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Ratio & Proportion, Alligation or Mixture

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উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ১১ / ১১ · ১,০০১–১,০৮০ / ১,০৮৬

১,০০১.
In a school the ratio of boys and girls is 3 : 4 respectively. When 50 girls leave the school the ratio becomes 4 : 5 respectively. How many boys are there in the school? 
  1. ক
    ক) 600
  2. খ
    খ) 625
  3. গ
    গ) 650
  4. ঘ
    ঘ) 675
ব্যাখ্যা
Question: In a school the ratio of boys and girls is 3 : 4 respectively. When 50 girls leave the school the ratio becomes 4 : 5 respectively. How many boys are there in the school? 

Solution: 
Let the number of boys and girls be 3x and 4x respectively

ATQ, 
3x/(4x - 50) = 4/5
⇒ 15x = 16x - 200
⇒ 16x - 15x = 200
∴ x = 200

∴ The number of boys = 3 × 200 = 600
১,০০২.
Time is taken by two trains running in opposite directions to cross a man standing on the platform in 28 seconds and 18 seconds respectively. It took 26 seconds for the trains to cross each other. What is the ratio of their speeds?
  1. ক
    ক) 2 : 3
  2. খ
    খ) 3 : 2
  3. গ
    গ) 1 : 4
  4. ঘ
    ঘ) 4 : 1
ব্যাখ্যা

Let the speed one train be x and the speed of the second train be y
Length of the first train = Speed × Time = 28x
Length of second train = Speed × Time = 18y

So, {(28x + 18y)/(x + y)} = 26
⇒ 28x + 18y = 26x + 26y
⇒ 2x = 8y

Therefore,
x : y = 4 : 1.

১,০০৩.
Kathy bought 4 times as many shares in Company X as Carl and Carl bought 3 times as many shares in the same company as Tom. Which of the following is the ratio of the number of shares bought by Kathy to the number of shares bought by Tom?
  1. ক
    3 : 4
  2. খ
    3 : 1
  3. গ
    4 : 1
  4. ঘ
    12 : 1
ব্যাখ্যা
Question: Kathy bought 4 times as many shares in Company X as Carl and Carl bought 3 times as many shares in the same company as Tom. Which of the following is the ratio of the number of shares bought by Kathy to the number of shares bought by Tom?

Solution:
Let,
Tom bought = a shares
Carl bought = 3a shares
Kathy bought = 4 × 3a = 12a shares

We're asked for the ratio of Kathy's shares to Tom's shares
Kathy : Tom = 12a : a = 12 : 1
১,০০৪.
What will be the ratio of the numbers which have a difference of 56 and the first number is 2/9 of the second?
  1. ক
    ক) 14 ∶ 56
  2. খ
    খ) 15 ∶ 56
  3. গ
    গ) 2 : 9
  4. ঘ
    ঘ) 16 ∶ 81
ব্যাখ্যা

Given that,

The difference between numbers = 56

first digit is = 2/9 of second number 

let second number = x

then first number = (2/9) x

the ratio of number = (2/9)x : x

 ∴ ratio is = 2: 9

১,০০৫.
In a proportion the product of 1st and 4th terms is 40 and that of 2nd and 3rd terms is 2.5x. Then the value of x is.
  1. ক
    ক) 16
  2. খ
    খ) 26
  3. গ
    গ) 75
  4. ঘ
    ঘ) 90
ব্যাখ্যা

Product of 1st and 4th terms (extremes) = product of 2nd and 3rd terms (means)
⇒ 2.5x = 40        
⇒ x = 40/2.5 = 16

১,০০৬.
A mixture of 20kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?
  1. ক
    ক) 2 kg
  2. খ
    খ) 4 kg
  3. গ
    গ) 3 kg
  4. ঘ
    ঘ) 1 kg
ব্যাখ্যা
20 কেজি মিশ্রনে পানি আছে = 20 এর 10% 
                                            = 20 এর 10/100
                                            = 2 কেজি 

স্পিরিট আছে = 20 - 2 = 18 কেজি 

ধরি, মিশ্রনে পানি মেশাতে হবে x কেজি 

প্রশ্নমতে,
⇒ 18/(2 + x) = 75/25
⇒18/(2 + x) = 3/1 
⇒3(2 + x)  = 18 
⇒6 + 3x  =18
⇒3x = 18 - 6 
⇒3x =12
⇒ x = 4
১,০০৭.
The fourth proportional to 5, 8, and 15 is -
  1. ক
    20
  2. খ
    22
  3. গ
    24
  4. ঘ
    25
ব্যাখ্যা

Question: The fourth proportional to 5, 8, and 15 is- 

Solution: 
Let, The fourth proportional is x.

So, 5/8 = 15/x
⇒ 5x = 120
∴ x = 24

১,০০৮.
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 white marbles are three in the jar?
  1. ক
    ক) 5
  2. খ
    খ) 6
  3. গ
    গ) 9
  4. ঘ
    ঘ) 10
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 white marbles are three in the jar?

Solution: 
let the amount of white, red and green marbels is 2x, 3x, 5x
after adding 6 more marbels it become 2x, 3x, 7x

so,
(2x + 3x + 7x) - (2x + 3x + 5x) = 6
12x - 10x = 6
2x = 6
x = 3

so, white marbels = 2 × 3 = 6
১,০০৯.
Rahim bought two varieties of sugar , costing 50tk/kg and 60tk/kg each, and mixed them in some ratio. Then he sold the mixture at 70tk/kg , making a profit of 20% . what was the ratio of the mixture?
  1. ক
    2 : 7
  2. খ
    1 : 5
  3. গ
    1 : 8
  4. ঘ
    3 : 8
ব্যাখ্যা
Question: Rahim bought two varieties of sugar , costing 50tk/kg and 60tk/kg each, and mixed them in some ratio. Then he sold the mixture at 70tk/kg , making a profit of 20% . what was the ratio of the mixture?

Solution:
ধরি, 50 টাকা দরে ক কেজির ক্রয়মূল্য = 50ক টাকা
এবং
60 টাকা দরে খ কেজির ক্রয়মূল্য 60ক টাকা

∴ (ক + খ) কেজির বিক্রয়মূল্য = (ক + খ)70 টাকা

এখন,
ক্রয়মূল্য 100 টাকা হলে 20% লাভে বিক্রয়মূল্য = 100 + 100 এর 20% = 100 + 20 = 120 টাকা

প্রশ্নমতে,
70(ক + খ)/ (50ক + 60খ) = 120/100
বা, (70ক + 70খ) / (50ক + 60খ) = 6/5
বা, 5 ×(70ক + 70খ) = 6×(50ক + 60খ) 
বা, 350ক + 350খ = 300ক + 360খ
বা, 350ক - 300ক = 360খ - 350খ
বা, 50ক = 10খ
বা, ক/খ = 10/50
বা, ক/খ = 1/5

সুতরাং মিশ্রণের অনুপাত= 1:5
১,০১০.
If Tk. 1564 be divided into three parts, proportional to 1/2 : 2/3 : 3/4, then the first part is:
  1. ক
    Tk. 532
  2. খ
    Tk. 182
  3. গ
    Tk. 408
  4. ঘ
    Tk. 204
ব্যাখ্যা
Question: If Tk. 1564 be divided into three parts, proportional to 1/2 : 2/3 : 3/4, then the first part is:

Solution:
Given ratio = 1/2 : 2/3 : 3/4
= 6 : 8 : 9

The first part = 1564 × (6/23)
= 408
১,০১১.
Tk. 23275 are divided among A, B and C in such a manner that the ratio of the amount of A to that of B is 3 : 7 and the ratio of the amount of B to that of C is 6 : 5. The amount of money received by B is = ?
  1. ক
    Tk. 11650
  2. খ
    Tk. 10290
  3. গ
    Tk. 12375
  4. ঘ
    Tk. 10780
  5. ঙ
    Tk. 14296
ব্যাখ্যা
Question: Tk. 23275 are divided among A, B and C in such a manner that the ratio of the amount of A to that of B is 3 : 7 and the ratio of the amount of B to that of C is 6 : 5. The amount of money received by B is = ?

Solution:
Given,
A : B = 3 : 7
= 18 : 42 (multiply with 6)

B : C = 6 : 5
= 42 : 35 (multiply with 7)

∴ A : B : C = 18 : 42 : 35

Let,
A = 18x, B = 42x, C = 35x

ATQ,
18x + 42x + 35x = 23275
⇒ 95x = 23275
⇒ x = 245

∴ Money received by B = 42 × 245
= Tk. 10290
১,০১২.
In what ratio must a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?
  1. ক
    ক) 1 : 4
  2. খ
    খ) 1 : 5
  3. গ
    গ) 2 : 5
  4. ঘ
    ঘ) 1 : 3
ব্যাখ্যা
Question: In what ratio must a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?

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

ATQ,
(30% of X) + (50% of Y) = 45% of (X + Y)
30X + 50Y = 45X + 45Y
15X = 5Y
X : Y = 1 : 3
১,০১৩.
How many litres of a 90% of concentrated acid needs to be mixed with a 75%. Solution of concentrated acid to get a 30 liter solution of 78% concentrated acid?
  1. ক
    ক) 8
  2. খ
    খ) 6
  3. গ
    গ) 7
  4. ঘ
    ঘ) 9
ব্যাখ্যা
Let V liters of 90% of conc acid is required to mix
Now to get 30 liters of 78% of conc acid solution we have total volume of solution = 30 liters
(30- v) liters of 75% of another conc acid to be mixed

Now we have
N1V + N2(30- V) = N × 30
90 × V + 75 × (30- V) = 78 × 30
V = 6 liters
১,০১৪.
A and B are in the ratio of 5 : 4 and B and C are in the ratio of 3 : 2. What is the ratio of A : C?
  1. ক
    12 : 8
  2. খ
    15 : 12
  3. গ
    20 : 8
  4. ঘ
    15 : 8
ব্যাখ্যা
Question:  A and B are in the ratio of 5 : 4 and B and C are in the ratio of 3 : 2. What is the ratio of A : C?

Solution:
Given the ratio of,
A : B = 5 : 4 = (5 × 3) : (4 × 3) = 15 : 12
And,
B : C = 3 : 2 = (3 × 4) : (2 × 4) = 12 : 8

∴ A : C = 15 : 8
১,০১৫.
If x : y = 4 : 5 & y : z = 7 : 9, find x : y : z.
  1. ক
    4 : 5 : 6
  2. খ
    5: 7 : 9
  3. গ
    9 : 15 : 21
  4. ঘ
    28 : 35 : 45
ব্যাখ্যা

Question: If x : y = 4 : 5 & y : z = 7 : 9, find x : y : z.

Solution: 
Given that, 
x : y = 4 : 5 = (4 × 7) : (5  × 7) = 28 : 35
∴ x : y = 28 : 35

And,
y : z = 7 : 9 = (7 × 5) : (9 × 5) = 35 : 45
∴ y : z = 35 : 45

∴ x : y : z = 28 : 35 : 45

১,০১৬.
Tea worth Tk. 240 per kg and Tk. 280 per kg are mixed with a third variety in the ratio 3 : 2 : 5. If the mixture is worth Tk. 300 per kg, the price of the third variety per kg will be:
  1. ক
    Tk. 324
  2. খ
    Tk. 344
  3. গ
    Tk. 368
  4. ঘ
    Tk. 410
ব্যাখ্যা

Question: Tea worth Tk. 240 per kg and Tk. 280 per kg are mixed with a third variety in the ratio 3 : 2 : 5. If the mixture is worth Tk. 300 per kg, the price of the third variety per kg will be:

Solution:
Let the price of the third variety be x Tk. per kg.
The given ratio of the three varieties is 3 : 2 : 5.
For calculation, let the quantities be 3 kg, 2 kg, and 5 kg respectively.

Total weight of the mixture = (3 + 2 + 5) = 10 kg
Total value of the mixture = 10 × 300 = Tk. 3000

According to the question (ATQ),
(3 × 240) + (2 × 280) + (5 × x) = 3000
⇒ 720 + 560 + 5x = 3000
⇒ 1280 + 5x = 3000
⇒ 5x = 3000 − 1280
⇒ 5x = 1720
⇒ x = 1720 / 5
∴ x = 344

∴ The price of the third variety is Tk. 344 per kg.

১,০১৭.
A milkman pays Tk. 6.40 per liter of milk. He adds water and sells the mixture at Tk. 8 per liter. By doing this, he makes 37.5% profit. Find the proportion of water to milk received by the customer.
  1. ক
    ক) 1 : 12
  2. খ
    খ) 1 : 10
  3. গ
    গ) 1 : 15
  4. ঘ
    ঘ) 1 : 20
ব্যাখ্যা

Let the quantity of milk purchased be x and quantity of water added be y.
Then, the ratio of water to milk is y : x.
CP = 6.4x
SP = 8(x+y)
Profit per cent = 37.5%
Therefore,
8(x+y) = 6.4x × 1.375
Or, 8x + 8y = 8.8x
Or, 8y = 0.8x
Or, y/x= 0.8/8
∴ y : x = 1 : 10

১,০১৮.
The average of two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. The larger number is -
  1. ক
    82
  2. খ
    80
  3. গ
    84
  4. ঘ
    48
ব্যাখ্যা
Question: The average of the two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. The larger number is -

Solution:
Let,
smaller number is x,
The larger number is y.

∴ (x + y)/2 = 62
⇒ x + y = 62 × 2
⇒ x + y = 124
∴ x = 124 - y .............. (1)

ATQ,
(x + 2)/y = 1/2
⇒ 2(x + 2) = y 
⇒ y = 2x + 4
⇒ y = 2(124 - y) + 4
⇒ y = 248 - 2y + 4
⇒ 3y = 252
⇒ y = 252/3
∴ x = 84

∴ The larger number is 84.
১,০১৯.
In a club 50% of the male voters and 80% of the female voters voted for candidate A. If candidate A received 70% of the total votes, what is the ratio of male to female voters?
  1. ক
    ক) 1/3
  2. খ
    খ) 3/4
  3. গ
    গ) 1/4
  4. ঘ
    ঘ) 1/2
ব্যাখ্যা

Let, Male voter = x  and  Female voter = y
50% of x + 80% of y = 70% of (x+y)
⇒ 50x/100 + 80y/100 = {70(x+y)}/100
⇒ (50x + 80y)/100 = (70x + 70y)/100
⇒ 80y – 70y= 70x – 50x
⇒ 10y = 20x
⇒ x/y = 10/20 = 1/2
∴ x : y = 1 : 2

১,০২০.
If A : B = 5 : 4 and A : C = 6 : 5 then, C : B = ?
  1. ক
    ক) 24 : 25
  2. খ
    খ) 25 : 24
  3. গ
    গ) 3 : 2
  4. ঘ
    ঘ) None
ব্যাখ্যা
Question: If A : B = 5 : 4 and A : C = 6 : 5 then, C : B = ?

Solution: 
A : B = 5 : 4 
⇒ A/B = 5/4

A : C = 6 : 5
⇒ A/C = 6/5
⇒ C/A = 5/6 

(A/B) × (C/A) = (5/4) × (5/6)
⇒ C/B = 25/24
⇒ C : B = 25 : 24 
১,০২১.
If Tk. 782 be divided into three parts, proportional to 1/2 : 2/3 : 3/4, then the first part is-
  1. ক
    Tk. 204
  2. খ
    Tk. 192
  3. গ
    Tk. 190
  4. ঘ
    Tk. 182
ব্যাখ্যা
Question: If Tk. 782 be divided into three parts, proportional to 1/2 : 2/3 : 3/4, then the first part is-

Solution:
Given ratio = 1/2 : 2/3 : 3/4
= 6 : 8 : 9

The first part = 782 × (6/23)
= 204
১,০২২.
The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?
  1. ক
    11 : 20
  2. খ
    17 : 22
  3. গ
    15 : 21
  4. ঘ
    21 : 22
ব্যাখ্যা
Question: The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?

Solution:
Originally, let the number of boys and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x) and (110% of 8x).
⇒ (120/100) × 7x and (110/100) × 8x
⇒ 42x/5 and 44x/5

∴ The required ratio = 42x/5 : 44x/5 = 21 : 22.
১,০২৩.
In an innings of a cricket match, three players A, B and C scored a total of 361 runs. If the ratio of the number of runs scored by A to that scored by B and also number of runs scored by B to that scored by C be 3 : 2, the number of runs scored by A was-
  1. ক
    151
  2. খ
    161
  3. গ
    171
  4. ঘ
    181
ব্যাখ্যা
Question: In an innings of a cricket match, three players A, B and C scored a total of 361 runs. If the ratio of the number of runs scored by A to that scored by B and also number of runs scored by B to that scored by C be 3 : 2, the number of runs scored by A was-

Solution:
A : B = 3 : 2
B : C = 3 : 2
= {3 × (2/3)} : {2 × (2/3)}
= 2 : (4/3)

A : B : C = 3 : 2 : (4/3)
= 9 : 6 : 4

∴ A's share = 361 × (9/19)
= 171
১,০২৪.
A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is:
  1. ক
    ক) 9 : 5
  2. খ
    খ) 2 : 3
  3. গ
    গ) 4 : 7
  4. ঘ
    ঘ) 5 : 6
  5. ঙ
    ঙ) 5 : 9
ব্যাখ্যা

Dog : Hare = (3 × 3) leaps of hare : 5 leaps of hare
= 9 : 5.

১,০২৫.
In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is-
  1. ক
    20 litres
  2. খ
    30 litres
  3. গ
    40 litres
  4. ঘ
    60 litres
ব্যাখ্যা
Question: In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is-

Solution:
Quantity of milk = 60 × (2/3) litres = 40 litres.
Quantity of water in it = (60 -  40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then,
milk : water = 40/(20 + x)
Now,
40/(20 + x) = 1/2
⇒ 20 + x = 80
∴ x = 60.
Quantity of water to be added = 60 litres.
১,০২৬.
A sum of money is divided among 6 males and some females in the ratio of the total money received by males to total money received by females as 3 : 1. If each male gets Tk. 600 and each female gets Tk. 1200, how many females are there? 
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    4
ব্যাখ্যা

Question: A sum of money is divided among 6 males and some females in the ratio of the total money received by males to total money received by females as 3 : 1. If each male gets Tk. 600 and each female gets Tk. 1200, how many females are there?

Solution:
Let the number of females be x.

Then,
(600 × 6)/1200x = 3/1
Or, 6/2x = 3/1
Or, 3/x = 3/1

So, 3x = 3
∴ x = 1

১,০২৭.
Which of the following represent ab = 64?
  1. ক
    ক) 32 : a = b : 2
  2. খ
    খ) a : 8 = b : 8
  3. গ
    গ) a : 16 = b : 4
  4. ঘ
    ঘ) 8 : a = 8 : b
ব্যাখ্যা
Question: Which of the following represent ab = 64?

Solution: 
32 : a = b : 2
ab = 64

a : 8 = b : 8
a = b

a : 16 = b : 4
4a = 16b
a = 4b

8 : a = 8 : b
8a = 8b 
a = b
১,০২৮.
Robi invests Tk. 40,000/- in a car wash center and starts a business. After 4 months, Rasel joins the business with an investment of Tk.50,000. At the end of the year, they make a profit of Tk. 1,87,000/-. What will be Rasel's share in this profit?
  1. ক
    ক) Tk. 38800
  2. খ
    খ) Tk. 64666.67
  3. গ
    গ) Tk. 85000
  4. ঘ
    ঘ) Tk. 97000
ব্যাখ্যা

We know,
The ratio of Investment x Time = Ratio of Profit
∴ (A's investment x Time) : (B's investment x Time) = Profit of A : Profit of B

∴ (Robi's Investment x Time) : (Rasel's Investment x Time) = Robi's Profit : Rasel's Profit
∴ 40000 x 12 : 50000 x 8 = Robi's Profit : Rasel's Profit
∴ Robi's Profit : Rasel's Profit = 4,80,000 : 4,00,000 = 6:5
∴ Rasel's profit = (5/11) × 187000 = Tk. 85000

১,০২৯.
A container is filled with a mixture of water and milk in the ratio of 3 : 5. Find the quantity of mixture to be drawn off and replaced with water, in order to get the mixture as half milk and half water.
  1. ক
    2/3
  2. খ
    1/2
  3. গ
    1/5
  4. ঘ
    1/4
ব্যাখ্যা

In whole mixture, there are:
Water = 3/(3 + 5) portion = 3/8 portion
And Milk = 5/(3 + 5) portion = 5/8 portion

Let, the portion of the mixture to be drawn off and replaced with water = x
So, in x portion mixture there are,
Water = 3/(3+5) portion of x = 3x/8 portion
And
Milk = 5/(3+5) portion of x = 5x/8 portion

As per the question,
(3/8 − 3x/8 + x) : (5/8 − 5x/8) = 1:1
Or, (3 − 3x + 8x)/8 = (5 − 5x)/8
Or, 3 + 5x = 5 − 5x
Or, 5x + 5x = 5 − 3
Or, 10x = 2
Or, x = 2/10
Or, x = 1/5

Answer: The 1/5 portion of the mixture to be drawn off and replaced with water, in order to get the mixture as half milk and half water.

১,০৩০.
A dishonest shopkeeper mixed cheaper quality rice, priced at Tk. 10/KG with good quality rice, priced at Tk. 25/KG, and sells the mixture at Tk. 15/KG. Find the ratio in which he mixes the two qualities of rice.
  1. ক
    4 : 1
  2. খ
    3 : 1
  3. গ
    3 : 2
  4. ঘ
    2 : 1
ব্যাখ্যা
Question: A dishonest shopkeeper mixed cheaper quality rice, priced at Tk. 10/KG with good quality rice, priced at Tk. 25/KG, and sells the mixture at Tk. 15/KG. Find the ratio in which he mixes the two qualities of rice.

Solution:



Thus, the ratio of quantities of cheaper and good quality rice = 10 : 5 = 2 : 1
১,০৩১.
The sides of a triangle are in the ratio (1/2) : (1/3) : (1/4), and its perimeter is 104 cm. The length of the long side is-
  1. ক
    28
  2. খ
    44
  3. গ
    48
  4. ঘ
    88
ব্যাখ্যা
Question: The sides of a triangle are in the ratio (1/2) : (1/3) : (1/4), and its perimeter is 104 cm. The length of the long side is-

Solution:
The sides of a triangle are in the ratio 1/2 : 1/3 : 1/4,
we multiply each term by 12 we get 6 : 4 : 3,
Let the sides be 6x, 4x, 3x 

Then, 13x=104
→ x= 8
• The length of the long side = 48.
১,০৩২.
A bag contains 50P, 25P and 10P coins in the ratio 5 : 9 : 4, amounting to Tk. 206. Find the number of coins of each type respectively.
  1. ক
    200,160,300
  2. খ
    200, 360,160
  3. গ
    160, 360, 200
  4. ঘ
    360, 160, 200
ব্যাখ্যা
Question: A bag contains 50P, 25P and 10P coins in the ratio 5 : 9 : 4, amounting to Tk. 206. Find the number of coins of each type respectively.

Solution:
Number of 50P coins = 5x
Number of 25P coins = 9x
Number of 50P coins = 4x

ATQ, 
5x (1/2) + 9x (1/4) + 4x(1/10) = 206
⇒ (50x + 45x + 8x)/20 = 206
⇒ 103x = 206 × 20
⇒ x = 40


The number of coins of each type respectively = 200, 360, 160 
১,০৩৩.
If two numbers are in the ratio 2 : 3 and the ratio becomes 3 : 4 when 8 is added to both the numbers, then the sum of the two numbers is-
  1. ক
    40
  2. খ
    56
  3. গ
    60
  4. ঘ
    80
ব্যাখ্যা
Question: If two numbers are in the ratio 2 : 3 and the ratio becomes 3 : 4 when 8 is added to both the numbers, then the sum of the two numbers is-

Solution:
Let, the numbers be 2x and 3x respectively.

ATQ,
(2x + 8)/(3x + 8) = 3/4
⇒ 9x + 24 = 8x + 32
⇒ 9x - 8x = 32 - 24
⇒ x = 8

∴ Sum of numbers = 2x + 3x
= 5x
= 5 × 8
= 40
১,০৩৪.
How much water be mixed in 36 litre of milk worth Tk. 4.80 per litre, so that value of mixture is Tk. 3.60 per litre?
  1. ক
    10 litres
  2. খ
    12 litres
  3. গ
    11 litres
  4. ঘ
    14 litres
ব্যাখ্যা
Question: How much water be mixed in 36 litre of milk worth Tk. 4.80 per litre, so that value of mixture is Tk. 3.60 per litre?

Solution:

Ratio of Milk to water will be = 18 : 6,
In 18 litre milk water added = 6 litre
In 1 litre milk water added = 6/18
In 36 litre milk water added = (6/18) × 36 = 12 litres
১,০৩৫.
A mixture of 20kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?
  1. ক
    4
  2. খ
    5
  3. গ
    6
  4. ঘ
    8
ব্যাখ্যা
Question: A mixture of 20kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?

Solution: 
20 কেজি মিশ্রনে পানি আছে = 20 এর 10% 
= 20 এর 10/100
= 2 কেজি 

স্পিরিট আছে = 20 - 2
= 18 কেজি 


ধরি,
মিশ্রনে পানি মেশাতে হবে x 

প্রশ্নমতে,
18/(2 + x) = 75/25
⇒ 18/(2 + x) = 3/1 
⇒ 3(2 + x)  = 18 
⇒ 6 + 3x  = 18
⇒ 3x = 18 - 6 
⇒ 3x =12
⇒ x = 4
১,০৩৬.
The ages of Sabiha and Suriya are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?
  1. ক
    6 years
  2. খ
    8 years
  3. গ
    10 years
  4. ঘ
    12 years
ব্যাখ্যা
Question:  The ages of Sabiha and Suriya are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?

Solution: 
Let, their ages are 7x, 3x 

ATQ, 
(7x + 6)/(3x + 6) = 5/3
⇒ 3(7x + 6) = 5(3x + 6)
⇒ 21x + 18 = 15x + 30 
⇒ 21x - 15x = 30 - 18 
⇒ 6x = 12 
∴ x = 2 

 The difference in their ages is = 7x - 3x 
= 4x
= 4 × 2
= 8 years
১,০৩৭.
The first number is 20% greater and the second number is 50% greater than a third number. What is the ratio of the two numbers?
  1. ক
    4 : 5
  2. খ
    2 : 3
  3. গ
    7 : 2
  4. ঘ
    5 : 2
  5. ঙ
    None of the above
ব্যাখ্যা
Question: The first number is 20% greater and the second number is 50% greater than a third number. What is the ratio of the two numbers?

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
১,০৩৮.
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?
  1. ক
    1/3
  2. খ
    1/4
  3. গ
    1/5
  4. ঘ
    1/7
  5. ঙ
    None of these
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 
⇒ (24 - 3x)/8 + x = (40 - 5x)/8
⇒ 24 - 3x + 8x = 40 - 5x
⇒ 5x + 24 = 40 - 5x 
⇒ 10x = 16 
∴ x = 8/5. 

So, part of the mixture replaced = (8/5) × (1/8) = 1/5. 
১,০৩৯.
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?
  1. ক
    2/7
  2. খ
    1/5
  3. গ
    2/3
  4. ঘ
    1/4
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
১,০৪০.
The ratio of boys and girls in a dance school is 7 ∶ 5. If, in the next session, 35 boys and 20 girls join the school, the ratio of boys and girls becomes 3 ∶ 2. How many girls are in the school now?
  1. ক
    90
  2. খ
    60
  3. গ
    65
  4. ঘ
    105
  5. ঙ
    70
ব্যাখ্যা
Question: The ratio of boys and girls in a dance school is 7 ∶ 5. If, in the next session, 35 boys and 20 girls join the school, the ratio of boys and girls becomes 3 ∶ 2. How many girls are in the school now?

Solution:
Given that,
The ratio of number of boys and girls = 7 ∶ 5
In the next session, 35 boys and 20 girls joined

Now,
Let the ratio of boys and girls be 7x ∶ 5x

According to the question,
(7x + 35) : (5x + 20) = 3 ∶ 2
⇒ (7x + 35)/(5x + 20) = 3/2
⇒ 14x + 70 = 15x + 60
⇒ 15x - 14x = 70 - 60 
∴ x = 10

∴ The number of girls now = (5 × 10) + 20 = 70

∴ The number of girls in the school now is 70.

 
১,০৪১.
A sum of tk. 427 is to be divided among A, B and C such that 3 times A's share, 4 times B's share and 7 times C's share are all equal. The share of C is:  
  1. ক
    ক) 140
  2. খ
    খ) 196
  3. গ
    গ) 84
  4. ঘ
    ঘ) 240
ব্যাখ্যা
Let, 
3A = 4B= 7C= x
A = x/3 , B = x/4 ,     C= x/7 
A: B: C = x/3 : x/4 : x/7 
           = 28 : 21 : 12 

The share of C is= (427 × 12)/ 61
                           = 84
১,০৪২.
In a can, there is a mixture of milk and water in the ratio 4 : 5. If it is filled with an additional 8 litres of milk the can would be full and ratio of milk and water would become 6 : 5. Find the capacity of the can?
  1. ক
    ক) 40
  2. খ
    খ) 44
  3. গ
    গ) 48
  4. ঘ
    ঘ) 52
ব্যাখ্যা

 Let the capacity of the can be T litres.
Quantity of milk in the mixture before adding milk = 4/9 (T - 8)
After adding milk, quantity of milk in the mixture = 6/11 T.
6T/11 - 8 = 4/9(T - 8)
10T = 792 - 352
=> T = 44

১,০৪৩.
A mixture contains 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  juice and sugar syrup in the new mixture is - 
  1. ক
    1 : 7 
  2. খ
    1 : 5
  3. গ
    2 : 7 
  4. ঘ
    2 : 5
ব্যাখ্যা
Question:  A mixture contains 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  juice and sugar syrup in the new mixture is - 

Solution: 
Let,
New mixture is = 12 liters
Old mixture = 12 × (1/4) = 3 liters 

Added sugar syrup = 12 - 9 = 9 liters 
New sugar syrup = 9 + (3/2) = 10.5 liters 

Ratio of  juice and sugar syrup = 1.5 : 10.5 
= 15 : 105 
= 1 : 7
১,০৪৪.
(3/4) : (1/2) :: 9y : ?
  1. ক
    ক) 3y
  2. খ
    খ) 6y
  3. গ
    গ) 9y
  4. ঘ
    ঘ) 27y
ব্যাখ্যা
Question: (3/4) : (1/2) :: 9y : ?

Solution: 
(3/4) : (1/2) :: 9y : p
⇒ (3/4) / (1/2) = 9y / p
⇒ 3/2 = 9y/p
⇒ p = 9y × 2/3
∴ p = 6y 
১,০৪৫.
A sum of money is divided among A, B, C and D in the ratio of 3 : 4 : 9 : 10 respectively. If the share of C is Tk. 2,530 more than the share of B, then what is the total amount of money of A and D together?
  1. ক
    ক) Tk. 6,078
  2. খ
    খ) Tk. 6,578
  3. গ
    গ) Tk. 6,478
  4. ঘ
    ঘ) Tk. 6,678
ব্যাখ্যা
Let
the shares of A, B, C and D be Tk. 3x, 4x, 9x and 10x respectively.
Now
= 9x - 4x = 2,530
⇒ 5x = 2,530
⇒ x = 2530/5
⇒ x = 506

∴ Required Amount
= 3x + 10x
= 13x
= Tk. (13 × 506)
= Tk. 6,578
১,০৪৬.
The ratio of boys to girls in a class is 2:5. If 2 boys leave and 4 girls join the class, the ratio of boys to the girls becomes 1:4. Originally, how many girls were in the class?
  1. ক
    16
  2. খ
    20
  3. গ
    21
  4. ঘ
    24
  5. ঙ
    None of these
ব্যাখ্যা

Boys = 2X
Girls = 5X
ATQ,
(2X - 2)/(5X + 4) = 1/4
or, 8X - 8 = 5X + 4
or, 3X = 12
or, X = 4
so, girls = 5 × 4 = 20.

১,০৪৭.
In a mixture with a 5:2 ratio of milk to water, adding 14 liters of water makes the ratio 5:4. What is the original quantity of milk in the mixture? 
  1. ক
    30 liters
  2. খ
    25 liters
  3. গ
    15 liters
  4. ঘ
    35 liters
  5. ঙ
    None
ব্যাখ্যা

Question: In a mixture with a 5:2 ratio of milk to water, adding 14 liters of water makes the ratio 5:4. What is the original quantity of milk in the mixture?

Solution:
The initial ratio is 5 : 2.
Let ‘b’ be the common ratio.

The initial quantity of milk = 5b liters
The initial quantity of water = 2b liters

Final quantity of milk = 5b liters
Final quantity of water = 2b + 14 liters

Final ratio = 5b : (2b + 14) = 5 : 4

⇒ 20b = 10b + 70
⇒ 10b = 70
⇒ b = 7

Therefore, the initial quantity of milk in the mixture = 5b
= 5 × 7
= 35 liters

১,০৪৮.
How much chicory at Tk. 5 a kg should be added to 20 kg of coffee at Tk. 12 a kg so that the mixture be worth Tk. 7.50 a kg?
  1. ক
    21 kg
  2. খ
    15 kg
  3. গ
    36 kg
  4. ঘ
    42 kg
ব্যাখ্যা
Question: How much chicory at Tk. 5 a kg should be added to 20 kg of coffee at Tk. 12 a kg so that the mixture be worth Tk. 7.50 a kg?

Solution:
Ratio in which coffee and chicory should be mixed
= 12 - 7.5 : 7.5 - 5 = 4.5 : 2.5 = 9 : 5.

Let x be quantity at Tk. 5 a kg
∴ 9 : 5 = x : 20
⇒ 5x = 180
⇒ x = 36
১,০৪৯.
A dealer has 1000 kg salt and he sells a part of it at 8% profit and the rest of it at 18% profit. The overall profit he earns is 14%. What is the quantity which is sold at 18% profit?
  1. ক
    ক) 400 Kg
  2. খ
    খ) 600 Kg
  3. গ
    গ) 800 Kg
  4. ঘ
    ঘ) 700 kg
ব্যাখ্যা
প্রশ্ন: A dealer has 1000 kg salt and he sells a part of it at 8% profit and the rest of it at 18% profit. The overall profit he earns is 14%. What is the quantity which is sold at 18% profit? 

সমাধান: 
ধরি,
৮% লাভে বিক্রয় করে ক কেজি
১৮% লাভে বিক্রয় করে ১০০০ - ক কেজি

৮% × ক + ১৮% × (১০০০ - ক) = ১৪% × ১০০০
বা, ৮ক + ১৮০০০ - ১৮ক = ১৪০০০
বা, ১০ক = ৪০০০
বা, ক = ৪০০

∴ ১৮% লাভে বিক্রয় করে ১০০০ - ক কেজি = ১০০০ - ৪০০ কেজি = ৬০০ কেজি
১,০৫০.
A started a business with a capital of Tk. 1,20,000. After some time, B joined the business with Tk. 80,000. At the end of one year, the profit was divided between A and B in the ratio 3 : 1. For how many months did B invest in the business?
  1. ক
    8 months
  2. খ
    4 months
  3. গ
    5 months
  4. ঘ
    6 months
  5. ঙ
    None
ব্যাখ্যা

Question: A started a business with a capital of Tk. 1,20,000. After some time, B joined the business with Tk. 80,000. At the end of one year, the profit was divided between A and B in the ratio 3 : 1. For how many months did B invest in the business?

Solution:
Let B join the business for X months.
A's investment is for 12 months,
B's for X months.

The ratio of profits is the ratio of (capital × time):
(1,20,000 × 12)/(80,000 × X) = 3/1
⇒ 1,20,000 × 12 = 80,000 × 3 × X
⇒ (1,20,000/80,000) × 12 = 3X
⇒ 1.5 × 12 = 3X
⇒ 18 = 3X
⇒ X = 6
Thus, B joined for 6 months.

১,০৫১.
How many litres of water should be added to a 65 litre mixture of milk and water containing milk and water in the ratio of 8 : 5 such that the resultant mixture contains 50% water?
  1. ক
    15
  2. খ
    20
  3. গ
    22
  4. ঘ
    25
  5. ঙ
    None
ব্যাখ্যা

Question: How many litres of water should be added to a 65 litre mixture of milk and water containing milk and water in the ratio of 8 : 5 such that the resultant mixture contains 50% water?

Solution: 
Given that, 
The mixture contains 65 litres of milk and water in the ratio 8 : 5.
Total parts = 8 + 5 = 13
Milk = (8/13) × 65 = 40 litres
Water = (5/13) × 65 = 25 litres

Let x litres of water be added.
∴ New water quantity = 25 + x litres
∴ New total mixture = 65 + x litres

The resultant mixture should contain 50% water.
(25 + x)/(65 + x) = 1/2
⇒ 2(25 + x) = 65 + x 
⇒ 50 + 2x = 65 + x 
⇒ 2x - x = 65 - 50 
∴ x = 15 

So 15 litres of water should be added.

১,০৫২.
The ratio of the present ages of Riyad and his father is 4 : 7. The father’s age at the time of Riyad’s birth was 18 years. Find the father’s present age.
  1. ক
    35 years
  2. খ
    48 years
  3. গ
    42 years
  4. ঘ
    32 years
ব্যাখ্যা
Question: The ratio of the present ages of Riyad and his father is 4 : 7. The father’s age at the time of Riyad’s birth was 18 years. Find the father’s present age.

Solution:
Present ratio is 7 : 4
Let actual ages are 7x and 4x.

∴ 7x - 4x = 18
⇒ 3x = 18
∴ x = 6

Hence the father’s present age = 7 × 6 = 42 years
১,০৫৩.
A mixture contains 2/5 of element A and 3/5 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?
  1. ক
    1 ml
  2. খ
    2 ml
  3. গ
    1.5 ml
  4. ঘ
    2.5 ml
ব্যাখ্যা
Question: A mixture contains 2/5 of element A and 3/5 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 = (2x)/5 ml.
B = (3x)/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,
(3x)/5 = (x + 5)/5
⇒ 15x = 5x + 25
⇒ 10x = 25
∴ x = 2.5

Original amount of A = (2 × 2.5)/5 ml = 1 ml.
১,০৫৪.
In what ratio must wheat A at Tk. 21.00 per kg be mixed with wheat B at Tk. 24.60 per kg, so that the mixture be worth of Tk. 22.00 per kg?
  1. ক
    ক) 13 : 5
  2. খ
    খ) 18 : 3
  3. গ
    গ) 17 : 5
  4. ঘ
    ঘ) 11 : 5
ব্যাখ্যা
Question: In what ratio must wheat A at Tk. 21.00 per kg be mixed with wheat B at Tk. 24.60 per kg, so that the mixture be worth of Tk. 22.00 per kg?

Solution: 
Let,
The quantity of wheat A = x kg 
The quantity of wheat B = y kg

ATQ,
21.00 x + 24.60 y = (x + y) × 22.00
⇒ 21.00 x + 24.60 y = 22.00 x + 22.00 y 
⇒ 22.00 x - 21.00 x = 24.60 y - 22.00 y
⇒ x = 2.60 y
⇒ x/y = 26/10
⇒ x/y = 13/5
x : y = 13 : 5 
১,০৫৫.
If X and Y are in the ratio 3 : 4, and Y and Z in the ratio 12 : 13, then X and Z will be in the ratio-
  1. ক
    ক) 9 : 23
  2. খ
    খ) 19 : 13
  3. গ
    গ) 9 : 13
  4. ঘ
    ঘ) 9 : 11
ব্যাখ্যা
Question: If X and Y are in the ratio 3 : 4, and Y and Z in the ratio 12 : 13, then X and Z will be in the ratio-

Solution: 
X : Y = 3 : 4
⇒ X/Y = 3/4

Y : Z = 12 : 13
⇒ Y/Z = 12/13

 X/Y ×  Y/Z = 3/4 × 12/13
⇒ X/Z = 9/13
∴ X : Z = 9 : 13 
১,০৫৬.
When 10% of a number is added to another number, the second number increases by 140%. What is the ratio between the first and the second number?
  1. ক
    9 : 2
  2. খ
    12 : 5
  3. গ
    14 : 1
  4. ঘ
    15 : 3
  5. ঙ
    None of the above
ব্যাখ্যা
Question: When 10% of a number is added to another number, the second number increases by 140%. What is the ratio between the first and the second number?

Solution:
Let,
The two numbers be x and y

ATQ,
y + 10% of x = y + 140% of y
⇒ 10x/100 = 140y/100
⇒ x/10 = 7y/5
⇒ 5x = 70y
⇒ x = 14y
⇒ x/y = 14/1
∴ x : y = 14 : 1
১,০৫৭.
The average of two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. What is the smaller number?
  1. ক
    40
  2. খ
    30
  3. গ
    60
  4. ঘ
    80
ব্যাখ্যা
Question: The average of two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. What is the smaller number?

Solution:
Let,
The numbers be x and y, x < y.
Then 
x + y = 124 ................(1)

(x + 2)/y = 1/2
⇒ y = 2x + 4 ..................(2)

x + y = 124
⇒ x + 2x + 4 = 124
⇒ 3x = 120
∴ x = 40
১,০৫৮.
Kamal and Kabir started a business investing Tk. 26000 and Tk. 30000 respectively. Out of a total profit of Tk. 7000, Kamal's share is -
  1. ক
    ক) 3750
  2. খ
    খ) 3250
  3. গ
    গ) 3150
  4. ঘ
    ঘ) 3050
ব্যাখ্যা
Question: Kamal and Kabir started a business investing Tk. 26000 and Tk. 30000 respectively. Out of a total profit of Tk. 7000, Kamal's share is -

Solution:
Given,
Investment ratio = 26000 : 30000
= 13 : 15
Sum of the ratio's = 13 + 15 = 28

∴ Kamal's share = 7000 × (13/28)
= 3250
১,০৫৯.
A bag contains Tk. 410 in the form of Tk. 5, Tk. 2, and Tk. 1 coins. The number of coins is in the ratio 4 : 6 : 9. So, find the number of 2 Taka’s coins -
  1. ক
    60
  2. খ
    56
  3. গ
    58
  4. ঘ
    62
  5. ঙ
    64
ব্যাখ্যা
ATQ,
if the ratio of coins = 4 : 6 : 9
That means Tk. 5 coins are 4, Tk. 2 coins are 6, and then Tk. 1 coins are 9.

According to the given ratio, the ratio of amounts = 5 × 4 : 6 × 2: 9 × 1 = 20 : 12 : 9
The sum of the ratios of the amounts = 20 + 12 + 9
= Tk. 41

But ATQ,
it is Tk. 410, which means multiply each ratio by 10
i.e., new ratio = 40 : 60 : 90
Now, 40 × 5 : 60 × 2 : 90 × 1 = 200 : 120 : 90

The total amount in the form of two taka coins = 120
So, the two taka coins = 120/2 = 60.
১,০৬০.
A Vessel is filled with liquid, 3 parts of which are water and 4 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?
  1. ক
    1/5
  2. খ
    1/6
  3. গ
    1/7
  4. ঘ
    1/8
ব্যাখ্যা

Question: A Vessel is filled with liquid, 3 parts of which are water and 4 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:
মনে করি,
পাত্রের মিশ্রণের পরিমাণ ৭ লিটার

মিশ্রণে পানির পরিমাণ ৩ লিটার
মিশ্রণে সিরাপের পরিমাণ  ৪ লিটার

পাত্রের পানি ও সিরাপের পরিমাণ অর্ধেক অর্ধেক করতে ক লিটার মিশ্রণ অপসারন করে পানি দিতে হবে।

ক লিটার মিশ্রণে পানির পরিমাণ ৩ক/৭ লিটার
ক লিটার মিশ্রণে সিরাপের পরিমাণ ৪ক/৭ লিটার

পানি মিশানোর পর,
নতুন মিশ্রণে পানির পরিমাণ হবে {(৩ - ৩ক/৭) + ক} লিটার
= (২১ + ৪ক)/৭ লিটার
নতুন মিশ্রণে সিরাপের পরিমাণ হবে (৪ - ৪ক/৭) লিটার
= (২৮ - ৪ক)/৭ লিটার

শর্তানুযায়ী,
(২১ + ৪ক)/৭ = (২৮ - ৪ক)/৭ 
বা, ২১ + ৪ক = ২৮ - ৪ক
বা, ৪ক + ৪ক = ২৮ - ২১
বা, ৮ক = ৭
∴ ক = ৭/৮

৭ লিটার মিশ্রণ ৭ লিটারের ১ বা সম্পূর্ণ অংশ
∴ ৭/৮ লিটার মিশ্রণ ৭ লিটারের (৭/৮)/৭ অংশ
= ১/৮ অংশ

১,০৬১.
The cost of 3 chairs and 2 tables is equal to the cost of 1 chair and 3 tables. Find the ratio of the price of the chair and the table.
  1. ক
    2 : 1
  2. খ
    1 : 3
  3. গ
    3 : 2
  4. ঘ
    1 : 2
ব্যাখ্যা
Question: The cost of 3 chairs and 2 tables is equal to the cost of 1 chair and 3 tables. Find the ratio of the price of the chair and the table.

Solution: 
Let, the price of the chair is = x 
the price of the table is = y

ATQ,
3x + 2y = x + 3y
3x - x = 3y - 2y
2x = y
x/y = 1/2

∴ x : y = 1 : 2
১,০৬২.
From 50 L wine, 10 L is removed and replaced with water. Process repeated twice. Wine left = ?
  1. ক
    31
  2. খ
    36
  3. গ
    34
  4. ঘ
    32
ব্যাখ্যা
Question: From 50 L wine, 10 L is removed and replaced with water. Process repeated twice. Wine left = ?

Solution:
A container has 50 liters of wine.
Each time, 10 liters of wine is removed and replaced with water.
This process is done twice.

After first removal of 10 L wine 
Wine left = 40
Water added 10 L
Total 50 L
Wine part = 40/50= 4/5
Water part = 1/5
Now, again 10 liters of wine is removed 
Wine removed = (4/5)×10 = 8
Water removed = (1/5) × 10 = 2

After removal:
Wine left = 40- 8 =32 L
Water left = 10- 2 = 8 L
10 L water added 
New water = 18 L
and Wine Left = 32 L
১,০৬৩.
A 35-liter mixture contains milk and water in a 3 : 4 ratio. How much milk should be added to the mixture to make the ratio equal?
  1. ক
    3 liters
  2. খ
    5 liters
  3. গ
    7 liters
  4. ঘ
    8 liters
ব্যাখ্যা
Question: A 35-liter mixture contains milk and water in a 3 : 4 ratio. How much milk should be added to the mixture to make the ratio equal?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 35 × (3/7) = 15 liters.
Quantity of water = 35 × (4/7) = 20 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(15 + x) : 20 = 1 : 1
⇒ (15 + x)/20 = 1/1
⇒ 15 + x = 20
⇒ x = 20 - 15
⇒ x = 5

∴ Quantity of milk to be added = 5 liters
১,০৬৪.
The ratio of milk to water in a mixture is 6 : 3. When adding 6 liters of water, the ratio becomes 5:5. What was the quantity of milk in the original mixture? 
  1. ক
    15 liters
  2. খ
    40 liters
  3. গ
    12 liters
  4. ঘ
    20 liters
ব্যাখ্যা

Question: The ratio of milk to water in a mixture is 6 : 3. When adding 6 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 = 6x liters
Water = 3x liters

When 6 liters of water are added, the new quantity of water becomes = (3x + 6) liters
The new ratio becomes 5 : 5, which simplifies to 1 : 1. This means the amount of milk and water are now equal. 
6x = 3x + 6
3x = 6 
∴ x = 2

So, the initial quantity of Milk = 6 × 2 = 12 liters

১,০৬৫.
Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 2 : 2. If the mixture is worth Tk. 153 per kg, the price of the third variety per kg will be:
  1. ক
    Tk. 184.5  
  2. খ
    Tk. 150
  3. গ
    Tk. 141.5
  4. ঘ
    None of these
ব্যাখ্যা
Question: Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 2 : 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 × 2y + x × 2y = 153 (y + 2y + 2y)
⇒ 126 + 270 + 2x = 765
⇒ 2x = 369
∴ x = 184.5 tk
১,০৬৬.
A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 72 minutes, will be covered by A in :
  1. ক
    ক) 9 minutes.
  2. খ
    খ) 10 minutes.
  3. গ
    গ) 12 minutes.
  4. ঘ
    ঘ) 14 minutes.
ব্যাখ্যা
The ratio of the speed of A, B and C = 6 ∶ 3 ∶ 1
The ratio of the time taken = 1/6 ∶ 1/3 ∶ 1 = 1 ∶ 2 ∶ 6

Time taken by C to cover the distance = 72 minutes

If C takes 6 min, then A takes 1 min.
If C takes 72 min, then A takes 72 × (1/6) min.
                                                = 12 minutes.
১,০৬৭.
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
    ক) 2 : 3 : 4
  2. খ
    খ) 3 : 4 : 2
  3. গ
    গ) 4 : 6 : 3
  4. ঘ
    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
১,০৬৮.
The drink for children has juice and water in the ratio 5:2 while the drink for adults has them in ratio 7:4. If both the mixes are poured in a jug, find the final ratio of water to juice in the jug.
  1. ক
    ক) 8:35
  2. খ
    খ) 25:52
  3. গ
    গ) 35:8
  4. ঘ
    ঘ) 52:25
ব্যাখ্যা

Children's drink → Juice : Water = 5 : 2 → Total 5 + 2 = 7 parts of liquid
Adult's drink → Juice : Water = 7 : 4 → Total 7 + 4 = 11 parts of liquid

Jug juice = Juice from children's mix + juice from adult mix = 5/7 + 7/11
= 104/77
Jug Water = Water from children's mix + Water from adult's mix = 2/7 + 4/11
= 50/77

Water to juice ratio in Jug = 50/77 : 104/77
= 50 : 104
= 25 : 52

১,০৬৯.
If a : b = 5 : 6, b : c = 4 : 7, then a : b : c = ?
  1. ক
    10 : 12 : 24
  2. খ
    6 : 12 : 21
  3. গ
    5 : 6 : 12
  4. ঘ
    10 : 12 : 21
ব্যাখ্যা
Question: If a : b = 5 : 6, b : c = 4 : 7, then a : b : c = ?

Solution:
a : b = 5 : 6
= (5 × 4) : (6 × 4)
= 20 : 24 

b : c = 4 : 7
= (4 × 6) : (7 × 6)
= 24 : 42

∴ a : b : c = 20 : 24 : 42 
= 10 : 12 : 21
১,০৭০.
What must be added to each term of the ratio 9 : 13, So as to make it equal to 3 : 4?
  1. ক
    ক) 7
  2. খ
    খ) 4
  3. গ
    গ) 3
  4. ঘ
    ঘ) 5
ব্যাখ্যা
Question: What must be added to each term of the ratio 9 : 13, So as to make it equal to 3 : 4?

Solution: 
ধরি,
যে সংখ্যাটি যোগ করতে হবে সেটি হল = ক

প্রশ্নমতে,
(৯ + ক) : (১৩ + ক) = ৩ : ৪
৩৯ + ৩ক = ৩৬ + ৪ক
ক = ৩
১,০৭১.
The ratio of investments of A, B, and C is 2 : 3 : 4, and their profit ratio is 1 : 2 : 3. If A invested for 12 months, find for how many months C invested.
  1. ক
    16 Months
  2. খ
    18 Months
  3. গ
    22 Months
  4. ঘ
    24 Months
ব্যাখ্যা

Question: The ratio of investments of A, B, and C is 2 : 3 : 4, and their profit ratio is 1 : 2 : 3. If A invested for 12 months, find for how many months C invested.

Solution:
Let the investments of A, B, and C be:
IA : IB : IC = 2 : 3 : 4

Let the time periods of investment be:
TA : TB : TC = ?

Profit = investment × time,
PA : PB : PC = 1 : 2 : 3

So:
IA × TA : IB × TB : IC × TC = 1 : 2 : 3

Substitute investments in ratio form:
⇒ 2 × 12 : 3 × TB : 4 × TC = 1 : 2 : 3
⇒ 24 : 3TB : 4TC = 1 : 2 : 3

Find multiplier
Let k be the factor:
⇒ 24 = 1 × k 
 ⇒ k = 24

Then:
3TB = 2 × k
⇒ 3TB = 2 × 24
⇒ 3TB = 48
⇒ TB = 48/3
⇒ TB = 16 months

Again,
4TC = 3 × k
⇒ 4TC = 3 × 24
⇒ 4TC = 72
⇒ TC = 72/4
⇒ TC = 18 months

১,০৭২.
Three containers A, B and C are having mixtures of milk and water in the ratio 20 : 28, 18 : 30 and 8 : 40 respectively. Find the ratio of milk to water, if the mixtures of all the three containers are mixed together?
  1. ক
    ক) 6 : 5
  2. খ
    খ) 5 : 6
  3. গ
    গ) 53 : 115
  4. ঘ
    ঘ) 23 : 49
  5. ঙ
    ঙ) None of the Above
ব্যাখ্যা

At, A ; M : W = 20 : 28 = 5 : 7 = 5/12 : 7/12
At, B; M : W = 18 : 30 = 3 : 5 = 3/8 : 5/8
At C ; M : W = 8 : 40 = 1 : 5 = 1/6 : 5/6
M : W = (5/12 + 3/8 + 1/6) : ( 7/12 + 5/8 + 5/6)
= (20 + 18 + 8)/48 : (28 + 30 + 40)/48
= 46 : 98
= 23 : 49

১,০৭৩.
A dishonest shopkeeper mixed cheaper quality rice, priced at Tk. 10/KG with good quality rice, priced at Tk. 25/KG, and sells the mixture at Tk. 15/KG. Find the ratio in which he mixes the two qualities of rice.
  1. ক
    2 : 1
  2. খ
    2 : 3
  3. গ
    3 : 2
  4. ঘ
    4 : 3
ব্যাখ্যা
Question: A dishonest shopkeeper mixed cheaper quality rice, priced at Tk. 10/KG with good quality rice, priced at Tk. 25/KG, and sells the mixture at Tk. 15/KG. Find the ratio in which he mixes the two qualities of rice.

Solution:

Thus, the ratio of quantities of cheaper and good quality rice = 10 : 5 = 2 : 1
১,০৭৪.
The ages of Sakil and Lamia are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their present ages?
  1. ক
    6 years
  2. খ
    8 years
  3. গ
    10 years
  4. ঘ
    12 years
ব্যাখ্যা
Question: The ages of Sakil and Lamia are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their present ages?

Solution: 
Let, their ages are 7x, 3x 

ATQ, 
(7x + 6)/(3x + 6) = 5/3
⇒ 3(7x + 6) = 5(3x + 6)
⇒ 21x + 18 = 15x + 30 
⇒ 21x - 15x = 30 - 18 
⇒ 6x = 12 
∴ x = 2 

 The difference in their ages is = 7x - 3x 
= 4x
= 4 × 2
= 8 years
১,০৭৫.
In what ratio must water be mixed with milk to gain 20% by selling the mixture at cost price?
  1. ক
    ক) 4 : 1
  2. খ
    খ) 5 : 1
  3. গ
    গ) 6 : 1
  4. ঘ
    ঘ) 8 : 1
ব্যাখ্যা
Question: In what ratio must water be mixed with milk to gain 20% by selling the mixture at cost price?

Solution:
Let he has to make milk to water ratio 1 : x to earn 20% profit
Let the cost price of milk be C/L

Then According to the question,
C × (x + 1) = C × (120/100)​
⇒ x + 1 = 6/5
⇒ x = 6/5 - 1
⇒ x = 1/5

So milk to water ratio = 1 : x
= 1 : 1/5
= 5 : 1
১,০৭৬.
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?
  1. ক
    ক) 1/3
  2. খ
    খ) 1/4
  3. গ
    গ) 1/5
  4. ঘ
    ঘ) 1/7
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: 
ধরি,
পাত্রের মিশ্রণের পরিমাণ ৮ লিটার।

মিশ্রণে পানির পরিমাণ ৩ লিটার
মিশ্রণে সিরাপের পরিমাণ ৫ লিটার

পাত্রের পানি ও সিরাপের পরিমাণ অর্ধেক অর্ধেক করতে ক লিটার মিশ্রণ অপসারণ করে পানি দিতে হবে।

ক লিটার মিশ্রণে পানির পরিমাণ ৩ক/৮ লিটার
ক লিটার মিশ্রণে সিরাপের পরিমাণ ৫ক/৮ লিটার

পানি মিশানোর পর,
নতুন মিশ্রণে পানির পরিমাণ হবে  (৩ - ৩ক/৮) + ক লিটার
=(২৪ - ৩ক)/৮ + ক লিটার
= (২৪ + ৫ক)/৮ লিটারনতুন মিশ্রণে সিরাপের পরিমাণ হবে (৫ - ৫ক/৮) লিটার
= (৪০ - ৫ক)/৮ লিটার

শর্তমতে,
(২৪ + ৫ক)/৮ = (৪০ - ৫ক)/৮
বা, ২৪ + ৫ক = ৪০ - ৫ক
বা, ১০ক = ১৬
বা, ক = ৮/৫

৮ লিটার মিশ্রণ ৮ লিটারের সম্পূর্ণ বা ১ অংশ
∴ ৮/৫ লিটার মিশ্রণ ৮ লিটারের (৮/৫)/৮ অংশ
= ১/৫ অংশ
১,০৭৭.
The ratio of present age Anik and Babul is 12 : 7 . After 3 years their ratio will be 30 : 20 then find the present age of Anik? 
  1. ক
    19 years 
  2. খ
    7 years 
  3. গ
    12 years 
  4. ঘ
    14 years 
ব্যাখ্যা
Question: The ratio of present age Anik and Babul is 12 : 7 . After 3 years their ratio will be 30 : 20 then find the present age of Anik? 

Solution:
Let,
present age of Anik = 12x
and present age of Babul = 7x 

ATQ,
(12x + 3)/(7x + 3) = 30/20
⇒(12x + 3) × 20 = (7x + 3) × 30
⇒ 240x + 60 = 210x + 90 
⇒ 30x = 30
⇒ x = 1

∴ Present age of Anik = 12 × 1 years
= 12 years
১,০৭৮.
Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to stamps?
  1. ক
    ক) 5 : 1
  2. খ
    খ) 10 : 5
  3. গ
    গ) 15 : 2
  4. ঘ
    ঘ) 25 : 2
ব্যাখ্যা
Question : Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to British stamps?

Solution:
US : Indian =  5 : 2  = 25 : 10
Indian : British = 5 : 1 = 10 : 2
US : British = 25 : 2
১,০৭৯.
If a : b = 2 : 3 and b : c = 6 : 7 than a : b : c = ?
  1. ক
    4 : 6 : 7
  2. খ
    2 : 4.5 : 7
  3. গ
    2 : 9 : 7
  4. ঘ
    4 : 9 : 7
ব্যাখ্যা
Question: If a : b = 2 : 3 and b : c = 6 : 7 than a : b : c = ?

Solution:
a : b = 2 : 3
= 2 × 2 : 3 × 2
= 4 : 6

b : c = 6 : 7

∴ a : b : c = 4 : 6 : 7
১,০৮০.
The monthly salaries of A, B and C are in the ratio 1 : 2 : 3. If C’s monthly salary is Tk. 1200 more than that of A, then what is B’s annual salary?
  1. ক
    Tk. 12000
  2. খ
    Tk. 14400
  3. গ
    Tk. 15000
  4. ঘ
    None of the above
ব্যাখ্যা

Question: The monthly salaries of A, B and C are in the ratio 1 : 2 : 3. If C’s monthly salary is Tk. 1200 more than that of A, then what is B’s annual salary?

Solution:
Let the monthly salary of A, B and C be x, 2x and 3x
If C’s monthly salary is Tk. 1200 more than that of A, then

ATQ,
3x = x + 1200
⇒ 2x = 1200
⇒ x = 1200/2 = 600
∴ x = 600

Then, B’s monthly salary = 2x = 2 × 600 = 1200

∴ B’s annual salary = 1200 × 12 = Tk. 14400