Bank Math

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: Out of three numbers, the first is twice the second and is half of the third. If the average of the three number is 56, then the difference of first and third number is:

Solution:
Let, 
the second number be x.
Then first number = 2x, third number = 4x.

∴ 2x + x + 4x = 56 × 3
⇒ 7x = 168
⇒ x = 168/7
⇒ x = 24

Required difference:
= 4x - 2x
= 2x
= 2 × 24
= 48.

Let Children = X
A/Q,
405/X = 20% of X
Or, X² = 2025
Or, x = 45
So each children receive = 405/45
= 9

Question: A school room is be built to accommodate 70 children so as to allow 2.2 m² of floor and 11 m³ of space for each child. If the room be 14 metres long, what must be its breadth and height ?
(৭০ জন ছাত্র-ছাত্রী ধারণ করার জন্য একটি বিদ্যালয় কক্ষ তৈরি করা হবে, যেখানে প্রতিটি শিশুর জন্য ২.২ বর্গমিটার মেঝে এবং ১১ ঘনমিটার স্থান থাকবে। কক্ষটির দৈর্ঘ্য ১৪ মিটার হলে, তার প্রস্থ এবং উচ্চতা কত হবে?)

Solution:
দেওয়া আছে,
ছাত্র সংখ্যা = ৭০
প্রতিটি ছাত্রের জন্য:
মেঝের ক্ষেত্রফল = ২.২ বর্গ মিটার
স্থান (ঘন মিটার): ১১ ঘন মিটার
রুমটির দৈর্ঘ্য = ১৪ মিটার

প্রতিটি ছাত্রের জন্য ২.২ বর্গ মিটার মেঝে স্থান প্রয়োজন। তাহলে, ৭০ জন ছাত্রের জন্য মোট মেঝে এলাকা হবে:
মোট মেঝের ক্ষেত্রফল = ৭০ × ২.২ = ১৫৪ বর্গ মিটার

প্রতিটি ছাত্রের জন্য ১১ ঘন মিটার স্থান প্রয়োজন। তাহলে, ৭০ জন ছাত্রের জন্য মোট স্থান হবে:
মোট স্থান = ৭০ × ১১ = ৭৭০ ঘন মিটার

রুমটির দৈর্ঘ্য দেওয়া আছে L = ১৪ মিটার, প্রস্থ b এবং উচ্চতা h বের করতে হবে।

মেঝের ক্ষেত্রফল হলো দৈর্ঘ্য এবং প্রস্থের গুণফল:
মেঝের ক্ষেত্রফল = দৈর্ঘ্য × প্রস্থ = ১৪ × b
১৪ × b = ১৫৪
এটি থেকে b বের করি:
b = ১৫৪ / ১৪ = ১১ মিটার

তাহলে, রুমটির প্রস্থ ১১ মিটার।

রুমটির মোট স্থান হলো দৈর্ঘ্য, প্রস্থ এবং উচ্চতার গুণফল:
স্থান = দৈর্ঘ্য × প্রস্থ × উচ্চতা = ১৪ × ১১ × h
১৪ × ১১ × h = ৭৭০

এটি থেকে h বের করি:
১৫৪ × h = ৭৭০
h = ৭৭০ / ১৫৪ = ৫ মিটার

তাহলে, রুমটির উচ্চতা ৫ মিটার।

চূড়ান্ত উত্তর:
রুমটির প্রস্থ ১১ মিটার।
রুমটির উচ্চতা ৫ মিটার।

Dividend on 1 share = (12.5 x 50)/100 = tk.6.25
tk.25 is income on an investment of tk.100
tk.6.25 is income on an investment of tk. (6.25 x 100)/25 = tk.25

Question: If 5 workers can harvest 60 kg of wheat in 3 days, how many kilograms of wheat will 8 workers harvest in 5 days?

Solution: 
5 workers 3 days harvest = 60 kg
1 worker 1 day harvest = (60/15) kg
8 workers 5 days harvest = ( 60 × 40 ) / 15 kg
= 160 kg

Question: If P and Q together can complete a work in 18 days, P and R together in 12 days, and Q and R together in 9 days, then Q alone can do the work in-

Solution:
One day's work of,
(P + Q) = 1/18 .......(1)

One day's work of,
(P + R) = 1/12 .......(2)

One day's work of,
(Q + R) = 9 .......(3)

Adding(1),(2)and(3),
⇒ 2 × (P + Q + R) = (1/18) + (1/12) + (1/9)
⇒ 2 × (P + Q + R) = 1/4
⇒ (P + Q + R) = 1/8

Now,
⇒ Q = (1/8) - (P + R)
⇒ Q = (1/8) - (1/12)
One day's work of Q = (3 - 2)/24 = 1/24
∴ Q need 24 days

Question: A garrison of 600 men had provisions for 30 days. After 3 days a reinforcement of 300 men arrived. How many days will the remaining food last now?

Solution:
After 3 days, food having for = (30 - 3) = 27 days
After arriving 300 men, total men = (600 + 300) = 900 men

600 men can eat the food for 27 days
∴ 1 man can eat the food for (27 × 600) days
∴ 900 men can eat the food for (27 × 600)/900 days
= 18 days

Question: What is the difference between simple and compound interest at 8% per annum on a sum of Tk. 12,500 at the end of 2 years?

Solution:
Here,
Principal, P = 12500 Tk.
Interest Rate, R = 8%
Time, T = 2 years

We know,
Simple Interest, SI = PRT/100
= (12500 × 8 × 2)/100
= 200000/100
= 2000 Tk.

For Compound Interest,
Amount, A = P{1 + (R/100)}^T
= 12500{1 + (8/100)}²
= 12500(1.08)²
= 12500 × 1.1664
= 14580 Tk.

∴ CI = A - P
= 14580 - 12500
= 2080 Tk.

∴ Difference between compound interest and simple interest,
= 2080 - 2000
= 80 Tk.

Question: In a certain code, LATER = 13579 and CHAIR = 20349, how is CHEAT coded in the language?

Solution:
Given that,
L  A  T  E  R
↓   ↓  ↓  ↓   ↓
1   3  5  7  9

and
C  H  A  I  R
↓   ↓   ↓  ↓   ↓
2   0   3  4   9

So
C  H  E  A  T
 ↓  ↓   ↓  ↓  ↓
 2  0   7  3  5

Question: A merchant has 1200 kg of rice, part of which he sells at 10% profit and the rest at 20% profit. If his overall gain is 16%, find the quantity of rice he sold at 20% profit.

Solution: 
Let the cost price per kg = Tk. x
∴ Total cost price = 1200x
Again,
Let the rice sold at 10% profit = y kg
Then the rice sold at 20% profit = (1200 - y) kg

Now,
Selling price of y kg at 10% profit,
SP₁ = (y × x) × (110/100)
= 110xy/100

Selling price of (1200 - y) kg at 20% profit,
SP₂ = (1200 - y) × (x) × (120/100)
= 120x(1200 - y)/100

Total Selling Price, SP = SP₁ + SP₂ 
= (110xy/100) + 120x(1200 - y)/100
= {110xy + 120x(1200 - y)}/100

Now,
Overall profit is 16%,
So, total SP =1200x × (116/100)
= 139200x/100

Now equating both total selling prices:
{110xy + 120x(1200 - y)}/100 = 139200x/100
⇒ 110xy + 120x(1200 - y) = 139200x
⇒ 110xy + 144000x - 120xy = 139200x
⇒ - 10xy = - 4800x
⇒ y = 4800x/10x
∴ y = 480 

So, rice sold at 20% profit = 1200 - y
= 1200 - 480 = 720 kg

Given that 1² + 2² + 3²..... + 10² = 385

Now, 3² + 6² + 9² + .....30²
= 3²×1² + 2²×3² + 3²×3² + ........ + 3²×10²
=  3²×(1² + 2² + 3²..... + 10²)
= 9 × 385
= 3465

Question: Rakib flipped a fair coin 6 times and got tails every time. What is the probability that he will get a tail on the 7th flip?

Solution:
মুদ্রা নিক্ষেপের প্রতিটি ট্রায়াল বা নিক্ষেপ একটি স্বাধীন ঘটনা (Independent Event)।

রাকিবের আগে ৬ বারই Tails পাওয়া, ৭ম নিক্ষেপের ফলাফলের ওপর কোনো প্রভাব ফেলবে না।

একটি নিরপেক্ষ মুদ্রার (Fair coin) ক্ষেত্রে কেবল দুটি সম্ভাব্য ফলাফল থাকে: Head অথবা Tail।
এখানে মোট ফলাফল সংখ্যা, n(S) = 2

৭ম নিক্ষেপে Tail পাওয়ার অনুকূল ফলাফল সংখ্যা, n(E) = 1

∴ P(E) = n(E)/n(S)
= 1/2

Question: If A = {x ∈ N : 3 ≤ x < 8} and B = {x ∈ N: x is an odd number and x < 10}, what is the value of A ∩ B?

Solution:
দেওয়া আছে,
A = {x ∈ N : 3 ≤ x < 8}
এখানে, x এর মান 3 এর সমান বা বড় এবং 8 এর ছোট স্বাভাবিক সংখ্যা।
∴ A = {3, 4, 5, 6, 7}

আবার,
B = {x ∈ N : x বিজোড় সংখ্যা এবং x < 10}
x স্বাভাবিক বিজোড় সংখ্যা যা 10 এর ছোট।
∴ B = {1, 3, 5, 7, 9}

প্রদত্ত রাশি, A ∩ B
= {3, 4, 5, 6, 7} ∩ {1, 3, 5, 7, 9}
= {3, 5, 7}

অতএব, A ∩ B এর মান হলো {3, 5, 7}।

We have: (l - b) = 23 and 2(l + b) = 206 or (l + b) = 103
Solving the two equations, we get:
l = 63 and b = 40
∴ Area = (l x b) = (63 x 40) m² = 2520 m²

Question: If (x/y) + (y/x) = √8 then what is the value of (x⁴/y⁴) + (y⁴/x⁴) ?

Solution:
Given that, 
(x/y) + (y/x) = √8

∴ x⁴/y⁴ + y⁴/x⁴
= (x/y)⁴ + (y/x)⁴
= {(x/y)²}² + {(y/x)²}²
= {(x/y)² + (y/x)²}² - 2.(x²/y²).(y²/x²)
= {(x/y)² + (y/x)²}² - 2
= [{(x/y) + (y/x)}² - 2.(x/y).(y/x)]² - 2
= {(√8)² - 2}² - 2
= (8 - 2)² - 2
= 6² - 2
= 36 - 2
= 34

Question: If θ = 30°, then sec²θ − tan²θ = ?

Solution:
Given, θ = 30°

Now,
sec²θ - tan²θ
= (sec30°)² - (tan30°)²
= (2/√3)² - (1/√3)²
= 4/3 - 1/3
= 3/3
= 1

B runs 45/2 m in 6 sec.
B covers 300 m in (6x2)/45 x300 sec
= 80 sec

Question: A man earns N dollars a month and spends S dollars a month on rent. If he then spends 3/8 of the remainder on food, how much, in dollars, is left over for other expenses, in terms of N and S?

Solution:
Given that,
Monthly income = N dollars
Rent = S dollars

∴ Remaining after rent = N - S

And, He spends 3/8​ of the remainder on food.
∴ Food expense = (3/8)(N - S)

∴ Left for other expenses = Remaining after rent - Food expense
= (N - S) - {(3/8)(N - S)}
= (8/8)(N - S) - {(3/8)(N - S)}
= {(8 - 3)/8}(N - S)
= (5/8)( N - S)

Question: The true discount on Tk. 2562 due 4 months hence is Tk. 122. What is the rate percent?

Solution: 
Given,
Amount (A) = 2562 Tk
True Discount (TD) = 122 Tk
Time (T) = 4 months = 1/3 of a year

Present Worth, P.W. = (2562−122) Tk
= 2440 Tk

True Discount (TD) = Simple Interest (SI) on the Present Worth (PW) for the given time period.
∴ S.I. on 2440 Tk for 4 months is 122 Tk.

We know, SI = PRT/100
∴ R = (SI × 100)/(P × T)
= [(100 × 122)/(2440 × (1/3)]
= 15

∴ Rate is 15%

Question: A glass when full of milk, weighs 1 kg. It weighs 0.75 kg when the glass is half full. What is weight of the empty glass?

Solution: 
Glass এর ওজন = x কেজি 
Milk এর ওজন = y  কেজি 

এখন 
x + y = 1..................(1)

x + y/2 = 0.75
⇒ (2x + y)/2 = 0.75
⇒  2x +y = 1.5..................(2)

(2) - (1) ⇒
2x + y - (x + y) = 1.5 - 1
2x + y - x - y = 0.5
x = 0.5 

Glass এর ওজন = 0.5 কেজি 

We need to SELECT people.
[SELECT = Combination = ⁿC_r = n!/r!(n-r)!

Here, we first have to select 10 ladies from 19.
Select = Combination
∴ Select 10 ladies = ¹⁹C₁₀

Arrange 10 ladies in circle = 10 - 1 = 9! ways
19 - 10 = 9 ladies remain.
Arrange the remaining 9 ladies in another circle = 9 - 1 = 8! ways

∴ Total ways to arrange 19 ladies in 2 required circles = ¹⁹C₁₀ × 9! × 8!

Let 4 consecutive even numbers be x, x + 2, x + 4 and x + 6 respectively.
3 consecutive odd numbers be y, y + 2 and y + 4
According to question,
(x + x + 2 + x + 4 + x + 6) - (y + y + 2 + y + 4) = 81
Or, (4x + 12) - (3y + 6) = 81
Or, 4x - 3y = 81 - 12 + 6
Or, 4x - 3y = 75 ..... (i)
And, x + 4 = 59 ..... (ii)
On solving equations (i) and (ii), we get
x = 36
y = 23
The sum of largest odd and even number = x + 6 + y + 4
= x + y + 10
= 36 + 23 + 10
= 69

Question: The compound interest on Tk. 10,000 at 10% per annum for a certain period is Tk. 2,100. The time period (in years) is -

Solution: 
Principal amount, P = 10,000 taka
Compound Interest, I = 2100 taka

So, Total amount, A = 12100 Taka
r = 10% = 0.1

Now, (1.1)² = 1.21
So, n = 2 (Time period)

Question: How many bricks, each measuring 25 cm × 12 cm × 6 cm, are required to build a wall measuring 5 m × 3 m × 12 cm?

Solution:
Wall Dimensions,
Length = 5 m = 500 cm
Width = 3 m = 300 cm
Height = 12 cm

Brick Dimensions,
Length = 25 cm
Width = 12 cm
Height = 6 cm

Volume of the wall = Length × Width × Height
= 500 × 300 × 12
= 1,800,000 cm³

Volume of one brick = Length × Width × Height
= 25 × 12 × 6
= 1,800 cm³

∴ Number of bricks = Volume of the wall ÷ Volume of one brick
= 1,800,000 ÷ 1,800
= 1,000

∴ The number of bricks needed to construct the wall is 1,000.

Question:
 

Solution:

Question: The ratio of two numbers is 2 : 5 and their H.C.F is 6. Their L.C.M is -

Solution:
ধরি, সংখ্যা দুটি হলো 2x এবং 5x
∴ গসাগু (H.C.F) = x = 6

∴ সংখ্যা দুটি হলো: 2 × 6 = 12 এবং 5 × 6 = 30

∴ সংখ্যাদ্বয়ের গুণফল = 12 × 30 = 360
এবং H.C.F = 6

আমরা জানি,
L.C.M = (Product of two numbers)/H.C.F
= 360/6
= 60

∴ সংখ্যা দুটির লসাগু (L.C.M) = 60

Question: The present ages of Kamal and Jamal are in the ratio 5 : 7. After 8 years, the ratio of their ages will be 3 : 4. What is the difference in their present ages?

Solution:
Let their present ages be 5x and 7x.

After 8 years, Kamal's age = 5x + 8
Jamal's age = 7x + 8

According to the question,
(5x + 8)/(7x + 8) = 3/4
⇒ 4(5x + 8) = 3(7x + 8)
⇒ 20x + 32 = 21x + 24
⇒ 20x - 21x = 24 - 32
⇒ - x = - 8
∴ x = 8

∴ Kamal's present age = 5 × 8 = 40 years
∴ Jamal's present age = 7 × 8 = 56 years

∴ Difference = 56 - 40 = 16 years

Pipe A can fill in 1 hour(60 minutes) is 1/1 or full of the tank.
Pipe B can empty in 1 hour (1/2) of the tank [120 mins= 2hrs]
Both pipes together can fill the tank in 1 hour = 1/1 - 1/2
= 1/2 of the tank.
Since 1/2 part of the tank is filled in 1 hour, the remaining part left is 1/2 of the tank.
The remaining 1/2 part will be filled in another 1 hour.
So both the pipes take 2 hours(120 minutes) to fill the tank.

Suppose he moves 4 km downstream in x hours.
Then,
Speed downstream = (4/x) km/hr
Speed upstream = (3/x) km//hr

So, 48/(4/x) + 48/(3/x) = 14
or, 12x + 16x = 14
or, 6x + 8x = 7
or, x = 1/2
So, Speed downstream = 8 km/hr
Speed upstream = 6km/hr

Rate of the stream = (1/2) (8-6) km/hr = 1km/hr

The letters of the word 'BAKERY' be arranged in 6! ways 
= 6!
= 6 × 5 × 4 × 3 × 2 × 1
= 720

Question: The value of 1001 ÷ 11 of 13 is :

Solution:
1001 ÷ (11 × 13)
= 1001 ÷ 143
= 7

Money spent by X = Tk. 150000
Money received by X = 105% of Tk. 150000 = Tk. 157500
C.P. to X = 98% of Tk. 157500 = Tk. 154350
∴ X gains Tk. (157500 - 154350) = Tk. 3150

16^3/4
= (2⁴)^3/4
= 2^4×3/4
= 2³
= 8

Question:  The supplement of an angle is twice the angle. Find the angle.

Solution:
Let the angle be x degrees.
The supplement of an angle = 180° - x

According to the question,
180° - x = 2x
⇒ 180° = 2x + x  
⇒ 180° = 3x  
⇒ x = 180°/3  
∴ x = 60°

∴ The angle is 60°

Question: Mamun invested 77500 tk in CT bank. In two years how much compound interest will he get, if the first year rate of interest was 10% and second year had 2% more than first year?

Solution:
Rate of Interest for Year 1 = 10% 
Rate of Interest for Year 2 = 10 + 2 = 12%

For compound Interest, Total Amount = P {1 + (R/100)}ⁿ
∴ Compound Interest = Amount - Principal = P {1 + (R/100)}ⁿ - P
= 77500{1 + (10/100)}¹ × {1 + (12/100)}¹ - 77500   [1st year 10% and 2nd year 12%]
= 77500[{1 + (10/100)}¹ × {1 + (10/100)}1 - 1]
= 77500 {(11/10) (112/100) - 1}
= 77500{(1232/1000) - 1}
= 77500(232/1000)
= 775(232/10)
= 17980 tk