IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

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.]

Question: If 2 : 5 :: 6 : x, then x is equal to:

Solution:
2 : 5 :: 6 : x
⇒ 2/5 = 6/x
⇒ 30 = 2x
∴ x = 15

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

Solution: 
Let’s assume,
The price of a larger quantity of lentils = x Tk.

Given,
Small quantity : Large quantity = 2 : 3
The sum of the ratios = 2 + 3 = 5

The purchase price of 5 kg = 14 × 2 + x × 3
= 28 + 3x

The selling price of 5 kg = 22 × 5 = 110

Thus, Profit = 110 - (28 + 3x)
= 110 - 28 - 3x
= 82 - 3x

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

Question: The first number is 20% greater and the second number is 50% greater than a third number. What is the ratio of the two numbers?

Solution:
Let the third number be x

Then, first number = 120% of x
= 120x/100
= 6x/5

Second number = 150% of x
= 150x/100
= 3x/2

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

Question: A 50 g gold-copper alloy contains 80% gold. How much additional gold is needed to raise the gold percentage to 90%?

Solution:
Gold in alloy =50 × 80% = 40gm
Copper in alloy =50 × 20% =10gm

Now,
(40 + x)/10 = 90/10
⇒ 40 + x = 90
⇒ x = 90 - 40
∴ x = 50gm

Question: If an article is on a 20% discount and costs Tk. 596, what was its original price?

Solution:
If the selling price of the article is A, then
A - 20% of A = 596
⇒ A - A/5 = 596
⇒ 4A/5 = 596
⇒ A = (596 × 5)/4
∴ A = 745

Question: If Tk. 782 is allocated into three portions according to the ratio 1/2 : 2/3 : 3/4, what is the amount of the first portion?

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

∴ The first portion = 782 × (6/23)
= 204 Tk.

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

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

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

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

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

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

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

Question: What is the ratio of 4 inches to 7 feet?

Solution: 
We know,
1 feet = 12 inches
So, 7 feet = 7 × 12
= 84 inches

Now, 
4 inches : 7 feet = 4 : 84 = 1 : 21

Question: After a 20% discount, the price of a television is Tk. 12000. What was the price before the discount?

Solution:
In 20% discount,
Discount price 80 taka when original price 100 taka

∴ Discount price 1 taka when original price 100/80 taka
∴ Discount price 12000 taka when original price
(12000 × 100)/80 taka
= 15000 taka

Question: How much sugar, costing Tk. 95 per kg, should be mixed with 17 kg of tea priced at Tk. 200 per kg to get a blend worth Tk. 130 per kg?

Solution:
Ratio in which tea and sugar should be mixed
= 200 - 130 : 130 - 95
= 70 : 35
= 10 : 5
= 2 : 1

Let x be the quantity at 95/kg.

∴ 2 : 1 = x : 17
⇒ 2/1 = x/17
⇒ x = 34

Hence x = 34 kg.

Question: Given that an office opens at 9 a.m. and closes at 5:30 p.m., with a 17-minute lunch break, what is the proportion of the lunch break to the total workday?

Solution:
The ratio of lunch breaks to the total period in the office
= 17/{(8 × 60) + 30}
= 17/510
= 1/30
= 1 : 30

Question: A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C?

Solution:
Let the initial investments of A and B be 3a and 5a.

A : B : C = (3a × 12) : (5a × 12) : (5a × 6)
= 36 : 60 : 30
= 6 : 10 : 5

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: 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 1^st discount = 200 - 40 = Tk. 160

Discount 2 = 160 × (20/100) = Tk. 32

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

Question: In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

Solution:
Quantity of Tk. 60 tea is x kg.
Quantity of Tk. 65 tea is y kg

S.P. of 1 kg of the mixture = Tk. 68.20,
Gain = 10%.
C.P of 1 kg of the mixture = Tk. (100/110 × 68.20) = Tk. 62

ATQ,
60x + 65y = (x + y)62
⇒ 60x + 65y = 62x + 62y
⇒ 62x - 60x = 65y - 62y
⇒ 2x = 3y

∴ x/y = 3/2

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: 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