IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

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

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

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

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.

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.

Question: If 3 : 7 :: 12 : x, then x is equal to:

Solution:
3 : 7 :: 12 : x
⇒ 3/7 = 12/x
⇒ 3x = 84
⇒ x = 28

∴ x = 28

Question: The first number is 25% greater than a third number, and the second number is 40% greater than the same third number. What is the ratio of the first number to the second number?

Solution:
Let the third number be x

Then,
First number = 125% of x
= 125x/100
= 5x/4

Second number = 140% of x
= 140x/100
= 7x/5

∴ Ratio of first two numbers
= 5x/4 : 7x/5
= 25x : 28x
= 25 : 28

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

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

Let the additional silver be x g.

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

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

Question: The marked price of a Footwear is Tk. 200, and it is sold after applying two successive 20% discounts. What is the final price at which it is sold?

Solution:
Discount 1 = 200 × (20/100) = Tk. 40

Selling price after 1st discount = 200 - 40 = Tk. 160

Discount 2 = 160 × (20/100) = Tk. 32

∴ Selling price after 2nd discount = 160 - 32 = Tk. 128

Question: If Tk. 945 is allocated into three portions according to the ratio (2/3) : (3/4) : (5/6), what is the amount of the first portion?

Solution:
The given ratio = 2/3 : 3/4 : 5/6

Take LCM of denominators 3, 4, 6 = 12

∴ The ratio = 8 : 9 : 10 (Multipy thr ratio with 12)

Sum of parts = 8 + 9 + 10 = 27

The first portion = 945 × (8/27) = 280 Tk

∴ First portion = 280 Tk.

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.

Question: What is the ratio of 5 inches to 9 feet?

Solution:
We know, 1 foot = 12 inches
So, 9 feet = 9 × 12 = 108 inches

Now,
5 inches : 9 feet = 5 : 108

∴ The ratio = 5 : 108

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

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

∴ Discount price 1 = 100/75 of original price

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

∴ Original price = Tk. 24,000

Question: Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?

Solution:
Let their investments be Tk. x for 14 months, Tk. y for 8 months and Tk. z for 7 months respectively.
Then, 14x : 8y : 7z = 5 : 7 : 8.
Now,
14x/8y = 5/7
⇒ 98x = 40y
∴ y = (49/20) x

And,
14x/7z = 5/8
⇒ 112x = 35z
∴ z = (112/35) x = (16/5) x.

x : y : z = x : (49/20) x : (16/5) x = 20 : 49 : 64.

Question: How much coffee, costing Tk. 100 per kg, should be mixed with 20 kg of cocoa priced at Tk. 300 per kg to get a blend worth Tk. 200 per kg?

Solution:
Ratio in which cocoa and coffee should be mixed
= 300 - 200 : 200 - 100
= 100 : 100
= 1 : 1

Let x be the quantity of coffee at 100/kg.

∴ 1 : 1 = x : 20
⇒ x = 20

Question: The marked price of a t-shirt was Tk. 800. A man bought the same for Tk. 420 after getting two successive discounts. the first being 25%. then the second discount rate is-

Solution:
Marked price = 800
Actual price = 420
First discount = 25%
Let the second discount be x%

Then, we can write

after 25% discount,
discounted amount = 800 × (25/100)
= 200

New price = 800 - 200
= 600 Tk

Again, second discount,
discounted amount = 600 × (x/100)
= 6x

New price = 600 - 6x

ATQ,
600 - 6x = 420
⇒ 600 - 420 = 6x
⇒ 6x = 180
⇒ x = 30

∴ Second discount = 30%

Question: Given that a shop opens at 10 a.m. and closes at 6:45 p.m., with a 20-minute tea break, what is the proportion of the break to the total working hours?

Solution:
Total working time = 6:45 - 10:00
= 8 hours 45 minutes
= (8 × 60) + 45
= 525 minutes

The ratio of the break to the total working period
= 20/525
= 4/105

Question: Find the difference of amount if 40% discount is given on Tk. 1000 and two consecutive discounts 30% and 10% are given on the same amount.

Solution:
40% discount on 1000 = 1000 × 40% = 400
 
Two consecutive discounts on 1000.
30% discount on 1000 = 30% of 1000
= 300

After 30% discount on 1000 = 1000 - 300
= 700

Again,
After 10% discount on 700 = 10% of 700
= 70

Total discount = 300 + 70
= Tk. 370
So, the difference = 400 - 370
= Tk. 30

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

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

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

∴ Required proportion - 12 : 21 : 4

Question: Copper is 9 times as heavy as water, and tin is 3 times as heavy as water. In what ratio should these be mixed to get an alloy 6 times as heavy as water?

Solution:
Let copper be 9x times as heavy and tin 3y times as heavy as water.

9x + 3y = 6(x + y)
⇒ 9x + 3y = 6x + 6y
⇒ 9x − 6x = 6y − 3y
⇒ 3x = 3y
⇒ x/y = 1/1

∴ Copper and tin should be mixed in the ratio 1 : 1.

Question: Two containers contain milk and water in the ratios 5 : 2 and 9 : 5. What ratio should the mixtures be combined in to achieve a final ratio of 2 : 1 milk to water?

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

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

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

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

∴ P : Q = 1 : 2

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

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

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

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

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

⇒ 20b = 10b + 70
⇒ 10b = 70
⇒ b = 7

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