Bank Math

Question: If Profit = 25 Taka and Cost Price = 150 Taka, what is the profit percentage?

Solution:
Here,
Cost Price = 150 Tk
Profit = 25 Tk

Now,
Cost Price is 150 and Profit is 25 Taka.
∴ When cost Price is 1 Taka then the Profit is = 25/150 Taka
∴ When cost Price is 100 Taka then the Profit is = (25× 100)/150 Taka
 = 16.6 %

Question: A couple has a son and a daughter. The age of the father is three times the daughter's age and the age of the mother is twice the son's age. The father is 9 years older than the mother and the daughter is 6 years younger than the son. What is the age of the mother?

Solution:
Let the ages of father, mother, son, and daughter be F, M, S, and D respectively.

According to the question (ATQ):
F = 3D .............(i)
M = 2S .............(ii)
F = M + 9 ...........(iii)
S = D + 6 ...........(iv)

From (ii) we get,
M = 2S
Or, M = 2(D + 6) [from (iv)]
Or, M = 2{(F/3) + 6} [from (i)]
Or, M = (2/3) × (F + 18)
Or, M = (2/3) × (M + 9 + 18) [from (iii)]
Or, M = (2/3) × (M + 27)
Or, 3M = 2M + 54
Or, M = 54

∴ The age of the mother is 54 years.

Let, the numbers be a and b where a > b.
According to the question,
a - b = 10 ....... (i)
And (a + b)/5 = 8
By cross multiplying, we get
⇒ a + b = 40 ...... (ii)
By subtracting equation (ii) from (i) we get
2b = 40 - 10 = 30
⇒ b = 30/2 = 15

And from (i)
a = 10 + 15 = 25
Answer: 15

Question: Two vessels A and B contain milk and water mixed in the ratio 4 : 3 and 2 : 3 respectively. What will be the new ratio of milk to water if these two mixtures are mixed together in equal quantities? Solution:

সমাধান:
ধরি, পাত্র A এবং B এর মিশ্রণের পরিমাণ সমান, অর্থাৎ ১ একক।
পাত্র A তে দুধের পরিমাণ = 4/(4 + 3) = 4/7
পাত্র A তে পানির পরিমাণ = 3/(4 + 3) = 3/7

পাত্র B তে দুধের পরিমাণ = 2/(2 + 3) = 2/5
পাত্র B তে পানির পরিমাণ = 3/(2 + 3) = 3/5

নতুন মিশ্রণে মোট দুধের পরিমাণ = (4/7) + (2/5)
= (20 + 14)/35 
​= 34/35

নতুন মিশ্রণে মোট পানির পরিমাণ = (3/7) + (3/5)
= (15 + 21)/35
​ = 36/35

∴ নতুন অনুপাত = (34/35) : (36/35)
= 34 : 36
= 17 : 18

speed of A : speed of B = 7/3:1 = 7:3
It means, in a race of 7 metres, A gains (7−3 ) = 4 metres.
If A needs to gain 80 metres, race should be of (7/4)×80 = 140 metres.

Question: Find the value of x if log_x 324 = 4.

Solution:
log_x324 = 4
⇒ x⁴ = 324
⇒ (x²)² = 18²
⇒ x² = 18²
⇒ x = √18
⇒ x = √3² × 2
⇒ x = 3√2

Length of the parallel sides of prism = 10 cm and 6 cm
Height of prism = 8 cm
∴ Volume of prism = (1/2){(10 + 6) × 5 × 8}
= (1/2) × 16 × 5 × 8
= 320 cm³

Question: What least value must be assigned to 'a' so that the number 197a5462 is divisible by 9?

Solution:
We know,
A number is divisible by 9 if the sum of its digits is divisible by 9.

The number is: 1 9 7 a 5 4 6 2
∴ Sum of the known digits = 1 + 9 + 7 + 5 + 4 + 6 + 2 = 34
∴ Total sum of digits = 34 + a
For (34 + x) to be divisible by 9, 
The multiples of 9 just above 34 are: 36, 45, 54, …
a = 36 - 34 = 2
a = 45 - 34 = 11 ; [wrong because a must be a single digit]

∴ So the least value of a = 2

Question: A man bought a book for Tk. 360 after getting a 20% discount. What was the original catalog price of the book?

Solution:
20% discount,
If the catalog price is Tk. 100 then the purchased price = (100 - 20) = Tk. 80

Now,
If the purchase price is Tk. 80, the catalog price = 100 taka.
If the purchase price is Tk. 1, the catalog price = 100/80 taka
If the purchase price is Tk. 360, the catalog price = (100 × 360)/80
= Tk. 450

Question: A train passes by a lamp post at platform in 6 sec. and passes by the platform completely in 24 sec. If the length of the platform is 360 meter, then length of the train is?

Solution:
Let, the length of the train is x meter.
The train passing the lamp post at a speed of x/6 meter/sec
And, the train passing the platform at a speed of (360 + x)/24 meter/sec

∴ x/6 = (360 + x)/24
⇒ x = (360 + x)/4
⇒ 4x = 360 + x 
⇒ 3x = 360
⇒ x = 360/3
⇒ x = 120

∴ Length of the train is 120 meter.

আমরা জানি,
বাক্সের উপরিতলের ক্ষেত্রফল = 2(ab + bc + ca)
= 2(2 × 3 + 3 × 4 + 4 × 2)
= 52 বর্গমিটার
∴ মোট খরচ = 52 × 3 = 156

Speed downstream = 15 km/hr
Rate of the current = 1(1/2) km/hr.
Speed in still water = {15 - 1(1/2)}
= 13(1/2) km/hr.
Rate against the current = {13(1/2) - 1(1/2)} km/hr
= 12 km/hr.

Question: A man borrowed some money for 90 days. He paid Tk. 270 as interest at the rate of 8% per annum. What was the amount he borrowed?

Solution:
এখানে,
সময়, n = 90 দিন
= 3 মাস = 3/12 বছর
= 1/4 বছর

মুনাফা, I = 270 টাকা
মুনাফার হার, r = 8% = 8/100 = 2/25
আসল, P = ?

আমরা জানি,
I = P × n × r
⇒ P × n × r = I
⇒ P = I/(n × r)
= 270/{(1/4) × (2/25)}
= 270/(1/50)
= 270 × 50
= 13500 টাকা

Question: How many diagonals can be drawn in an octagon?

Solution:
An octagon has n = 8 vertices and 8 sides.
Total number of lines formed by joining 2 vertices out of 8 is: ⁸C₂
∴ Total lines = ⁸C₂
= 8!/{2! × (8 - 2)!}
= (8 × 7)/2 
= 28

These 28 lines include both the sides and the diagonals of the octagon.

Number of diagonals = Total lines - Number of sides
= 28 - 8
= 20

∴ The number of diagonals in an octagon is 20.

3 pumps in 2 days empties by working 8 hrs
1      ''         2  ''           ''             ''           ''   (8 × 3)  hrs
1      ''         1  ''           ''             ''           ''   (8 × 3 × 2)  hrs
∴ 4   ''         1  ''           ''             ''           ''   (8 × 3 × 2) /4 hrs
                                                                 = 12 hrs

Question: 

Solution: 

Let the numbers be 37a and 37b.
Then,
37a x 37b = 4107
=> ab = 3.
Now co-primes with product 3 are (1, 3)
So, the required numbers are (37 x 1, 37 x 3) i.e, (37,111).
Therefore, Greater number = 111.

Question: The speed of a boat in still water is 15 km/h, and the speed of the current is 5 km/h. In how much time (in hours) will the boat travel a distance of 60 km upstream and the same distance downstream?

Solution: 
Given that, 
Speed of boat in still water = 15 km/h
Speed of current = 5 km/h
Distance (each way) = 60 km

∴ Downstream speed = 15+ 5 = 20 km/h
∴ Upstream speed = 15 - 5 = 10 km/h

∴ Time to travel 60 km downstream = 60/20 = 3 hours
∴ Time to travel 60 km upstream = 60/10 = 6 hours

∴ Total time = 3 + 6 = 9 hours

∴ The boat will take 9 hours in total.

Question: Solve: 3(2x - 1) ≥ 4(x + 5), Then what is the solution set?

Solution:
Given the inequality,
3(2x − 1) ≥ 4(x + 5)
⇒ 6x - 3 ≥ 4x + 20
⇒ 6x - 4x - 3 ≥ 20 ; [Subtract 4x from both sides]
⇒ 2x - 3 ≥ 20
⇒ 2x ≥ 23 ; [Add 3 to both sides] 
⇒ x ≥ 23/2 ; [Divide both sides by 2]
∴ x ≥ 11.5

Solution set: In interval notation: [11.5, ∞)

Question: Two successive discounts of 25% and 15% are equal to a single discount of___

Solution:
Formula for successive discounts
Single equivalent discount = d₁ + d₂ - (d₁ × d₂)/100
⇒ d = (25 + 15) - (25 × 15)/100
⇒ d = 40 - (375/100)
⇒ d = 40 - 3.75
d = 36.25

∴ Single discount = 36.25%

Question: x² + y² + z² = 2(x + z - 1), then the value of x³ + y³ + z³ = ?

Solution:
Given that,
x² + y² + z² = 2(x + z - 1)
⇒ x² + y² + z² = 2x + 2z - 2
⇒ x² + y² + z² = 2x + 2z - 1 - 1
⇒ (x² + 1 - 2x) + y² + (z² + 1 - 2z) = 0
⇒ (x - 1)² + y² + (z - 1)² = 0
We know,
The sum of three squares of real numbers can only be zero if each individual square is zero.
So,
(x - 1)² = 0
∴ x = 1

y² = 0
∴ y = 0
And,
(z - 1)² = 0
∴ z = 1

Substitute the values of x, y, z into the expression,
x³ + y³ + z³
= 1³ + 0 + 1³
= 1 + 1
= 2

P + C + M = C + 120
⇒ P + M = 120
∴ Required average = (P + M)/2 = 120/2
= 60

Question: In a mixture of milk and water, the ratio is 4 : 3. If 5 liters of water is added, the new ratio becomes 4 : 4. What was the original amount of milk in the mixture?

Solution:
ধরি, শুরুতে দুধ ছিল = 4x লিটার,
পানি ছিল = 3x লিটার।

এখন 5 লিটার পানি যোগ করলে, নতুন পানি = 3x + 5 লিটার

ATQ,
4x/(3x + 5) = 4/4
⇒ 4x/(3x + 5) = 1
⇒ 4x = 1 × (3x + 5)
⇒ 4x = 3x + 5
⇒ 4x - 3x = 5
⇒ x = 5

∴ দুধের পরিমাণ = 4x = 4 × 5 = 20 লিটার

Here,
(28+14+7+4+2+1)/ 28 = 56/ 28 =2

Question: Write an equation of the line with slope 2 and x-intercept (- 4, 0).

Solution: 
Given that, 
Slope m = 2
x-intercept (- 4, 0)

We know, 
y - y₁ ​= m(x - x1​)
⇒ y - 0 = 2{x - (- 4)}. ; [Here, (x₁, y₁) = (- 4, 0) and m = 2]
⇒ y = 2(x + 4)
∴ y = 2x + 8

So the equation of the line is  y = 2x + 8.

3x + 2y = 10 .... (i)
3x - 2y = 8 .... (ii)

(i) + (ii), 6x = 18
Or, x = 3

From, (i), y = 1/2

So, xy = 3.1/2 = 3/2

Question: Two numbers have LCM = 180 and HCF = 6. If one number is 30, find the other.

Solution:
Let the numbers be a=30 and b=?, with HCF = 6 and LCM = 180.
Use the formula connecting LCM and HCF:
a × b = HCF × LCM
30 × b = 6 × 180
b = 1080/30 
b = 36

Question: Having incurred a 20% loss on a saree sold for TK 3200, a shopkeeper now wishes to set a marked price that allows for a 25% profit, even after applying an 10% discount. What is that price?

Solution:
After selling a saree for TK. 3200 a shopkeeper suffers a loss of 20%.

Selling price TK. 80 when cost price TK. 100
∴ selling price TK. 3200 when cost price = TK. (100 × 3200)/80
= TK. 4000

25% profit,
cost price TK. 100 then selling price TK. 125
∴ cost price TK. 4000 then selling price = TK. (125 × 4000)/100
= TK. 5000

discount 10%
selling price TK. 90 when marked price = TK. 100
∴ selling price TK. 5000 when marked price = (100 × 5000)/90
= TK. 5555.55 

এখানে,
প্রথম রাশি, (1 - a)² = (1 - a)(1 - a)
দ্বিতীয় রাশি, (1 - a)
তৃতীয় রাশি, (a - 1)² = {-(1 - a)}² = (1 - a)² = (1 - a)(1 - a)
∴ LCM = (1 - a)(1 - a) = (1 - a)²

Question: If 40 workers working at 80% efficiency can complete a task in 15 days, how many workers working at 100% efficiency would be needed to complete the same task in 10 days?

Solution: 
Case-1: 
W = 40 × 0.8 × 15
= 480

Case-2:
Let the required number of workers be x.
W = x × 1 × 10
480 = 10x
∴ x = 48

Question: Find the odd one out.

Solution: 
Here, 'first number × 2 = second number ; second number × 3 = third number' 

Look at the pattern in each option.
a) 2 : 4 : 12
2 × 2 = 4
4 × 3 = 12

c) 7 : 14 : 42 
7 × 2 = 14
14 × 3 = 42

d) 5 : 10 : 30 
5 × 2 = 10
10 × 3 = 30

But,
b) 9 : 18 : 72
9 × 2 = 18, but 18 × 4 = 72 (not × 3)
Here the multiplier changes to × 4 instead of × 3

Therefore, option b) breaks the consistent pattern of × 2 followed by × 3.

So the odd one out is b) 9 : 18 : 72

Let,
Speed of boat = x and, Speed of stream = y
ATQ,
x - y = 24/6 = 4 ...... (i)
x + y = 24/4 = 6 ....... (ii)
From,
(ii) – (i),
x + y - x + y = 6 - 4
Or, 2y = 2
∴ y = 1
So, Speed of the stream = 1 km/hr

Question: In a class of 100 students, 45 are taking Mathematics, 35 are taking English and 15 are taking both courses. How many students are not enrolled in either course?

Solution:
এখানে মোট শিক্ষার্থী সংখ্যা N = 100
 গণিত নেওয়া শিক্ষার্থীর সংখ্যা n(M) = 45, 
ইংরেজি নেওয়া শিক্ষার্থীর সংখ্যা n(E) = 35
উভয় বিষয় নেওয়া শিক্ষার্থীর সংখ্যা n(M ∩ E) = 15

আমরা জানি, 
অন্তত একটি কোর্স নেওয়া শিক্ষার্থীর সংখ্যা, n(M ∪ E) = n(M) + n(E) - n(M ∩ E)
=  45 + 35 - 15
= 65

∴ কোনো কোর্সেই ভর্তি না হওয়া শিক্ষার্থীর সংখ্যা = N - n(M ∪ E)
= 100 - 65
= 35

Question: Find an equation of the vertical line containing the point (9, - 3).

Solution:
দেওয়া আছে, 
প্রদত্ত বিন্দুটি হলো (9, -3)।

উল্লম্ব রেখার একটি প্রধান বৈশিষ্ট্য হলো, এই রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক সর্বদা একই থাকে। যেহেতু রেখাটি (9, -3) বিন্দু দিয়ে যায়, তাই রেখাটির উপর অবস্থিত প্রতিটি বিন্দুর x-এর মান হবে 9।

সুতরাং, নির্ণেয় সমীকরণটি হবে x = 9.

Let the total investment be Tk. z
Then, 20% of z = 98000
⇒ z = 98000 × 100)/20
= 490000.
Let the capitals of P, Q and R be Tk. 5x, Tk. 6x and Tk. 6x respectively.
Then, (5x × 12) + (6x 12) + (6x × 6) = 490000 × 12
⇔ 168x = 490000 × 12 ⇔ x = (490000 × 12)/168 = 35000.
∴ R's investment = 6x = Tk. (6 × 35000)
= Tk. 210000.