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Ratio & Proportion, Alligation or Mixture

মোট প্রশ্ন১,০৮৬এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ৬ / ১১ · ৫০১–৬০০ / ১,০৮৬

৫০১.
A, B, C enter into partnership. A invests 3 times as much as B invests and B invests two third of what C invests. At the end of the year, profit earned is TK. 6600. What is the share of B?
  1. ক
    1100
  2. খ
    1150
  3. গ
    1175
  4. ঘ
    1200
সঠিক উত্তর: ঘ
1200
উত্তর
সঠিক উত্তর: ঘ
1200
ঘ
ব্যাখ্যা
Question: A, B, C enter into partnership. A invests 3 times as much as B invests and B invests two third of what C invests. At the end of the year, profit earned is TK. 6600. What is the share of B?

Solution:
A invests 3 times as much as B invests and B invests two third of what C invests.
Let,
C invests = x
B invests = 2x/3
A invests = 3 × (2x/3) = 2x

∴ The ratio of invests or profit = 2x : 2x/3 : x
= 6x : 2x : 3x
= 6 : 2 : 3

The share of B = (2/11) × 6600
= 1200
৫০২.
If A : B = 4 : 5, B: C = 7 : 9 and C : D = 3 : 4, and if A’s share is Tk. 1680, the share of D is -
  1. ক
    ক) Tk. 2100
  2. খ
    খ) Tk. 2700
  3. গ
    গ) Tk. 2900
  4. ঘ
    ঘ) Tk. 3600
সঠিক উত্তর: ঘ
ঘ) Tk. 3600
উত্তর
সঠিক উত্তর: ঘ
ঘ) Tk. 3600
ঘ
ব্যাখ্যা

A : B = 4 : 5;

B : C = 7 : 9
= (7 × 5/7) : (9 × 5/7)
= 5 : 45/7

C : D = 3 : 4
= (3 × 15/7) : (4 × 15/7)
= 45/7 : 60/7.

A : B : C : D
= 4 : 5 : 45/7 : 60/7
= 28 : 35 : 45 : 60.

Let the shares of A, B, C, D be 28x, 35x, 45x, 60x respectively.

Then,
28x = 1680
⇒ x = 1680/28
= 60

∴ D's Share = 60x = Tk. (60 × 60)
= Tk. 3600.

৫০৩.
P, Q and R investment in a partnership in the ratio of 5 : 6 : 8. The ratio of their profit is 5 : 3 : 12. Find the ratio of time for their investment?
  1. ক
    ক) 4 : 2 : 3
  2. খ
    খ) 2 : 1 : 3
  3. গ
    গ) 3 : 2 : 5
  4. ঘ
    ঘ) 6 : 1 : 4
সঠিক উত্তর: খ
খ) 2 : 1 : 3
উত্তর
সঠিক উত্তর: খ
খ) 2 : 1 : 3
খ
ব্যাখ্যা
The required ratio of time
= ratio of (profit/investment)
= (5/5) : (3/6) : (12/8)
= 1 : 1/2 : 3/2
 = (1 × 2) : (1/2 × 2) : (3/2 × 2)
= 2 : 1 : 3
৫০৪.
In what ratio water must be mixed with milk costing Tk. 50 per liter to get a mixture worth Tk. 35 per liter?
  1. ক
    3 : 5
  2. খ
    3 : 2
  3. গ
    3 : 7
  4. ঘ
    2 : 7
সঠিক উত্তর: গ
3 : 7
উত্তর
সঠিক উত্তর: গ
3 : 7
গ
ব্যাখ্যা
Question: In what ratio water must be mixed with milk costing Tk. 50 per liter to get a mixture worth Tk. 35 per liter?

Solution:
Cost Price of water Tk. 0
Cost Price of milk TK. 50
Mean Price of mixure Tk. 35
Let,
Quantity of water be x
Quantity of milk be y
Using the formula,

x/y = (50-35)/(35-0)
x/y = 15/35
x/y = 3/7
∴x : y = 3 : 7
৫০৫.
Three numbers A, B, and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, by what amount should C be increased so that the new average becomes 3% higher? 
  1. ক
    110 Tk.
  2. খ
    140 Tk.
  3. গ
    144 Tk.
  4. ঘ
    124 Tk.
সঠিক উত্তর: গ
144 Tk.
উত্তর
সঠিক উত্তর: গ
144 Tk.
গ
ব্যাখ্যা

Question: Three numbers A, B, and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, by what amount should C be increased so that the new average becomes 3% higher?

Solution:
Let
A = x, B = 2x, C = 3x.
Then,
x + 2x + 3x = 600 × 3
Or, 6x = 1800
Or, x = 300

So, A = 300, B = 600, C = 900.
New value of A = 110% of 300 = 330
New value of B = 80% of 600 = 480 
New average = 103% of 600 = 618

∴ New value of C = (618 × 3) - (330 + 480)
= 1854 - 810
= 1044

∴ Increase in value of C = 1044 - 900
= 144 Tk.

৫০৬.
A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallons?
  1. ক
    20 gallons
  2. খ
    10 gallons
  3. গ
    15 gallons
  4. ঘ
    18 gallons
সঠিক উত্তর: ক
20 gallons
উত্তর
সঠিক উত্তর: ক
20 gallons
ক
ব্যাখ্যা
Question: A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallons?

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

ATQ, 
(x/2) - 8 = x/10 
⇒ (x - 16)/2 = x/10 
⇒ 10(x - 16) = 2x 
⇒ 10x - 160 = 2x
⇒ 10x - 2x = 160 
⇒  8x = 160 
∴ x = 160/8 = 20 gallons
৫০৭.
A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4 : 5. Find the quantity of alcohol in the given mixture.
  1. ক
    10
  2. খ
    12
  3. গ
    15
  4. ঘ
    18
  5. ঙ
    None of these
সঠিক উত্তর: ক
10
উত্তর
সঠিক উত্তর: ক
10
ক
ব্যাখ্যা
Question: A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4 : 5. Find the quantity of alcohol in the given mixture.

Solution:
Let the quantity of alcohol and water be 4x litres and 3x litres respectively
4x/(3x + 5) = 4/5
⇒ 20x = 4(3x+5)
⇒ 8x = 20
∴ x = 2.5

Quantity of alcohol = (4 × 2.5) litres = 10 litres.
৫০৮.
Three friends A, B and C invest in a business in the ratio 6 : 4: 10 respectively. They share the profit in the ratio 8 : 7 : 9 respectively. Find the ratio of time periods of their investment.
  1. ক
    ক) 3 : 2 : 5
  2. খ
    খ) 24 : 14 : 45
  3. গ
    গ) 80 : 75 : 54
  4. ঘ
    ঘ) 80 : 105 : 54
সঠিক উত্তর: ঘ
ঘ) 80 : 105 : 54
উত্তর
সঠিক উত্তর: ঘ
ঘ) 80 : 105 : 54
ঘ
ব্যাখ্যা

We know that Ratio of Investment x Time = Ratio of Profit
∴ To find the ratio of time just divide respective profits ratios by respective investment ratios
So if the profit ratio is P1: P2 : P3 and the investment ratio is I1 : I2 : I3 then,
Time ratio is found by = (P1/I1) : (P2/I2) : (P3/I3)

6 : 4 : 10 can be reduced to 3 : 2 : 5. So, their investments are actually in the ratio 3 : 2 : 5.
The ratio of time periods for A, B, and C is found by = 8/3 : 7/2 : 9/5
Making denominators common by multiplying by 30
∴ The ratio of Time periods for A, B, and C = (30 × 8)/3 : (30 × 7)/2 : (30 × 9)/5
= 80 : 105 : 54

৫০৯.
The salaries of A, B and C are in the ratio 1 : 3 : 4. If the salaries are increased by 5%, 10% and 10% respectively, then the increased salaries will be in the ratio - 
  1. ক
    11 : 23 : 34
  2. খ
    21 : 66 : 88
  3. গ
    22 : 36 : 28
  4. ঘ
    12 : 61 : 18
সঠিক উত্তর: খ
21 : 66 : 88
উত্তর
সঠিক উত্তর: খ
21 : 66 : 88
খ
ব্যাখ্যা

Question: The salaries of A, B and C are in the ratio 1 : 3 : 4. If the salaries are increased by 5%, 10% and 10% respectively, then the increased salaries will be in the ratio -

Solution:
Given that
Salary has  in 1 : 3 : 4 ratio
Let
A's Salary = Tk. 100
B's Salary = Tk. 300
C's Salary = Tk. 400

Now,
5% increase in A's Salary,
A's new Salary = (100 + 5% of 100) = Tk. 105

B's Salary increases by 10%, Then,
B's new Salary = (300 + 10% of 300) = Tk. 330

C's Salary increases by 10%,
C's new Salary = (400 + 10% of 400) = Tk. 440

Then, ratio of increased Salary,
A : B : C = 105 : 330 : 440 = 21 : 66 : 88

৫১০.
The earnings of Rahim is 12000 Tk every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 Tk. Find the ratio of the savings.
  1. ক
    12 : 30
  2. খ
    17 : 18
  3. গ
    8 : 25
  4. ঘ
    17 : 50
সঠিক উত্তর: ঘ
17 : 50
উত্তর
সঠিক উত্তর: ঘ
17 : 50
ঘ
ব্যাখ্যা
Question: The earnings of Rahim is 12000 Tk every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 Tk. Find the ratio of the savings.

Solution: 
Savings of Rohan per month = Tk. (12000-9960) = Tk. 2040

Yearly income of Anish = Tk. 191520
Hence, the monthly income of Anish = Tk. 191520/12 = Tk. 15960.
So, the savings of Anish per month = Tk. (15960 – 9960) = Tk. 6000

Thus, the ratio of savings of Rohan and Anish is 2040 : 6000 = 17 : 50.
৫১১.
Kalam bought two varieties of rice, costing Tk. 50/kg and Tk. 60/kg each, and mixed them in some ratio. Then he sold the mixture at Tk. 70/kg. Making a profit of 20%. What was the ratio of the mixture?
  1. ক
    ক) 1 : 10
  2. খ
    খ) 1 : 5
  3. গ
    গ) 2 : 7
  4. ঘ
    ঘ) 3 : 8
  5. ঙ
    ঙ) None
সঠিক উত্তর: খ
খ) 1 : 5
উত্তর
সঠিক উত্তর: খ
খ) 1 : 5
খ
ব্যাখ্যা
ধরি, কালাম 50 টাকায় চাল কেনে x পরিমাণ এবং 60 টাকায় কেনে y পরিমাণ চাল।
অর্থাৎ, তার মোট খরচ হয় (50x + 60y) টাকা।
যেহেতু সে 20% লাভে বিক্রয় করে চালের মিশ্রণ সেহেতু,
প্রশ্নমতে, 120/100(50x+60y) = 70(x+y)
বা, 60x + 72y = 70x + 70y
বা, 10x = 2y
বা, x:y = 1:5
৫১২.
Two vessels of equal capacity contain juice and water in the ratio of 7 : 2 and 11 : 7 respectively. The mixture of both vessels is mixed and transferred into a bigger vessel. What is the ratio of juice and water in the new mixture?
  1. ক
    18 : 9
  2. খ
    21 : 6
  3. গ
    22 : 7
  4. ঘ
    25 : 11
সঠিক উত্তর: ঘ
25 : 11
উত্তর
সঠিক উত্তর: ঘ
25 : 11
ঘ
ব্যাখ্যা
Question: Two vessels of equal capacity contain juice and water in the ratio of 7 : 2 and 11 : 7 respectively. The mixture of both vessels is mixed and transferred into a bigger vessel. What is the ratio of juice and water in the new mixture?

Solution:
The ratio of juice and water in the first vessel = 7 : 2  ................(1)
Total capacity of first vessel = 7 + 2 = 9 units

The ratio of juice and water in the second vessel = 11 : 7 ...............(2)
Total capacity of second vessel = 11 + 7 = 18 units

We will have to equal the total capacity of both vessels, so multiply by 2 in equation (1).
The ratio of juice and water in the first vessel = 14 : 4  ................(3)

∴ Ratio of juice and water in bigger vessel = (14 + 11) : (4 + 7) = 25 : 11
৫১৩.
If 2/3 of A = 75% of B = 0.6 of C, then A : B : C is- 
  1. ক
    27 : 24 : 40
  2. খ
    9 : 8 : 10 
  3. গ
    9 : 8 : 4
  4. ঘ
    None of these
সঠিক উত্তর: খ
9 : 8 : 10 
উত্তর
সঠিক উত্তর: খ
9 : 8 : 10 
খ
ব্যাখ্যা
Question: If 2/3 of A = 75% of B = 0.6 of C, then A : B : C is- 

Solution: 
2/3 of A=75%B
⇒ 2A/3 = 75B/100 = 3B/4
⇒ A/B = 9/8
⇒ A : B = 9 : 8 

75% of B = 0.6 of C
⇒ 3B/4 = 3C/5
⇒ B/C = 12/15 = 4/5
 ⇒ B : C = 4 : 5 = 8 : 10 

A : B : C = 9 : 8 : 10 
৫১৪.
The cost of a table and a chair are in the ratio of 5 : 7. If the cost of a chair and table is increased by 20% and 10% respectively, then what will be the new ratio?
  1. ক
    16 : 27
  2. খ
    60 : 77
  3. গ
    55 : 84
  4. ঘ
    None of these
সঠিক উত্তর: গ
55 : 84
উত্তর
সঠিক উত্তর: গ
55 : 84
গ
ব্যাখ্যা
Question: The cost of a table and a chair are in the ratio of 5 : 7. If the cost of a chair and table is increased by 20% and 10% respectively, then what will be the new ratio?

Solution:
Let, the cost of the table and chair be Tk. 5x and Tk. 7x respectively.

New cost of chair = 120% of 7x
= (120 × 7x)/100
= 42x/5

New cost of table = 110% of 5x
= (110 × 5x)/100
= 55x/10

∴ New ratio = 55x/10 : 42x/5
= 55 : 84
৫১৫.
In a school, the ratio of boys to girls is 7 : 5. If 20 more girls join and 14 boys leave, the new ratio becomes 6 : 5. How many boys were there?
  1. ক
    250
  2. খ
    310
  3. গ
    180
  4. ঘ
    290
  5. ঙ
    266
সঠিক উত্তর: ঙ
266
উত্তর
সঠিক উত্তর: ঙ
266
ঙ
ব্যাখ্যা
Question: In a school, the ratio of boys to girls is 7 : 5. If 20 more girls join and 14 boys leave, the new ratio becomes 6 : 5. How many boys were there?

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

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

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

So the number of the boys = 7x = 7 × 38 = 266
৫১৬.
The monthly incomes of two brothers are in the ratio 11 : 7 and their monthly expenditures are in the ratio 9 : 5. Each of them saves Tk. 4,800 per month. Find the monthly income of the first brother.
  1. ক
    Tk. 28800
  2. খ
    Tk. 25200
  3. গ
    Tk. 24000
  4. ঘ
    Tk. 27600
  5. ঙ
    Tk. 26400
সঠিক উত্তর: ঙ
Tk. 26400
উত্তর
সঠিক উত্তর: ঙ
Tk. 26400
ঙ
ব্যাখ্যা

Question: The monthly incomes of two brothers are in the ratio 11 : 7 and their monthly expenditures are in the ratio 9 : 5. Each of them saves Tk. 4,800 per month. Find the monthly income of the first brother.

Solution: 
Let the monthly income of the two brothers are 11x and 7x
Let the monthly expenses of the two brothers are 9y and 5y
Since each saves Tk. 4800 per month.

Then we get,
11x - 9y = 7x - 5y
⇒ 4x = 4y
∴  x = y

Now, 
11x - 9y = 4800
⇒ 11x - 9x = 4800
⇒ 2x = 4800
⇒ x = 4800/2
∴ x = 2400
∴ y = 2400

Therefore, 
Monthly income of the first brother = 11x = 11 × 2400 = Tk. 26400

So the monthly incomes are Tk. 26400.

৫১৭.
The present ages of John and Mary are In the ratio of 6 : 4. Five years ago their ages were in the ratio of 5 : 3. How old is John now?
  1. ক
    ক) 42
  2. খ
    খ) 36
  3. গ
    গ) 30
  4. ঘ
    ঘ) 24
সঠিক উত্তর: গ
গ) 30
উত্তর
সঠিক উত্তর: গ
গ) 30
গ
ব্যাখ্যা

ATQ,
6x - 5 : 4x - 5 = 5 : 3
Or, 20x - 25 = 18x - 15
Or, 2x = 10
Or, x = 5
So, John's age now is = 6×5 = 30 years

৫১৮.
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 third variety per kg will be:
  1. ক
    ক) Tk. 169.50
  2. খ
    খ) Tk. 170
  3. গ
    গ) Tk. 175.50
  4. ঘ
    ঘ) Tk. 180
সঠিক উত্তর: গ
গ) Tk. 175.50
উত্তর
সঠিক উত্তর: গ
গ) Tk. 175.50
গ
ব্যাখ্যা

Assume that the Third Variety of Tea quantity is X.
Then (126 X 1 + 135 X 1 + 2 X x)/(1+1+2) = 153
On solving,
261 + 2x = 4 X 153
2x = 612-261
x = 351/2
∴ x = 175.50

৫১৯.
A dishonest trade mixes 2kg of vegetable ghee costing Tk. 45 a kg with 3kg of standard ghee costing TK. 70 per kg. He sells the mixed ghee at TK. 65 per kg. What is his percentage of profit?
  1. ক
    8.33%
  2. খ
    16.67%
  3. গ
    25%
  4. ঘ
    6%
সঠিক উত্তর: ক
8.33%
উত্তর
সঠিক উত্তর: ক
8.33%
ক
ব্যাখ্যা
Question: A dishonest trade mixes 2kg of vegetable ghee costing Tk. 45 a kg with 3kg of standard ghee costing TK. 70 per kg. He sells the mixed ghee at TK. 65 per kg. What is his percentage of profit?

Solution:
Costing for ghee is (2 × 45 + 3 × 70) Taka
= 90 + 210 Taka
= 300 Taka

Selling Price = (2 + 3) × 65 taka 
= 5 × 65 Taka
= 325 Taka

∴ Profit 325 - 300 Taka 
= 25 Taka

∴ Percentage of profit = (25 × 100)/300 %
= 8.33%
৫২০.
If A : B = 3 : 4, C : B = 5 : 4, C : D = 10 : 9 then A : B : C : D is -
  1. ক
    ক) 8 : 6 : 9 : 10
  2. খ
    খ) 8 : 6 : 10 : 9
  3. গ
    গ) 6 : 8 : 10 : 9
  4. ঘ
    ঘ) 6 : 8 : 9 : 10
সঠিক উত্তর: গ
গ) 6 : 8 : 10 : 9
উত্তর
সঠিক উত্তর: গ
গ) 6 : 8 : 10 : 9
গ
ব্যাখ্যা

A: B = 3 : 4 = (3 × 4) : (4 × 4) = 12 : 16
C : B = 5 : 4 = (5 × 4) : (4 × 4) = 20 : 16
C : D = 10 : 9 = (10 × 2) : (9 × 2) = 20 : 18
∴ A : B : C : D = 12 : 16 : 20 : 18 = 6 : 8 :10 : 9

৫২১.
A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B tk. 5000 more than C. Out of a total profit of tk. 35,000, A receives
  1. ক
    ক) 8,400
  2. খ
    খ) 11,900
  3. গ
    গ) 13,600
  4. ঘ
    ঘ) 14,700
সঠিক উত্তর: ঘ
ঘ) 14,700
উত্তর
সঠিক উত্তর: ঘ
ঘ) 14,700
ঘ
ব্যাখ্যা

Let C = x.
Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000
So, x + x + 5000 + x + 9000 = 50000
⇒ 3x = 36000
⇒ x = 12000
A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12
So A's Share 
= Tk 35000× (21/50)
= Tk 14700

৫২২.
Of 132 examinees of a certain school, the ratio of successful to unsuccessful candidates is 9 : 2. If 4 more students passed, what would have been the ratio of successful to unsuccessful students?
  1. ক
    21 : 5
  2. খ
    18 : 5
  3. গ
    23 : 5
  4. ঘ
    28 : 5
সঠিক উত্তর: ঘ
28 : 5
উত্তর
সঠিক উত্তর: ঘ
28 : 5
ঘ
ব্যাখ্যা
Question: Of 132 examinees of a certain school, the ratio of successful to unsuccessful candidates is 9 : 2. If 4 more students passed, what would have been the ratio of successful to unsuccessful students?

Solution: 
successful candidates = (9/11) × 132
= 108 

unsuccessful candidates = (2/11) × 132
= 24 

new successful candidates = 108 + 4 
= 112 
new unsuccessful candidates = 24 - 4 = 20

new ratio = 112 : 20 
= 28 : 5 
৫২৩.
A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture becomes 1 : 1 = ?
  1. ক
    3/4
  2. খ
    1/5
  3. গ
    1/2
  4. ঘ
    1/3
সঠিক উত্তর: ঘ
1/3
উত্তর
সঠিক উত্তর: ঘ
1/3
ঘ
ব্যাখ্যা
Question: A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture becomes 1 : 1 = ?

Solution:
Original ratio of the mixture 3 : 1

Taking out the mixture actually means just taking out the wine because the water anyways is going to be added back.

If we remove 1 part of wine, it makes the ratio 2 : 2 (1 part water is added to keep the volume constant )
so, what we have actually done is remove 1 part wine from 3 part Wine. i.e. 1/3.

So, 1/3 mixture drawn.
৫২৪.
Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-
  1. ক
    3600
  2. খ
    7200
  3. গ
    8100
  4. ঘ
    None
সঠিক উত্তর: গ
8100
উত্তর
সঠিক উত্তর: গ
8100
গ
ব্যাখ্যা
Question: Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-

Solution:
Mr. Power's average charge is 3/2 as much as that Mr. Fixit

Let,
Mr. Power got 3x Taka per call
Mr. Fixit got 2x Taka per call

ATQ,
For Mr. Fixit
16 × 30 × 2x = 7200
⇒ x = 7200/960
∴ x = 7.5

∴ Mr. Power erned = 12 × 30 × 3x = 360 × 3 × 7.5 = 8100 taka
৫২৫.
A bag contains 50p, 25p, and 10p coins in the ratio 2 : 5 : 3, amounting to Tk. 510. Find the number of 50p coins.
  1. ক
    400
  2. খ
    500
  3. গ
    600
  4. ঘ
    1000
  5. ঙ
    480
সঠিক উত্তর: ক
400
উত্তর
সঠিক উত্তর: ক
400
ক
ব্যাখ্যা
Question: A bag contains 50p, 25p, and 10p coins in the ratio 2 : 5 : 3, amounting to Tk. 510. Find the number of 50p coins.

Solution:
Let the common ratio be 100k.
Number of 50p coins = 200k
Number of 25p coins = 500k
Number of 10p coins = 300k

Value of 50p coins = 0.5 × 200k = 100k
Value of 25p coins = 0.25 × 500k = 125k
Value of 10p coins = 0.1 × 300k = 30k

∴ Total value of all coins = 100k + 125k + 30k = 255k = 510 (given)
∴ k = 2

Therefore, Number of 50p coins = 200k = 200 × 2 = 400
৫২৬.
The sum of the square of three numbers is 532 and the ratio of the first and the second as also of the second and the third is 3:2. Then the first number is-
  1. ক
    18
  2. খ
    6
  3. গ
    12
  4. ঘ
    2x+3
  5. ঙ
    None of above
সঠিক উত্তর: ক
18
উত্তর
সঠিক উত্তর: ক
18
ক
ব্যাখ্যা

The first number : The second Number = 3:2 = 3 × 3:2 × 3 = 9:6
The second Number : 3rd Number = 3:2 =3 × 2:2 × 2 = 6:4
1st:2nd:3rd = 9:6:4
Let,
The 1st, 2nd & 3rd are consecutively = 9x, 6x, 4x
Therefore,
(9x)2 + (6x)2 + (4x)2 = 532
So, X = 2 and 1st number is 9 × 2 = 18

৫২৭.
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 part
  2. খ
    1/4 part
  3. গ
    1/5 part
  4. ঘ
    1/7 part
সঠিক উত্তর: গ
1/5 part
উত্তর
সঠিক উত্তর: গ
1/5 part
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: 
The initial ratio of water : syrup is 3:5.
This means the total mixture has 3 parts water and 5 parts syrup → 8 parts in total.
We need to make the final ratio 1:1 (half water, half syrup).

Let x be the fraction of the mixture drawn off and replaced with water.
Water removed = 3x/8
Water left = (3/8) - 3x/8

Syrup removed = 5x/8
Syrup left = (5/8) - 5x/8

Since the removed mixture is replaced with water, the new amount of water is = (3/8) - 3x/8 + x

ATQ, 
(3/8) - 3x/8 + x = (5/8) - 5x/8
3 - 3x + 8x = 5 - 5x
10x = 2 
∴ x = 1/5

৫২৮.
The ratio of milk to water in 60 liters of a mixture is 7 : 5. The milk to be added to it make the ratio 2 : 1 is-
  1. ক
    ক) 10 liters
  2. খ
    খ) 12 liters
  3. গ
    গ) 15 liters
  4. ঘ
    ঘ) 16 liters
সঠিক উত্তর: গ
গ) 15 liters
উত্তর
সঠিক উত্তর: গ
গ) 15 liters
গ
ব্যাখ্যা
Question: The ratio of milk to water in 60 liters of a mixture is 7 : 5. The milk to be added to it make the ratio 2 : 1 is-

Solution:
Quantity of milk = 60 × (7/12) = 35 liters
Quantity of water = 60 - 35 = 25 liters

Let, x liters of milk be added to the mixture.

ATQ,
(35 + x)/25 = 2/1
⇒ 35 + x = 50
⇒ x = 50 - 35
⇒ x = 15
৫২৯.
In a garden, a man worked 3 days, his wife 2 days, and daughter 4 days. The daily wage ratio between the man and the woman is 5 : 4, and between the man and daughter is 5 : 3. Their total income is Tk. 105. Find how much the daughter earns per day.
  1. ক
    Tk. 17
  2. খ
    Tk. 15
  3. গ
    Tk. 10
  4. ঘ
    Tk. 13
  5. ঙ
    Tk. 9
সঠিক উত্তর: ঙ
Tk. 9
উত্তর
সঠিক উত্তর: ঙ
Tk. 9
ঙ
ব্যাখ্যা

Question: In a garden, a man worked 3 days, his wife 2 days, and daughter 4 days. The daily wage ratio between the man and the woman is 5:4, and between the man and daughter is 5:3. Their total income is Tk. 105. Find how much the daughter earns per day.
 
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

৫৩০.
A piece of ornament is made by mixing bronze, silver and gold. In that piece of ornament, the ratio of bronze and silver is 1 : 2 and the ratio of silver and gold is 3 : 5. Find how many grams of gold there are in an ornament weighing 19 grams.
  1. ক
    5 grams
  2. খ
    3 grams
  3. গ
    12 grams
  4. ঘ
    10 grams
সঠিক উত্তর: ঘ
10 grams
উত্তর
সঠিক উত্তর: ঘ
10 grams
ঘ
ব্যাখ্যা
Question : A piece of ornament is made by mixing bronze, silver and gold. In that piece of ornament, the ratio of bronze and silver is 1 : 2 and the ratio of silver and gold is 3 : 5. Find how many grams of gold there are in an ornament weighing 19 grams.

Solution :
Given,
The ratio of bronze and silver is = 1 : 2
= 3 : 6
The ratio of silver and gold is = 3 : 5
= 6 : 10

So the  ratio of bronze, silver and gold = 3 : 6 : 10

∴ Total unit of bronze, silver and gold = 3 + 6 +10
= 19

So the weight of gold = 10/19 parts of 19 grams
= 10 grams
৫৩১.
Find two numbers such that their mean proportional is 6 and third proportional is 20.25.
  1. ক
    ক) 4 and 9
  2. খ
    খ) 3 and 5
  3. গ
    গ) 5 and 7
  4. ঘ
    ঘ) 2 and 5
সঠিক উত্তর: ক
ক) 4 and 9
উত্তর
সঠিক উত্তর: ক
ক) 4 and 9
ক
ব্যাখ্যা
Question: Find two numbers such that their mean proportional is 6 and third proportional is 20.25.

সমাধান:
Let, the two numbers be x and y 
Mean proportional between x and y = 6
∴ x : 6 = 6 : y
⇒ xy = 36
⇒ x = 36/y  ..............(1)

Third proportional to x and y = 20.25 = 2025/100 = 81/4
∴ x : y = y : (81/4)
⇒ y2 = (81/4) × (36/y)     [From equation(1)]
⇒ y3 = 81 × 9
⇒ y3 = 9 x 9 x9
⇒ y3 = 93
∴ y = 9

Again, From equation(1), we get,
x = 36/y = 36/9 = 4

∴ The two numbers are 4 and 9.
৫৩২.
To fill a tank, 25 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 two-fifth of its present capacity?
  1. ক
    62.5
  2. খ
    35
  3. গ
    10
  4. ঘ
    Cannot be determined
সঠিক উত্তর: ক
62.5
উত্তর
সঠিক উত্তর: ক
62.5
ক
ব্যাখ্যা
Question: To fill a tank, 25 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 two-fifth of its present capacity?

Solution:
Let,
the capacity of 1 bucket = x
the capacity of tank = 25x 
Capacity of the new bucket = 2x/5

∴ Required number of buckets = 25x/(2x/5)
= (25x × 5)/2x
= 125/2
= 62.5
৫৩৩.
An iron rod that weights 24 kg is cut into two pieces so that one of these pieces weights 16 kg and 34 m long. If the weight of each piece is proportional to its length, how long is the other one is-
  1. ক
    17 m
  2. খ
    21 m
  3. গ
    34 m
  4. ঘ
    68 m
সঠিক উত্তর: ক
17 m
উত্তর
সঠিক উত্তর: ক
17 m
ক
ব্যাখ্যা
Question: An iron rod that weights 24 kg is cut into two pieces so that one of these pieces weights 16 kg and 34 m long. If the weight of each piece is proportional to its length, how long is the other one is-

Solution:
Total weight = 24 kg
1st piece weight = 16 kg and length = 34 m
2nd piece weight = (24 - 16) kg = 8kg 
Let, length = x m

According to the question,
16 : 8 = 34 : x
⇒ 34/x = 16/8
⇒ 34/x = 2
⇒ 2x = 34
∴ x = 17
৫৩৪.
40% of a number A is equal to 1/5 of another number B. Find the ratio A : B. 
  1. ক
    1 : 5
  2. খ
    1 : 3
  3. গ
    1 : 2
  4. ঘ
    2 : 1
সঠিক উত্তর: গ
1 : 2
উত্তর
সঠিক উত্তর: গ
1 : 2
গ
ব্যাখ্যা

Question: 40% of a number A is equal to 1/5 of another number B. Find the ratio A : B.

Solution:
ATQ,
40% of A = 1/5 of B
Or, (40/100) × A = (1/5) × B
Or, (2/5) A = (1/5) B
Or, 2A = B
∴ Ratio A : B = 1 : 2

৫৩৫.
An amount of Tk. 83 is divided among A, B and C such that A gets Tk. 7 more than B and B gets Tk. 8 more than C . What is the ratio of their shares? 
  1. ক
    ক) 36 : 19 : 21
  2. খ
    খ) 28 : 26 : 21
  3. গ
    গ) 30 : 27 : 20
  4. ঘ
    ঘ) 35 : 28 : 20
সঠিক উত্তর: ঘ
ঘ) 35 : 28 : 20
উত্তর
সঠিক উত্তর: ঘ
ঘ) 35 : 28 : 20
ঘ
ব্যাখ্যা
Let
C gets =x  Tk..
B gets=x + 8  Tk.
A gets=x + 8 + 7 Tk.


Then according to the question
x + x + 8 + x + 15=83
3x=83 - 23
3x = 60
x= 20
Then C's share=20 Tk.
B's share=20 + 8= 28 Tk.
A's share=20 + 15= 35 Tk.
Hence the ratio of their share=35 : 28 : 20
৫৩৬.
In a map, 4 cm represents 80 km. The distance between two cities is 9 cm on the map. The actual distance between the cities is -
  1. ক
    299.5 km 
  2. খ
    180 km 
  3. গ
    99 km 
  4. ঘ
    200 km 
সঠিক উত্তর: খ
180 km 
উত্তর
সঠিক উত্তর: খ
180 km 
খ
ব্যাখ্যা

Question: In a map, 4 cm represents 80 km. The distance between two cities is 9 cm on the map. The actual distance between the cities is - 

Solution: 
Since 4 cm = 80 km,
Actual Distance = (80/4) × 9
= 20 × 9 km
= 180 km 

৫৩৭.
In a certain zoo, the ratio of tigers to zebras to giraffes in stock is 3 : 5 : 7. If there are 96 zebras and giraffes total in stock, how many tigers are there?
  1. ক
    24
  2. খ
    12
  3. গ
    28
  4. ঘ
    18
সঠিক উত্তর: ক
24
উত্তর
সঠিক উত্তর: ক
24
ক
ব্যাখ্যা
Question: In a certain zoo, the ratio of tigers to zebras to giraffes in stock is 3 : 5 : 7. If there are 96 zebras and giraffes total in stock, how many tigers are there?

Solution:
5x + 7x = 96
⇒ 12x = 96
⇒ x = 96/12 = 8

number of tigers = 3x
= 3 × 8
= 24
৫৩৮.
In two types of stainless steel, the ratio of chromium and steel are 2 : 11 and 5 : 21 respectively. In what proportion should the two types be mixed so that the ratio of chromium to steel in the mixed type becomes 7 : 32?
  1. ক
    ক) 2 : 3
  2. খ
    খ) 3 : 4
  3. গ
    গ) 1 : 2
  4. ঘ
    ঘ) 1 : 3
  5. ঙ
    ঙ) 3 : 1
সঠিক উত্তর: গ
গ) 1 : 2
উত্তর
সঠিক উত্তর: গ
গ) 1 : 2
গ
ব্যাখ্যা

According to the question,
Chromium : Steel -
Type 1 - 2 : 11
Type 2 - 5 : 21
Now using alligation,

Ratio of quantity → 1 : 2

৫৩৯.
A sum of money is divided among 150 males and some females in the ratio 8 : 11. Individually each male gets Tk. 4 and a female Tk. 3. What is the number of females? 
  1. ক
    ক) 220
  2. খ
    খ) 250
  3. গ
    গ) 275
  4. ঘ
    ঘ) 300
সঠিক উত্তর: গ
গ) 275
উত্তর
সঠিক উত্তর: গ
গ) 275
গ
ব্যাখ্যা
Question: A sum of money is divided among 150 males and some females in the ratio 8 : 11. Individually each male gets Tk. 4 and a female Tk. 3. What is the number of females? 

Solution:
Let the number of females be x
ATQ,
(150 × 4)/3x = 8/11
⇒ x = (150 × 4 × 11)/(3 × 8)
∴ x = 275
৫৪০.
From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
  1. ক
    580
  2. খ
    624
  3. গ
    756
  4. ঘ
    812
সঠিক উত্তর: গ
756
উত্তর
সঠিক উত্তর: গ
756
গ
ব্যাখ্যা
Question: From a group of 7 men and 6 women, five persons are to be selected to form a committee so that atl east 3 men are there on the committee. In how many ways can it be done?

Solution: 
Ways in which at least 3 men are selected;
• 3 men + 2 women
• 4 men + 1 woman
• 5 men + 0 woman

Number of ways = 7C3 × 6C2 + 7C4 × 6C1 + 7C5 × 6C0
= 35 × 15 + 35 × 6 + 21
= 735 + 21
= 756

∴ The required no of ways = 756
৫৪১.
In a certain English class, (1/4) of the number of girls is equal to (1/6) of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
  1. ক
    1 : 2
  2. খ
    2 : 1
  3. গ
    1 : 4
  4. ঘ
    2 : 3
সঠিক উত্তর: ক
1 : 2
উত্তর
সঠিক উত্তর: ক
1 : 2
ক
ব্যাখ্যা
Question: In a certain English class, (1/4) of the number of girls is equal to (1/6) of the total number of students. What is the ratio of the number of boys to the number of girls in the class?

Solution:
Let,
g = number of girls,
b = number of boys

∴ Total number of students = b + g

ATQ,
g/4 = (b + g)/6
⇒ 6g = 4b + 4g
⇒ 2g = 4b
⇒ 2/4 = b/g
⇒ b/g = 1/2
∴ b : g = 1 : 2
৫৪২.
A, B and C play cricket. The ratio of A’s runs to B’s runs is 5 : 3 and B’s runs to C’s is 3 ∶ 2. They made a total of 350 runs. How many runs did "B" make?
  1. ক
    95
  2. খ
    105
  3. গ
    120
  4. ঘ
    150
সঠিক উত্তর: খ
105
উত্তর
সঠিক উত্তর: খ
105
খ
ব্যাখ্যা
Question: A, B and C play cricket. The ratio of A’s runs to B’s runs is 5 : 3 and B’s runs to C’s is 3 ∶ 2. They made a total of 350 runs. How many runs did "B" make?

Solution:
Given,
A + B + C = 350
and,
A : B = 5 : 3
B : C = 3 : 2

∴ A : B : C = 5 : 3 : 2

∴ Runs made by "B" = 3/10 × 350 = 105
৫৪৩.
A blend consists of an equal amount of lemon juice and sugar syrup. When mixed with extra sugar syrup in a 1 : 3 ratio, what is the final ratio of lemon juice to sugar syrup?
  1. ক
    1 : 2
  2. খ
    1 : 7
  3. গ
    1 : 4
  4. ঘ
    1 : 9
  5. ঙ
    None of the above
সঠিক উত্তর: খ
1 : 7
উত্তর
সঠিক উত্তর: খ
1 : 7
খ
ব্যাখ্যা

Question: A blend consists of an equal amount of lemon juice and sugar syrup. When mixed with extra sugar syrup in a 1 : 3 ratio, what is the final ratio of lemon juice to sugar syrup?

Solution: 
Let, the new mixture is 12 litres

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

The new sugar syrup = 9 + (3/2) = 10.5 

∴ The final ratio of lemon juice and sugar syrup = 1.5 : 10.5 
= 15 : 105 
= 1 : 7

[The statement "Let, the new mixture is 12 liters" is used as an assumption to make the calculations easier. By assuming the total amount of the new mixture is 12 liters, it simplifies the process of determining the amounts of lemon juice and sugar syrup in the mixture. This assumption is just a way to set up the problem in a manageable way.

12 liters is a convenient number because it is divisible by 4 (the ratio of lemon juice and sugar syrup in the original mixture), and it helps make the calculation easier for the final ratio.]

৫৪৪.
A man takes twice as long to row a distance against the stream as to row the same distance in favor of the stream. The ratio of the speed of the boat (in still water) and the stream is: 
  1. ক
    ক) 2 : 1
  2. খ
    খ) 3 : 2
  3. গ
    গ) 4 : 3
  4. ঘ
    ঘ) 3 : 1
সঠিক উত্তর: ঘ
ঘ) 3 : 1
উত্তর
সঠিক উত্তর: ঘ
ঘ) 3 : 1
ঘ
ব্যাখ্যা
Let man's rate upstream be x kmph
Then, his rate downstream = 2x kmph
∴ (speed in still water) : (Speed of stream)
=(2x+x)/2 : (2x−x)/2
= 3x/2 : x/2 = 3 : 1
 
৫৪৫.
Gold is 19 times as heavy as water and copper 9 times as heavy as water. The ratio in which these two metals be mixed so that the mixture is 15 times as heavy as water is :
  1. ক
    ক) 1:2
  2. খ
    খ) 2:3
  3. গ
    গ) 3:2
  4. ঘ
    ঘ) 19:135
  5. ঙ
    ঙ) None of the above
সঠিক উত্তর: গ
গ) 3:2
উত্তর
সঠিক উত্তর: গ
গ) 3:2
গ
ব্যাখ্যা


Required ratio 6 : 4 = 3 : 2
৫৪৬.
Monthly income of B is Tk. 3000 more than the monthly income of A. Monthly savings of A and B are Tk. 8000 and Tk. 17000 respectively. Find the monthly income of A if the monthly expenditures of A and B are in the ratio 4 : 3 respectively.
  1. ক
    Tk. 27000
  2. খ
    Tk. 32000
  3. গ
    Tk. 35000
  4. ঘ
    Tk. 37000
সঠিক উত্তর: খ
Tk. 32000
উত্তর
সঠিক উত্তর: খ
Tk. 32000
খ
ব্যাখ্যা
Question: Monthly income of B is Tk. 3000 more than the monthly income of A. Monthly savings of A and B are Tk. 8000 and Tk. 17000 respectively. Find the monthly income of A if the monthly expenditures of A and B are in the ratio 4 : 3 respectively.

Solution:
Let,
The monthly income of A is Tk. x
∴ The monthly income of B is Tk. x + 3000

Monthly expenditures of A is Tk. x - 8000
Monthly expenditures of B is Tk. x + 3000 - 17000 = Tk. x - 14000

ATQ,
(x - 8000)/(x - 14000) = 4/3
⇒ 3x - 24000 = 4x - 56000
⇒ 4x - 3x = 56000 - 24000
∴ x = 32000
৫৪৭.
A football team has a ratio of win to loss of 3 : 1. After winning 6 games in a row, the team's ratio of win to loss became 5 : 1. How many games had the team won before it played the last six games?
  1. ক
    9
  2. খ
    12
  3. গ
    15
  4. ঘ
    6
  5. ঙ
    None of these
সঠিক উত্তর: ক
9
উত্তর
সঠিক উত্তর: ক
9
ক
ব্যাখ্যা
Quesion: A football team has a ratio of win to loss of 3 : 1. After winning 6 games in a row, the team's ratio of win to loss became 5 : 1. How many games had the team won before it played the last six games?

Solution:
Let,
3x be the number of games won and x be the number of games lost.

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

∴ Initial wins = 3x = 3 × 3 = 9
So, the team had won 9 games before the last 6 wins.
৫৪৮.
Given that 24 carat gold is pure gold, 18 carat gold is 3/4th of pure gold and 20 carat gold is 5/6th of pure gold, the ratio of the pure gold in 18 carat gold to the pure gold in 20 carat gold is
  1. ক
    ক) 5 : 8
  2. খ
    খ) 10 : 9
  3. গ
    গ) 8 : 5
  4. ঘ
    ঘ) 9 : 10
সঠিক উত্তর: ঘ
ঘ) 9 : 10
উত্তর
সঠিক উত্তর: ঘ
ঘ) 9 : 10
ঘ
ব্যাখ্যা
Question: Given that 24 carat gold is pure gold, 18 carat gold is 3/4 of pure gold and 20 carat gold is 5/6 of pure gold, the ratio of the pure gold in 18 carat gold to the pure gold in 20 carat gold is-

Solution:
18 ক্যারেট সোনায় খাঁটি সোনার পরিমাণ = (24 × 3)/4 ক্যারেট 
= 18 ক্যারেট 

20 ক্যারেট সোনায় খাঁটি সোনার পরিমাণ = (24 × 5)/6 ক্যারেট 
= 20 ক্যারেট 

নির্ণেয় অনুপাত = 18 : 20 
= 9 : 10 
৫৪৯.
In what ratio must rice at Tk 45 per kg be mixed with rice at Tk 60 per kg so that the mixture must be worth Tk 48 per kg?
  1. ক
    3 : 1
  2. খ
    5 : 2
  3. গ
    4 : 1
  4. ঘ
    7 : 3
  5. ঙ
    None
সঠিক উত্তর: গ
4 : 1
উত্তর
সঠিক উত্তর: গ
4 : 1
গ
ব্যাখ্যা
Question: In what ratio must rice at Tk 45 per kg be mixed with rice at Tk 60 per kg so that the mixture must be worth Tk 48 per kg?

Solution:
Let the quantity of rice that is Tk 45 per kg be x kg and rice that is Tk 60 per kg be y kg.
Hence, the required ratio is x : y

Price of x kg rice = Tk 45x
Price of y kg rice = Tk 60y

For the mixture:
Quantity of mixture = x + y [This mixture is Tk 48 per kg]
Total price of mixture = Tk 48(x + y)

ATQ,
45x + 60y = 48(x + y)
⇒ 45x + 60y = 48x + 48y
⇒ 45x - 48x = 48y - 60y
⇒ - 3x = -12y
⇒ x/y = 12/3
⇒ x/y = 4
∴ x : y = 4 : 1
Therefore, the required ratio is 4 : 1
৫৫০.
In a class the number of girls is 25% more than that of the boys. The strength of the class is 63. If 5 more girls are admitted to the class, the ratio of the number of boys to that of girls is- 
  1. ক
    ক) 3 : 5
  2. খ
    খ) 4 : 7 
  3. গ
    গ) 6 : 11 
  4. ঘ
    ঘ) 7 : 10 
সঠিক উত্তর: ঘ
ঘ) 7 : 10 
উত্তর
সঠিক উত্তর: ঘ
ঘ) 7 : 10 
ঘ
ব্যাখ্যা
Question: In a class the number of girls is 25% more than that of the boys. The strength of the class is 63. If 5 more girls are admitted to the class, the ratio of the number of boys to that of girls is- 

Solution: 
Let the number of boys be x.
Then number of girls = 125% of x = 5x/4

ATQ,
x + (5x/4) = 63
⇒ (4x + 5x)/4 = 63
⇒ 9x/4 = 63
⇒ 9x = 63 × 4
⇒ x = (63 × 4)/9
∴ x = 28

So, the number of boys is 28
and the number of girls = (63 - 28) + 5 = 40

∴ Required ratio = 28 : 40 = 7 : 10 
৫৫১.
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-
  1. ক
    Tk. 169.50
  2. খ
    Tk. 170
  3. গ
    Tk. 175.50
  4. 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:
Since first and second varieties are mixed in equal proportions.
So, their average price = Tk. (126 + 135)/2 = Tk. 130.50
So, the mixture is formed by mixing two varieties, one at Tk. 130.50 per kg and the other at say, Tk. x per kg in the ratio 2 : 2 = 1 : 1. We have to find x.
We know,

⇒ Quantity of cheaper : Quantity of Dearer  = (CP of of Dearer - Mean Price) : (Mean Price - CP of Cheaper)
⇒ (CP of of Dearer - Mean Price) : (Mean Price - CP of Cheaper) = Quantity of cheaper : Quantity of Dearer
⇒ (x - 153) : (153 - 130.50) = 1 : 1
⇒ x - 153 = 153 - 130.50
⇒ x = 22.5 + 153
∴ x = 175.5
৫৫২.
A, B and C entered into a partnership by investing Tk 15400, Tk.18200 and Tk. 12600 respectively. B left after 6 months. If after 8 months, there was a profit of Tk. 28790, then what is the share of C in the profit?
  1. ক
    ক) 8710
  2. খ
    খ) 9432
  3. গ
    গ) 8352
  4. ঘ
    ঘ) 8568
সঠিক উত্তর: ক
ক) 8710
উত্তর
সঠিক উত্তর: ক
ক) 8710
ক
ব্যাখ্যা

Investment of A for 8 months=Tk 15400
Investment of B for 6 months=Tk. 18200
Investment of C for 8 months= Tk.12600
Ratio of the share of A, B and C
= 15400×8:18200×6:12600×8
= 154×8:182×6:126×8
= 44:39:36
Sum of the terms of ratio
= 44+39+36 = 119
∴ Share of C
= Tk.(36/119 × 28790) ≈ Tk 8710 

৫৫৩.
If (3/2)P = (5/7)Q = (6/5)R, then what is P : Q : R?
  1. ক
    20 : 42 : 25
  2. খ
    17 : 39 : 42
  3. গ
    15 : 21 : 25
  4. ঘ
    24 : 35 : 48
সঠিক উত্তর: ক
20 : 42 : 25
উত্তর
সঠিক উত্তর: ক
20 : 42 : 25
ক
ব্যাখ্যা
Question: If (3/2)P = (5/7)Q = (6/5)R, then what is P : Q : R?

Solution:
(3/2)P = (5/7)Q = (6/5)R .................(1)

LCM of their numerators = LCM of (3, 5, 6) = 30

Divide the eq. (1) by 30.
3P/(2 × 30) =5Q/(7 × 30) = 6R/(5 × 30)
⇒ P/20 = Q/42 = R/25

∴ P : Q : R = 20 : 42 : 25
৫৫৪.
The number of passengers in 3 train is in the ratio 3 : 4 : 5. If 20 passengers are increased in each train this ratio changes to 4 : 5 : 6. The number of passengers in the third train in the beginning was -
  1. ক
    ক) 240
  2. খ
    খ) 100
  3. গ
    গ) 80
  4. ঘ
    ঘ) 60
সঠিক উত্তর: খ
খ) 100
উত্তর
সঠিক উত্তর: খ
খ) 100
খ
ব্যাখ্যা
Question: The number of passengers in 3 train is in the ratio 3 : 4 : 5. If 20 passengers are increased in each train this ratio changes to 4 : 5 : 6. The number of passengers in the third train in the beginning was -

Solution: 
Let the number of passengers in the trains be 3x, 4x and 5x respectively;
Total passengers = 3x + 4x + 5x = 12x
According to the question,
(3x + 20) : (4x + 20) = 4 : 5
or, 5(3x + 20) = 4(4x+20)
or, 15x + 100 = 16x + 80
or, x = 20

Hence,
the number of passengers in the third train in the beginning was = 5x = 5 × 20 = 100
৫৫৫.
The average daily wages of female workers in a factory is Tk. 30 and that of male workers is Tk. 42. If the average wages of all the workers is Tk. 37, what is the ratio of male to female workers?
  1. ক
    3 : 2
  2. খ
    4 : 3
  3. গ
    7 : 5
  4. ঘ
    None
সঠিক উত্তর: গ
7 : 5
উত্তর
সঠিক উত্তর: গ
7 : 5
গ
ব্যাখ্যা
Question: The average daily wages of female workers in a factory is Tk. 30 and that of male workers is Tk. 42. If the average wages of all the workers is Tk. 37, what is the ratio of male to female workers?

Solution:
Let's denote:

The number of female workers as F
The number of male workers as M
Average daily wages of female workers = Tk. 30
Average daily wages of male workers = Tk. 42
Average daily wages of all workers = Tk. 37

The total wages for female workers is 30F and for male workers is 42M
The average wage of all workers is given by:
(30F + 42M)/(F + M) = 37
⇒ 30F + 42M = 37(F + M)
⇒ 30F + 42M = 37F + 37M
⇒ 42M - 37M = 37F - 30F
⇒ 5M = 7F
⇒ M/F ​= 7/5
∴ M : F = 7 : 5

The ratio of male to female workers is 7 : 5.
৫৫৬.
The ratio between two numbers is 3:4 and their sum is 420. The greater one of the two numbers is-
  1. ক
    ক) 360
  2. খ
    খ) 240
  3. গ
    গ) 180
  4. ঘ
    ঘ) 120
সঠিক উত্তর: খ
খ) 240
উত্তর
সঠিক উত্তর: খ
খ) 240
খ
ব্যাখ্যা

Let, the number is 3x and 4x
Then 3x + 4x = 420
Or, 7x = 420
Or, x = 60
So, the greater number is 4 × 60 = 240

৫৫৭.
Price of 2 pencils is less than 20% of price of a pen. What is the ratio of the price of a pen to that of a pencil?
  1. ক
    ক) 5 : 2
  2. খ
    খ) 5 : 3
  3. গ
    গ) 7 : 5
  4. ঘ
    ঘ) None
সঠিক উত্তর: ক
ক) 5 : 2
উত্তর
সঠিক উত্তর: ক
ক) 5 : 2
ক
ব্যাখ্যা

ধরি,একটি কলমের মূল্য = 100 টাকা
তাহলে 2 টি কলমের মূল্য = 100 - 100×20% = 100 - 20 = 80 টাকা
সুতরাং 1 টি কলমের মূল্য = 80/2 = 40 টাকা
সুতরাং কলমঃপেন্সিল = 100:40 = 5:2.

৫৫৮.
Which number when added to each of the numbers 24, 32 and 42 would make the sums to be in continued proportion?
  1. ক
    4
  2. খ
    5
  3. গ
    6
  4. ঘ
    8
সঠিক উত্তর: ঘ
8
উত্তর
সঠিক উত্তর: ঘ
8
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.
৫৫৯.
In a 120-liter mixture of milk and water, the ratio of milk to water is 3 : 2. How many liters of water must be added to make the ratio become 1 : 3? 
  1. ক
    168 liters
  2. খ
    118 liters
  3. গ
    100 liters
  4. ঘ
    180 liters
  5. ঙ
    None
সঠিক উত্তর: ক
168 liters
উত্তর
সঠিক উত্তর: ক
168 liters
ক
ব্যাখ্যা

Question: In a 120-liter mixture of milk and water, the ratio of milk to water is 3 : 2. How many liters of water must be added to make the ratio become 1 : 3?

Solution:
Total mixture = 120 litres
Given ratio (milk : water) = 3 : 2

Milk = 120 × (3/5) = 72 liters
Water = 120 - 72 = 48 liters

To make the ratio 1 : 3, x liters of water need to be added.
milk : water = 72 : (48 + x)

So,
72/(48 + x) = 1/3

Cross-multiplying,
3 × 72 = 48 + x
⇒ 216 = 48 + x
⇒ x = 216 - 48
∴ x = 168

Quantity of water to be added = 168 liters.

৫৬০.
u : v = 4 : 7 and v : w = 9 : 7. If u = 72, then what is the value of w?
  1. ক
    94
  2. খ
    96
  3. গ
    98
  4. ঘ
    None of these
সঠিক উত্তর: গ
98
উত্তর
সঠিক উত্তর: গ
98
গ
ব্যাখ্যা
Question: u : v = 4 : 7 and v : w = 9 : 7. If u = 72, then what is the value of w?

Solution: 
u = 7

u : v = 4 : 7
⇒ u/v = 4/7
⇒ v = (7/4) 72 = 126

v : w = 9 : 7
⇒ v/w = 9/7
⇒ w = (7/9) 126 = 98  
৫৬১.
Salaries of Rakib and Sumon are in the ratio 2 : 3. If the salary of each is increased by Tk. 4000, the new ratio becomes 40 : 57. What is Sumon's new salary?
  1. ক
    Tk. 17,000
  2. খ
    Tk. 20,000
  3. গ
    Tk. 25,500
  4. ঘ
    Tk. 38,000
সঠিক উত্তর: ঘ
Tk. 38,000
উত্তর
সঠিক উত্তর: ঘ
Tk. 38,000
ঘ
ব্যাখ্যা
Question: Salaries of Rakib and Sumon are in the ratio 2 : 3. If the salary of each is increased by Tk. 4000, the new ratio becomes 40 : 57. What is Sumon's new salary?

Solution:
Let the original salaries of Rakib and Sumon be Tk. 2x and Tk. 3x respectively.
Then,
(2x + 4000)/(3x + 4000) = 40/57
⇒ 57(2x + 4000) = 40(3x + 4000)
⇒ 6x = 68,000
⇒ 3x = 34,000

Sumon's present salary = (3x + 4000) = (34000 + 4000) = 38,000.
৫৬২.
A is greater than B by 13; C is less than B by 7. If the ratio of A and C is 9 : 5, what is the value of B?
  1. ক
    ক) 25
  2. খ
    খ) 32
  3. গ
    গ) 26
  4. ঘ
    ঘ) 36
সঠিক উত্তর: খ
খ) 32
উত্তর
সঠিক উত্তর: খ
খ) 32
খ
ব্যাখ্যা
Question: A is greater than B by 13; C is less than B by 7. If the ratio of A and C is 9 : 5, what is the value of B?

Solution:
Given,
A - B = 13
B - C = 7
and A : C = 9 : 5
⇒ A/C = 9/5
⇒ A = 9C/5

Now,
A - B + B - C = 13 + 7
⇒ A - C = 20
⇒ 9C/5 - C = 20
⇒ (9C - 5C)/5 = 20
⇒ 4C = 100
⇒ C = 25

So, B - 25 = 7
⇒ B = 32
৫৬৩.
In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4, amounting to Tk 18,800. Find the number of each note respectively.
  1. ক
    20, 36, 16
  2. খ
    37, 1, 1
  3. গ
    10, 18, 8
  4. ঘ
    25, 10, 7
সঠিক উত্তর: ক
20, 36, 16
উত্তর
সঠিক উত্তর: ক
20, 36, 16
ক
ব্যাখ্যা
Question: In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4, amounting to Tk 18,800. Find the number of each note respectively.

Solution:
In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4
Let,
Number of Tk. 500 note is 5x
Number of Tk. 200 note is 9x
Number of Tk. 100 note is 4x

ATQ,
500 × 5x + 200 × 9x + 100 × 4x = 18800
⇒ 2500x + 1800x + 400x = 18800
⇒ 4700x = 18800
⇒ x = 18800/4700
∴ x = 4

Number of Tk. 500 note is 5x = 5 × 4 = 20
Number of Tk. 200 note is 9x = 9 × 4 = 36
Number of Tk. 100 note is 4x = 4 × 4 = 16
৫৬৪.
একটি কারখানার মহিলা কর্মচারীদের দৈনিক গড় মজুরি ৩০ টাকা এবং পুরুষ কর্মচারীদের দৈনিক গড় মজুরী ৪২ টাকা। সকল কর্মচারীর গড় মজুরী ৩৭ টাকা হলে পুরুষ ও মহিলা কর্মচারীর অনুপাত কত?
  1. ক
    ৬ : ৫
  2. খ
    ৫ : ৬
  3. গ
    ৫ : ৭
  4. ঘ
    ৭ : ৫
  5. ঙ
    কোনটিই নয়
সঠিক উত্তর: ঘ
৭ : ৫
উত্তর
সঠিক উত্তর: ঘ
৭ : ৫
ঘ
ব্যাখ্যা
প্রশ্ন: একটি কারখানার মহিলা কর্মচারীদের দৈনিক গড় মজুরি ৩০ টাকা এবং পুরুষ কর্মচারীদের দৈনিক গড় মজুরী ৪২ টাকা। সকল কর্মচারীর গড় মজুরী ৩৭ টাকা হলে পুরুষ ও মহিলা কর্মচারীর অনুপাত কত?

সমাধান:
মনে করি,
পুরুষের সংখ্যা = ক
মহিলার সংখ্যা = খ

প্রশ্নমতে,
৪২ক + ৩০খ = ৩৭(ক + খ)
বা, ৪২ক - ৩৭ক = ৩৭খ - ৩০খ
বা, ৫ক = ৭খ
বা, ক/খ = ৭/৫
ক : খ = ৭ : ৫
∴ পুরুষ ও মহিলা ক‍‍র্মচারীর অনুপাত ৭ : ৫
৫৬৫.
An iron rod that weights 30 kg is cut into two pieces so that one of these pieces weights 16 kg and 40 m long. If the weight of each piece is proportional to its length, how long is the other one is
  1. ক
    ক) 32
  2. খ
    খ) 35
  3. গ
    গ) 30
  4. ঘ
    ঘ) 38
সঠিক উত্তর: খ
খ) 35
উত্তর
সঠিক উত্তর: খ
খ) 35
খ
ব্যাখ্যা
Question: An iron rod that weights 30 kg is cut into two pieces so that one of these pieces weights 16 kg and 40 m long. If the weight of each piece is proportional to its length, how long is the other one is

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

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

Now,
16 : 14 = 40 : x
⇒ 16/14 = 40/x
⇒ x = (40 × 14)/16
⇒ x = 35
৫৬৬.
When 30% of one number is subtracted from another number, the second number reduces to its four-fifths. What is the ratio of the first to the second number?
  1. ক
    3 : 2
  2. খ
    2 : 3
  3. গ
    2 : 5
  4. ঘ
    4 : 7
সঠিক উত্তর: খ
2 : 3
উত্তর
সঠিক উত্তর: খ
2 : 3
খ
ব্যাখ্যা
Question: When 30% of one number is subtracted from another number, the second number reduces to its four-fifths. What is the ratio of the first to the second number? 

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

ATQ,
y - 30% of x = y (4/5)
⇒ y - (30x/100) = 4y/5
⇒ y - (3x/10) = 4y/5
⇒ (10y - 3x)/10 = 4y/5
⇒ (10y - 3x)/2 = 4y
⇒ 10y - 3x = 8y
⇒ - 3x = 8y - 10y
⇒ 3x = 2y
⇒ x/y = 2/3
∴ x : y = 2 : 3
৫৬৭.
A Water reservoir is (1/5)th full and requires 20 liters more to make it (3/5)th full. What is the capacity of the reservoir?
  1. ক
    ক) 40
  2. খ
    খ) 50
  3. গ
    গ) 60
  4. ঘ
    ঘ) None
সঠিক উত্তর: খ
খ) 50
উত্তর
সঠিক উত্তর: খ
খ) 50
খ
ব্যাখ্যা

Question: A Water reservoir is (1/5)th full and requires 20 liters more to make it (3/5)th full. What is the capacity of the reservoir?

Solution: 
একটি চৌবাচ্চার ১/৫ অংশ পূর্ণ।
আরো ২০ লিটার যোগ করে ৩/৫ অংশ পূর্ণ হয়। 

অর্থাৎ, ২০ লিটার, (৩/৫) - (১/৫) = ২/৫ অংশের সমান 

চৌবাচ্চার ২/৫ অংশের সমান = ২০ লিটার 
∴ চৌবাচ্চায় পানি ধরে = ২০ × ৫/২ লিটার 
= ৫০ লিটার 

৫৬৮.
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
    12 litres
  2. খ
    11 litres
  3. গ
    10 litres
  4. ঘ
    9 litres
সঠিক উত্তর: ক
12 litres
উত্তর
সঠিক উত্তর: ক
12 litres
ক
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:
Here,
Milk = Tk. 4.80
Mixture = Tk. 3.60
Water = Tk. 0

ATQ,
Ratio of Milk and water will be = 3.60 : 1.20
= 360 : 120
= 18 : 6

∴ In 18 litres milk water added = 6 litres
∴ In 1 litre milk water added = 6/18 litres
∴ In 36 litres milk water added = {(6/18) × 36} litres
= 12 litres
৫৬৯.
If three numbers in the ratio 3:2:5 be such that the sum of their squares is 1862, the middle number will be-
  1. ক
    ক) 10
  2. খ
    খ) 14
  3. গ
    গ) 18
  4. ঘ
    ঘ) 22
সঠিক উত্তর: খ
খ) 14
উত্তর
সঠিক উত্তর: খ
খ) 14
খ
ব্যাখ্যা

Let the numbers be 3x, 2x and 5x.
Then,
9x2 + 4x2 + 25x2=1862
=> 38x2 = 1862
=> x2 = 49
so, here x = 7.
So, middle number = 2x = 14

৫৭০.
Pure ghee costs Tk.100 per kg. A shopkeeper mixes vegetable oil costing Tk. 50 per kg and sells the mixture at Tk. 96 per kg, making a profit of 20%. In what ratio does he mix the pure ghee with the vegetable oil.
  1. ক
    3 : 2
  2. খ
    2 : 3
  3. গ
    4 : 3
  4. ঘ
    3 : 1
সঠিক উত্তর: ক
3 : 2
উত্তর
সঠিক উত্তর: ক
3 : 2
ক
ব্যাখ্যা
Question: Pure ghee costs Tk.100 per kg. A shopkeeper mixes vegetable oil costing Tk. 50 per kg and sells the mixture at Tk. 96 per kg, making a profit of 20%. In what ratio does he mix the pure ghee with the vegetable oil.

Solution:
Cost price of 1 kg of mixure = Tk. (100/120) × 96 = Tk. 80 = mean price
Cost of 1 kg of pure ghee = Tk. 100 = dearer quantity.  
Cost of 1 kg of vegetable = Tk. 50 = cheaper quantity. 
And, the mean price = m = Tk. 20

Therefore,
(Dearer quantity) : (Cheaper quantity) = (m - c) : (d - m) = (80 - 50) : (100 - 80) = 30 : 20 = 3 : 2 

∴ Required ratio = 3 : 2
৫৭১.
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
    21 : 22
  2. খ
    17 : 18
  3. গ
    8 : 9
  4. ঘ
    7 : 8
  5. ঙ
    Cannot be determined
সঠিক উত্তর: ক
21 : 22
উত্তর
সঠিক উত্তর: ক
21 : 22
ক
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:
let the number of boys and girls in the college be 7x and 8x

their increased number is (120% of 7x) = (120/100) × 7x = 42x/5

And (10% of 8x).= (110/100) ×8x = 44x/5

∴ Required Ratio = 42x/5 : 44x/5 = 21 : 22
৫৭২.
What is the correct ratio of mixing sugar at Tk 15 per kg and sugar at Tk 25 per kg to obtain a mixture that costs Tk 22 per kg?
  1. ক
    2 : 5
  2. খ
    3 : 7
  3. গ
    5 : 7
  4. ঘ
    2 : 7
সঠিক উত্তর: খ
3 : 7
উত্তর
সঠিক উত্তর: খ
3 : 7
খ
ব্যাখ্যা
Question: What is the correct ratio of mixing sugar at Tk 15 per kg and sugar at Tk 25 per kg to obtain a mixture that costs Tk 22 per kg?

Solution:
Let the amount of sugar at Tk 15 per kg be x kg
and the amount of sugar at Tk 25 per kg be y kg.

ATQ,
15x + 25y = 22(x + y)
⇒ 15x + 25y = 22x + 22y
⇒ 7x = 3y
⇒ x/y = 3/7
⇒ x : y = 3 : 7
৫৭৩.
Himu invested 10% more than Tushar. Tushar invested 10% less than Rubel. If the total sum of their investments is Tk. 5780, how much amount did Rubel invest ?
  1. ক
    ক) Tk. 2010
  2. খ
    খ) Tk. 2200
  3. গ
    গ) Tk. 2100
  4. ঘ
    ঘ) Tk. 2000
সঠিক উত্তর: ঘ
ঘ) Tk. 2000
উত্তর
সঠিক উত্তর: ঘ
ঘ) Tk. 2000
ঘ
ব্যাখ্যা

Let money invested by Rubel = Tk. x
Money invested by Tushar = 9/10 x = 0.9x
Money invested by Himu = 9/10x × 110/100 = 0.99x
Also, x + 0.9x + 0.99x = 5780
= x= 5780/2.89
= 2000

Therefore, amount invested by Rubel is Tk. 2000.

৫৭৪.
Oranges of type A that cost 60 Tk./Kg is mixed with the orange of type B that costs 80 Tk./Kg in a ratio of 3 : 5. What will be the price of the resultant mixture of oranges?
  1. ক
    ক) Tk. 72
  2. খ
    খ) Tk. 72.5 
  3. গ
    গ) Tk. 74.5 
  4. ঘ
    ঘ) Tk. 75
সঠিক উত্তর: খ
খ) Tk. 72.5 
উত্তর
সঠিক উত্তর: খ
খ) Tk. 72.5 
খ
ব্যাখ্যা
প্রশ্ন: Oranges of type A that cost 60 Tk./Kg is mixed with the orange of type B that costs 80 Tk./Kg in a ratio of 3 : 5. What will be the price of the resultant mixture of oranges?

সমাধান: 
Let,
Type A oranges is 3x Kg
Type B oranges is 5x Kg 
The price  Of the resultant mixure is P

Now,
60 × 3x + 80 × 5x = (3x + 5x) × P
⇒ 180x + 400x = 8xP
⇒ 8xP = 580x
⇒ P = 580x/8x
∴ P = 72.5

∴ The price  Of the resultant mixure is Tk. 72.5 
৫৭৫.
A grocer mixes two types of pulses, one costing 20 taka per kg and the other 25 taka per kg.If the mixture is sold at 22 taka per kg, what is the ratio of two types of pulses in the mixture?
  1. ক
    2 : 3
  2. খ
    3 : 2
  3. গ
    4 : 5
  4. ঘ
    5 : 4
সঠিক উত্তর: খ
3 : 2
উত্তর
সঠিক উত্তর: খ
3 : 2
খ
ব্যাখ্যা

Question: A grocer mixes two types of pulses, one costing 20 taka per kg and the other 25 taka per kg.If the mixture is sold at 22 taka per kg, what is the ratio of two types of pulses in the mixture?

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

প্রথম প্রকার ডালের x কেজির মূল্য = 20x টাকা
দ্বিতীয় প্রকার ডালের y কেজির মূল্য = 25y টাকা

প্রশ্নমতে,
20x + 25y = 22(x + y)
⇒ 20x + 25y = 22x + 22y
⇒ 22x - 20x= 25y - 22y
⇒ 2x = 3y
∴ x/y = 3 : 2

৫৭৬.
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) 4:5
  2. খ
    খ) 9:5
  3. গ
    গ) 8:5
  4. ঘ
    ঘ) 9:2
সঠিক উত্তর: খ
খ) 9:5
উত্তর
সঠিক উত্তর: খ
খ) 9:5
খ
ব্যাখ্যা

Dog : Hare
= (3 × 3) leaps of hare : (5×1) leaps of hare
= 9 : 5

৫৭৭.
A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Tk.1000 more than D, what is one-third of B's share?
  1. ক
    Tk. 2000
  2. খ
    Tk. 6000
  3. গ
    Tk. 1333.33
  4. ঘ
    Tk. 666.67
সঠিক উত্তর: ঘ
Tk. 666.67
উত্তর
সঠিক উত্তর: ঘ
Tk. 666.67
ঘ
ব্যাখ্যা
Question: A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Tk. 1000 more than D, what is one-third of B's share?

Solution:
Let,
The shares of A be Tk. 5x
The shares of B be Tk. 2x
The shares of C be Tk. 4x
The shares of D be Tk. 3x

Then,
4x - 3x = 1000
∴ x = 1000

∴ B's share is Tk. 2 × 1000 = Tk. 2000

∴ One-third of B's share is Tk. (2000/3) = Tk. 666.67
৫৭৮.
How much coffee of variety A, costing Tk. 5 a kg should be added to 20 kg of Type B coffee at Tk. 12 a kg so that the cost of the two coffee variety mixture be worth Tk. 7 a kg?
  1. ক
    ক) 80 kg
  2. খ
    খ) 60 kg
  3. গ
    গ) 50 kg
  4. ঘ
    ঘ) 70 kg
সঠিক উত্তর: গ
গ) 50 kg
উত্তর
সঠিক উত্তর: গ
গ) 50 kg
গ
ব্যাখ্যা
The basic formula which is used to find the ratio in which the ingredients are mixed is :
 
As per the rule of Allegation, 
Quantity of Dearer: Quantity of Cheaper = (12-7) : (7-5) = 5 : 2
Quantity of Variety A coffee that needs to be mixed
⇒ 5 : 2 = x : 20
⇒ x/20 = 5/2
⇒ x =50 kg
৫৭৯.
The present ratio of ages of A and B is 4 : 5. 18 years ago, this ratio was 11 : 16. Find the sum of their present ages.
  1. ক
    ক) 70 years
  2. খ
    খ) 64 years
  3. গ
    গ) 90 years
  4. ঘ
    ঘ) 75 years
সঠিক উত্তর: গ
গ) 90 years
উত্তর
সঠিক উত্তর: গ
গ) 90 years
গ
ব্যাখ্যা
Let present age of A and B be 4x and 5x
18 years ago their ages
(4x - 18)/(5x - 18) = 11/16
x = 10
Sum of the present ages = 40 + 50 = 90 years
৫৮০.
What is the ratio of 3/4 to the product  ?
  1. ক
    ক) 1/3
  2. খ
    খ) 1/4
  3. গ
    গ) 4/9
  4. ঘ
    ঘ) 9/4
সঠিক উত্তর: খ
খ) 1/4
উত্তর
সঠিক উত্তর: খ
খ) 1/4
খ
ব্যাখ্যা
প্রশ্ন: What is the ratio of 3/4 to the product  ?

সমাধান: 

৫৮১.
A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?
  1. ক
    ক) 10 Litres
  2. খ
    খ) 20 Litres
  3. গ
    গ) 30 Litres
  4. ঘ
    ঘ) 40 Litres
সঠিক উত্তর: ক
ক) 10 Litres
উত্তর
সঠিক উত্তর: ক
ক) 10 Litres
ক
ব্যাখ্যা

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

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

Thus, (30 + P) = 25% of (150 + P)
⇒ 30 + P = 25/100 of (150 + P)
⇒ 30 + P = (1/4) × (150 + P)
⇒ 120 + 4P = 150 + P
⇒ 4P - P = 150 - 120
⇒ 3P = 30
⇒ P = 10

10 Litres of water is added to the mixture to make water 25% of the new mixture.

৫৮২.
The angels of a pentagon are in ratio 9 : 10 : 12 : 14 : 15. What is the sum of measure of the smallest and largest angels?
  1. ক
    ক) 54
  2. খ
    খ) 81
  3. গ
    গ) 135
  4. ঘ
    ঘ) 216
সঠিক উত্তর: ঘ
ঘ) 216
উত্তর
সঠিক উত্তর: ঘ
ঘ) 216
ঘ
ব্যাখ্যা

আমরা জানি, পঞ্চভূজের সমষ্টি = {(5 - 2) × 180}° = 540°
পঞ্চভুজের কোণের অনুপাতের সমষ্টি = 9 + 10 + 12 + 14 + 15 = 60
∴ পঞ্চভুজের বৃহত্তম এবং ক্ষুদ্রতম কোণের সমষ্টি = [{(9+15)/60} × 540]° = 216°

৫৮৩.
How much water must be added to 20 L of 30% salt solution to make it 20%?
  1. ক
    5 L
  2. খ
    10 L
  3. গ
    7.5 L
  4. ঘ
    15 L
সঠিক উত্তর: খ
10 L
উত্তর
সঠিক উত্তর: খ
10 L
খ
ব্যাখ্যা
Question: How much water must be added to 20 L of 30% salt solution to make it 20%?

Solution:
Total solution 20 L
30% salt solution has salt of = (20×30)/100 = 6 L

Now,
Let the amount of water to be added to make 20% salt solution = x liters
Then, new total solution = 20 + x
Salt remains = 6 L

New salt percentage = 
→ 400+ 20x= 600
→ 20x= 200
→ x= 10
৫৮৪.
How many kg of Sugar at Tk. 50 per kg must a man mix with 25 kg of sugar at Tk. 34 per kg so that by selling the mixture at Tk. 44 per kg he gains 10% on the outlay?
  1. ক
    20 kg
  2. খ
    18 kg
  3. গ
    16 kg
  4. ঘ
    15 kg
সঠিক উত্তর: ঘ
15 kg
উত্তর
সঠিক উত্তর: ঘ
15 kg
ঘ
ব্যাখ্যা
Question: How many kg of Sugar at Tk. 50 per kg must a man mix with 25 kg of sugar at Tk. 34 per kg so that by selling the mixture at Tk. 44 per kg he gains 10% on the outlay?

Solution:
Cost price of mixture = (44/110) × 100 = 40


Ratio of sugar of Tk. 50 and Tk. 34 is = 6 : 10 = 3 : 5
∴ sugar of Tk. 50 per kg to be mixed = (3/5) × 25 = 15kg
৫৮৫.
If m : 5 = n : 7 = r : 8, then (m + n + r)/m =?
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 4
  4. ঘ
    ঘ) 5
সঠিক উত্তর: গ
গ) 4
উত্তর
সঠিক উত্তর: গ
গ) 4
গ
ব্যাখ্যা
Question: If m : 5 = n : 7 = r : 8, then (m + n + r)/m =?

Solution: 
m : 5 = n : 7
⇒ m/5 = n/7
⇒ n = (7/5)m

m : 5 = r : 8
⇒ m/5 = r/8
∴ r = 8m/5

(m + n + r)/m = {m + (7/5)m + 8m/5}/m
= 1 + (7/5) + (8/5)
= 20/5
= 4
৫৮৬.
Cost of two types of dates is Tk.15 and Tk. 20 per KG, respectively. If both the dates are mixed together in the ratio 2 : 3, then what should be the price with Tk. 2 profit of mixed variety of dates per KG?
  1. ক
    ক) 22 Taka
  2. খ
    খ) 20 Taka
  3. গ
    গ) 15 Taka
  4. ঘ
    ঘ) 18 Taka
সঠিক উত্তর: খ
খ) 20 Taka
উত্তর
সঠিক উত্তর: খ
খ) 20 Taka
খ
ব্যাখ্যা
প্রশ্ন: Cost of two types of dates is Tk.15 and Tk. 20 per KG, respectively. If both the dates are mixed together in the ratio 2 : 3, then what should be the price with Tk. 2 profit of mixed variety of dates per KG?

সমাধান: 
ধরি,
১৫ টাকার খেজুর ছিল ২ক কিলোগ্রাম
২০ টাকার খেজুর ছিল ৩ক কিলোগ্রাম
∴ মোট খেজুর ৫ক কিলোগ্রাম 

১৫ টাকার খেজুর আছে ৩০ক টাকার 
২০ টাকার খেজুর আছে ৬০ক টাকার 
∴ মোট দাম ৯০ক টাকা 

৫ক কিলোগ্রামের দাম ৯০ক টাকা 
∴ ১ কিলোগ্রামের দাম ৯০ক/৫ক টাকা 
= ১৮ টাকা

২ টাকা লাভে প্রতি কেজি খেজুরের দাম = ১৮ + ২ = ২০ টাকা
৫৮৭.
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 ∶ 12
  2. খ
    21 ∶ 22
  3. গ
    13 ∶ 15
  4. ঘ
    27 ∶ 29
সঠিক উত্তর: খ
21 ∶ 22
উত্তর
সঠিক উত্তর: খ
21 ∶ 22
খ
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: 
ধরি,
কলেজে ছেলে এবং মেয়ের সংখ্যা যথাক্রমে 7x এবং 8x

∴ ছেলেদের বর্ধিত সংখ্যা = 120% × 7x 
= (120/100) × 7x 
= (6/5) × 7x 
= (42x)/5

এবং, মেয়েদের বর্ধিত সংখ্যা = 110% × 8x
= (110/100) × 8x
= (11/5) × 4x
= (44x)/5

 ∴ বর্ধিত সংখ্যার অনুপাত (New ratio) = (42x)/5 ∶ (44x)/5
= 21 ∶ 22

৫৮৮.
A can contains a mixture of two liquids A and B in the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?
  1. ক
    10
  2. খ
    20
  3. গ
    21
  4. ঘ
    25
সঠিক উত্তর: গ
21
উত্তর
সঠিক উত্তর: গ
21
গ
ব্যাখ্যা
Question: A can contains a mixture of two liquids A and B in the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?

Solution:
Suppose the can initially contains 7x and 5x litres of mixtures A and B respectively
Quantity of A in mixture left = 7x - (7/12) × 9 = (7x - 2/14) litres.

Quantity of B in mixture left = 5x - (5/12) ×9 = (5x - 15/4) litres.
ATQ,
(7x - 2/14)/{(5x - 15/4) + 9} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ x = 3

So, the can contained 21 litres of A.
৫৮৯.
If p : q = 7 : 9, then 2p + 5q : 5q - 2p = ?
  1. ক
    59 : 37
  2. খ
    53 : 31
  3. গ
    59 : 31
  4. ঘ
    57 : 31
সঠিক উত্তর: গ
59 : 31
উত্তর
সঠিক উত্তর: গ
59 : 31
গ
ব্যাখ্যা
p : q = 7 : 9
⇒ 2p : 5q = 14 : 45
⇒ (2p + 5q)/(5q - 2p) = (14 + 45)/(45 - 14) = 59/31
⇒ (2p + 5q) : (5q - 2p) =59 : 31
৫৯০.
Ratio of two number A and B is 3 : 2. If we decrease 60 from A and Add 60 with B then ratio of A and B is 18 : 17. Find original value of A?  
  1. ক
    480
  2. খ
    475
  3. গ
    420
  4. ঘ
    450
সঠিক উত্তর: গ
420
উত্তর
সঠিক উত্তর: গ
420
গ
ব্যাখ্যা

Question: Ratio of two number A and B is 3 : 2. If we decrease 60 from A and Add 60 with B then ratio of A and B is 18 : 17. Find original value of A?  

solution:
Given that,
Ratio of A : B = 3 : 2
If A decreased by 60 and B increased by 60
And new ratio = 18 : 17

Let A = 3x and B = 2x

ATQ,
{3x - 60}/{2x + 60} = {18}/{17}
⇒ 17(3x - 60) = 18(2x + 60)
⇒ 51x - 1020 = 36x + 1080
⇒ 51x - 36x = 1080 + 1020
⇒ 15x = 2100
∴ x = 140

∴ Original value of A = 3x = 3 × 140 = 420

∴ The original value of A = 420.

৫৯১.
If the ratio between the areas of two circles is 4 : 1 then the ratio between their radii will be-
  1. ক
    ক) 1 : 4
  2. খ
    খ) 1 : 1
  3. গ
    গ) 2 : 1
  4. ঘ
    ঘ) 2 : 2
সঠিক উত্তর: গ
গ) 2 : 1
উত্তর
সঠিক উত্তর: গ
গ) 2 : 1
গ
ব্যাখ্যা
⇒ πr21/πr22 = 4/1
⇒ r21/r22 = 4/1
⇒ (r1/r2)2 = (2/1)2
⇒r1/r2 = 2/1
     r1 : r2 = 2 : 1
৫৯২.
A and B together have Tk 1820. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?
  1. ক
    Tk. 828
  2. খ
    Tk. 518
  3. গ
    Tk. 748
  4. ঘ
    Tk. 728
সঠিক উত্তর: ঘ
Tk. 728
উত্তর
সঠিক উত্তর: ঘ
Tk. 728
ঘ
ব্যাখ্যা
Question: A and B together have Tk 1820. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?

Solution:
ATQ,
4A/15 = 2B/5
⇒ A = (2B × 15)/(5 × 4) 
⇒ A = 3B/2
⇒ A/B = 3/2
⇒ A : B = 3 : 2

B's share = 1820 × (2/5) = 728 Tk
৫৯৩.
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?
  1. ক
    12
  2. খ
    10
  3. গ
    8
  4. ঘ
    5
সঠিক উত্তর: খ
10
উত্তর
সঠিক উত্তর: খ
10
খ
ব্যাখ্যা
Question: 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?

Solution:
Let, the number of banana of the basket 3x and apple 2x.

According to the question,
(3x - 5)/2x = 1/1
⇒ 3x - 5 = 2x
⇒ 3x - 2x = 5
∴ x = 5

There were apples in the basket = 2x = 2 . 5 = 10
৫৯৪.
The angles of triangle are in the ratio 3 : 4 : 5. The largest angle is
  1. ক
    ক) 50°
  2. খ
    খ) 66°
  3. গ
    গ) 70°
  4. ঘ
    ঘ) 75°
সঠিক উত্তর: ঘ
ঘ) 75°
উত্তর
সঠিক উত্তর: ঘ
ঘ) 75°
ঘ
ব্যাখ্যা
Question: The angles of triangle are in the ratio  3 : 4 : 5. The largest angle is 

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

Hence, 
The largest angle is (180° × 5/12) or, 75°
৫৯৫.
Adhira, Enayet, Rocky enter into a partnership. Adhira initially invests 25 lakh & adds another 10 lakhs after one year. Enayet initially invests 35 lakh & withdrawal 10 lakh after 2 years and Rocky invests 30 Lakhs . In what ratio should the profit be divided at the end of 3 years?
  1. ক
    18 : 18 : 19
  2. খ
    19 : 18 : 18
  3. গ
    19 : 19 : 18
  4. ঘ
    1 : 1 : 1
সঠিক উত্তর: গ
19 : 19 : 18
উত্তর
সঠিক উত্তর: গ
19 : 19 : 18
গ
ব্যাখ্যা
Question: Adhira, Enayet, Rocky enter into a partnership. Adhira initially invests 25 lakh & adds another 10 lakhs after one year. Enayet initially invests 35 lakh & withdrawal 10 lakh after 2 years and Rocky invests 30 Lakhs . In what ratio should the profit be divided at the end of 3 years?

Solution: 
 ratio should the profit be divided at the end of 3 years = (25 + 35 × 2) : (35 × 2 + 25) : (30 × 3)} lakhs
= 95 : 95 : 90 
= 19 : 19 : 18 [5 দ্বারা ভাগ করে]
৫৯৬.
A jar contains only marbles of three colour: red, green and yellow. The red and green marbles are in the ratio of 2 : 5 and the yellow and red marbles are in ration of 5 : 6. Which of the following could be the total number of marbles?
  1. ক
    52
  2. খ
    64
  3. গ
    100
  4. ঘ
    None of these
সঠিক উত্তর: ক
52
উত্তর
সঠিক উত্তর: ক
52
ক
ব্যাখ্যা

Question: A jar contains only marbles of three colour: red, green and yellow. The red and green marbles are in the ratio of 2 : 5 and the yellow and red marbles are in ration of 5 : 6. Which of the following could be the total number of marbles?

Solution:
Red : Green = 2 : 5 = 6 : 15
Yellow : Red = 5 : 6 = 5 : 6
Red : Green : Yellow = 6 : 15 : 5

অনুপাতের রাশিগুলোর সমষ্টি = 6 + 15 + 5 = 26
মার্বেল সংখ্যা হবে 26 এর গুণিতক।
মার্বেল সংখ্যা হতে পারে 26, 52, 78, 104,..............

৫৯৭.
Solution A is made up of alcohol and water mixed in the ratio of 21 : 4 by volume; Solution B is made up of alcohol and water mixed in the ratio of 2 : 3 by volume. If Solution A and Solution B are mixed in the ratio of 5 : 6 by volume, what percent of the resultant mixture is alcohol?
  1. ক
    32.5%
  2. খ
    60%
  3. গ
    40%
  4. ঘ
    52.5%
সঠিক উত্তর: খ
60%
উত্তর
সঠিক উত্তর: খ
60%
খ
ব্যাখ্যা
Question: Solution A is made up of alcohol and water mixed in the ratio of 21 : 4 by volume; Solution B is made up of alcohol and water mixed in the ratio of 2 : 3 by volume. If Solution A and Solution B are mixed in the ratio of 5 : 6 by volume, what percent of the resultant mixture is alcohol?

Solution:
25 units of A contain 21 units of alcohol and 4 units of water.
So, 5 units of A contain (21/5) units of alcohol and (4/5) units of water
5 units of B contain 2 units of alcohol and 3 units of water

If we mix 5 × 5 = 25 units of A with 5 × 6 = 30 units of B (so that A and B are mixed in the ratio of 5 : 6), the resultant 55 units solution will contain:
Alcohol: (21/5) × 5 + (2 × 6) = 33 units
Water: (4/5) × 5 + (3 × 6) = 22 units

% of alcohol in the resultant solution = (33/55) × 100 = 60
৫৯৮.
Find the ratio of purchase price to sell price if there is loss of 19%?
  1. ক
    81 : 19
  2. খ
    19 : 81
  3. গ
    100 : 81
  4. ঘ
    19 : 100
সঠিক উত্তর: গ
100 : 81
উত্তর
সঠিক উত্তর: গ
100 : 81
গ
ব্যাখ্যা
ধরি, ক্রয়মূল্য 100 টাকা
19% ক্ষতিতে বিক্রয়মূল্য (100 - 19) = 81 টাকা
ক্রয়মূল্য : বিক্রয়মূল্য = 100 : 81
৫৯৯.
The ratio of milk and water in a solution is 7 : 4. After adding 8 liters of water, the ratio of milk and water becomes 3 : 2. Find the final amount of water in the solution.
  1. ক
    48 liters
  2. খ
    54 liters
  3. গ
    56 liters
  4. ঘ
    60 liters
সঠিক উত্তর: গ
56 liters
উত্তর
সঠিক উত্তর: গ
56 liters
গ
ব্যাখ্যা

Question: The ratio of milk and water in a solution is 7 : 4. After adding 8 liters of water, the ratio of milk and water becomes 3 : 2. Find the final amount of water in the solution.

Solution:
Let the initial amount of milk = 7x liters
Let the initial amount of water = 4x liters

According to the question,
7x/(4x + 8) = 3/2
⇒ 2 × 7x = 3 × (4x + 8)
⇒ 14x = 12x + 24
⇒ 14x - 12x = 24
⇒ 2x = 24
⇒ x = 12

∴ Final amount of water = 4x + 8
= 4 × 12 + 8
= 48 + 8 = 56 liters

৬০০.
200 litres of a mixture contains milk and water in the ratio 17 : 3. After the addition of some more milk to it, the ratio of milk to water in the resulting mixture becomes 7 : 1. The quantity of milk added to it was?
  1. ক
    ক) 20 Litres
  2. খ
    খ) 40 Litres
  3. গ
    গ) 60 Litres
  4. ঘ
    ঘ) 80 Litres
সঠিক উত্তর: খ
খ) 40 Litres
উত্তর
সঠিক উত্তর: খ
খ) 40 Litres
খ
ব্যাখ্যা

Milk : water = 17:3 = 17x : 3x
∴ 17x + 3x = 200
⇒ x =10litre
so, Milk = 170 litre and water = 30 litre in initial mixture.
Let 'y' litre of milk added in mixture
i.e. (170+y) : 30 = 7:1
⇒ (170 + y)/ 30= 7/1
∴ y = 210 - 170 = 40 litres