Bank Math Master

Question: A 35-liter mixture contains milk and water in a 3 : 4 ratio. How much milk should be added to the mixture to make the ratio equal?

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

Quantity of milk = 35 × (3/7) = 15 liters.
Quantity of water = 35 × (4/7) = 20 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(15 + x) : 20 = 1 : 1
⇒ (15 + x)/20 = 1/1
⇒ 15 + x = 20
⇒ x = 20 - 15
⇒ x = 5

∴ Quantity of milk to be added = 5 liters

Question: Jisan cycled halfway at 3 km/h and the rest at 6 km/h. What was his average speed for the whole trip?

Solution:
ধরি,
3 কি.মি./ঘণ্টা বেগে অতিক্রম করে = x কি.মি.
এবং 6 km/h বেগে অতিক্রম করে = x কি.মি.
∴ পথের মোট দূরত্ব = x + x = 2x কি.মি.

যাত্রাপথের অর্ধেক দূরত্বে প্রয়োজনীয় সময় = x/3 ঘণ্টা
বাকি অর্ধেক দূরত্বে প্রয়োজনীয় সময় = x/6 ঘণ্টা

∴ সম্পূর্ণ যাত্রায় গড় গতিবেগ = মোট দূরত্ব/মোট সময়
= 2x/{(x/3) + (x/6)}
= 2x/{(2x + x)/6}
= 2x/(3x/6)
= 2x/(x/2)
= 2x × (2/x)
= 4 কি.মি./ঘণ্টা

Question: A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

Solution:
ধরি,
শুরুতে জারের মধ্যে তেলের পরিমাণ = 7x লিটার 
পানির পরিমাণ = 5x লিটার 
∴ মোট অংশ = 12x

9 লিটার মিশ্রণ ফেলে দিলে,
ফেলে দেওয়া মিশ্রণে তেলের পরিমাণ = 9 এর (7x/12x) = 21/4 লিটার 
এবং পানির পরিমাণ = 9 এর (5x/12x) = 15/4 লিটার 

বাকি মিশ্রণে,
তেলের পরিমাণ = 7x - (21/4) = (28x - 21)/4
পানির পরিমাণ = 5x - (15/4) = (20x - 15)/4

মিশ্রণে 9 লিটার পানি যোগ করা হলে পানির নতুন পরিমাণ = {(20x - 15)/4} + 9 = (20x - 15 + 36)/4 = (20x + 21)/4

প্রশ্নমতে,
{(28x - 21)/4}/{(20x + 21)/4} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 9(28x - 21) = 7(20x + 21)
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 189 + 147
⇒ 112x = 336
⇒ x = 336/112
⇒ x = 3

∴ শুরুতে মিশ্রণে তেলের পরিমাণ ছিলো = (7 × 3) লিটার = 21 লিটার

Question: In a 500-meter race, B starts 50 meters ahead of A, yet A defeats B by a margin of 25 meters. What distance did B cover when A reached the finish line?

Solution:
500 মিটার রেসে B 50 মিটার এগিয়ে থেকে দৌড় শুরু করায় B কে দূরত্ব অতিক্রম করতে হবে = (500 - 50) মিটার = 450 মিটার 
A এর অতিক্রান্ত দূরত্ব = 500 মিটার 

কিন্তু A, B-কে 25 মিটার দূরত্বে পরাজিত করে।
∴ A যখন শেষপ্রান্ত স্পর্শ করে তখন B এর অতিক্রান্ত দূরত্ব = (450 - 25) মিটার = 425 মিটার

Question: A train crosses two bridges that are 420 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 + 420) meters
And,
For the second bridge, the distance covered by the train = (x + 200) meters

According to the question,
(x + 420)/60 = (x + 200)/40
⇒ 40(x + 420) = 60(x + 200)
⇒ 40x + 16800 = 60x + 12000
⇒ 60x - 40x = 16800 - 12000
⇒ 20x = 4800
⇒ x = 4800/20 
⇒ x = 240

∴ The length of the train is 240 meters

Question: The speed of a boat in still water is 6 km/h. The time it takes to travel downstream is half 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 = (6 + x) km/h
Upstream speed = (6 − x) km/h

According to the question:
(6 + x) = 2(6 − x)
⇒ 6 + x = 12 - 2x
⇒ 2x + x = 12 - 6
⇒ 3x = 6
⇒ x = 6/3
⇒ x = 2

∴ The speed of the current = 2 km/h

Question: A grocer mixes two types of pulses, one costing 15 taka per kg and the other 20 taka per kg. If the mixture is sold at 16.50 Taka per kg, what is the ratio of the two types of pulses in the mixture? 

Solution:
ধরি,
প্রথম প্রকার ডালের পরিমাণ = x কেজি
দ্বিতীয় প্রকার ডালের পরিমাণ = y

প্রথম প্রকার ডালের x কেজির মূল্য = 15x টাকা 
দ্বিতীয় প্রকার ডালের y কেজির মূল্য = 20y টাকা 

প্রশ্নমতে,
15x + 20y = 16.50(x + y)
⇒ 15x + 20y = 16.50x + 16.50y
⇒ 20y - 16.50y = 16.50x - 15x
⇒ 3.50y = 1.50x
⇒ x/y = 3.50/1.50 
⇒ x/y = 7 : 3

শর্টকাট: 

∴ অনুপাত = 3.50 : 1.50
= 7 : 3

Question: A swimmer swims in a river against the current and takes 30 minutes to swim 2 km. With the current, he takes 20 minutes to swim the same distance. What is his speed in still water?

Solution:
Given that,
distance covered= 2 km 
time consumed = 30 min = 30/60 hr = 1/2 hr
∴ speed of the swimmer against current (upstream) = 2/(1/2) = 4 km/hr

Again,
With the current,
distance covered = 2 km
time consumed= 20 min = 20/60 hr = 1/3 hr
∴ speed of the swimmer with current (downstream)= 2/(1/3) = 6 km/hr

∴ speed in still water = (speed in downstream + speed in upstream)/2
 = (6 + 4)/2
=10/2
= 5
So, the swimmer’s speed in still water is 5 km/h

Question: A car moves 550 meters in a minute, and a train travels 72 km in 45 minutes. How much faster is one than the other?

Solution:
Speed of the car = Distance/Time
= (550/1) meters/minute
= (550/1000)/(1/60) km/h
= (550 × 60)/1000 km/h
= 33 km/h

Speed of the train = Distance/Time
= (72/45) km/minute
= 72/(45/60) km/h
= (72 × 60)/45 km/h
= 96 km/h

∴ Difference in speed between the train and the bus = (96 - 33) km/h = 63 km/h

Question: 400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Amount of sugar = 400 × (30/100) = 120 grams
let,
x gm sugar to be added

According to the question,
(120 + x)/(400 + x) = 50%
⇒ (120 + x)/(400 + x) = 50/100
⇒ (120 + x)/(400 + x) = 1/2
⇒ 2 × (120 + x) = (400 + x)
⇒ 240 + 2x = 400 + x
⇒ 2x - x = 400 - 240
⇒ x = 160 grams

Question: A train running at a certain speed crosses a 496-meter-long platform in 56 seconds. If the length of the train is 560 meters, how long will it take to cross a bridge that is 100 meters in length?

Solution:
We know,
When a train crosses any object, it covers a distance equal to the sum of the object's length and its own length.

So, when crossing a platform, the distance covered by the train = (496 + 560) meters = 1056 meters
And
when crossing a bridge, the distance covered by the train = (100 + 560) meters = 660 meters

Now,
The train covers 1056 meters in = 56 seconds
∴ It covers 1 meter in = (56/1056) seconds
∴ It covers 660 meters in = (56 × 660)/1056 seconds = 35 seconds

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

Question: Two trains are running in the same direction at 80 km/h and 60 km/h. The faster train crosses a man in the slower train in 36 seconds. What is the length of the faster train?

Solution:
Relative speed of train = (80 - 60) km/h
= 20 km/h
= (20 × 1000)/3600
= (50/9) m/s

∴ Distance covered  = Speed × Time
= (50/9) × 36
= 200 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 =  200 meters.

Question: A boat goes 13 km upstream in 39 minutes. The speed of stream is 3 km/h. What is the speed of the boat in still water?

Solution:
Against the stream,
Speed of the boat = Distance/Time
= 13/(39/60) km/h
= (13 × 60)/39 km/h
= 20 km/h

We know,
Speed of the boat upstream = Speed in still water − Speed of the stream
⇒ Speed in still water = Speed of the boat upstream + Speed of the stream
⇒ Speed in still water = (20 + 3) km/h = 23 km/h

∴ Speed of the boat in still water = 23 km/h.

Question: A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

Solution:
Let, the actual distance travelled be x km.

Then,
x/60 = (x + 100)/80 
⇒ x/6 = (x + 100)/8
⇒ 6(x + 100) = 8x
⇒ 6x + 600 = 8x
⇒ 8x - 6x = 600
⇒ 2x = 600
⇒ x = 600/2
⇒ x = 300 km

Question: A person has two acid solutions — one containing 40% acid and the other 60% acid. In what quantities should he mix each to obtain 10 liters of a 50% acid solution?

Solution:
ধরি,
প্রথম দ্রবণটি মেশাতে হবে = x লিটার 
দ্বিতীয় দ্রবণটি মেশাতে হবে = (10 - x) লিটার 

প্রশ্নমতে,
x লিটার 40% এসিডের দ্রবণ + (10 - x) লিটার 60% এসিডের দ্রবণ  = 10 লিটার 50% এসিডের দ্রবণ
⇒ x × (40/100) + (10 - x) × (60/100) = 10 × (50/100)
⇒ (2x/5) + {3(10 - x)/5} = 5
⇒ (2x/5) + {(30 - 3x)/5} = 5
⇒ (2x + 30 - 3x)/5 = 5
⇒ (30 - x)/5 = 5
⇒ 30 - x = 25
⇒ x = 30 - 25
⇒ x = 5

40% এসিডের প্রথম দ্রবণটি মেশাতে হবে = 5 লিটার 
60% এসিডের দ্বিতীয় দ্রবণটি মেশাতে হবে = (10 - 5) লিটার = 5 লিটার 

অর্থাৎ 40% ও 60% এসিডের প্রতিটি দ্রবণ 5 লিটার করে মিশ্রিত করলে 50% এসিডের দ্রবণ পাওয়া যাবে।

Question: A boat travels at 12 km/h in still water, and the stream's speed is 3 km/h. If a boatman rows 90 km to a destination and comes back, how much time does he take in total?

Solution:
Given,
Speed of the boat in still water = 12 km/h
Speed of the current = 3 km/h

Speed of the boat downstream = (Speed in still water + Speed of current) = (12 + 3) km/h = 15 km/h
Speed of the boat upstream = (Speed in still water − Speed of current) = (12 − 3) km/h = 9 km/h

Time taken to go downstream = Distance / Speed = (90/15) hours = 6 hours
Time taken to go upstream = Distance / Speed = (90/9) hours = 10 hours

Total time for the round trip = (6 + 10) hours = 16 hours

Question: A MiG-29 fighter jet covers a certain distance at a speed of 1240 km/h in 5 hours. What speed must it maintain to cover the same distance in 250 minutes?

Solution:
Total distance = Speed × Time
= (1240 × 5) km
= 6200 km

Given time = 250 minutes = (250/60) hours= 25/6 hours

∴ Required speed = Distance/Time
= {6200/(25/6)} km/h
= {6200 × (6/25)} km/h
= (248 × 6) km/h
= 1488 km/h

Question: A 180-meter-long train takes 18 seconds to pass a pole. How long will it take to pass a 330-meter-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: How much water should be added to 50 liters of pure milk to gain extra 12% profit when selling the mixture at the price of pure milk?

Solution:
Let’s assume,
Price of pure milk per liter = 100 Taka
So, the price of 50 liters of pure milk = 100 × 50 = 5000 Taka

Now assume,
Water added to the milk = x liters
Then the total quantity of the milk-water mixture = (50 + x) liters

Since the mixture is sold at the price of pure milk,
The selling price of (50 + x) liters = 100(50 + x) Taka

According to the question,
100(50 + x) = 5000 + 5000 এর 12%
⇒ 5000 + 100x = 5000 + {5000 × (12/100)}
⇒ 5000 + 100x = 5000 + 600
⇒ 5000 - 5000 + 100x = 600
⇒ 100x = 600
⇒ x = 600/100
⇒ x = 6

∴ Amount of water to be added = 6 liters

Shortcut:
Amount of water to be added = (Profit%/100) × Quantity of pure milk
= (12/100) × 50
= 6 liters