Bank Math

Question: Rahman spends 40% of his monthly income on household items, 25% of his monthly income on buying clothes, 10% of his monthly income on medicines and saves the remaining Tk. 9,000. What is Rahman’s monthly income?

Solution:
Total spends = (40 + 25 + 10) = 75%

∴ Saves = (100 - 75) = 25%

Let total income of Rahman be x. Then,
⇒ x × 25% = 9000
⇒ 25x/100 = 9000
⇒ x/4 = 9000
⇒ x = 9000 × 4
∴ x = 36000

∴ Rahman’s monthly income is Tk. 36,000

Question: A sum of Taka 50,000 is invested at 6% simple interest for the first 3 years and 8% compound interest for the next 2 years. What is the total amount after 5 years?

Solution: 
Given,
Tk. 50,000 is invested.
6% simple interest for the first 3 years.
8% compound interest for the next 2 years.

Simple Interest, I = Pnr
= 50000 × 3 × (6/100)
= 9000

New Principal = 50,000 + 9,000 = 59,000

Compound Principal = P × (1 + r)ⁿ
= 59000 × (1 + (8/100))²
= 59000 × (1.08)²
= 68,817.60

Question: The difference in Taka between simple and compound interest at 4% annually on a sum of Tk. 8,000 after 2 years is:

Solution:
Given that, Principal, P = Tk. 8,000
Rate, r = 4%
Time, n = 2 years

We know that,
Simple Interest = Prn/100
= (8000 × 4 × 2)/100
= 640

And Compound Interest = P(1 + r/100)ⁿ - P
= 8000{1 + (4/100)}² - 8000 
= 8000(104/100)² - 8000
= {8000 × (104/100) × (104/100)} - 8000 
= 8652.8 - 8000
= 652.8

∴ Difference = 652.8 - 640 = 12.8

The difference between compound and simple interest is Tk. 12.8

Dividend on Tk.20 = Tk. (9/100 × 20) = Tk. 9/5
Tk.12 is income of Tk.100.
∴Tk. 9/5 is an income of = Tk (100/12 × 9/5) = Tk.15

Question: A clock is started at noon. By 20 minutes past 4, through how many degrees has the hour hand turned?

From noon (12.00 pm) to 4 pm = 4 hours
and 20 minutes = 20/60 hours = 1/3 hours

Total time = (4 + 1/3) = 13/3 hours

We know,
Angle traced by hour hand in 12 hours =  360°

∴ Angle in 13/3 hours,
= (360/12) × (13/3)
= 130°

অপশন a থেকে, 
a = {300/(9 + 1)} × 1 = 30 [100 থেকে ছোট, তাই বাদ]

অপশন b থেকে,
a = {300/(5 + 2)} × 5 = 128.7  [ভগ্নাংশ, তাই বাদ]

অপশন c থেকে,
a = {300/(5 + 3)} × 5 = 187.5 [ভগ্নাংশ, তাই বাদ]
b = {300/(5 + 3)} × 3 = 112.5 [ভগ্নাংশ, তাই বাদ]

অপশন d থেকে,
a = {300/(3 + 2)} × 3 = 180
এবং, b = {300/(3 + 2)} × 2 = 120

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

Relative speed = 120 + 80
= 200 km/hr.
= 200 × (5/18)
= 500/9
Time = 9 seconds
Distance covered = (500/9) × 9
= 500 meter.
Length of the train = (500 - 270) meter.
= 230 meter.

Let the distance be x
Speed upstream = (40-10) = 30 kmph
Speed downstream = (40+10) = 50 kmph
Total time taken = 1 hr
⇒ x/50 + x/30 = 1
⇒ 8x/150 = 1
⇒ x = 150/8 = 18.75 km

Let the sum be x
Then,
Simple interest = x/4
T = 3(1/8)
= 25/8 years
R = {100 × (x/4)}/{x × (25/8)}
= 8
Hence Required interest rate = 8%

ধরি,
একটি জাহাজ A speed - এ যাচ্ছে এবং অপরটি B speed এ যাচ্ছে।
নদীর প্রস্থ w
At time = t₁
A (t₁) = 750
B(t₁) =(w - 750)
⇒ B/A = (w - 750)/750
750B = (w - 750)A
At time = t₂
A(t₂) = w + 250
B(t₂) = w +(w - 250)
= 2w - 250
⇒ B/A = (2w - 250)/(w + 250)
(w + 250)B = (2w - 250)A .... Eq.[1]
750B = (w - 750/A
B = [(w - 750)/750]A
Plug into Eq. [1]
(w + 250)[(w - 750)/750]A = (2w - 250) A
(w + 250)(w - 750)/750 =750(2w -250)
w² - 500w - 187,500 = 1500w - 187500
w2 - 2000w = 0
w2 = 2000w
w = 2000 yards.

Let, buying price is x Taka
ATQ, 5% of x = 350 - 340 = 10
⇒ x = (100 × 10) / 5
∴ x = 200

Question: An observer 1.5 m tall stands 10√3 meters away from a flagpole. The angle of elevation from his eye to the top of the flagpole is 30°. What is the height of the flagpole?

Solution:

Here,
Flagpole Height = AB

Now,
tan∠c = AE/CE
⇒ tan30° = AE/10√3
⇒ 1/√3 = AE/10√3 
∴ AE = 10

∴ AB = AE + BE 
= 10 + 1.5
= 11.5 m

মনে করি,
radius = 10
সুতরাং area = πr²
= 100π
radius 100% বাড়ল।
নতুন radius = 20
area = πr²
= 400π
area বাড়ল 300%

Let the large number be x.
Then,
x - 20 is a 20% of x = 20x/100 = x/5
Or, x - x/5 = 20
Or, 4x = 100
∴ x = 25

Question: If one-third of one-fourth of a number is 15, then three-tenth of that number is - 

Solution:
Let the number be x.
The equation:
⇒ 1/3 × 1/4 × x = 15
⇒ 1/12 × x = 15
⇒ x = 12 × 15
⇒ x = 180

three-tenth of the number:
= 3/10 × 180
= 54

∴ That number is 54

Question: A and B are centers of two circles that touch each other externally, as shown in the figure. What is the area of the circle whose diameter is AB?


Solution:
যেহেতু বৃত্ত দুটি পরস্পরকে বহিস্থভাবে স্পর্শ করে, তাই তাদের কেন্দ্রবিন্দুদ্বয়ের মধ্যবর্তী দূরত্ব (AB) হবে তাদের ব্যাসার্ধের যোগফলের সমান।

এখন, নতুন বৃত্তের ব্যাস, AB = (4 + 3) সেমি = 7 সেমি।
সুতরাং, নতুন বৃত্তের ব্যাসার্ধ, r = 7/2 সেমি।

∴নতুন বৃত্তের ক্ষেত্রফল = πr²
 = π(7/2)²
= π(49/4)
= 49π/4 বর্গ সেমি।

সুতরাং, নতুন বৃত্তের ক্ষেত্রফল হবে 49π/4 বর্গ সেমি।

30% ক্ষতিতে দাম, 70% = 6384
30% লাভে দাম, 130% = (6384×130) / 70 = 11856

Given, the diagonal of a square measures, a√2 = 16√2 cm
So, side of the square is = 16 cm
Area of the square = 16² = 256 cm²

দেয়া আছে, y = x - 2
যেহেতু, y = mx + c
সুতরাং, ঢাল = 1

a, c, d এই তিনটি অপশনের সমীকরণ দেখলে বুঝা যায় যে এগুলোর সমাধান করলে x এর সহগ 1 হবে না

সমান্তরাল হতে হলে দুটির ঢাল ই সমান হতে হবে। অপশনের মধ্য থেকে
2y = 2x - 6
⇒ y = x - 3

সুতরাং, ঢাল = 1

Question: The average of a natural number and Its cube Is 13 times the number. The cube of the number is:
(Janata RC 2022 অনুযায়ী) 

Solution:
let the natural number be = x

According to the Question,
(x + x³)/2 = 13x
⇒ x + x³ = 26x
⇒ x³ = 26x - x
⇒ x³ = 25x
⇒ x³/x = 25
⇒ x² = 25
⇒ x = ± 5

But since x is a natural number, the value of x must be positive.

Therefore, x = 5.
Hence, x³ = 5³ = 125

Question: The square root of (6 + 5√2)(6 - 5√2) is:

Solution:
√{(6 + 5√2)(6 - 5√2)}
= √{6² - (5√2)²}
= √{36 - (25 × 2)}
= √(36 - 50)
= √(- 14)
= √{14 × (- 1)}
= √(14 × i²) [∵ i² = - 1]
= i√14

Question: If x and y are positive integers satisfying x + y = 7, what is the probability that x < y?

Solution: 
Since both x and y must be positive integers,
total possible ways = (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 pairs.
Next, we identify the pairs where x < y,
(1, 6), (2, 5), (3, 4) ; There are 3 pairs satisfying x < y.

∴ Probability = Number of pairs where x < y/Total number of pairs = 3/6 = 1/2

So the probability that x < y is 1/2

Question: A car reaches from City A to City B in 9 hours travelling at a speed of 40 km/hr. If its speed is increased by 20 km/hr, then the time of journey is reduced by-

Solution: 
Original speed = 40 km/hr
Time taken = 9 hours

∴ Distance between City A and City B = speed × time
= 40 × 9 = 360 km

∴ New speed = 40 + 20 = 60 km/hr

∴ New time taken = distance/new speed
= 360/60
= 6 hours

∴ Reduction in time = original time - new time
= 9 - 6
= 3 hours

So the time of journey is reduced by 3 hours.

Question: The radius of a circular plate is 20 cm. If the radius is decreased by 20%, what is the percentage decrease in its area?

Solution:
Given that, 
Original radius = 20 cm
And decreased by 20%
 ∴ Decrease = 20% of 20 = 4 cm
∴ New radius = 20 - 4 = 16 cm

We know,
Area of circle = πr²
∴ Original area = π × (20)² = 400π cm²
∴ New area = π × (16)² = 256π cm²

∴ Decrease in area = original area - new area = 400π - 256π = 144π cm²

∴ Percentage decrease in area,
= (decrease/original area) × 100%
= (144π/400π) × 100%
= (144/400) × 100%
= 0.36 × 100%
= 36%

So the area decreases by 36%.

Question: If the simple interest on a sum of money is 40% of the principal over 8 years, what is the annual interest rate?

Solution: 
We know, Simple Interest, I = Pnr

Given, 
Simple Interest = 0.4 × Principal (since 40% of the principal)
= 0.4 × P

Where, Principal = P
Rate = r
Time, n = 8 years

So, 0.4 × P = P × 8 × r
⇒ r = 0.4/8
⇒ r = 0.05 × 100%
∴ r = 5%

A team of 6 members has to be selected from the 10 players.
This can be done in 10C6 or 210 ways.
Now, the captain can be selected from these 6 players in 6 ways.
Therefore, total ways the selection can be made is = 210 × 6
= 1260

Question: Today is Monday. After 59 days, it will be-

Solution:
We know that each day of the week is repeated after 7 days.

59 ÷ 7 = 8 (remainder 3)

So, after (7 × 8) = 56 days, it will be Monday.
∴ After 59 days, it will be (Monday + 3 days) = Thursday

Let be the number of students n
ATQ, 4n +20 = (n-3)5
⇒ n = 35
So, No. of books = 4×35 + 20 = 160

Let, y = f(x) = 2x - 1
or, y = 2x - 1
or, 2x = y + 1
or, x = (y + 1)/2
∴ y = f(x)
Or, f^-1(y) = x
or, f^-1(y) = (y + 1)/2
∴ f^-1(x) = (x + 1)/2

Given that a single dice is rolled.
Sample space = {1, 2, 3, 4, 5, 6}
Let P(A) be the probability of getting an odd number, where A = {1, 3, 5}
Let P(B) be the probability of getting a number less than 4, where B = {1,2,3}
A ⋂ B ={1, 3}
So, P(A) = 3/6 = 1/2
P(B) = 3/6 = 1/2
P(A ⋂ B) = 2/6 = 1/3

Let P be the required probability of getting an odd number or a number less than 4
P = P(A ⋃ B) = P(A) + P(B) - P(A ⋂ B)
Or, P(A ⋃ B) = 1/2 + 1/2 - 1/3
Or, P(A ⋃ B) = 1 - 1/3
Or, P(A ⋃ B) = 2/3

Therefore, the probability of rolling an odd number or a number less than 4 is 2/3.

There are 52 cards, out of which there are 12 face (Jack, King and Queen)) cards.
∴ Probability = 12/52 = 3/13

Question: A ladder is leaning against a wall. It makes a 60° angle with the ground. If the distance between the foot of the ladder and the wall is 8 meters, what is the length of the ladder?

Solution:

ধরি, দেয়ালটি হলো AB এবং মইটি হলো AC
মইটি ভূমির সাথে ∠ACB = 60° কোণ তৈরি করে।
মইয়ের গোড়া থেকে দেয়ালের দূরত্ব, BC = 8 মিটার।
মইয়ের দৈর্ঘ্য হলো AC

এখন, ΔABC -এ
cos 60° = BC/AC
⇒ 1/2 = 8/AC
⇒ AC = 8 × 2
⇒ AC = 16
∴ মইয়ের দৈর্ঘ্য 16 মিটার।