Bank Math Master

Question: In what ratio must sugar at Tk. 12 per kg be mixed with sugar at Tk. 18 per kg so that the mixture be worth Tk. 15 per kg?

Solution:
Let x kg of sugar at Tk. 12 and y kg of sugar at Tk. 18 be mixed.

ATQ,
⇒ 12x + 18y = 15(x + y)
⇒ 12x + 18y = 15x + 15y
⇒ 15x - 12x = 18y - 15y
⇒ 3x = 3y
∴ x : y = 1 : 1

∴ required ratio 1 : 1.

Question: Rahim ran half the distance at 5 km/h and the remaining half at 10 km/h. What was his average speed for the entire run?

Solution:
Let the total distance = 2x km, so each half = x km.

Time for first half = x / 5 hours
Time for second half = x / 10 hours

∴ Total time = (x/5) + (x/10) = (2x + x)/10 = 3x/10 hours

∴Average speed = Total distance / Total time
= 2x ÷ (3x/10)
= 2x × (10/3x)
= 20/3
= 6.67 km/h 

Question: In a 600 meter race, C starts 60 meters ahead of D, yet D defeats C by a margin of 30 meters. What distance did C cover when D reached the finish line?

Solution:
In a 600 meter race,
C starts 60 meters ahead. 
so C needs to cover = 600 - 60 = 540 meters
D covers the full distance = 600 meters

D defeats C by 30 meters

∴ When D finishes,
C’s distance = 540 - 30 = 510 meters.

Question: A train crosses two bridges that are 600 meters and 200 meters long in 60 seconds and 40 seconds respectively. What is the length of the train?

Solution:
We know,
To cross a bridge, a train must cover the length of the bridge along with its own length.

Let the length of the train = x meters

Then,
For the first bridge, the distance covered by the train = (x + 600) meters
And,
For the second bridge, the distance covered by the train = (x + 200) meters

According to the question,
(x + 600)/60 = (x + 200)/40
⇒ 40(x + 600) = 60(x + 200)

⇒ 40x + 24000 = 60x + 12000
⇒ 60x - 40x = 24000 - 12000
⇒ 20x = 12000
⇒ x = 12000/20 
⇒ x = 600

∴ The length of the train is 600 meters.

Question: In two identical glasses, one is half full of milk and the other is three-fifth full of milk. Both glasses are then filled with water and poured into a tumbler. What is the ratio of milk to water in the tumbler?

Solution:
In the 1st glass,
Milk = 1/2
Water = 1 − 1/2 = 1/2

In the 2nd glass,
Milk = 3/5
Water = 1 − 3/5 = 2/5

Now total milk in tumbler = (1/2 + 3/5)
= {(5 + 6)/10}
= 11/10

Total water in tumbler = (1/2 + 2/5)
= {(5 + 4)/10}
= 9/10

∴ Ratio of milk and water
= (11/10) : (9/10)
= 11 : 9

Question: The speed of a boat in still water is 10 km/h. The time it takes to travel downstream is one-third the time it takes to travel upstream. What is the speed of the stream?

Solution:
Let the speed of the current be = x km/h

Then,
Downstream speed = (10 + x) km/h
Upstream speed = (10 − x) km/h

We know, time = distance/speed

According to the question:
distance/(10 + x) = distance/{3 × (10 - x)}
⇒ (10 + x) = 3(10 - x)
⇒ 4x = 20
⇒ x = 5
∴ The speed of the current = 5 km/h

Question: A swimmer covers 3 km against the current in 30 minutes and the same distance with the current in 15 minutes. What is the speed of the swimmer in still water?

Solution:
Given that,
Distance covered = 3 km
Time consumed = 30 minutes = 1/2 hour.
∴ speed of the swimmer against current (upstream) = 3/(1/2) = 6 km/hr

Again,
With the current,
Distance covered = 3 km
Time consumed = 15 minutes = 15/60 = 1/4 hr
∴ speed of the swimmer with current (downstream) = 3/(1/4) = 12 km/hr

let,
swimmer's speed in still water = x km/hr
speed of current = y km/hr

According to question,
x + y = 12 ------(1)
x - y = 6 -------(2)

(1) + (2) ⇒
2x = 18
⇒ x = 9  

∴ swimmer's speed in still water = 9 km/hr

Question: A bus moves 600 meters in a minute, and a car travels 90 km in 60 minutes. How much faster is the car than the bus? 

Solution:
Speed of the bus = Distance/Time
= (600/1) meters/minute
= (600/1000)/(1/60) km/h
= (600 × 60)/1000 km/h
= 36 km/h

A car travels 90 km in 60 minutes
So, speed of the car = Distance/Time
= 90 km/h [60 minutes = 1 hour]

∴ Difference in speed between the car and the bus = (90 - 36) km/h = 54 km/h.

Question: Two trains are running in the same direction at 90 km/h and 60 km/h. The faster train crosses a man in the slower train in 18 seconds. What is the length of the faster train?

Solution:
Relative speed of train = (90 - 60) km/h
= 30 km/h
= (30 × 1000)/3600
= 25/3 m/s

∴ Distance covered = Speed × Time
= (25/3) × 18
= 150 meters

To overtake the person, the train has to travel a distance equal only to its own length.

So, Length of faster train = Distance covered = 150 meters.

Question: A car travels at 50 km/h. If it had traveled 10 km/h faster, it would have reached its destination 30 minutes earlier. What is the distance of the journey?

Solution:
Let the distance be x km.

Time at 50 km/h = x/50 hours
Time at 60 km/h = x/60 hours

According to the question, the difference in time = 0.5 hours
⇒ x/50 − x/60 = 0.5
⇒ (6x − 5x) / 300 = 0.5
⇒ x / 300 = 0.5
⇒ x = 0.5 × 300
⇒ x = 150 km

∴ Distance of the journey = 150 km

Question: A 180 m long train takes 18 seconds to pass a pole. How long will it take to pass a 330 m long platform?

Solution:
আমরা জানি,
কোনো ট্রেন একটি খুঁটিকে অতিক্রম করলে ট্রেনটি তার নিজের দৈর্ঘ্যকে অতিক্রম করে।
আবার,
কোনো প্ল্যাটফর্মকে অতিক্রম করতে হলে ট্রেনকে নিজের দৈর্ঘ্য এবং প্ল্যাটফর্মের দৈর্ঘ্যের সমান দূরত্ব অতিক্রম করতে হয়।

∴ প্ল্যাটফর্ম অতিক্রম করলে ট্রেনের অতিক্রান্ত দূরত্ব = (ট্রেনের দৈর্ঘ্য + প্ল্যাটফর্মের দৈর্ঘ্য) = (180 + 330) মিটার = 510 মিটার 

এখন,
180 মিটার অতিক্রম করে = 18 সেকেন্ডে
∴ 1 মিটার অতিক্রম করে = 18/180 সেকেন্ডে
∴ 510 মিটার অতিক্রম করে = (18 × 510)/180 = 51 সেকেন্ডে

Question: A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 50 meters, and the speed of the train is 45 km/hr. In 4 hours, how many pillars will he count?

Solution:
Distance covered by the train in 4 hours = (Speed × Time)
= (45 × 4) km
= 180 km
= 180000 m

∴ Number of pillars counted by the man = (180000/50) + 1
= (3600 + 1)
= 3601

Question: The ratio of orange and water in a mixture is 2 : 3, then the percentage of water in mixture is-

Solution: 
The ratio of orange and water in a mixture is 2 : 3
Total = 5

∴ percentage of water = (3/5) × 100%
= 60%

Question: A man completes a journey of 120 km by a car. He covers 60 km at 30 km/h, 40 km at 40 km/h, and the remaining 20 km at 20 km/h. What is his average speed for the whole journey?

Solution:
A man completes a journey of 120 km by a car.
First case, 
Distance covered: 60 km at 30 km/h
∴ Time = 60 ÷ 30 = 2 hours

Second case,
40 km at 40 km/h
∴ Time = 40 ÷ 40 = 1 hour

Third case,
the remaining 20 km at 20 km/h
 ∴ Time = 20 ÷ 20 = 1 hour

∴ Total distance = 60 + 40 + 20 = 120 km
∴ Total time = 2 + 1 + 1 = 4 hours

∴ Average speed = Total distance ÷ Total time = 120 ÷ 4 = 30 km/h

Question: How much water should be added to 40 liters of pure milk to gain an extra 20% profit when selling the mixture at the price of pure milk?

Solution:
Assume,
Price of pure milk per liter = 100 Taka
So, the price of 40 liters of pure milk = 40 × 100 = 4000 Taka

Let, the amount of water added = x liters
Total mixture = (40 + x) liters

Since the mixture is sold at the price of pure milk,
Selling price of (40 + x) liters = 100(40 + x) Taka

According to the question,
100(40 + x) = 4000 + 4000 of 20%
⇒ 4000 + 100x = 4000 + (4000 × 20/100)
⇒ 4000 + 100x = 4000 + 800
⇒ 100x = 800
⇒ x = 800/100
⇒ x = 8

∴ Required water: 8 liters.

Question: The speed of P and Q are in the ratio 5 : 8. Q takes 24 minutes less than P to reach a destination. Time in which Q reaches the destination?

Solution:
Given, speed of P and Q = 5 : 8

So, ratio of time taken = 8 : 5 [Time ∝ 1/Speed]

Let time taken by P and Q be 8x and 5x minutes respectively.

According to the question,
8x - 5x = 24
⇒ 3x = 24
⇒ x = 8

Hence, time taken by Q = 5 × 8 = 40 minutes

Question: A train running at the speed of 72 km/h crosses a pole in 20 seconds. What is the length of the train?

Solution:
Given,
Speed = 72 km/h
= (72 × 5/18) m/sec
= 20 m/sec

Length of the train = Speed × Time
= 20 × 20 m
= 400 m

∴ Length of the train = 400 m

Question: A 42 liter mixture contains milk and water in the ratio 3 : 4. How many liters of milk must be added to the mixture so that the ratio of milk to water becomes 1 : 1?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 42 × (3/7) = 18 liters.
Quantity of water = 42 × (4/7) = 24 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(18 + x) : 24 = 1 : 1
⇒ (18 + x)/24 = 1/1
⇒ 18 + x = 24
⇒ x = 24 - 18
⇒ x = 6

∴ Quantity of milk to be added = 6 liters