Bank Math

time = 1 hr 40 min 48 sec
= 1 hr + 40/60 hr + 48/3600 hr
= 1 + 2/3 + 1/75 = 126/75 hr
distance = 42 km
speed = distance/time = 42/( 126/75 ) = (42 × 75)/126 = 25 km/hr
⇒ 5/7 of the actual speed = 25
⇒ Actual speed = 25 × 7/5 = 35 km/hr

Question: A, B and C together invest Tk. 53000 in a business. A invests Tk. 5000 more than B and B invests Tk. 6000 more than C. Out of a total profit of Tk. 31800, find the share of A.

Solution:
Let C's investment = x
 Then,
B invests Tk. 6,000 more than C
∴ B = x + 6000
A invests Tk. 5000 more than B
∴ A = (x + 6000) + 5000 = x + 11000

Total investment,
A + B + C = 53000
⇒ (x + 11000) + (x + 6000) + x = 53000
⇒ 3x + 17000 = 53000
⇒ 3x = 36000  
∴  x = 12000 

Now,
C = x = 12000
B = 12000 + 6000 = 18000
A = 12000 + 11000 = 23000

∴ Profit sharing ratio is,
A : B : C = 23000 : 18000 : 12000 = 23 : 18 : 12

∴ Share of A = 31800 × (23/53) = Tk. 13800

Question: A fair coin is tossed 3 times. What is the probability of getting exactly two tails?

Solution:
একটি মুদ্রা 3 বার ছুঁড়লে মোট সম্ভাব্য ফলাফল (Total outcomes) হলো 2³ = 8 টি।
মোট ফলাফল (S) = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
মোট ফলাফলের সংখ্যা, n(S) = 8

'ঠিক দুটি টেইল' (exactly two tails) এর অনুকূল ফলাফল, E = {HTT, THT, TTH}
∴ অনুকূল ফলাফলের সংখ্যা, n(E) = 3

∴ সম্ভাবনা (Probability) = অনুকূল ফলাফলের সংখ্যা/মোট ফলাফল
P(E) = n(E)/n(S)
P(E) = 3/8

অতএব, ঠিক দুটি টেইল পাওয়ার সম্ভাবনা হলো 3/8।

ATQ, x + x + 3x + 25 = 180°
Or, 5x = 155°
∴ x = 31°

Question: In January, the value of a stock increased by 50%; and in February, it decreased by 20%. In March, it increased by 25%; and in April, it decreased by 10%. If a person invested Tk. 200 in the stock on January 1 and sold it at the end of April, what was the percentage change in the price of the stock?

Solution:
At the end of January,
The value of the stock is = Tk. 200 + 50% of (Tk. 200)
= Tk. 200 + Tk. 100 = Tk. 300.

At the end of February,
The value of the stock is = Tk. 300 - 20% of (Tk. 300)
= Tk. 300 - Tk. 60 = Tk. 240.

At the end of March,
The value of the stock is = Tk. 240 + 25% of (Tk. 240)
= Tk. 240 + Tk. 60 = Tk. 300.

At the end of April,
The value of the stock is = Tk. 300 - 10% of (Tk. 300)
= Tk. 300 - Tk. 30 = Tk. 270.

Now, the percentage change in price is,
= (Change in price/Original price) × 100%
= (270 - 200)/200 × 100%
= (70/200) × 100%
= 35%

Question: A pipe can fill a tank in p hours and another pipe can empty it in q hours (q > p). If both pipes are open together, in how many hours will the tank be filled?

Solution:

Let the tank capacity = 1 unit.
Filling pipe rate = 1/p (tank per hour)
Emptying pipe rate = 1/q (tank per hour → negative)

Net rate when both pipes are open = 1/p – 1/q = (q – p) / (pq)
Time to fill the tank = Total tank ÷ Net rate = 1 ÷ [(q – p)/ (pq)] = pq / (q – p) hours

Let present age of the elder person = x
The present age of the younger person = x - 16
According to the question,
x - 6 = 3(x - 16 - 6)
⇒ x - 6 = 3x - 66
⇒ 2x = 60
⇒ x = 30
Hence, the present age of the elder person is 30 years.

Question: A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is:

Solution:
Total quantity of sugar = 1000 kg

Let x kg be sold at 18% profit
Then (1000 - x) kg be sold at 8% profit

Profit from x kg at 18% = 18x/100
Profit from (1000 - x) kg at 8% = 8(1000 - x)/100

Total profit = 14% of 1000 = 14000/100 = 140

According to the question,
18x/100 + 8(1000 - x)/100 = 140
⇒ 18x + 8(1000 - x) = 14000
⇒ 18x + 8000 - 8x = 14000
⇒ 10x = 14000 - 8000
⇒ 10x = 6000
⇒ x = 600

∴ 600 kg of sugar was sold at 18% profit.

বিকল্প সমাধান:
By the rule of alligation,

Ration of 1^st and 2^nd parts = 4 : 6 = 2 : 3
∴ Quantity of 2^nd kind = (3/5) × 1000 = 600 kg

Question: A clock loses (falls behind) 15 minutes each day. How many days will it take to reach a point where the clock will indicate the correct time?

Solution:
একটি ঘড়িকে পুনরায় সঠিক সময় দেখাতে হলে তাকে 12 ঘণ্টা সময় হারাতে হবে। কারণ একটি 12 ঘণ্টার ঘড়িতে যখন সঠিক সময় থেকে 12 ঘণ্টা সময় পিছিয়ে পড়বে, তখন এটি আবার সঠিক সময় নির্দেশ করবে।
12 ঘণ্টা = 12 × 60 মিনিট = 720 মিনিট

এখন, যেহেতু ঘড়িটি প্রতিদিন 15 মিনিট করে সময় হারায়,

15 মিনিট সময় হারায় 1 দিনে
∴720 মিনিট সময় হারাবে = (1/15) × 720 দিনে
= 48 দিনে
সুতরাং, 48 দিন পর ঘড়িটি আবার সঠিক সময় দেখাবে।

Question: What will come at the place of question mark?
3, 6, 11, 18, 27, ?

Solution:
Look at the differences between consecutive terms.
6 - 3 = 3
11 - 6 = 5
18 - 11 = 7
27 - 18 = 9
The first differences are 3, 5, 7, 9
These increase by + 2 each time (3 ⇒ 5 ⇒ 7 ⇒ 9).
So the next difference should be = 9 + 2 = 11

Therefore next number = 27 + 11 = 38

When at least 2 women are included.
The committee may consist of 3 women, 2 men : It can be done in ⁴C₃ × ⁶C₂ : ⁴C₃ × ⁶C₂ ways.
or, 4 women, 1 man : It can be done in ⁴C₄ × ⁶C₁ : ⁴C₄ × ⁶C₁ ways.
or, 2 women, 3 men : It can be done in ⁴C₂ × ⁶C₃ : ⁴C₂ × ⁶C₃ ways.
Total number of ways of forming the committees,
= ⁴C₂ × ⁶C₃ + ⁴C₃ × ⁶C₂ + ⁴C₄ × ⁶C₁
= 6 × 20 + 4 × 15 + 1 × 6
= 120 + 60 + 6
= 186

We know,
The ratio of Investment x Time = Ratio of Profit
∴ (A's investment x Time) : (B's investment x Time) = Profit of A : Profit of B

The total value of investment of Arman, Ghalib, and Jishan after 12 months is =
Tk. 40000 x 12 months : Tk. 80000 x 7 months : Tk. 144000 x (12 - 7)months = 480000 : 560000 : 720000
∴ Profit ratio = 240000 : 280000 : 360000 = 6 : 7 : 9
∴ Share of Ghalib = 7/(6 + 7 + 9) x 50600 = Tk. 16100

The difference in interest rate = 6(1/4)% - 4%
= (25/4)% - 4%
= (9/4)%
Gain per year
= simple interest on Tk. 5000 at = (9/4)% for 1 year.
= {5000 × (9/4) × 1}/100
= 225/2
= Tk. 112.5

Question: A train moving at speed of 108 km/hr crosses a pole in 12 seconds. Find the length of the train.

Solution:
Length of the train is equal to the distance covered by train to cross the pole.
So, we will find the distance travelled by the train in 12 seconds.

Now,
Speed is given in Km/hr so we will convert it into m/s
Speed = 108 × (5/18) = 30 m/s
Time = 12 seconds

We know, 
Distance = Speed × Time
∴ Length of train = 30 × 12 = 360 meters

Question: In how many years will Tk. 15,000 amount to Tk. 18,600 at 12% per annum simple interest?

Solution:
Principal, P = Tk. 15,000
Amount, A = Tk. 18,600

∴ Simple interest, I = A - P
= 18,600 - 15,000
= Tk. 3600

Rate of interest, r = 12%

We know,
I = Pnr
⇒ 3,600 = 15,000 × n × (12/100)
⇒ 3,600 = 15,000 × n × (3/25)
⇒ 3,600 = 1,800n
⇒ n = 3,600/1,800
∴ n = 2 years

Question: If a shopkeeper buys eggs at Tk. 42 per sext (six eggs) and sells them at Tk. 56 per two quads (eight eggs), what percentage profit does he make?

Solution:
Cost price of 6 eggs = Tk. 42
Cost price of 1 egg = 42/6
= 7 Tk.

Cost price of 8 eggs = 8 × 7
= 56
Selling price of 8 eggs = Tk. 56

∴ Profit = 56 − 56
= 0 

∴ As the selling and cost prices are the same, there is no profit. 

Question: The square root of (7 + 3√5) (7 - 3√5) is ?

Solution:
Given that, 
{(7 + 3√5) (7 - 3√5)}
= √{(7)² - (3√5)²}
= √(49 - 45)
= √4
= 2

Question: What is the minimum value of sin θ?

Solution:
sinθ এর সর্বনিম্ন মান - 1 এবং সর্বোচ্চ মান 1
cosθ এর সর্বনিম্ন মান - 1 এবং সর্বোচ্চ মান 1

ধরি, সুদের হার r %
প্রশ্নমতে, (400 × 2 × r)/100 + (550 × 4 × r)/100 + (1200 × 6 × r)/100 = 1020
⇒ 8r + 22r + 72r = 1020
⇒ 102r = 1020
⇒ r = 10%

Suppose
a = 3, b = 2
Option a) a² > b² = (3)² > (2)² = 9 > 4; True
Option b) a² < ab = (3)² < 3×2 = 9 < 6; False
Option c) a - b < 0 = 3 - 2 < 0 = 1 < 0; False
Option d) b + a > 2a = 2 + 3 > 2×3 = 5 > 6; False

Let the age of Ali, Gazi and Masud be respectively 2x, 3x and 4x years

ATQ, 10 years ago, 2x + 3x + 4x = 93 - 30
9x = 63
x = 7
So, 10 years ago Masud's age was, 4x = 28 years

Now, Masud's age is = 38 years

যেহেতু aþb = ab/(a + b)
∴ 10þ2 = (10 × 2)/(10 + 2)
= 20/12
= 5/3

Question: There is a dishonest shopkeeper whose claim is that he sells a certain product at a cost of Tk 23/kg, which actually costs him Tk 25/kg. The shopkeeper says that he is taking the loss to let his customers get a better deal. When examined thoroughly, a policeman finds that the shopkeeper is actually using an 800 gms weight in place of a 1 kg weight. How much does he gain or lose?

Solution: 
selling price of 0.8 kg = 23 taka

selling price of 1 kg = 23/0.8 taka 
= 115/4 taka
= 28.75 taka 

percentage profit = {(28.75 - 25)/25} × 100%
= (3.75/25) × 100%
= 15% 

According to the question,
∴ 50 m = 20 m
∴ 1m = 20/50 m
∴ 1000 m = (20/50)×1000 m
= 400 metres

Parts of B = 72 × (3/24) kg
= 9 kg

Question: Which of the following is irrational?

Solution:
একটি সংখ্যা অমূলদ (irrational) হয় যদি এটি p/q আকারে প্রকাশ করা না যায়, যেখানে p এবং q পূর্ণসংখ্যা এবং q ≠ 0।

ক) 0.75 = 75/100 = 3/4 = এটি p/q আকারে প্রকাশ করা যায়, তাই এটি মূলদ সংখ্যা।

খ) √289 = 17 একটি পূর্ণবর্গ সংখ্যা (17² = 289), তাই √289 = 17 একটি মূলদ সংখ্যা।

গ) 3/5 = এটি ইতিমধ্যে p/q আকারে আছে, তাই এটি মূলদ সংখ্যা।

ঘ) √18 = √(9 × 2) = 3√2 একটি পূর্ণবর্গ সংখ্যা নয়, তাই √18 একটি অমূলদ সংখ্যা। এটি p/q আকারে প্রকাশ করা যায় না।

উত্তর: ঘ) √18 একটি অমূলদ (irrational) সংখ্যা।

Question: An observer who is 1.5 meters tall is standing 10√3 meters away from a flagpole. If the angle of elevation from his eye to the top of the flagpole is 30°, what is the height of the flagpole?

Solution:

এখানে,
পর্যবেক্ষকের উচ্চতা, CD = 1.5 মিটার
এখানে, CD = EB
পতাকা দণ্ডের (Flagpole) উচ্চতা, = 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

Question: What is the average (arithmetic mean) of the values 36, 42, 40, 38, 44, 40, and 40?

Solution:
Find the sum of all values:
36 + 42 + 40 + 38 + 44 + 40 + 40 = 280

Divide the sum by the total number of values:
Average = 280 ÷ 7 = 40