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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 ২ / ১১ · ১০১–২০০ / ১,০৮৬

১০১.
Ibrahim has 20 ounces of a 20% flavoured solution. How much salt should he add to make it a 25% solution?
  1. ক
    ক) 10.3
  2. খ
    খ) 50
  3. গ
    গ) 1.33
  4. ঘ
    ঘ) 5
সঠিক উত্তর: গ
গ) 1.33
উত্তর
সঠিক উত্তর: গ
গ) 1.33
গ
ব্যাখ্যা

Let y be the amount of flavour.
(0.2 × 20) + 1 × y = 0.25 (20 + y)
Or, 4 + y = 5 + 0.25y
Or, 0.75y = 1
So, y = 1.33

১০২.
In what ratio a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?
  1. ক
    ক) 1 : 2
  2. খ
    খ) 1 : 3
  3. গ
    গ) 2 : 1
  4. ঘ
    ঘ) 3 : 1
সঠিক উত্তর: খ
খ) 1 : 3
উত্তর
সঠিক উত্তর: খ
খ) 1 : 3
খ
ব্যাখ্যা
Question: In what ratio a mixture of 30%  alcohol strength be mixed with that of 50%  alcohol strength so as to get a mixture of 45% alcohol strength?

Solution:
Let the ratio be 1:x
Then according to the question
⇒ 30 × 1 + 50x = (1+x) 45
⇒ 30 + 50x = 45 + 45x
⇒ 50x - 45x = 45 - 30
⇒ 5x = 15
⇒ x = 3

∴ the ratio is 1 : 3
১০৩.
A jug when full of water, weights 2 kg. It weights 1.5 kg when the jug is half full. What is weight of the empty jug?
  1. ক
    1 kg
  2. খ
    0.5 kg
  3. গ
    0.75 kg
  4. ঘ
    0.80 kg
সঠিক উত্তর: ক
1 kg
উত্তর
সঠিক উত্তর: ক
1 kg
ক
ব্যাখ্যা
Question: A jug when full of water, weights 2 kg. It weights 1.5 kg when the jug is half full. What is weight of the empty jug?

Solution: 
ধরি,
Jug এর ওজন = x কেজি 
Water এর ওজন = y  কেজি 

এখন 
x + y = 2..................(1)

x + y/2 = 1.5
⇒ 2x + y = 3 ..................(2)

(2) - (1) ⇒
2x + y - (x + y) = 3 - 2
⇒ 2x + y - x - y = 1
∴ x = 1 

∴ Jug এর ওজন = 1 কেজি
১০৪.
If a : b = 3 : 1, find ratio (3a + 5b) : (3a - 5b).
  1. ক
    11 : 4
  2. খ
    4 : 1
  3. গ
    7 : 2
  4. ঘ
    1 : 4
সঠিক উত্তর: গ
7 : 2
উত্তর
সঠিক উত্তর: গ
7 : 2
গ
ব্যাখ্যা

Question: If a : b = 3 : 1, find ratio (3a + 5b) : (3a - 5b).

Solution: 
(3a + 5b) : (3a - 5b)
= b(3a/b + 5) : b (3a/b - 5)
= (3 × 3/1 + 5) : (3 × 3/1 - 5)
= (9 + 5) : (9 - 5)
= 14 : 4
= 7 : 2

১০৫.
Abrar, Aziz, Vijay enter into a partnership. Abrar initially invests 25 lakh & adds another 10 lakhs after one year. Aziz initially invests 35 lakh & withdrawal 10 lakh after 2 years and Vijay invests 30 Lakhs . In what ratio should the profit be divided at the end of 3 years?
  1. ক
    19 : 17 : 18
  2. খ
    19 : 19 : 15
  3. গ
    13 : 19 : 18
  4. ঘ
    19 : 19 : 18
সঠিক উত্তর: ঘ
19 : 19 : 18
উত্তর
সঠিক উত্তর: ঘ
19 : 19 : 18
ঘ
ব্যাখ্যা
Question: Abrar, Aziz, Vijay enter into a partnership. Abrar initially invests 25 lakh & adds another 10 lakhs after one year. Aziz initially invests 35 lakh & withdrawal 10 lakh after 2 years and Vijay 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]
১০৬.
The total age of A, B and C is 90 years. Ten years ago the ratio of their ages was 1 : 2 : 3. What is the present age of B?
  1. ক
    ক) 40 years
  2. খ
    খ) 30 years
  3. গ
    গ) 20 years
  4. ঘ
    ঘ) 18 years
  5. ঙ
    ঙ) None of these
সঠিক উত্তর: খ
খ) 30 years
উত্তর
সঠিক উত্তর: খ
খ) 30 years
খ
ব্যাখ্যা
Let, 10 years ago there age was = x, 2x, 3x
∴ x + 2x + 3x = 90 - 30
⇒ 6x = 60
⇒x = 10
⇒ Present age of B is, 2x + 10 = 2×10 + 10 = 30 years
১০৭.
The ratio of two numbers is 4 : 3. The sum of the numbers is 140. The difference between the two numbers is:
  1. ক
    ক) 20
  2. খ
    খ) 18
  3. গ
    গ) 15
  4. ঘ
    ঘ) 10
সঠিক উত্তর: ক
ক) 20
উত্তর
সঠিক উত্তর: ক
ক) 20
ক
ব্যাখ্যা
Question: The ratio of two numbers is 4 : 3. The sum of the numbers is 140. The difference between the two numbers is:

Solution: 
ধরি,
ছোট সংখ্যাটি 3x 
বড় সংখ্যাটি 4x

প্রশ্নমতে,
4x + 3x = 140
7x = 140
x = 140/7
x = 20

ছোট সংখ্যাটি 3x  = 3 × 20 = 60
বড় সংখ্যাটি 4x = 4 × 20 = 80 

সংখ্যা দুইটির পার্থক্য 80 - 60 = 20
১০৮.
800 grams of sugar solution has 40% sugar in it. How much sugar should be added to make 60% in the solution?
  1. ক
    440 gram
  2. খ
    410 gram
  3. গ
    450 gram
  4. ঘ
    400 gram
সঠিক উত্তর: ঘ
400 gram
উত্তর
সঠিক উত্তর: ঘ
400 gram
ঘ
ব্যাখ্যা
Question: 800 grams of sugar solution has 40% sugar in it. How much sugar should be added to make 60% in the solution? 

Solution:
Amount of sugar = 800 × 40/100
= 320 grams

Let,
x gm sugar to be added

ATQ,
(320 + x)/(800 + x) = 60%
⇒ (320 + x)/(800 + x) = 3/5
⇒ 1600 + 5x = 2400 + 3x
⇒ 5x - 3x = 2400 - 1600
⇒ 2x = 800
∴ x = 400
১০৯.
A shopkeeper mixes 15 kg of tea costing Tk. 280 per kg with 10 kg of tea costing Tk. 400 per kg. He then adds some inferior tea costing Tk. 200 per kg so that the average price of the mixture becomes Tk. 300 per kg. How many kg of inferior tea is added?
  1. ক
    5 kg
  2. খ
    7 kg
  3. গ
    8 kg
  4. ঘ
    10 kg
  5. ঙ
    6 kg
সঠিক উত্তর: খ
7 kg
উত্তর
সঠিক উত্তর: খ
7 kg
খ
ব্যাখ্যা

Question: A shopkeeper mixes 15 kg of tea costing Tk. 280 per kg with 10 kg of tea costing Tk. 400 per kg. He then adds some inferior tea costing Tk. 200 per kg so that the average price of the mixture becomes Tk. 300 per kg. How many kg of inferior tea is added?

Solution:
Let the quantity of inferior tea added be X kg.
Total cost = (15 × 280) + (10 × 400) + (X × 200)
= 4200 + 4000 + 200X 
= 8200 + 200X Tk.

Total weight = 15 + 10 + X
= 25 + X kg.

∴ Average price = 300 Tk. per kg.

(8200 + 200X)/(25 + X) = 300
⇒ 8200 + 200X = 300 × (25 + X)
⇒ 8200 + 200X = 7500 + 300X
⇒ 8200 - 7500 = 300x - 200X
⇒ 700 = 100X
⇒ X = 7

Thus, 7 kg of inferior tea is added.

১১০.
A cricket team has a ratio of win to loss 4 : 3. After losing 5 games in a row, the team's ratio of win to loss became 9 : 8. How many games had the team loss before it played the last five games?
  1. ক
    24
  2. খ
    27
  3. গ
    30
  4. ঘ
    33
  5. ঙ
    35
সঠিক উত্তর: খ
27
উত্তর
সঠিক উত্তর: খ
27
খ
ব্যাখ্যা

Question: A cricket team has a ratio of win to loss 4 : 3. After losing 5 games in a row, the team's ratio of win to loss became 9 : 8. How many games had the team loss before it played the last five games?

Solution:
মনে করি,
দলটির জেতা খেলার সংখ্যা ছিল 4p
দলটির হারা খেলার সংখ্যা ছিল 3p
পরপর 5 টি খেলা হারার পর,
জেতা খেলার সংখ্যা 4p
হারা খেলার নতুন সংখ্যা (3p + 5)

প্রশ্নমতে,
4p : (3p + 5) = 9 : 8
⇒ 4p/(3p + 5) = 9/8
⇒ 32p = 27p + 45
⇒ 32p - 27p = 45
⇒ 5p = 45
⇒ p = 45/5
⇒ p = 9
∴ p = 9

∴ ১ম অবস্থায় হারা খেলার সংখ্যা ছিল = 3 × 9 = 27
অতএব, শেষ 5 টি খেলা খেলার আগে দলটি 27 খেলায় হেরেছিল।

১১১.
Which ratio will come next in the following series?
1 : 2, 3 : 6, 4 : 8, ?
  1. ক
    ক) 7: 21
  2. খ
    খ) 8 : 32
  3. গ
    গ) 6 : 8
  4. ঘ
    ঘ) 5 : 10
সঠিক উত্তর: ঘ
ঘ) 5 : 10
উত্তর
সঠিক উত্তর: ঘ
ঘ) 5 : 10
ঘ
ব্যাখ্যা
The given series is: 1 : 2, 3 : 6, 4 : 8, ?

The logic followed is:
All the quantities in the above series maintain the ratio of 1 : 2 with each other.

⇒ 1 : 2
⇒ 3 : 6 = 1 : 2
⇒ 4 : 8 = 1 : 2

Now, check from the options which option maintain the ratio of 1: 2 with each other.

1) 7 : 21  ⇒ 1 : 3

2) 8 : 32 ⇒ 1 : 4

3) 6 : 8 ⇒ 2 : 3

4) 5 : 10 ⇒ 1 : 2

Only option 4) maintains the ratio of 1 : 2 with each other.

Hence, the correct answer is "5 : 10".
১১২.
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
    ক) 1/4
  2. খ
    খ) 1/3
  3. গ
    গ) 3/4
  4. ঘ
    ঘ) 2/3
সঠিক উত্তর: খ
খ) 1/3
উত্তর
সঠিক উত্তর: খ
খ) 1/3
খ
ব্যাখ্যা

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.

১১৩.
If P : 6 = Q : 8 = R : 10, then (P + Q + R)/P =?
  1. ক
    2
  2. খ
    3
  3. গ
    4
  4. ঘ
    5
  5. ঙ
    8
সঠিক উত্তর: গ
4
উত্তর
সঠিক উত্তর: গ
4
গ
ব্যাখ্যা

Question: If P : 6 = Q : 8 = R : 10, then (P + Q + R)/P =?

Solution:
P : 6 = Q : 8 
or, P/6 = Q/8
or, 6Q = 8P
or, Q = 8P/6
or, Q = 4P/3

P : 6 = R : 10
or, P/6 = R/10
or, 6R = 10P
or, R = 10P/6
or, R = 5P/3

∴ (P + Q + R)
= P + (4P/3) + (5P/3)
= (3P + 4P + 5P)/3
= 12P/3
= 4P

∴ (P + Q + R)/P
= 4P/P
= 4

১১৪.
Silver is 12 times as heavy as water, and lead is 8 times as heavy as water. In what ratio should these be mixed to get an alloy 9 times as heavy as water? 
  1. ক
    1 : 4
  2. খ
    1 : 3 
  3. গ
    2 : 3
  4. ঘ
    3 : 1 
সঠিক উত্তর: খ
1 : 3 
উত্তর
সঠিক উত্তর: খ
1 : 3 
খ
ব্যাখ্যা

Question: Silver is 12 times as heavy as water, and lead is 8 times as heavy as water. In what ratio should these be mixed to get an alloy 9 times as heavy as water?

Solution:
Let silver be 12x times as heavy and lead 8y times as heavy as water.

12x + 8y = 9(x + y)
⇒ 12x + 8y = 9x + 9y
⇒ 12x - 9x = 9y - 8y
⇒ 3x = y
∴ x/y = 1/3 = 1 : 3

১১৫.
The ratio of two numbers is 3 : 4 and their sum is 630. The smaller one of the numbers is -
  1. ক
    ক) 360
  2. খ
    খ) 270
  3. গ
    গ) 180
  4. ঘ
    ঘ) 120
সঠিক উত্তর: খ
খ) 270
উত্তর
সঠিক উত্তর: খ
খ) 270
খ
ব্যাখ্যা

ধরি,
ক্ষুদ্রতর সংখ্যাটি 3x এবং বৃহত্তর সংখ্যাটি 4x
প্রশ্নমতে,
3x + 4x = 630
⇒ 7x = 630
⇒ x = 630/7
⇒ x = 90
অতএব, ক্ষুদ্রতর সংখ্যাটি = 3 × 90
= 270.
Answer: 270.

১১৬.
If a, b and c are positive numbers such that (a2 + b2) : (b2 + c2) : (c2 + a2) = 34 : 61 : 45, then (b - a) : (c - b) : (c - a) = ?
  1. ক
    6 : 5 : 3
  2. খ
    4 : 3 : 2
  3. গ
    2 : 1 : 3
  4. ঘ
    5 : 2 : 7
সঠিক উত্তর: গ
2 : 1 : 3
উত্তর
সঠিক উত্তর: গ
2 : 1 : 3
গ
ব্যাখ্যা
Question : If a, b and c are positive numbers such that (a2 + b2) : (b2 + c2) : (c2 + a2) = 34 : 61 : 45, then (b - a) : (c - b) : (c - a) = ?

Solution :
Given,
(a2 + b2) : (b2 + c2) : (c2 + a2) = 34 : 61 : 45

Add all
2(a2 + b2 + c2) = 34 + 61 + 45
⇒ 2(a2 + b2 + c2) = 140
⇒ (a2 + b2 + c2) = 70
⇒ c2 = 70 - 34
⇒ c2 = 36
∴ c = 6

b2 = 70 - 45
⇒ b2 = 25
∴ b = 5

a2 = 70 - 61
⇒a2 = 9
∴ a = 3

(b - a) : (c - b) : (c - a)
= (5 - 3) : (6 - 5) : (6 - 3)
= 2 : 1 : 3
১১৭.
How much water be mixed in 45 litre of milk worth Tk. 6.40 per litre, so that value of mixture is Tk. 4.80 per litre?
  1. ক
    10 litres
  2. খ
    12 litres
  3. গ
    15 litres
  4. ঘ
    20 litres
সঠিক উত্তর: গ
15 litres
উত্তর
সঠিক উত্তর: গ
15 litres
গ
ব্যাখ্যা

Question: How much water be mixed in 45 litre of milk worth Tk. 6.40 per litre, so that value of mixture is Tk. 4.80 per litre?

Solution:
মনে করি, x লিটার পানি মেশাতে হবে।

মিশ্রণের মোট পরিমাণ = (45 + x) লিটার
45 লিটার দুধের দাম = 45 × 6.40 = 288 টাকা
পানির দাম = 0 টাকা

প্রশ্নমতে, মিশ্রণের মোট দাম = (45 + x) × 4.80
যেহেতু পানি বিনামূল্যে পাওয়া যায়, তাই মোট দাম অপরিবর্তিত থাকবে:
288 = (45 + x) × 4.80
⇒ 288/4.80 = 45 + x [উভয় পক্ষকে 4.80 দ্বারা ভাগ করে পাই]
⇒ 60 = 45 + x
⇒ x = 60 - 45
⇒ x = 15 litres 

∴ 15 লিটার পানি মেশাতে হবে।

১১৮.
If √2 : (1 + √3 ) :: √6 : x, then x is equal to-
  1. ক
    ক) √2 + 3
  2. খ
    খ) √3 + 3
  3. গ
    গ) √3 - 3
  4. ঘ
    ঘ) √3 + 1
সঠিক উত্তর: খ
খ) √3 + 3
উত্তর
সঠিক উত্তর: খ
খ) √3 + 3
খ
ব্যাখ্যা
√2 : (1 + √3 ) :: √6 : x
√2/ (1 + √3) = √6 / x
x√2 = √6(1 + √3)
x = √6(1 + √3)/√2 
x = √3(1 + √3)
x =√3 + 3
১১৯.
The cost of a table and a chair are in the ratio of 10 : 7. If the cost of table and a chair is increased by 8% and 20% respectively, then what will be the new ratio?
  1. ক
    12 : 13
  2. খ
    7 : 12
  3. গ
    9 : 7
  4. ঘ
    24 : 55
সঠিক উত্তর: গ
9 : 7
উত্তর
সঠিক উত্তর: গ
9 : 7
গ
ব্যাখ্যা
Question : The cost of a table and a chair are in the ratio of 10 : 7. If the cost of table and a chair is increased by 8% and 20% respectively, then what will be the new ratio?

Solution :
Let the cost of the table and chair be Tk. 10x and Tk. 7x respectively

New cost of table = 108% of 10x
= (108 × 10x)/100
= 54x/5

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

So the new ratio = 54x/5 : 42x/5
= 54 : 42
= 9 : 7
১২০.
The ages of Sabit and Sumit are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the sum of their ages?
  1. ক
    20 years
  2. খ
    8 years
  3. গ
    16 years
  4. ঘ
    10 years
সঠিক উত্তর: ক
20 years
উত্তর
সঠিক উত্তর: ক
20 years
ক
ব্যাখ্যা
Question: The ages of Sabit and Sumit are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the sum of 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
= 10x
= 10 × 2
= 20 years
১২১.
The ratio of spirit and water in two mixtures of 20 litres and 36 litres is 3 : 7 and 7 : 5 respectively. Both the mixtures are mixed together. Now the ratio of the spirit and water in the new mixture is ?
  1. ক
    ক) 18 : 29
  2. খ
    খ) 20 : 29
  3. গ
    গ) 23 : 25
  4. ঘ
    ঘ) 27 : 29
সঠিক উত্তর: ঘ
ঘ) 27 : 29
উত্তর
সঠিক উত্তর: ঘ
ঘ) 27 : 29
ঘ
ব্যাখ্যা
In 20 litres mixture - 1; sum of ratio = 3 + 7 = 10
10 units = 20 litres
1 unit = 20/10 = 2

In 36 litres mixture - 1; sum of ratio = 5 + 7 = 12
12 units = 36 litres
1 unit = 36/12 = 3

In 20 litres mixture - 1; spirit : water = 3 × 2 : 7 × 2 = 6 : 14
In 36 litres mixture - 2; spirit : water = 7 × 3 : 5 × 3 = 21 : 15

spirit : water = (6 + 21) : (14 + 15) = 27 : 29
১২২.
The fourth proportional to 5, 8, 15 is
  1. ক
    ক) 24
  2. খ
    খ) 18
  3. গ
    গ) 21
  4. ঘ
    ঘ) 27
সঠিক উত্তর: ক
ক) 24
উত্তর
সঠিক উত্তর: ক
ক) 24
ক
ব্যাখ্যা
Question: The fourth proportional to 5, 8, 15 is 

Solution: 
Let the fourth proportion is X.
then,
5 : 8 : : 15 : X
5X = 120
X = 24
১২৩.
A bucket contains a mixture of two liquids A & B in the proportion 5 : 3. If 16 liters of the mixture is replaced by 16 liters of liquid B, then the ratio of the two liquids becomes 3 : 5. How much of the liquid B was there in the bucket?
  1. ক
    ক) 4
  2. খ
    খ) 7
  3. গ
    গ) 10
  4. ঘ
    ঘ) None of these
সঠিক উত্তর: ঘ
ঘ) None of these
উত্তর
সঠিক উত্তর: ঘ
ঘ) None of these
ঘ
ব্যাখ্যা
Let bucket contains 5x and 3x of liquids A and B respectively.
Quantity of A in 16 liters = 16 × (5x / 8x) = 10
Quantity of B in 16 liters = 16 - 10 = 6
(5x - 10)/(3x - 6 + 16) = 3/5
Upon solving, x = 5
So initial quantity of B = 15 liters
১২৪.
The students in three batches in school is in the ratio of 2 : 3 : 5. If 20 students in each batch are increased than the ratio changes to 4 : 5 : 7. The total number of students in the three before the increase was-
  1. ক
    100
  2. খ
    120
  3. গ
    160
  4. ঘ
    20
সঠিক উত্তর: ক
100
উত্তর
সঠিক উত্তর: ক
100
ক
ব্যাখ্যা
Question: The students in three batches in school is in the ratio of 2 : 3 : 5. If 20 students in each batch are increased than the ratio changes to 4 : 5 : 7. The total number of students in the three before the increase was-

Solution:
Let,  the students in the three before the increase were 2x, 3x, 5x

After increase, 2x + 20, 3x + 20, 5x + 20

(2x + 20)/ (3x + 20) = 4/5
⇒ 10x + 100 = 12x + 80
⇒ 2x = 20
⇒ x = 10

The total number of students in the three before the increase was = (2x + 5x + 3x)
= 10x
= 10 × 10
= 100
১২৫.
In a 729 liters mixture of milk and water, the ratio of milk to water 7 : 2. To get a new mixture containing milk and water in the ratio 7 : 3, the amount of water to be added is
  1. ক
    71 liters
  2. খ
    81 liters
  3. গ
    56 liters
  4. ঘ
    50 liters
সঠিক উত্তর: খ
81 liters
উত্তর
সঠিক উত্তর: খ
81 liters
খ
ব্যাখ্যা

Question: In a 729 liters mixture of milk and water, the ratio of milk to water 7 : 2. To get a new mixture containing milk and water in the ratio 7 : 3, the amount of water to be added is- 

Solution: 
Quantity of milk in 729 litre of mixture
= 7 × 729/9 = 567 litres

Quantity of water
= 729 - 567
= 162 litres

Let x litre of water be added to become ratio 7 : 3

According to the question,
7/3 = 567 / (162 + x)
Or, 162 × 7 + 7x = 567 × 3
Or, 7x = 1701 - 1134 = 567
Or, x = 567 / 7 = 81
Therefore, 81 litres of water is to be added.

১২৬.
The present age of Habib and Shikha are in the ratio of 6 : 4. Five years ago their ages were in the ratio of 5 : 3. How old is Habib now?
  1. ক
    ক) 24
  2. খ
    খ) 30
  3. গ
    গ) 36
  4. ঘ
    ঘ) 42
সঠিক উত্তর: খ
খ) 30
উত্তর
সঠিক উত্তর: খ
খ) 30
খ
ব্যাখ্যা

Let,
Habib’s age is = 6x
And Shikha’s age = 4x
ATQ,
(6x - 5)/(4x - 5) = 5/3
⇒ 18x - 15 = 20x - 25
⇒ 2x = 10
⇒ x = 5
So, Habib’s age is = 6×5 = 30 year

১২৭.
Two numbers are in the ratio 3 : 4. If the difference of their squares is 63, then the smallest number is -
  1. ক
    ক) 12
  2. খ
    খ) 9
  3. গ
    গ) 21
  4. ঘ
    ঘ) 18
সঠিক উত্তর: খ
খ) 9
উত্তর
সঠিক উত্তর: খ
খ) 9
খ
ব্যাখ্যা
Question: Two numbers are in the ratio 3 : 4. If the difference of their squares is 63, then the smallest number is -

Solution:
Let
the numbers are 3x, 4x

Now 
(4x)2 - (3x)2 = 63
16x2 - 9x2 = 63
7x2 = 63 
x2 = 9
x = 3

The smallest number is = 3 × 3 = 9
১২৮.
The ratio of the measures ∠A and ∠B of a triangle ABC is 3 : 2. The ratio of the measures of ∠B and ∠C is 4 : 5. Find the measure of largest angle of the triangle ABC.
  1. ক
    72°
  2. খ
    78°
  3. গ
    60°
  4. ঘ
    48°
সঠিক উত্তর: ক
72°
উত্তর
সঠিক উত্তর: ক
72°
ক
ব্যাখ্যা
Question: The ratio of the measures ∠A and ∠B of a triangle ABC is 3 : 2. The ratio of the measures of ∠B and ∠C is 4 : 5. Find the measure of largest angle of the triangle ABC.

Solution:
∠ A : ∠ B = 3 : 2.
∠ B : ∠ C = 4 : 5.
∴ ∠ A : ∠ B : ∠ C = 6 : 4 : 5.
Let actual values are 6x, 4x and 5x.

So
6x + 4x + 5x = 180° 
⇒ 15x = 180°
∴ x = 12°

So largest angle is  6(12°) = 72°
১২৯.
A chemist has two solutions, one containing 40% acid and the other containing 80% acid. How many liters of each solution should be mixed to get 12 liters of a solution containing 60% acid? 
  1. ক
    4 liters of 40% and 8 liters of 80%
  2. খ
    5 liters of 40% and 7 liters of 80%
  3. গ
    6 liters of 40% and 6 liters of 80%
  4. ঘ
    3 liters of 40% and 9 liters of 80% 
সঠিক উত্তর: গ
6 liters of 40% and 6 liters of 80%
উত্তর
সঠিক উত্তর: গ
6 liters of 40% and 6 liters of 80%
গ
ব্যাখ্যা

Question: A chemist has two solutions, one containing 40% acid and the other containing 80% acid. How many liters of each solution should be mixed to get 12 liters of a solution containing 60% acid?

Solution:
Let x liters of 40% solution be used. Then (12 - x) liters of 80% solution will be used.

According to the problem:

40% × x + 80% × (12 - x) = 60% × 12
⇒ 40x + 80(12 - x) = 720
⇒ 40x + 960 - 80x = 720
⇒ -40x + 960 = 720
⇒ -40x = -240
⇒ x = 6

∴ 6 liters of 40% solution and 6 liters of 80% solution are needed.

১৩০.
ধানে চাল ও তুষের অনুপাত 7 : 3 হলে, এতে শতকরা কী পরিমাণ চাল আছে?
  1. ক
    70%
  2. খ
    30%
  3. গ
    7%
  4. ঘ
    3%
সঠিক উত্তর: ক
70%
উত্তর
সঠিক উত্তর: ক
70%
ক
ব্যাখ্যা
প্রশ্ন: ধানে চাল ও তুষের অনুপাত 7 : 3 হলে, এতে শতকরা কী পরিমাণ চাল আছে?

সমাধান:
ধানের চাল ও তুষের অনুপাত 7 : 3
অনুপাতের যোগফল = 7 + 3 = 10

এতে শতকরা চালের পরিমাণ = (7/10) × 100%
= 70%
১৩১.
Two numbers are in ratio 4:5 and their LCM is 180. The smaller number is-
  1. ক
    ক) 9
  2. খ
    খ) 15
  3. গ
    গ) 36
  4. ঘ
    ঘ) 45
সঠিক উত্তর: গ
গ) 36
উত্তর
সঠিক উত্তর: গ
গ) 36
গ
ব্যাখ্যা

Numbers are in the ratio 4:5
Let the numbers be 4x and 5x
Hence, LCM = 20x
Hence, 20x = 180
Hence, x = 180/20 = 9
Hence the numbers are 36 and 45

১৩২.
The ratio of milk and water in a solution is 20 : 7 and after adding 5 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.
  1. ক
    ক) 7 liters
  2. খ
    খ) 10 liters
  3. গ
    গ) 11 liters
  4. ঘ
    ঘ) 12 liters
সঠিক উত্তর: ঘ
ঘ) 12 liters
উত্তর
সঠিক উত্তর: ঘ
ঘ) 12 liters
ঘ
ব্যাখ্যা
প্রশ্ন: The ratio of milk and water in a solution is 20 : 7 and after adding 5 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.

সমাধান: 
Let the initial amount of milk be 20x liters and amaount of water 7x liters

Ratio of milk and water after adding 5 liters = 20x/(7x + 5) = 5/3
⇒ 60x = 35x + 25
⇒ 25x = 25
⇒ x = 1.

∴ Final amount of water in solution = 7x + 5 = 7 + 5 = 12 liters.
১৩৩.
The ratio of milk and water in a solution is 20 : 7 and after adding 5 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.
  1. ক
    9 litres
  2. খ
    10 litres
  3. গ
    11 litres
  4. ঘ
    12 litres
সঠিক উত্তর: ঘ
12 litres
উত্তর
সঠিক উত্তর: ঘ
12 litres
ঘ
ব্যাখ্যা
Question: The ratio of milk and water in a solution is 20 : 7 and after adding 5 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.

Solution:
Let the initial amount of milk be 20x and of water 7x.
Ratio of milk and water after adding 5 litres,
20x/ (7x + 5) = 5/3
⇒ 60x = 35x + 25
⇒ 25x = 25
⇒ x = 1.

∴ Final amount of water in solution = 7x + 5 = 7 + 5 = 12 litres.
১৩৪.
If 0.75 : x :: 5 : 8, then x is equal to:
  1. ক
    1.25
  2. খ
    1.12
  3. গ
    1.2
  4. ঘ
    1.30
সঠিক উত্তর: গ
1.2
উত্তর
সঠিক উত্তর: গ
1.2
গ
ব্যাখ্যা
Question: If 0.75 : x :: 5 : 8, then x is equal to:

Solution:
0.75 : x :: 5 : 8
⇒ 0.75/x = 5/8
⇒ 5x = 0.75 × 8
⇒ 5x = 6
⇒ x = 6/5
∴ x = 1.2
১৩৫.
The ratio of cost price and selling price is 6 : 8. The profit percent is
  1. ক
    33.33%
  2. খ
    28.57%
  3. গ
    30.50%
  4. ঘ
    34.34%
  5. ঙ
    25%
সঠিক উত্তর: ক
33.33%
উত্তর
সঠিক উত্তর: ক
33.33%
ক
ব্যাখ্যা
Question: The ratio of cost price and selling price is 6 : 8. The profit percent is

Solution:
Given,
the ratio of cost price (C.P.) to selling price (S.P.) is 6 : 8
Let the cost price be 6x and the selling price be 8x.

Profit = S.P - .C.P. = 8x - 6x = 2x
Profit percent = (Profit/CP) × 100
= (2x/6x) × 100
= (1/3) × 100
= 33.33%
১৩৬.
600 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 75% sugar in the solution?
  1. ক
    1000 grams
  2. খ
    1400 grams
  3. গ
    1200 grams
  4. ঘ
    1080 grams
সঠিক উত্তর: ঘ
1080 grams
উত্তর
সঠিক উত্তর: ঘ
1080 grams
ঘ
ব্যাখ্যা

Question: 600 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 75% in the solution?

Solution:
Amount of sugar =600 × 30/100
=180 grams
Let,
x gram sugar to be added

ATQ,
(180 + X)/(600 + X) = 75%
⇒ (180 + X)/(600 + X) = 75/100
⇒ (180 + X)/(600 + X) = 3/4
⇒ 4(180 + X) = 3(600 + X)
⇒ 4X + 720 = 3X + 1800
⇒ 4X - 3X= 1800 - 720
∴ X = 1080 grams

১৩৭.
If a : b : c = 2 : 3 : 4 and 2a - 3b + 4c = 33, then the value of c is-
  1. ক
    22
  2. খ
    12
  3. গ
    18
  4. ঘ
    10
সঠিক উত্তর: খ
12
উত্তর
সঠিক উত্তর: খ
12
খ
ব্যাখ্যা

Question: If a : b : c = 2 : 3 : 4 and 2a - 3b + 4c = 33, then the value of c is-

Solution:
Let,
 a = 2x, b = 3x and c = 4x  ; (since a : b : c = 2 : 3 : 4)

Now substitute into the equation,
2a - 3b + 4c = 33
⇒ 2(2x) - 3(3x) + 4(4x) = 33 
⇒ 4x - 9x + 16x = 33 
⇒ 11x = 33
∴ x = 3

Since c = 4x = 4 × 3 = 12

১৩৮.
Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-
  1. ক
    4 : 3
  2. খ
    1 : 5
  3. গ
    4 : 9
  4. ঘ
    4 : 5
সঠিক উত্তর: ঘ
4 : 5
উত্তর
সঠিক উত্তর: ঘ
4 : 5
ঘ
ব্যাখ্যা
Question: Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-

Solution:
Let the third number be x

Then, first number = 120% of x = 120x/100 = 6x/5
Second number = 150% of x = 150x/100 = 3x/2

∴ Ratio of first two numbers = 6x/5 : 3x/2
= 12x : 15x
= 4 : 5
১৩৯.
Fresh grapes contain 80% water by weight, while dry grapes contain only 20% water. If a person has 100 kg of fresh grapes, how many kilograms of dry grapes can be obtained from them after drying?
  1. ক
    25 kg
  2. খ
    20 kg
  3. গ
    30 kg
  4. ঘ
    40 kg
সঠিক উত্তর: ক
25 kg
উত্তর
সঠিক উত্তর: ক
25 kg
ক
ব্যাখ্যা
Question: Fresh grapes contain 80% water by weight, while dry grapes contain only 20% water. If a person has 100 kg of fresh grapes, how many kilograms of dry grapes can be obtained from them after drying?

Solution:
Fresh grapes: 80% water → 20% solid
So, in 100 kg fresh grapes:
Water = 80 kg
Solid = 20 kg
When grapes dry, water evaporates, but the solid remains unchanged.

Dry grapes: 20% water → 80% solid
That means the 20 kg solid must now be 80% of the dry grape weight
So, let total weight of dry grapes = x

Then:
0.8x=20
⇒x=200/0.8 =25 kg
১৪০.
If a : b : c = 3 : 4 : 7, then the ratio (a + b + c) : 5c is equal to
  1. ক
    ক) 1 : 3
  2. খ
    খ) 2 : 1
  3. গ
    গ) 4 : 1
  4. ঘ
    ঘ) 2 : 5
সঠিক উত্তর: ঘ
ঘ) 2 : 5
উত্তর
সঠিক উত্তর: ঘ
ঘ) 2 : 5
ঘ
ব্যাখ্যা
(a + b + c) : 5c = (3 + 4 + 7) : 5 × 7 = 14 : 35 = 2 : 5
১৪১.
A mixture of 50 liters of milk and water contains 10% of water. How much water must be added to make the water 20% in the new mixture?
  1. ক
    ক) 6.25 litres
  2. খ
    খ) 7 litres
  3. গ
    গ) 7.25 litres
  4. ঘ
    ঘ) 8.25 litres
সঠিক উত্তর: ক
ক) 6.25 litres
উত্তর
সঠিক উত্তর: ক
ক) 6.25 litres
ক
ব্যাখ্যা
Quantity of water in given mixture
=10 / 100 x 50=5 litres.

Quantity of milk in given mix.
= 50 - 5= 45 litres.

Let x litres of water be added to it.
Milk : 45 litres, water (5 + x) litres
Total mix = (50 + x) litres
(5+x)/ (50+ x ) = 20/100
⇒ (5 + x)5 = 50 + x
⇒ 4x = 25
⇒ x= 6.25
6.25 litres of water must be added.
১৪২.
A vessel contains milk and water in the ratio 7 : 4. If 15 liters of milk are added to it, the ratio of milk to water becomes 10 : 4. Find the final amount of milk in the new mixture. 
  1. ক
    50 liters
  2. খ
    68 liters
  3. গ
    75 liters
  4. ঘ
    78 liters
সঠিক উত্তর: ক
50 liters
উত্তর
সঠিক উত্তর: ক
50 liters
ক
ব্যাখ্যা

Question: A vessel contains milk and water in the ratio 7 : 4. If 15 liters of milk are added to it, the ratio of milk to water becomes 10 : 4. Find the final amount of milk in the new mixture.

Solution:
Let the initial amount of milk be 7x liters
and the amount of water 4x liters.

According to the question, 
(7x + 15)/4x = 10/4
⇒ 4(7x + 15) = 10 × 4x
⇒ 28x + 60 = 40x
⇒ 60 = 12x
⇒ x = 60/12
⇒ x = 5

∴ Final amount of milk in mixture = 7x + 15
= 7 × 5 + 15
= 35 + 15
= 50 liters.

১৪৩.
Which of the following is the lowest ratio?
  1. ক
    ক) 15 : 23
  2. খ
    খ) 7 : 15
  3. গ
    গ) 17 : 25
  4. ঘ
    ঘ) 21 : 39
সঠিক উত্তর: খ
খ) 7 : 15
উত্তর
সঠিক উত্তর: খ
খ) 7 : 15
খ
ব্যাখ্যা
Question: Which of the following is the lowest ratio?

Solution: 
15 : 23 = 15/23 = 0.652
7 : 15 = 7/15 = 0.467
17 : 25 = 17/25 = 0.68
21 : 39 = 21/39 = 0.538

এখানে, 7 : 15 = 7/15 = 0.467 এর মান সবচেয়ে ক্ষুদ্রতম।
১৪৪.
A, B, and C are boxes containing marbles in the ratio 2 : 3 : 4. Total number of marbles is 90. The above ratio can be changed to 4 : 5 : 6 by transferring-
  1. ক
    4 marbles from C to A
  2. খ
    3 marbles from A to C
  3. গ
    6 marbles from B to C
  4. ঘ
    3 marbles from C to B
সঠিক উত্তর: ক
4 marbles from C to A
উত্তর
সঠিক উত্তর: ক
4 marbles from C to A
ক
ব্যাখ্যা
Question: A, B, and C are boxes containing marbles in the ratio 2 : 3 : 4. Total number of marbles is 90. The above ratio can be changed to 4 : 5 : 6 by transferring-

Solution:
A's share = 90 × (2/9) = 20
B's share = 90 × (3/9) = 30
C's share = 90 × (4/9) = 40

When marbles are shared in the ratio of 4 : 5 : 6

A's share = 90 × (4/15) = 24
B's share = 90 × (5/15) = 30
C's share = 90 × (6/15) = 36

Clearly,
From C (40 - 4) = 36 marbles have been transferred from to A (20 + 4) = 24

So, The above ratio can be changed to 4 : 5 : 6 by transferring 4 marbles from C to A.
১৪৫.
The ratio of milk to water in 84 liters of mixture is 5 : 2. The water (in liters) to be added to it to make the the ratio 2 : 1 is -
  1. ক
    ক) 3 liters
  2. খ
    খ) 6 liters
  3. গ
    গ) 8 liters
  4. ঘ
    ঘ) 10 liters
সঠিক উত্তর: খ
খ) 6 liters
উত্তর
সঠিক উত্তর: খ
খ) 6 liters
খ
ব্যাখ্যা
Question: The ratio of milk to water in 84 liters of mixture is 5 : 2. The water (in liters) to be added to it to make the the ratio 2 : 1 is -

Solution:
Quantity of milk = {84 × (5/7)} litres = 60 liters
Quantity of water = 84 - 60 = 24 liters

Let the quantity of water to be added be x liters

Then, 
{60/(24 + x)} = 2/1
⇒ 2x + 48 = 60
⇒ 2x = 60 - 48
⇒ 2x = 12
∴ x = 6
১৪৬.
An office opens at 9 a.m. and closes at 5.30 p.m. with a lunch break of 30 minutes. What is the ratio of lunch breaks to the total period in the office?
  1. ক
    1 : 3
  2. খ
    1 : 6
  3. গ
    1 : 9
  4. ঘ
    1 : 17
সঠিক উত্তর: ঘ
1 : 17
উত্তর
সঠিক উত্তর: ঘ
1 : 17
ঘ
ব্যাখ্যা
Question: An office opens at 9 a.m. and closes at 5.30 p.m. with a lunch break of 30 minutes. What is the ratio of lunch breaks to the total period in the office?

Solution: 
The ratio of lunch breaks to the total period in the office = 30/{(8 × 60) + 30}
= 30/510
= 1/17
১৪৭.
Which of the following is the lowest ratio?
  1. ক
    ক) 2 : 3
  2. খ
    খ) 1 : 3
  3. গ
    গ) 1 : 5
  4. ঘ
    ঘ) 2 : 5
সঠিক উত্তর: গ
গ) 1 : 5
উত্তর
সঠিক উত্তর: গ
গ) 1 : 5
গ
ব্যাখ্যা
Solution: 
2/3 = 0.66
1/3 = 0.33
1/5 = 0.2
2/5 = 0.4

hance the lowest ratio is 1 : 5
১৪৮.
Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 10 times as heavy as water? 
  1. ক
    4 : 9
  2. খ
    2 : 9
  3. গ
    1 : 8 
  4. ঘ
    1 : 9
  5. ঙ
    None of the above
সঠিক উত্তর: ঘ
1 : 9
উত্তর
সঠিক উত্তর: ঘ
1 : 9
ঘ
ব্যাখ্যা

Question: Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 10 times as heavy as water? 

Solution: 
let, gold is 19x tme heavy and copper 9y times heavy as water 

19x + 9y = 10 (x + y)
⇒ 19x + 9y = 10x + 10y 
⇒ 19x - 10x = 10y - 9y
 ⇒ 9x = y

∴ x/y = 1/9
= 1 : 9 

১৪৯.
A trader mixes 6ltr of milk costing 500 TK. with 7ltr of milk costing 600 TK. per litre. The trader also mixes some quantity of water to the mixture so as to bring the price to 480 TK. per litre. How many litres of water is added?
  1. ক
    3 litre
  2. খ
    2.5 litre
  3. গ
    4 litre
  4. ঘ
    2 litre
সঠিক উত্তর: ঘ
2 litre
উত্তর
সঠিক উত্তর: ঘ
2 litre
ঘ
ব্যাখ্যা
Question: A trader mixes 6ltr of milk costing 500 TK. with 7ltr of milk costing 600 TK. per litre. The trader also mixes some quantity of water to the mixture so as to bring the price to 480 TK. per litre. How many litres of water is added?

Solution:
Let us consider, the water be 'W' litre

Here,
(6 × 500 + 7 × 600)/(13 + W) = 480 
⇒ 3000 + 4200 = 6240 + 480W
⇒ 480W = 7200 - 6240
⇒ W = 960/480
∴ W = 2

W is the amount of water added, W = 2 litre
১৫০.
In a kilometre race, A can beat B by 100 metres and B can beat C by 60 metres. In the same race A can beat C by -
  1. ক
    154 metres
  2. খ
    144 metres
  3. গ
    124 metres
  4. ঘ
    164th metres
সঠিক উত্তর: ক
154 metres
উত্তর
সঠিক উত্তর: ক
154 metres
ক
ব্যাখ্যা

A : B = 1000 : (1000 - 100)
= 1000 : 900
= 10 : 9
B : C = 1000 : (1000 - 60)
= 1000 : 940
100 : 94
= 9 : 846/100
Therefor, A : C = 10 : 846/100
= 10000 : 846
hence A can beat C by = (10000 - 846)
= 154 metres.

১৫১.
If a : b : c = 5 : 2 : 7, then the ratio (a + b + c) : c is equal to
  1. ক
    2 : 5
  2. খ
    3 : 7
  3. গ
    2 : 1
  4. ঘ
    5 : 2
সঠিক উত্তর: গ
2 : 1
উত্তর
সঠিক উত্তর: গ
2 : 1
গ
ব্যাখ্যা
Question : If a : b : c = 5 : 2 : 7, then the ratio (a + b + c) : c is equal to

Solution :
Here,
a : b : c = 5 : 2 : 7

Let, a : b : c = 5x : 2x : 7x
So, a + b + c = 5x + 2x + 7x
= 14x

and, c = 7x

(a + b + c) : c = 14x : 7x
= 2 : 1
১৫২.
The value of a fraction is 2/5. If the numerator decreased by 2 and the denominator increased by 1, the resulting fraction is equivalent to 1/4. Find the numerator of the original fraction.
  1. ক
    ক) 3
  2. খ
    খ) 4
  3. গ
    গ) 6
  4. ঘ
    ঘ) 8
সঠিক উত্তর: গ
গ) 6
উত্তর
সঠিক উত্তর: গ
গ) 6
গ
ব্যাখ্যা
Question: The value of a fraction is 2/5. If the numerator decreased by 2 and the denominator increased by 1, the resulting fraction is equivalent to 1/4. Find the numerator of the original fraction.

Solution: 
let, the original fraction = x/y

x/y = 2/5
⇒ 2y = 5x
⇒ y = (5/2)x

(x - 2)/(y + 1) = 1/4
⇒ 4 (x - 2) = y + 1
⇒ 4x - 8 = (5x/2) + 1
⇒ 4x - (5x/2) = 1 + 8
⇒ 3x/2 = 9
∴ x = 6

So, the numerator is 6.
১৫৩.
If A : B : C = 2 : 3 : 4, then the ratio (A/B) : (B/C) : (C/A) = ?
  1. ক
    ক) 8 : 9 : 24
  2. খ
    খ) 6 : 9 : 22
  3. গ
    গ) 5 : 8 : 25
  4. ঘ
    ঘ) 3 : 6 : 20
সঠিক উত্তর: ক
ক) 8 : 9 : 24
উত্তর
সঠিক উত্তর: ক
ক) 8 : 9 : 24
ক
ব্যাখ্যা
Question: If A : B : C = 2 : 3 : 4, then the ratio (A/B) : (B/C) : (C/A) = ?

Solution:
Let, 
A = 2k
B = 3k
C = 4k

Then,
A/B = 2k/3k = 2/3
B/C = 3k/4k = 3/4
C/A = 4k/2k = 2

(A/B) : (B/C) : (C/A) = (2/3) : (3/4) : 2
= (2/3) × 12 : (3/4) × 12 : 2 × 12
= 8 : 9 : 24
১৫৪.
Paint Pro makes pink paint by mixing red paint and white paint in the ratio 3 : 4. Colour Co makes pink paint by mixing red paint and white paint in the ratio 5 : 7. Which company uses a higher proportion of red paint in their mixture?
  1. ক
    They are the same
  2. খ
    Paint Pro
  3. গ
    Colour Co
  4. ঘ
    It is impossible to tell
সঠিক উত্তর: খ
Paint Pro
উত্তর
সঠিক উত্তর: খ
Paint Pro
খ
ব্যাখ্যা
Question: Paint Pro makes pink paint by mixing red paint and white paint in the ratio 3 : 4. Colour Co makes pink paint by mixing red paint and white paint in the ratio 5 : 7. Which company uses a higher proportion of red paint in their mixture?

Solution:
The proportion of red paint for Paint Pro is 3/7
The proportion of red paint for Colour Co is 5/12

We can compare fractions by putting them over a common denominator using equivalent fractions
3/7 = 36/84
5/12 = 35/84

3​/7 is a bigger fraction so Paint Pro uses a higher proportion of red paint.
১৫৫.
In your wallet, there are Tk 1000, Tk 500, and Tk 100 notes in the ratio 3 : 5 : 2, amounting to Tk 22,800. Find the number of each note respectively.
  1. ক
    12, 20, 8
  2. খ
    16, 12, 14
  3. গ
    15, 16, 10
  4. ঘ
    12, 18, 14
সঠিক উত্তর: ক
12, 20, 8
উত্তর
সঠিক উত্তর: ক
12, 20, 8
ক
ব্যাখ্যা

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

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

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

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

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

১৫৬.
There are pieces of mangoes in a bowl is 8/15. Calculate the ratio of mangoes to other pieces of fruit in the bowl.
  1. ক
    ক) 8 : 7
  2. খ
    খ) 8 : 15
  3. গ
    গ) 7 : 15
  4. ঘ
    ঘ) None of these.
সঠিক উত্তর: ক
ক) 8 : 7
উত্তর
সঠিক উত্তর: ক
ক) 8 : 7
ক
ব্যাখ্যা
প্রশ্ন: There are pieces of mangoes in a bowl is 8/15. Calculate the ratio of mangoes to other pieces of fruit in the bowl.

সমাধান: 
Let the total number of pieces of fruit be 15.
The number of mangoes is 8.
The number of other pieces of fruit is therefore 7.

∴ The ratio of mangoes to other pieces of fruit is therefore 8 : 7
১৫৭.
What is the ratio of 6 inches to 6 feet?
  1. ক
    1 : 10
  2. খ
    1 : 100
  3. গ
    1 : 120
  4. ঘ
    1 : 12
সঠিক উত্তর: ঘ
1 : 12
উত্তর
সঠিক উত্তর: ঘ
1 : 12
ঘ
ব্যাখ্যা
Question: What is the ratio of 6 inches to 6 feet?

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

Now, 
8 inches : 6 feet = 6 : 72 = 1 : 12
১৫৮.
Find the rate of discount being given on a shirt whose selling price is Tk.546 after deducting a discount of Tk.104 on its marked price.
  1. ক
    11%
  2. খ
    12%
  3. গ
    13%
  4. ঘ
    15%
  5. ঙ
    None of the above
সঠিক উত্তর: ঙ
None of the above
উত্তর
সঠিক উত্তর: ঙ
None of the above
ঙ
ব্যাখ্যা
Question: Find the rate of discount being given on a shirt whose selling price is Tk.546 after deducting a discount of Tk.104 on its marked price.

Solution:
The price written on the item = 546 + 104 Tk.
= 650 Tk.

On 650 Taka, the commission is 104 Taka.
∴ Therefore, the commission on 100 Taka is (104 × 100)/650 Tk.
= 16 Tk.
১৫৯.
A solution contains 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-
  1. ক
    ক) 20%
  2. খ
    খ) 10.18%
  3. গ
    গ) 20.18%
  4. ঘ
    ঘ) 18.18%
সঠিক উত্তর: ঘ
ঘ) 18.18%
উত্তর
সঠিক উত্তর: ঘ
ঘ) 18.18%
ঘ
ব্যাখ্যা
Question: A solution contains 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

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

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

percent of sugar = 20 × 100%/110
= 18.18%
১৬০.
A mixture of 60 kg contains sand and stone in the ratio 7 : 5. Find the quantity of sand to be added to the mixture so that the ratio of sand to stone becomes 2 : 1.
  1. ক
    10 liters
  2. খ
    15 liters
  3. গ
    18 liters
  4. ঘ
    25 liters
সঠিক উত্তর: খ
15 liters
উত্তর
সঠিক উত্তর: খ
15 liters
খ
ব্যাখ্যা
Question: A mixture of 60 kg contains sand and stone in the ratio 7 : 5. Find the quantity of sand to be added to the mixture so that the ratio of sand to stone becomes 2 : 1.

Solution:
Quantity of sand = 60 × (7/12) = 35 kg
Quantity of stone= 60 - 35 = 25 kg
Let, x kg of sand be added to the mixture.

According to the question,
(35 + x)/25 = 2/1
⇒ 35 + x = 50
⇒ x = 50 - 35
⇒ x = 15
১৬১.
Find the third proportional to 25 and 30
  1. ক
    ক) 36
  2. খ
    খ) 32
  3. গ
    গ) 34
  4. ঘ
    ঘ) 38
সঠিক উত্তর: ক
ক) 36
উত্তর
সঠিক উত্তর: ক
ক) 36
ক
ব্যাখ্যা

Let third proportional be x
⇒ 25 : 30 : : 30 : x
⇒ 25 × x = 30 x 30
⇒ x = (30 x 30)/25
= 36.

১৬২.
In a college, the ratio of foreign to local students is 3 : 7. If three-fourths of the local students are female and one-quarter of the foreign students is female. What fraction of the combined students is female?
  1. ক
    50%
  2. খ
    60%
  3. গ
    64%
  4. ঘ
    65%
  5. ঙ
    None
সঠিক উত্তর: খ
60%
উত্তর
সঠিক উত্তর: খ
60%
খ
ব্যাখ্যা
Question: In a college, the ratio of foreign to local students is 3 : 7. If three-fourths of the local students are female and one-quarter of the foreign students is female. What fraction of the combined students is female?

Solution:
Since the ratio of foreign : local is 3 : 7,
let's use a total of 10 parts
So if we have a total of 100 students, that means
Foreign students = 30 students
Local students = 70 students

Foreign female students = (1/4) × 30 = 15/2 students
Local female students = (3/4) × 70 = 105/2 students

Total female students = (15/2) + (105/2)
= (15 + 105)/2
= 60 students or 60% of the combined students are female.
১৬৩.
When 120 guests guests take seat in auditorium, only 3/4 of the seats occupied. What is the total number of seats in the auditorium?
  1. ক
    ক) 160
  2. খ
    খ) 180
  3. গ
    গ) 180
  4. ঘ
    ঘ) 200
সঠিক উত্তর: ক
ক) 160
উত্তর
সঠিক উত্তর: ক
ক) 160
ক
ব্যাখ্যা
Question: When 120 guests guests take seat in  auditorium, only 3/4 of the seats occupied. What is the total number of seats in the auditorium?

Solution: 
ধরি, মোট আসন সংখ্যা x টি 

x × 3/4 = 120 
⇒ x = 120 × 4/3
∴ x = 160 

মোট আসন সংখ্যা ১৬০ টি 
১৬৪.
35% of Nabila's income is equal to 25% of Nuru's income. The ratio of their income is
  1. ক
    ক) 5 : 7
  2. খ
    খ) 4 : 7
  3. গ
    গ) 7 : 3
  4. ঘ
    ঘ) 4 : 3
সঠিক উত্তর: ক
ক) 5 : 7
উত্তর
সঠিক উত্তর: ক
ক) 5 : 7
ক
ব্যাখ্যা

ধরি, Nabila's income = T
এবং Nuru's income = N
প্রশ্নমতে, T এর 35% = S এর 25%
⇒ T(35/100) = N (25/100)
⇒ T/N = 25/100 × 100/35 = 5/7
T : N = 5 : 7

১৬৫.
If x + y : y + z : z + x  = 6 : 7 : 8 and x + y + z = 14, find z 
  1. ক
    ক) 5
  2. খ
    খ) 6
  3. গ
    গ) 7
  4. ঘ
    ঘ) 8
সঠিক উত্তর: খ
খ) 6
উত্তর
সঠিক উত্তর: খ
খ) 6
খ
ব্যাখ্যা
Given that 
x + y : y + z : z + x  = 6 : 7 : 8
x + y + z = 14

Let
x + y = 6k
y + z = 7k
z + x = 8k

adding
2( x + y + z) = 21k
2 × 14 = 21k
k = (2 × 14)/21 
k = 4/3

x + y = 6k
x + y = 6(4/3)
x + y = 8 

Now
x + y + z = 14
8 + z = 14 
z = 14 - 8 
z = 6
১৬৬.
A circular floor with a radius of 7 cm is 60% carpeted. What is the area of the uncovered section?
  1. ক
    60.60  cm2
  2. খ
    61.60  cm2
  3. গ
    63.60  cm2
  4. ঘ
    62.60  cm2
সঠিক উত্তর: খ
61.60  cm2
উত্তর
সঠিক উত্তর: খ
61.60  cm2
খ
ব্যাখ্যা
Question: A circular floor with a radius of 7 cm is 60% carpeted. What is the area of the uncovered section?

Solution : 
Area of the floor = πr2 
= (22/7) × 72 cm2
= (22/7) × 49 cm2
= 154 cm2

If 60% of its area is carpeted, then uncovered area = (100 - 60)%
= 40% 

∴ the area of the uncovered section = 154 × 40%  cm2
= 154 × (40/100) cm2
= 6160/100 cm2
= 61.60  cm2
১৬৭.
200 kg of sugar solution has 30% sugar in it. How much sugar should be added to make it 50% in the solution?
  1. ক
    40 kg
  2. খ
    60 kg
  3. গ
    80 kg
  4. ঘ
    120 kg
সঠিক উত্তর: গ
80 kg
উত্তর
সঠিক উত্তর: গ
80 kg
গ
ব্যাখ্যা
Question: 200 kg of sugar solution has 30% sugar in it. How much sugar should be added to make it 50% in the solution?

Solution: 
30% suger = 30% of 200 = 60
let, X kg of sugar should be added,

ATQ,
60 + X/200 + X = 1/2
or, 200 + X = 120 + 2X
or, X = 80 kg
১৬৮.
A deceitful dairy seller claims to sell his milk at a cost price but he mixes water with it and thereby gains 20%. The percentage of water in the mixture is:
  1. ক
    16.67%
  2. খ
    10.67%
  3. গ
    19%
  4. ঘ
    None of the above
সঠিক উত্তর: ক
16.67%
উত্তর
সঠিক উত্তর: ক
16.67%
ক
ব্যাখ্যা
Question: A deceitful dairy seller claims to sell his milk at a cost price but he mixes water with it and thereby gains 20%. The percentage of water in the mixture is:

Solution:
Let,
The cost price = 100
The selling price = 120

∴ The amount of milk = 100/120
= 5/6

∴ The amount of water is = (1 - 5/6) × 100%
= 16.67%
১৬৯.
A and B invested in a business. The profit earned was divided in the ratio 2 : 3. If A invested Tk. 35500, the amount invested by B is -
  1. ক
    ক) Tk. 54250
  2. খ
    খ) Tk. 53250
  3. গ
    গ) Tk. 53500
  4. ঘ
    ঘ) Tk. 53350
সঠিক উত্তর: খ
খ) Tk. 53250
উত্তর
সঠিক উত্তর: খ
খ) Tk. 53250
খ
ব্যাখ্যা
Question: A and B invested in a business. The profit earned was divided in the ratio 2 : 3. If A invested Tk. 35500, the amount invested by B is -

Solution:
ধরি,
A বিনিয়োগ করেছিল 2x টাকা এবং B বিনিয়োগ করেছিল 3x টাকা

প্রশ্নমতে, 2x = 35500
∴ x = 17750

∴ B বিনিয়োগ করেছিল = 3x = 3 × 17750
= 53250 টাকা।
১৭০.
In a mixture 60 liters, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then estimate the quantity of water in liter to be further added in the mixture.
  1. ক
    30 liters
  2. খ
    40 liters
  3. গ
    80 liters
  4. ঘ
    60 liters
সঠিক উত্তর: ঘ
60 liters
উত্তর
সঠিক উত্তর: ঘ
60 liters
ঘ
ব্যাখ্যা
Question: In a mixture 60 liters, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then estimate the quantity of water in liter to be further added in the mixture.

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

Quantity of milk = 60 × (2/3) = 40 liters
Hence, quantity of water will be
= 60 - 40 = 20 liters

Assume,
x liters water needs to be added.
ATQ,
40/(x + 20) = 1/2
⇒ 80 = x + 20
∴ x = 60

∴ 60 liters water needs to be added.
১৭১.
If 2A = 3B and 4B = 6C, then A : C equal to -
  1. ক
    9 : 4
  2. খ
    3 : 2
  3. গ
    4 : 9
  4. ঘ
    2 : 3
সঠিক উত্তর: ক
9 : 4
উত্তর
সঠিক উত্তর: ক
9 : 4
ক
ব্যাখ্যা
Question: If 2A = 3B and 4B = 6C, then A : C equal to -

Solution:
A : B = 3 : 2
B : C = 3 : 2

A : C = A/B × B/C
= 3/2 × 3/2
= 9 : 4
১৭২.
In a business, the ratio of the capitals of Arun and Babul is 2 : 1, that of Babul and Chandan is 4 : 3, and that of Dipu and Chandan is 6 : 5. What is the ratio of the capitals of Arun and Dipu? 
  1. ক
    9 : 18
  2. খ
    22 : 9
  3. গ
    10 : 9
  4. ঘ
    20 : 9
সঠিক উত্তর: ঘ
20 : 9
উত্তর
সঠিক উত্তর: ঘ
20 : 9
ঘ
ব্যাখ্যা

Question: In a business, the ratio of the capitals of Arun and Babul is 2 : 1, that of Babul and Chandan is 4 : 3, and that of Dipu and Chandan is 6 : 5. What is the ratio of the capitals of Arun and Dipu?

Solution:
Given,
Arun : Babul = 2 : 1
⇒ Arun/Babul = 2/1

Babul : Chandan = 4 : 3
⇒ Babul/Chandan = 4/3

Dipu : Chandan = 6 : 5
⇒ Chandan/Dipu = 5/6

Now,
Arun/Dipu = (Arun/Babul) × (Babul/Chandan) × (Chandan/D)
= (2/1) × (4/3) × (5/6)
= 20/9

∴ Arun/Dipu = 20 : 9

১৭৩.
A mixture contains sugar solution and colored water in the ratio of 4 : 3. If 10 liters of colored water is added to the mixture, the ratio becomes 4: 5. Find the initial quantity of sugar solution in the given mixture.
  1. ক
    15 liters
  2. খ
    20 liters
  3. গ
    25 liters
  4. ঘ
    30 liters
  5. ঙ
    10 liters
সঠিক উত্তর: খ
20 liters
উত্তর
সঠিক উত্তর: খ
20 liters
খ
ব্যাখ্যা
Question: A mixture contains sugar solution and colored water in the ratio of 4 : 3. If 10 liters of colored water is added to the mixture, the ratio becomes 4: 5. Find the initial quantity of sugar solution in the given mixture.

Solution:
The initial ratio is 4 : 3.
Let ‘k’ be the common ratio.
Initial quantity of sugar solution = 4k liters
Initial quantity of colored water = 3k liters

Final quantity of sugar solution = 4k liters
Final quantity of colored water = 3k + 10 liters

Final ratio = 4k : (3k + 10) = 4 : 5
⇒ 20k = 12k + 40
⇒ 8k = 40
∴ k = 5

Therefore, the initial quantity of sugar solution in the given mixture = 4k = 4 × 5 = 20 liters
১৭৪.
80% of a number is equal to the 4/5th of the other number. What is the ratio between the first number and the second number respectively?
  1. ক
    ক) 3 : 4
  2. খ
    খ) 3 : 5
  3. গ
    গ) 5 : 3
  4. ঘ
    ঘ) None of these
সঠিক উত্তর: ঘ
ঘ) None of these
উত্তর
সঠিক উত্তর: ঘ
ঘ) None of these
ঘ
ব্যাখ্যা
Question: 80% of a number is equal to the 4/5th of the other number. What is the ratio between the first number and the second number respectively?

Solution:
Let the first number be x and the second number be y
According to the question,
80% of x = 4/5 of y
⇒ 80x/100 = 4y/5
⇒ 4x/5 = 4y/5
⇒ x = y
⇒ x : y =1 : 1
১৭৫.
A butler stole wine from a butt of Rony which contained 40% of spirit and he replaced what he had stolen with wine containing only 16% spirit. The butt was then 24% strength only. How much of the butt did he steal?
  1. ক
    1/6
  2. খ
    2/3
  3. গ
    3/5
  4. ঘ
    4/5
সঠিক উত্তর: খ
2/3
উত্তর
সঠিক উত্তর: খ
2/3
খ
ব্যাখ্যা
Question: A butler stole wine from a butt of Rony which contained 40% of spirit and he replaced what he had stolen with wine containing only 16% spirit. The butt was then 24% strength only. How much of the butt did he steal?

Solution:
ধরি,
ওয়াইনের মোট পরিমাণ ক লিটার 
চুরি করা ওয়াইনের পরিমাণ খ লিটার 

চুরি করা খ লিটারে স্পিরিটের পরিমাণ = খ এর ৪০% 
= ৪০খ/১০০
= ২খ/৫ 

চুরি করার পর স্পিরিট অবশিষ্ট থাকে = ক এর ৪০% - ২খ/৫
= ২ক/৫ - ২খ/৫
= ২/৫ (ক - খ) 

সে খ লিটার ওয়াইন পুনরায় যুক্ত করে যেখানে ১৬% স্পিরিট থাকে।

∴ পুনরায় যুক্ত করার পর স্পিরিট হয় = ২/৫(ক - খ) + খ এর ১৬%
= ২/৫(ক - খ) + ৪খ/২৫ 

প্রশ্নমতে,
২/৫(ক - খ) + ৪খ/২৫ = ক এর ২৪% 
বা, ২/৫(ক - খ) + ৪খ/২৫ = ৬ক/২৫
বা, ১০(ক - খ) + ৪খ = ৬ক 
বা, ১০ক - ১০খ + ৪খ = ৬ক 
বা, ৪ক - ৬খ = ০
বা, ৬খ = ৪ক 
বা, খ = ৪ক/৬ 
∴ খ = ২ক/৩ 

∴ সে ওয়াইনের ২/৩ অংশ চুরি করেছিল। 
১৭৬.
In what ratio should a mixture of 20% alcohol strength be mixed with a mixture of 60% alcohol strength to get a mixture of 50% alcohol strength?
  1. ক
    1 : 2
  2. খ
    3 : 2
  3. গ
    1 : 3
  4. ঘ
    2 : 3
সঠিক উত্তর: গ
1 : 3
উত্তর
সঠিক উত্তর: গ
1 : 3
গ
ব্যাখ্যা
Question: In what ratio should a mixture of 20% alcohol strength be mixed with a mixture of 60% alcohol strength to get a mixture of 50% alcohol strength?

Solution:
Let the ratio be 1 : x
Then according to the question
⇒ 20 × 1 + 60x = (1 + x) 50
⇒ 20 + 60x = 50 + 50x
⇒ 60x - 50x = 50 - 20
⇒ 10x = 30
⇒ x = 3

∴ the ratio is 1 : 3
১৭৭.
The Mathematics textbook for Class VII has 320 pages. The chapter ‘Symmetry’ runs from page 261 to page 272. The ratio of the number of pages of this chapter to the total number of pages of the book is
  1. ক
    3 : 8
  2. খ
    31 : 80
  3. গ
    3 : 2
  4. ঘ
    3 : 80
সঠিক উত্তর: ঘ
3 : 80
উত্তর
সঠিক উত্তর: ঘ
3 : 80
ঘ
ব্যাখ্যা
Question: The Mathematics textbook for Class VII has 320 pages. The chapter ‘Symmetry’ runs from page 261 to page 272. The ratio of the number of pages of this chapter to the total number of pages of the book is

Solution: 
The chapter ‘Symmetry’ runs for pages = 272 - 261 + 1
= 12

The ratio of the number of pages of this chapter to the total number of pages of the book is = 12 : 320
= 3 : 80
১৭৮.
The cost of a diamond varies as the square of its weight. A diamond weighing 20 decigrams costs 4800 tk. Find the cost of a diamond of the same kind weighing 8 decigrams.
  1. ক
    642 tk
  2. খ
    768 tk
  3. গ
    812 tk
  4. ঘ
    836 tk
সঠিক উত্তর: খ
768 tk
উত্তর
সঠিক উত্তর: খ
768 tk
খ
ব্যাখ্যা
Question: The cost of a diamond varies as the square of its weight. A diamond weighing 20 decigrams costs 4800 tk. Find the cost of a diamond of the same kind weighing 8 decigrams.

Solution:
Let's denote the weight of the diamond by w and the cost by C.
According to the problem,
C ∝ w2
⇒ c = k × w2  [where k is a constant]
⇒ 4800 = k × 202
⇒ k = 4800/400
∴ k = 12

Now, for w = 8 decigrams
C = 12 × 82
∴ C = 768 tk
১৭৯.
In a mixture with a 4:3 ratio of sugar solution to coloured water, adding 10 litres of coloured water makes the ratio 4:5. What is the original quantity of sugar solution in the mixture?
  1. ক
    10 liters
  2. খ
    20 liters
  3. গ
    25 liters
  4. ঘ
    30 liters
  5. ঙ
    None of the above
সঠিক উত্তর: খ
20 liters
উত্তর
সঠিক উত্তর: খ
20 liters
খ
ব্যাখ্যা
Question: In a mixture with a 4:3 ratio of sugar solution to coloured water, adding 10 litres of coloured water makes the ratio 4:5. What is the original quantity of sugar solution in the mixture?

Solution:
The initial ratio is 4 : 3.
Let ‘a’ be the common ratio.
The initial quantity of sugar solution = 4a liters
The initial quantity of coloured water = 3a liters

Final quantity of sugar solution = 4a liters
Final quantity of colored water = 3a + 10 liters

Final ratio = 4a : (3a + 10) = 4 : 5
⇒ 20a = 12a + 40
⇒ 8a = 40
∴ a = 5

Therefore, the initial quantity of sugar solution in the given mixture = 4a
= 4 × 5
= 20 litres
১৮০.
In what ratio must a grocer mix two varieties of pulses costing Tk. 15 and Tk. 20 per kg respectively so as to get a mixture worth Tk. 16.50 per kg?
  1. ক
    3 : 5
  2. খ
    5 : 7
  3. গ
    2 : 5
  4. ঘ
    7 : 3
সঠিক উত্তর: ঘ
7 : 3
উত্তর
সঠিক উত্তর: ঘ
7 : 3
ঘ
ব্যাখ্যা

Question: In what ratio must a grocer mix two varieties of pulses costing Tk. 15 and Tk. 20 per kg respectively so as to get a mixture worth Tk. 16.50 per kg?

Solution:
Let the ratio be x : y

Cost of first variety = Tk. 15 per kg
Cost of second variety = Tk. 20 per kg 
Cost of mixture = Tk. 16.50 per kg

For x kg of first variety and y kg of second variety:
Total cost of mixture = (x × 15) + (y × 20) = 15x + 20y
Total quantity of mixture = (x + y) kg

According to the question,
(15x + 20y)/(x + y) = 16.50
⇒ 15x + 20y = 16.50(x + y) 
⇒ 15x + 20y = 16.50x + 16.50y 
⇒ 20y - 16.50y = 16.50x - 15x 
⇒ 3.50y = 1.50x 
⇒ y/x = 1.50/3.50 
⇒ y/x = 150/350 
⇒ y/x = 3/7

Therefore, x : y = 7 : 3

বিকল্প সমাধান:
By the rule of alligation,

 ∴ Required rate = 3.50 : 1.50 = 7 : 3

১৮১.
A solution contains 25% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-
  1. ক
    33.33%
  2. খ
    32%
  3. গ
    40%
  4. ঘ
    23.27%
সঠিক উত্তর: গ
40%
উত্তর
সঠিক উত্তর: গ
40%
গ
ব্যাখ্যা
Question: A solution contains 25% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

Solution: 
let, solution is 100 unit
amount of sugar = 100 × 25%
= 100 × (25/100)
= 25 unit 

by doubling, amount of sweet = 50 unit

∴ solution now = 100 + 25 = 125 unit 

∴ percent of sugar = {(50 × 100)/125}% = 40%
১৮২.
In a box, the ratio of red marbles to blue marbles is 7 : 4. Which of the following could be the total number of marbles in the box?
  1. ক
    18
  2. খ
    19
  3. গ
    20
  4. ঘ
    22
সঠিক উত্তর: ঘ
22
উত্তর
সঠিক উত্তর: ঘ
22
ঘ
ব্যাখ্যা

Question: In a box, the ratio of red marbles to blue marbles is 7 : 4. Which of the following could be the total number of marbles in the box?

Solution: 
18, 19, 21 কোনটিই 7 + 4 বা 11 দ্বারা নি:শেষে বিভাজ্য নয়। 
অতএব, সঠিক উত্তর 22

১৮৩.
A mixture contains acid 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 acid in the given mixture.
  1. ক
    5 liters
  2. খ
    7.5 liters
  3. গ
    10 liters
  4. ঘ
    12 liters
সঠিক উত্তর: গ
10 liters
উত্তর
সঠিক উত্তর: গ
10 liters
গ
ব্যাখ্যা
Question: A mixture contains acid 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 acid in the given mixture.

Solution:
Let the quantity of acid and water be 4x liters and 3x liters respectively

ATQ,
4x/(3x + 5) = 4/5
⇒ 20x = 4(3x + 5)
⇒ 20x = 12x + 20
⇒ 8x = 20
∴ x = 2.5

Quantity of acid
= (4 × 2.5) liters
= 10 liters
১৮৪.
In a cricket tournament 30,000 tickets were sold. One-fourth of the tickets were sold at Tk. 30 each, 1/3 were sold at Tk. 25 each, and the remaining tickets were sold at Tk. 20 each. How many tickets were sold at Tk. 20 each?
  1. ক
    ক) 10000
  2. খ
    খ) 12000
  3. গ
    গ) 12500
  4. ঘ
    ঘ) 13500
সঠিক উত্তর: গ
গ) 12500
উত্তর
সঠিক উত্তর: গ
গ) 12500
গ
ব্যাখ্যা
Question: In a cricket tournament 30,000 tickets were sold. One-fourth of the tickets were sold at Tk. 30 each, 1/3 were sold at Tk. 25 each, and the remaining tickets were sold at Tk. 20 each. How many tickets were sold at Tk. 20 each?

Solution: 
মোট টিকিট বিক্রি করা হয়েছে ৩০০০০ টি 
১/৪ অংশ বিক্রি করা হয়েছে প্রতিটি ৩০ টাকায় 
১/৩ অংশ বিক্রি করা হয়েছে প্রতিটি ২৫ টাকায় 

বাকি থাকে = ১ - (১/৪) - (১/৩) অংশ 
= (১২ - ৩ - ৪)/১২ অংশ 
= ৫/১২ অংশ 

প্রতিটি ২০ টাকায় বিক্রি করা হয়েছে = ৩০০০০ এর ৫/১২ অংশ 
= ১২৫০০  টি 
১৮৫.
If A varies directly as B and inversely as C and A = 6, when B = 2 and C= 3, what is the value of A when B = 8 and C = 6?
  1. ক
    ক) 6
  2. খ
    খ) 18
  3. গ
    গ) 12
  4. ঘ
    ঘ) 24
সঠিক উত্তর: গ
গ) 12
উত্তর
সঠিক উত্তর: গ
গ) 12
গ
ব্যাখ্যা

Let the constant be x,
so putting the first scenario in equation
A = x × (B/C)
⇒ 6 = x × 2/3
⇒ 2x = 18
⇒ x = 9
We can find out A in the second scenario by putting the value of x as 9
A = 9 × 8/6
⇒ A = 12.

১৮৬.
If A : B = (1/2) : (1/3) and B : C = (1/2) : (1/3), then A : B : C =? 
  1. ক
    3 : 6 : 4
  2. খ
    9 : 6 : 5
  3. গ
    9 : 6 : 4
  4. ঘ
    None of these
সঠিক উত্তর: গ
9 : 6 : 4
উত্তর
সঠিক উত্তর: গ
9 : 6 : 4
গ
ব্যাখ্যা
Question: If A : B = (1/2) : (1/3) and B : C = (1/2) : (1/3), then A : B : C =? 

Solution: 
A : B = (1/2) : (1/3)
⇒ A : B = (6/2) : (6/3)
= 3 : 2
= (3 × 3) : (3 × 2)
= 9 : 6

B : C = (1/2) : (1/3)
⇒ B : C = (6/2) : (6/3)
= 3 : 2
= (3 × 2) : (2 × 2)
= 6 : 4

then A : B : C = 9 : 6 : 4
১৮৭.
A seller bought 120 pens for 600 taka. How many pens does he need to sell for 600 taka to make a profit of 20%?
  1. ক
    80 pens
  2. খ
    110 pens
  3. গ
    120 pens
  4. ঘ
    100 pens
সঠিক উত্তর: ঘ
100 pens
উত্তর
সঠিক উত্তর: ঘ
100 pens
ঘ
ব্যাখ্যা

Question: A seller bought 120 pens for 600 taka. How many pens does he need to sell for 600 taka to make a profit of 20%?

Solution:
Cost price of 120 pens = Tk. 600 Cost price per pen = 600 ÷ 120
= Tk. 5

∴ Selling price per pen for 20% profit = 5 × (120/100)
= 5 × 1.2
= Tk. 6

∴ Number of pens to be sold for Tk. 600
= 600 ÷ 6
= 100 pens

১৮৮.
The ratio of X : Y is 3 : 5, and the ratio of Y : Z is 6 : 7. If X = 18, what is the value of Z? 
  1. ক
    28
  2. খ
    30
  3. গ
    35
  4. ঘ
    25
সঠিক উত্তর: গ
35
উত্তর
সঠিক উত্তর: গ
35
গ
ব্যাখ্যা

Question: The ratio of X : Y is 3 : 5, and the ratio of Y : Z is 6 : 7. If X = 18, what is the value of Z?

Solution:
Given that,
X : Y = 3 : 5
Y : Z = 6 : 7
And X = 18

Now,
X : Y = 3 : 5
⇒ X/Y = 3/5
⇒ 5X = 3Y
⇒ Y = (5 × 18)/3 ; [X = 18]
∴ Y = 30

And,
Y : Z = 6 : 7
Y/Z = 6/7
⇒ 30/Z = 6/7
⇒ Z = (30 × 7)/6
∴ Z = 35

So the value of Z is 35.

১৮৯.
A and B are two partners in a firm sharing the profit in the ratio 4 : 5. If the firm earns a profit of Tk. 72000, then profit to be received by B ?
  1. ক
    ক) Tk. 40,000
  2. খ
    খ) Tk. 32,000
  3. গ
    গ) Tk. 48,000
  4. ঘ
    ঘ) Tk. 56,000
সঠিক উত্তর: ক
ক) Tk. 40,000
উত্তর
সঠিক উত্তর: ক
ক) Tk. 40,000
ক
ব্যাখ্যা
Sum of 4 and 5 is 9
Share of B = 72000 × 5/9 = Tk. 40,000
১৯০.
To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 100 kg of pure milk is -
  1. ক
    2.5 kg
  2. খ
    5 kg
  3. গ
    7.5 kg
  4. ঘ
    10 kg
সঠিক উত্তর: ঘ
10 kg
উত্তর
সঠিক উত্তর: ঘ
10 kg
ঘ
ব্যাখ্যা
Question: To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 100 kg of pure milk is -

Solution:
Let the quantity of water mixed be x kg.
Let,
CP of 1 kg of pure milk = Tk. 1
CP of 100 kg of pure milk = Tk. 100

Hence,
% gain = (x/100) × 100
⇒ 10 = x
১৯১.
What is the third proportional to 9 and 75?
  1. ক
    ক) 675
  2. খ
    খ) 625
  3. গ
    গ) 695
  4. ঘ
    ঘ) 745
সঠিক উত্তর: খ
খ) 625
উত্তর
সঠিক উত্তর: খ
খ) 625
খ
ব্যাখ্যা
If a , b, c are in proportion then
a/b = b/c

Let the third proportional be x
  9/75 = 75/x
⇒ x = 75 × 75/9
⇒ x = 625

∴ The third proportional is 625.


১৯২.
The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 5000. The sum of the numbers is:
  1. ক
    100
  2. খ
    60
  3. গ
    120
  4. ঘ
    50
সঠিক উত্তর: গ
120
উত্তর
সঠিক উত্তর: গ
120
গ
ব্যাখ্যা
Question: The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 5000. The sum of the numbers is-

Solution:
Let, the number be 3x, 4x, 5x

According to the question,
(3x)2 + (4x)2 + (5x)2 = 5000
⇒ 9x2 + 16x2 + 25x2 = 5000
⇒ 50x2 = 5000
⇒ x2 = 5000/50
⇒ x2 = 100
∴ x = 10

∴ The sum of the numbers = 3x + 4x + 5x
= 30 + 40 + 50
= 120
১৯৩.
A businessman has 1000 Kg of rice, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. How much Kg of rice did he sell at 18% profit?
  1. ক
    600
  2. খ
    620
  3. গ
    640
  4. ঘ
    None
সঠিক উত্তর: ক
600
উত্তর
সঠিক উত্তর: ক
600
ক
ব্যাখ্যা
Question: A businessman has 1000 Kg of rice, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. How much Kg of rice did he sell at 18% profit?

Solution:
Let the quantity sold at 8% profit = x
And the quantity sold at 18% profit = y.

8x +18y =14(x + y)
⇒ 4y = 6x
∴ x : y = 2 : 3

Quantity sold at 18% profit = (3/5) × 1000
= 600
১৯৪.
A bucket contains 64 liters of petrol. 16 liters of petrol is removed and replaced with kerosene. 16 liters of this mixture is removed and replaced with kerosene. How much kerosene (in liters) is present now?
  1. ক
    28 liters
  2. খ
    36 liters
  3. গ
    16 liters
  4. ঘ
    64 liters
সঠিক উত্তর: ক
28 liters
উত্তর
সঠিক উত্তর: ক
28 liters
ক
ব্যাখ্যা
Question: A bucket contains 64 liters of petrol. 16 liters of petrol is removed and replaced with kerosene. 16 liters of this mixture is removed and replaced with kerosene. How much kerosene (in liters) is present now?

Solution:
বালতিতে পেট্রোল ছিল ৬৪ লিটার 
১৬ লিটার পেট্রোল সরিয়ে কেরোসিন দিলে মিশ্রণে পেট্রোল থাকে (৬৪ - ১৬) = ৪৮ লিটার 
মিশ্রণে কেরোসিন হয় ১৬ লিটার 

∴ ১৬ লিটার পেট্রোল সরিয়ে কেরোসিন দিলে মিশ্রণে পেট্রোল ও কেরোসিনের অনুপাত হয় = ৪৮ : ১৬ 
= ৩ : ১

এখন,
১৬ লিটার মিশ্রণে কেরোসিন থাকে = ১৬ × (১/৪) লিটার 
= ৪ লিটার 

সুতরাং ১৬ লিটার মিশ্রণ তোলে নেয়ার পর কেরোসিন থাকে = ১৬ - ৪ লিটার = ১২ লিটার 

আবার মিশ্রণে ১৬ লিটার কেরোসিন দিলে মোট কেরোসিন হয় = (১২ + ১৬) লিটার 
= ২৮ লিটার 
১৯৫.
The ratio of the total amount distributed in all the males and females as salary is 6 : 5. The ratio of the salary of each male and female is 2 : 3. Find the ratio of the number of males and females.
  1. ক
    5 : 9
  2. খ
    5 : 7
  3. গ
    7 : 5
  4. ঘ
    9 : 5
সঠিক উত্তর: ঘ
9 : 5
উত্তর
সঠিক উত্তর: ঘ
9 : 5
ঘ
ব্যাখ্যা
Question: The ratio of the total amount distributed in all the males and females as salary is 6 : 5. The ratio of the salary of each male and female is 2 : 3. Find the ratio of the number of males and females.

Solution:
Let,
The number of males be x
The number of females be y

Each male's salary 2a
Each female's salary 3a

Total salary of males 2a × x = 2ax
Total salary of females 3a × y = 3ay

ATQ,
2ax/3ay = 6/5
⇒ 2x/3y = 6/5
⇒ x/y = 18/10
⇒ x/y = 9/5
∴ x : y = 9 : 5
১৯৬.
A relief truck carries enough food to feed 180 women or 270 children. If 90 children have already been fed, how many women can be fed with the remaining food?
  1. ক
    90 women
  2. খ
    100 women
  3. গ
    120 women
  4. ঘ
    130 women
সঠিক উত্তর: গ
120 women
উত্তর
সঠিক উত্তর: গ
120 women
গ
ব্যাখ্যা
Question: A relief truck carries enough food to feed 180 women or 270 children. If 90 children have already been fed, how many women can be fed with the remaining food?

Solution:
Total food = for 270 children
Already taken = 90 children

Remaining = (270 − 90) = 180 children

ATQ,
270 children = 180 women
1 children = 180/270 women
180 children = (180 × 180)/270 women
= 120 women
১৯৭.
A and B are two alloys in which ratios of gold and copper are 5 : 3 and 5 : 11 respectively. If these equally amount of two alloys are melted and made alloy C. What will be the ratio of gold and copper in alloy C?
  1. ক
    12 : 19
  2. খ
    15 : 17
  3. গ
    20 : 27
  4. ঘ
    25 : 34
সঠিক উত্তর: খ
15 : 17
উত্তর
সঠিক উত্তর: খ
15 : 17
খ
ব্যাখ্যা
Question: A and B are two alloys in which ratios of gold and copper are 5 : 3 and 5 : 11 respectively. If these equally amount of two alloys are melted and made alloy C. What will be the ratio of gold and copper in alloy C?

Solution:
Ratio of Gold and Copper in Alloy A = 5 : 3
Ratio of Gold and Copper in Alloy B = 5 : 11

Amount of Gold in Alloy A = 5/8
Amount of Gold In Alloy B = 5/16

Amount of Copper in A = 3/8
Amount of Copper in B = 11/16

Amount of Gold In C,
= (Amount of gold in A + Amount of gold in B) = (5/8) + (5/16)
= (10 + 5)/16
= 15/16
Amount of Copper in C,
= Amount of Copper in A + Amount of Copper in B = (3/8) + (11/16)
= 17/16

So, Ratio of Gold and Copper in C,
= (15/16) : (17/16) = 15 : 17
১৯৮.
Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg.
  1. ক
    1 : 3
  2. খ
    2 : 3
  3. গ
    3 : 4
  4. ঘ
    4 : 5
সঠিক উত্তর: খ
2 : 3
উত্তর
সঠিক উত্তর: খ
2 : 3
খ
ব্যাখ্যা
Question: Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg.

Solution:
Quantity of rice at Tk. 7.20 per kg = x
Quantity of rice at Tk. 5.70 per kg = y
By the rule of alligation:

⇒ x : y = (7.20 - 6.30) : (6.30 - 5.70) = 0.9 : 0.6 = 9 : 6 = 3 : 2


১৯৯.
The ratio of A : B is 2 : 3, and the ratio of B : C is 4 : 5. If A 16, what is the value of C?
  1. ক
    15
  2. খ
    20
  3. গ
    24
  4. ঘ
    30
সঠিক উত্তর: ঘ
30
উত্তর
সঠিক উত্তর: ঘ
30
ঘ
ব্যাখ্যা

Question: The ratio of A : B is 2 : 3, and the ratio of B : C is 4 : 5. If A 16, what is the value of C?

Solution:
Given that,
A : B = 2 : 3
B : C = 4 : 5
And A = 16

Now, 
A : B = 2 : 3 
⇒ A/B = 2/3
⇒ 2B = 3A
⇒ B = (16 × 3)/2  ; [A = 16]
∴ B = 24

And,
B : C = 4 : 5
B/C = 4/5
⇒ 24/C = 4/5
⇒ C = (24 × 5) / 4
∴ C = 30

So the value of C is 30.

২০০.
A salesman usually makes 45% profit on every TV he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%, What is the ratio of his new profit to his usual profit?
  1. ক
    9 : 8
  2. খ
    9 : 10
  3. গ
    44 : 45
  4. ঘ
    10 : 11
  5. ঙ
    .
সঠিক উত্তর: গ
44 : 45
উত্তর
সঠিক উত্তর: গ
44 : 45
গ
ব্যাখ্যা
Question: A salesman usually makes 45% profit on every TV he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%, What is the ratio of his new profit to his usual profit?

Solution:
প্রথমে টিভি বিক্রয় করেন = x টি

45% লাভে মোট লাভ = 45x/100
10% বিক্রি বৃদ্ধিতে নতুন বিক্রির পরিমাণ = x + x এর 10%
= x + 10x/100
= 1.1x

নতুন লাভ = 1.1x × 40/100
= 44x/100

নতুন লাভ : পুরাতন লাভ = 44x/100 : 45x/100
= 44 : 45