noyna and Moyna's exam scores?

ক
2 : 1
খ
11 : 12
গ
3 : 4
ঘ
4 : 3
ঙ
None

সঠিক উত্তর: গ

3 : 4

উত্তর

সঠিক উত্তর: গ

3 : 4

গ

ব্যাখ্যা

Question: Moyna and Noyna respectively got 20% more and 10% less marks than Jolin in exam. What is the ratio of Noyna and Moyna's exam scores?

Solution:
Let,
Jolin got 100
∴ Moyna got 120
∴ Noyna got 90

Noyna : Moyna = 90 : 120 = 9 : 12 = 3 : 4

২৯২.

One type of liquid contains 25% of benzene, the other contains 30% of benzene. A can is filled with 6 parts of the first liquid and 4 parts of the second liquid. Find the percentage of benzene in the new mixture.

ক
28%
খ
25%
গ
30%
ঘ
27%
ঙ
None of these

সঠিক উত্তর: ঘ

27%

উত্তর

সঠিক উত্তর: ঘ

27%

ঘ

ব্যাখ্যা

Question: One type of liquid contains 25% of benzene, the other contains 30% of benzene. A can is filled with 6 parts of the first liquid and 4 parts of the second liquid. Find the percentage of benzene in the new mixture.

Solution:
Let the percentage of benzene in new mixure = X
(30 - X)/(X - 25) = 6/4 = 3/2
⇒ 60 - 2X = 3x - 75
⇒ 5X = 135
∴ X = 27

required percentage of benzene = 27%

২৯৩.

If p : q = 3 : 2, find ratio (4p + 5q) : (4p - 5q).

ক
5 : 2
খ
3 : 1
গ
1 : 11
ঘ
11 : 1

সঠিক উত্তর: ঘ

11 : 1

উত্তর

সঠিক উত্তর: ঘ

11 : 1

ঘ

ব্যাখ্যা

Question: If p : q = 3 : 2, find ratio (4p + 5q) : (4p - 5q).

Solution:
(4p + 5q) : (4p - 5q)
= q(4p/q + 5) : q(4p/q - 5)
= (4 × 3/2 + 5) : (4 × 3/2 - 5)
= (6 + 5) : (6 - 5)
= 11 : 1

২৯৪.

How many kilograms of sugar costing Tk. 8 per kg must be mixed with 24 kg of sugar costing Tk. 6 per kg so that a gain of 10% by selling the mixture at Tk. 8.5 per kg?

ক
ক) 97 kg
খ
খ) 113 kg
গ
গ) 152 kg
ঘ
ঘ) 167 kg

সঠিক উত্তর: গ

গ) 152 kg

উত্তর

সঠিক উত্তর: গ

গ) 152 kg

গ

ব্যাখ্যা

Question: How many kilograms of sugar costing Tk. 8 per kg must be mixed with 24 kg of sugar costing Tk. 6 per kg so that a gain of 10% by selling the mixture at Tk. 8.5 per kg?

Solution:
Le the sugar of 8Tk. per kg is = x kg

ATQ,
(110/100) × (6 × 24 + 8 × x) = 8.5(24 + x)
⇒ (11/10) × (144 + 8x) = 204 + 8.5x
⇒ 1584 + 88x = 2040 + 85x
⇒ 3x = 2040 - 1584
⇒ 3x = 456
∴ x = 152

২৯৫.

A man, his wife and daughter worked in a garden. The man worked for 3 days, his wife for 2 days and daughter for 4 days. The ratio of daily wages for man to women is 5 : 4 and the ratio for man to daughter is 5 : 3. If their total earnings is mounted to Tk. 105, then find the daily wage of the daughter.

ক
Tk. 9
খ
Tk. 13
গ
Tk. 10
ঘ
Tk. 15
ঙ
None

সঠিক উত্তর: ক

Tk. 9

উত্তর

সঠিক উত্তর: ক

Tk. 9

ক

ব্যাখ্যা

Question: A man, his wife and daughter worked in a garden. The man worked for 3 days, his wife for 2 days and daughter for 4 days. The ratio of daily wages for man to women is 5 : 4 and the ratio for man to daughter is 5 : 3. If their total earnings is mounted to Tk. 105, then find the daily wage of the daughter.

Solution:
Assume that the daily wages of man, women and daughter are Tk 5x, Tk 4x, Tk 3x respectively.
Multiply (no. of days) with (assumed daily wage) of each person to calculate the value of x.
[3 × (5x)] + [2 × (4x)] + [4 × (3x)] = 105
⇒ 15x + 8x + 12x = 105
⇒ 35x = 105
⇒ x = 3
Hence, man's daily wage = 5x = 5 × 3 = Tk. 15
Wife's daily wage = 4x = 4 × 3 = Tk. 12
Daughter's daily wage = 3x = 3 × 3 = Tk. 9

২৯৬.

Monir buys Tk. 40 shares in a company which pays 10% dividend. If the man gets 12.5% on his investment, at what price did he buy the shares?

ক
ক) Tk. 32
খ
খ) Tk. 30
গ
গ) Tk. 34
ঘ
ঘ) None of the above

সঠিক উত্তর: ক

ক) Tk. 32

উত্তর

সঠিক উত্তর: ক

ক) Tk. 32

ক

ব্যাখ্যা

Question: Monir buys Tk. 40 shares in a company which pays 10% dividend. If the man gets 12.5% on his investment, at what price did he buy the shares?

Solution:
Dividend on 40 share = 10%
∴ Dividend on 1 share = (10 × 40)/100
= Tk. 4

Tk. 12.50 is an income on an investment of Tk. 100
Tk. 4 is an income on an investment of = (100 × 4)/12.50
= (100 × 4 × 10)/125
= Tk. 32

২৯৭.

Which number when added to each of the numbers 24, 32 and 42 would make the sums to be in continued proportion?

ক
4
খ
5
গ
6
ঘ
8

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Which number when added to each of the numbers 24, 32 and 42 would make the sums to be in continued proportion?

Solution:
Let the number to be added is x.
∴(24 + x) / (32 + x) = (32 + x) / (42 + x).
⇒ (24 + x)(42 + x) = (32 + x)²
⇒ 1008 + 66x + x² = 1024 + 64x + x²
⇒ 2x = 16
∴ x = 8

২৯৮.

P and Q are two alloys of gold and copper prepared by mixing metals in the ratio 7:2 and 7:11 respectively. If equal amounts of both alloys are melted to form a third alloy R, then the ratio of gold and copper in R will be -

ক
7 : 5
খ
7 : 11
গ
7 : 13
ঘ
5 : 13
ঙ
5 : 9

সঠিক উত্তর: ক

7 : 5

উত্তর

সঠিক উত্তর: ক

7 : 5

ক

ব্যাখ্যা

Question: P and Q are two alloys of gold and copper prepared by mixing metals in the ratio 7:2 and 7:11 respectively. If equal amounts of both alloys are melted to form a third alloy R, then the ratio of gold and copper in R will be -

Solution:
Alloy P (Gold:Copper) = 7:2
Gold fraction in P = 7/9
Copper fraction in P = 2/9

Alloy Q (Gold:Copper) = 7:11
Gold fraction in Q = 7/18
Copper fraction in Q = 11/18

Let, 1 kg of each alloy is mixed;
Gold in Alloy R = 7/9 + 7/18 = 21/18

Copper in Alloy R = 2/9 + 11/18 = 15/18

∴ The ratio of Gold:Copper in R = (21/18) : (15/18)
= 21 : 15
= 7 : 5

২৯৯.

A, B and C invested capitals in the ratio of 4 : 6 : 9. At the end of the business term, they received the profit in the ratio of 2 : 3 : 5. Find the ratio of their time for which they contributed their capitals .

ক
ক) 1 : 1 : 9
খ
খ) 2 : 2 : 9
গ
গ) 10 : 10 : 9
ঘ
ঘ) 9 : 9 : 10

সঠিক উত্তর: ঘ

ঘ) 9 : 9 : 10

উত্তর

সঠিক উত্তর: ঘ

ঘ) 9 : 9 : 10

ঘ

ব্যাখ্যা

Here,
P₁ : P₂ : P₃ = 2 : 3 : 5 [profit's ratio] and
x₁ : x₂ : x₃ = 4 : 6 : 9 [investment's ratio]
According to the rule, Required ratio = P₁/x₁ : P₂/x₂ : P₃/x₃ = 2/4 : 3/6 : 5/9 = 1/2 : 1/2 : 5/9 = 9 : 9 : 10

৩০০.

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
5 : 11
খ
3 : 10
গ
10 : 4
ঘ
4 : 7

সঠিক উত্তর: খ

3 : 10

উত্তর

সঠিক উত্তর: খ

3 : 10

খ

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