Bank Math Master

Question: If the price of an article is increased by 25% and then decreased by 20%, the net change in the price is -

Solution: 
Let,
The price of an article is 100 Tk.

If the price increased by 25%,
So, the new price will be after increase = 100 + {100 × (25/100)} Tk.
= 100 + 25 Tk.
= 125 Tk.

Then the new price decreased by 20%,
So, the new price will be after decrease = 125 - {125 × (20/100)} Tk.
= 125 - 25 Tk.
= 100 Tk.

∴ The net effect on the price of the article is = (100 - 100) 
= 0 Taka

So, after both the increase and the decrease, the price remains the same.

Question: A merchant mixes two varieties of tea costing Taka 250/kg and Taka 350/kg in the ratio 3:2. If he sells the mixture at Taka 360/kg, his profit percentage is - 

Solution: 
Cost of first variety = 250 Taka/kg
Cost of second variety = 350 Taka/kg
Ratio = 3:2

Cost price of mixture = {(3×250) +(2×350)}/(3+2)
= 1450/5
= 290 Taka/kg

Profit = Selling price - Cost price
= 360 - 290 
= 70 Taka/kg

Profit percentage = (70/290) × 100%
= 24.13%
= 25%

Question: The compound interest on a sum of money for 2 years at 10% per annum is Taka 525. The simple interest on the same sum for the same period and rate is - 

Solution: 
Compound Interest, 

For Simple Interest, 
I = Pnr
= 2500 × 2 × (10/100)
= 500 Taka

Question: The ratio of milk to water in a mixture is 5:3. When adding 4 liters of water, the ratio becomes 5:5. What was the quantity of milk in the original mixture? 

Solution:
Let the initial quantity of 
Milk = 5x liters
Water = 3x liters

When 4 liters of water are added, the new quantity of water becomes = (3x+4) liters
The new ratio becomes 5:5, which simplifies to 1:1. This means the amount of milk and water are now equal. 
5x=3x+4
2x = 4 
∴ x = 2

So, the initial quantity of Milk = 5 × 2 = 10 liters

Question: A container has 64 liters of milk. From this container, 16 liters of milk was taken out and replaced with water. This process was repeated twice. What is the final ratio of milk and water in the container? 

Solution:
Initial quantity of milk = 64 liters
Removed quantity = 16 liters
Total quantity = 64 liters
Number of times the process is repeated = 2

Remaining milk = Initial quantity × {1 - (Removed Quantity/Total Quantity)}ⁿ
= 64 × {1 - (16/64)}²
= 64 × (3/4)²
= 36 liters

So, Final quantity of milk = 36 liters
Final quantity of water = (64 - 36) = 28 liters

Therefore, Ratio of milk to water = 36/28 = 9 : 7

Question: A person can swim at a speed of 4 km/h in still water. If the speed of the current is 2 km/h, then the time taken to swim 12 km downstream and return to the starting point is - 

Solution: 
Speed of the swimmer in still water = 4 km/h
Speed of the current = 2 km/h
Distance one way = 12 km

Downstream Speed = 4 + 2 = 6 km/h
Upstream Speed = 4 - 2 = 2 km/h 

Time taken downstream = 12/6 = 2 hours
Time taken upstream = 12/2 = 6 hours

Total time = 6 + 2 = 8 hours

Question: A and B are partners in a business. A invests Taka 25,000 for 8 months and B invests Taka 30,000 for 6 months. If the profit at the end of the year is Taka 15,200, what is B's share in the profit?

Solution: 
A’s Contribution = 25000×8 = 200000 Taka
B’s Contribution = 30000×6 = 180000 Taka

The ratio of A's and B's investment = 200000 : 180000
= 10 : 9

So, B's share in profit = {9/(10+9)} × 15200
= 7200 Taka

Question: A certain sum of money becomes 2.5 times itself in 10 years at simple interest. What is the rate of interest per annum? 

Solution: 
Let the principal amount be P.
The sum becomes 2.5 times itself in 10 years.

So, A = 2.5P
∴ Simple Interest, I = 2.5P - P = 1.5P

We know, I = Pnr/100
1.5P = P × 10 × r/100
10r = 150
∴ r = 15%

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

Solution: 
Let, The fourth proportional is x.

So, 5/8 = 15/x
⇒ 5x = 120
∴ x = 24

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

Question: A train 240 meters long passes a pole in 15 seconds. How long will it take to pass through a platform 300 meters long?

Solution: 
Train length = 240 meters
Time to pass a pole = 15 seconds

Speed = (240/15) m/s = 16 m/s

Total distance to be covered (train + platform) = 240 + 300 = 540 meters
So, Time = 540/16
= 33.75 seconds

Question: The investment ratio of two partners, A and B, is 11 : 12, while their profit ratio is 2:3. Given that A invested his funds for 8 months, determine the duration of B's investment.

Solution:
Let, A invested Tk 11x for 8 months
and B invested Tk 12x for y months

Now,
(11x × 8) : (12x × y) = 2 : 3
⇒ (11x × 8) / (12x × y) = 2/3
⇒ 88x/12xy = 2/3
⇒ 24y = 264
⇒ y = 11

So, B invested for 11 months. 

Question: A shopkeeper offers two successive discounts of 10% and 20% on an item marked at 1,000 Taka. What is the final selling price?

Solution: 
New Price after First Discount:
1000 - 10% of 1000
= 1000 - 100
= 900 Taka

∴ Final selling price = 900 - 20% of 900
= 900 - 180
= 720 Taka

Question: The difference between compound interest and simple interest on a sum of Taka 10,000 for 2 years at 10% per annum compounded annually is - 

Solution: 
Simple interest = 10000 × 2 × (10/100)
= 2000 Taka

Compound Interest, 

So, difference = 2100 - 2000 = 100 Taka

Question: In a map, 2 cm represents 85 km. The distance between two cities is 9.4 cm on the map. The actual distance between the cities is - 

Solution: 
Since 2 cm = 85 km,
Actual Distance = (85/2) × 9.4 
= 799/2
= 399.5 km 

Question: Two trains, each 100 meters long, are running on parallel tracks in opposite directions. If their speeds are 72 km/h and 54 km/h, how long will it take them to cross each other completely?

Solution: 
Train 1 length = 100 meters
Train 2 length = 100 meters
Total distance = (100 + 100) = 200 meters

Speed of Train 1 = 72 km/h = 72 × (5/18) m/s = 20 m/s
Speed of Train 2 = 54 km/h = 54 × (5/18) m/s = 15 m/s

Relative speed = (20 + 15) = 35 m/s

Time = (200/35) = 5.71 seconds
= 5.7 seconds

Question: A man can row 12 km/h in still water. He finds that it takes him twice as long to row upstream as to row downstream. What is the speed of the stream?

Solution: 
Speed in still water = 12 km/h
Speed of the stream = x km/h

Let the distance travelled is = D km
Time taken upstream = 2 × (Time taken downstream)
D/(12-x) = 2 × {D/(12+x)}
⇒ 1/(12-x) = 2/(12+x)
⇒ 12 + x = 24 - 2x
⇒ 3x = 12
∴ x = 4

So, Speed of the stream = 4 km/h