Bank Math

Question: The average age of a group is 28. If a 42-year-old person leaves, the average becomes 26. How many people are in the group?

Solution: 
Let
The original number of people be x.

Total age of the group:
= 28 × x
= 28x

After the 42-year-old leaves, total age:
= 28x - 42

Number of people remaining = x - 1
New average = 26

Accordingly, 
(28x - 42) / (x - 1) = 26
⇒ 28x - 42 = 26(x - 1)
⇒ 28x - 42 = 26x - 26
⇒ 28x - 26x = - 26 + 42
⇒ 2x = 16
⇒ x = 8

Question: A started a business with a capital of Tk. 1,20,000. After some time, B joined the business with Tk. 80,000. At the end of one year, the profit was divided between A and B in the ratio 3 : 1. For how many months did B invest in the business?

Solution:
Let B join the business for X months.
A's investment is for 12 months,
B's for X months.

The ratio of profits is the ratio of (capital × time):
(1,20,000 × 12)/(80,000 × X) = 3/1
⇒ 1,20,000 × 12 = 80,000 × 3 × X
⇒ (1,20,000/80,000) × 12 = 3X
⇒ 1.5 × 12 = 3X
⇒ 18 = 3X
⇒ X = 6
Thus, B joined for 6 months.

Question: A man completes a journey in 5 hours. He travels the first half of the journey at the rate of 20 km/hr and the second half at the rate of 30 km/hr. Find the total journey in km.

সমাধান:
ধরা যাক, মোট যাত্রার দূরত্ব হলো D কিমি।
তাহলে, যাত্রার প্রথম অর্ধেকের দূরত্ব হবে D/2 কিমি এবং দ্বিতীয় অর্ধেকের দূরত্বও হবে D/2 কিমি।

প্রথম অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (D/2)/20 ঘন্টা
= D/40 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (D/2)/30 ঘন্টা
= D/60 ঘন্টা

প্রশ্নমতে,
D/40 + D/60 = 5
⇒ (3D + 2D)/120 = 5
⇒ 3D + 2D = 5 × 120
⇒ 5D = 600
⇒ D = 600/5
⇒ D = 120 কিমি

সুতরাং, মোট যাত্রার দূরত্ব হলো 120 কিমি।

Question: x - y = 4 and xy = 45, find the value of x² + y² = ?

Solution:
Given that,
$x\; -\; y=4$
and xy = 45

We know,
x² + y² = (x - y)2 + 2xy
= 4² + 2 × 45
= 16 + 90
= 106​

Question: A ferry can carry 30 trucks or 50 motorcycles at a time. If there are 18 trucks on the ferry, how many motorcycles can be loaded onto it?

Explanation:
Given,
30 trucks = 50 motorcycles
∴ 1 truck = 50/30 = 5/3 motorcycles
∴ 18 trucks = 18 × 5/3 = 30 motorcycles

Maximum motorcycles on ferry = 50

∴ Remaining motorcycles that can be loaded = 50 - 30 = 20
So, the ferry can carry 20 more motorcycles along with the 18 trucks.

Question: Find the least number which will leaves remainder 5 when divided by 8, 12, 16 and 20.

Solution: 
We have to find the Least number, therefore we find out the LCM of 8, 12, 16 and 20.
8 = 2 × 2 × 2;
12 = 2 × 2 × 3;
16 = 2 × 2 × 2 × 2;
20 = 2 × 2 × 5

∴ LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240

This is the least number which is exactly divisible by 8, 12, 16 and 20.
So, Required number which leaves remainder 5 is = 240 + 5 = 245.

Question: If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is the value of P + 2Q = ? 

Solution:
Given that,
Five-digit number 750PQ is divisible by 3, 7 and 11

Now, The LCM of 3, 7, and 11 is 231.

By taking the largest 5-digit number 75099 and dividing it by 231.
If we divide 75099 by 231 we get 325 as the quotient and 24 as the remainder.

Then, the five-digit number is 75099 - 24 = 75075.

The number = 75075 and P = 7, Q = 5

Now, P + 2Q = 7 + 2 × 5
= 7 + 10
= 17

∴ The value of P + 2Q is 17.

Question: If the price of Onion is increased by 50% by what fraction must its consumption be reduced so as to keep the same expenditure on its consumption?

Solution: 
let, x kg onion costs 100 taka 

increased by 50%, x kg costs = 100 + 100 × .5 = 150 taka

in 100 taka, new amount = 100x/150 = 2x/3 kg 


fraction must its consumption be reduced = {x - (2x/3)}/x
= 1/3

Question: A rectangular prism is 24 inches by 15 inches by 10 inches. Inside it is a cylinder with radius 5 inches and height 7 inches. Rounded to the nearest cubic inch, what is the volume of the prism not taken up by the cylinder?

Solution: 
Volume of the rectangular prism.
Volume = length × width × height
= 24 × 15 × 10
= 3600 cubic inches

And
Volume of the cylinder.
Volume = π × r² × h
= (22/7) × 5² × 7 ; [r = 5 in, h = 7 in] 
= 22 × 25
= 550 cubic inches

∴ Empty space = prism volume - cylinder volume
= (3600 - 550) cubic inches
= 3050 cubic inches

Friends, we know we will need gain amount to get gain percent, right.
So, lets get gain first.
Let the cost price of 1 pen is Tk 1
Cost of 8 pens = Tk 8
Selling price of 8 pens = 12
Gain = 12 - 8 = 4
Gain% = (gain/cost × 100)%
= (4/8 × 100)%
= 50%

In the question, the first piece of relevant information given is that eighteen years ago a father was three times as old as his son. Mathematically an equation can be developed from this but some variables needs to be assigned before we can do this.
Let father’s current age = x (in years)
Let son’s current age = y (in years)
x - 18 = 3(y - 18)
⇒ 3y - 54 = x - 18
⇒ 3y = x + 36
∴ y = x/3 + 12 … (1)

The second piece of information is that currently the father is twice as old as his son. Mathematically a second equation can be derived from this and simultaneous equations can be used where there are two variables and two equations.
x = 2y … (2)
put (1) into (2)
⇒ x = 2(x/3 + 12)
⇒ x = 2x/3 + 24
⇒ x/3 = 24
⇒ x = 72
⇒ x = 2y
⇒ y = x/2
⇒ y = 72/2
∴ y = 36
The father’s age is 72 years old and the son’s age is 36 years old.
Therefore, sum of present ages of son and father = 36 + 72 = 108 years.

Question: With an average speed of 50 km/hr, a train reaches its destination in time. If it goes with an average speed of 40 km/hr, it is late by 24 min. The total journey is:

Solution:
Difference between timings = 24 min = 24/60 hr = 2/5 hr.
Let the length of the journey be x km.

Then,
(x/40) - (x/50) = 2/5
Or, (5x - 4x)/200 = 2/5
Or, x/200 = 2/5
Or, x = (25 × 200)
∴ x = 80 km.

∴ The total journey is 80 km.

Question: How many litres of water should be added to a 65 litre mixture of milk and water containing milk and water in the ratio of 8 : 5 such that the resultant mixture contains 50% water?

Solution: 
Given that, 
The mixture contains 65 litres of milk and water in the ratio 8 : 5.
Total parts = 8 + 5 = 13
Milk = (8/13) × 65 = 40 litres
Water = (5/13) × 65 = 25 litres

Let x litres of water be added.
∴ New water quantity = 25 + x litres
∴ New total mixture = 65 + x litres

The resultant mixture should contain 50% water.
(25 + x)/(65 + x) = 1/2
⇒ 2(25 + x) = 65 + x 
⇒ 50 + 2x = 65 + x 
⇒ 2x - x = 65 - 50 
∴ x = 15 

So 15 litres of water should be added.

Let original length = x and original breadth = y.
Original area = xy.
New length = x/2 and New breadth = 3y
New area = 3/2xy
Therefore,
Increase in area = New area-original area = 3/2xy - xy = 1/2xy
Therefore,
Increase % = increase in area/original area × 100 = {(1/2)xy/xy} × 100 = 50%

Let the average of 8 men be x.
Age of new man be N.
Total age of 8 men = 8x
⇒ x + 4 = ((8x – 30) + N)/8
⇒ 8x + 32 = 8x – 30 + N
⇒ N = 32 + 30
⇒ N = 62

Question: 25 men can complete a job in 36 days. After 12 days, 5 men left. How many days will the remaining men take to finish the job?

Solution:
25 men can complete a job in 36 days
∴ Total work = 25 × 36 = 900 man-days

∴ Work done in first 12 days = 25 × 12 = 300 man-days

And,
Remaining work = 900 - 300 = 600 man-days
Remaining days = 25 - 5 = 20 men

So,
Remaining days = 600/20 = 30 days 

∴ The remaining men will take 30 days to finish the job.

Question: Fresh grapes contain 80% water, while dried grapes contain 20% water. If the weight of the dried grapes is 400 kg, what was the total weight of the grapes when fresh? 

Solution:
Given that,
Dried grapes contain 20% water
∴ They contain 80% solid matter. 
Fresh grapes contain 80% water
∴ They contain 20% solid matter.

The amount of solid matter never changes during drying.

Let the weight of fresh grapes = x kg
Solid matter in fresh grapes = 20% of x = 0.20x kg
Solid matter in dried grapes = 80% of 400 kg
= (80/100) × 400
= 320 kg

Since the solid matter remains the same,
0.20x = 320
⇒ x = 320 / 0.20
⇒ x = (320 × 100)/20
∴ x = 1600 kg

The total weight of the grapes when fresh was 1600 kg.

Question: The average age of a group of 15 employees is 28 years. When 5 more employees join the group, the average age increases by 2 years. The average age of the new employees is?

Solution: Average age of 15 employees = 28 years
Total age of 15 employees = 15 × 28 = 420 years

After 5 more employees join:
Total number of employees = 15 + 5 = 20 employees
New average age = 28 + 2 = 30 years
Total age of 20 employees = 20 × 30 = 600 years

Total age of 5 new employees = 600 - 420 = 180 years

Average age of 5 new employees = 180 ÷ 5 = 36 years

Let the ten-digit be x, the unit digit is y.

According to the question,
(10x + y) - (10y + x) = 36
⇒ 9x - 9y = 36
⇒ x - y = 4.

Question: In a right-angled triangle ABC, where B is 90° and  tanC = 3/4, what is the length of the hypotenuse of triangle ABC?
 
Solution:
We know,
tanC = Perpendicular/Base
= 3/4

So,
Perpendicular of triangle ABC = 3
Base of triangle ABC = 4

By Pythagoras theorem,
(Hypotenuse)² = (Perpendicular)² + (Base)²
⇒ (Hypotenuse)² = 3²+ 4²
⇒ (Hypotenuse)² = 9 + 16
⇒ (Hypotenuse)² = 25
∴ Hypotenuse = 5

Let the average weight of the 59 students be A.
So the total weight of 59 of them will be = 59 × A = 59A

The questions state that when the weight of this student who left is added,
the total weight of the class=59A + 45
When this student is also included, the average weight decreases by 0.2 kgs

(59A + 45)/60 = (A - 0.2)
⇒ 59A + 45 = 60A - 12
⇒ 45 + 12 = 60A - 59A
⇒ A = 57.

আমরা জানি,
Fibonacci সংখ্যা = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... পরপর দুটি সংখ্যার যােগফল পরবর্তী সংখ্যার সমান।
সুতরাং, এই ধারার পরবর্তী সংখ্যা হবে = 21 + 34 = 55

Let the marked price of the item be Tk. 100x
Discount % = 20%
Changes for delivery and packaging
= 10% on discounted price

Then
20% discount of 100x = 80x
Charges 10% on the discount price 80x = 88x
⇒ 88x + 88x × (25/100) - 88x = 2046
⇒ 110x - 88x = 2046
⇒ 22x = 2046
⇒ x = 93
Hence, 100x = 93 × 100
= 9300
Marked price = Tk. 9300

Question: What will be the number in the question mark?
3, 7, 14, 24, 37, ?

Solution:
প্রদত্ত ধারাটি হলো: 3, 7, 14, 24, 37, ?

ধারার সংখ্যাগুলোর মধ্যে পার্থক্য নির্ণয় করি:
7 - 3 = 4
14 - 7 = 7
24 - 14 = 10
37 - 24 = 13

এখানে, প্রতিবার পার্থক্য 3 করে বৃদ্ধি পাচ্ছে।
∴ পরবর্তী পার্থক্য হবে = 13 + 3 = 16
∴ পরবর্তী সংখ্যাটি হবে = 37 + 16 = 53

Question: A couple has a son and a daughter. The age of the father is four times the son and the age of the mother is three times the daughter. The mother is 6 years younger to the father and the daughter is 3 years older than the son. What is the age of the mother?

Solution:
Let, father, mother, son and daughter have age of F, M, S and D respectively.
ATQ,
F = 4S ………….(i)
M = 3D………….(ii)
M = F - 6………..(iii)
D = S + 3………..(iv)

From (ii) we get,
M = 3D
Or, M = 3(S + 3)  [from (iv)]
Or, M = 3{(F/4) + 3}  [from (i)]
Or, M = ¾(F + 12)
Or, M = ¾(M + 6 + 12)  [from(iii)]
Or, M = ¾(M + 18)
Or, 4M = 3M + 54
Or, M = 54

∴ the age of the mother is 54 years.

Question: If 8x - 3y = 24 and y = 0, then x = ?

Solution:
Here,
8x - 3y = 24  ............(1)
y = 0  .............(2)

Putting y = 0 into equation (1) we get,
⇒ 8x - (3 × 0) = 24
⇒ 8x = 24
⇒ x  = 24/8
∴ x = 3

From the given data,
when my age is 24, my mothers age is double of my age 
=> 48 years is my mothers age
=> Difference is 48 - 24 = 24 year.
When my age is 44
=> My mother is 44 + 24 = 68 year aged.

Question: 5 - [4 - {3 - (3 - 3 - 6)}] is equal to

Solution:
Given,
5 - [4 - {3 - (3 - 3 - 6)}]
= 5 - [4 - {3 - (-6)}]
= 5 - [4 - {3 +6}]
= 5 - [4 - {9}]
= 5 - [4 - 9]
= 5 - [-5]
= 5 + 5
= 10

Given, Principal, P  = 2800
Compound rate, R  =  10% per annum  =  10/2  =  5% half - yearly
Amount  =  [2800 × (1  +  10/100) × (1  +  5/100)] 
=  (2800 × 11/10 × 21/20)
=  3234
∴ C.I. = (3234 − 2800)  =  434

Question: What is the angle between the hour and minute hands of a clock at 14 : 50?

Solution: 
At 14 : 50, the clock is in the second hour of the afternoon. We use the 12-hour format for calculation.
Then we get, 14 : 50 = 2 : 50

We know, 
Angle = |11M - 60H|/2 = |(11 × 50) - (60 × 2)|/2
= |550 - 120|/2
= 430/2
= 215°

∴ The smaller angle between hands = 360° - 215° = 145°

So the angle between the hour and minute hands at 14 : 50 is 145°.

Question: In an examination, 80% of the candidates passed algebra and 85% passed mechanics. If 73% of the candidates passed both subjects, what percentage of candidates failed in both algebra and mechanics?

Solution:
Students passed in algebra = 80%
Students passed in mechanics = 85%
Students passed in both subjects = 73%
Then, number of students passed in at least one subject
= (80 + 85) - 73
= 92%. 

Thus, students failed in both subjects = 100 - 92 = 8%.

Question: The total marks obtained by a student in Bangla, English, and Mathematics together are 150 more than the marks obtained by him in Bangla. What is the average mark obtained by him in English and Mathematics together?

Solution:
Bangla + English + Mathematics = Bangla + 150
⇒ English + Mathematics = Bangla + 150 - Bangla
⇒ English + Mathematics = 150

∴ Required average = 150/2
= 75

Question: Four metal rods of lengths 78 cm, 104 cm, 117 cm and 169 cm are to be cut into parts of equal length. Each part must be as long as possible. What is the maximum number of pieces that can be cut?

Solution:
Four metal rods of lengths 78 cm, 104 cm, 117 cm and 169 cm 
78 = 2 × 3 × 13
104 = 2 × 2 × 2 × 13 
117 = 3 × 3 × 13
169 = 13 × 13 

HCF of 78, 104, 117 and 169 = 13 

Maximum length of each part = HCF of 78 cm, 104 cm, 117 cm, 169 cm = 13 cm

The maximum number of pieces, 
78/13 = 6
104/13 = 8
117/13 = 9
169/13 = 13

The maximum number of pieces = 6 + 8 + 9 + 13 = 36 

∴ The maximum number of pieces is 36.

Question: A monkey climbs a 70 m high pole. In first minute he climbs 8 m and slips down 4 in the next minute. How much time is required by it to reach the top?

সমাধান:
১ম মিনিটে উঠে ৮ মিটার
২য় মিনিটে নামে ৪ মিটার

∴  ২ মিনিটে উঠবে ৪ মিটার 
৩ মিনিটে উঠবে ৪ + ৮ = ১২ মিটার 
৪ মিনিটে উঠবে ১২ - ৪ = ৮ মিটার 

প্যাটার্ন অনুসারে, 
১, ৩, ৫, ৭তম,............ মিনিটে উঠবে যথাক্রমে ৮, ১২, ১৬, ২০...........মিটার
২, ৪, ৬, ৮তম.......... মিনিটে উঠবে যথাক্রমে ৪, ৮, ১২, ১৬......... মিটার

এভাবে, ৩০ মিনিটে উঠবে ৬০ মিটার
৩১ মিনিটে উঠবে ৬৮ মিটার
৩২ মিনিটে উঠবে = ৬৮ - ৪ = ৬৪ মিটার

∴ উঠা বাকি আছে = ৭০ - ৬৪ = ৬ মিটার

৮ মিটার উঠতে সময় লাগে ১ মিনিট বা ৬০ সেকেন্ড
৬ মিটার উঠতে সময় লাগে (৬০ × ৬)/৮ সেকেন্ড
= ৪৫ সেকেন্ড

∴ মোট সময় লাগবে = ৩২ মিনিট ৪৫ সেকেন্ড

[অপশনে ৩২ মিনিট ৪৫ সেকেন্ড না থাকায় সঠিক উত্তর হবে ঘ) None of the above]

===============================

প্রচলিত সমাধান:

যেহেতু ১ মিনিটে ৮ মিটার উঠে এবং শেষ মিনিটে এই দূরত্ব উঠবে সেহেতু এটা বিয়োগ করলে থাকে ৭০ - ৮ = ৬২ মিটার

১ম মিনিটে উঠে ৮ মিটার
২য় মিনিটে নামে ৪ মিটার

∴  ২ মিনিটে উঠবে ৪ মিটার 

অর্থাৎ ৪ মিটার উঠে = ২ মিনিটে
∴ ৬২ মিটার উঠে = (২ × ৬২)/৪
= ৩১ মিনিটে 

∴ খুঁটি বেয়ে উঠতে সময় লাগবে = ৩১ + ১ = ৩২ মিনিট

Let the number be x
Then,
⇔x−4= 21/x
⇔x²−4x−21=0
⇔(x−7)(x+3)=0
⇔x=7

দেওয়া আছে, লোকটি স্রোতের অনুকূলে 4 ঘন্টায় যায় 18 km
অর্থাৎ, স্রোতের অনুকূলে 1 ঘন্টায় যায় 18/4 km = 4.5 km
এবং লোকটি স্রোতের প্রতিকূলে 12 ঘন্টায় যায় 18km
অর্থাৎ, লোকটি স্রোতের প্রতিকূলে 1 ঘন্টায় যায় 18/12 = 1.5km
ধরি, নৌকার বেগ x km/hr এবং স্রোতের বেগ y km/hr

প্রশ্নমতে,
x + y = 4.5 ... (i) এবং x - y = 1.5 ... (ii)
(i) থেকে (ii) বিয়োগ করে পাই,
 2y = 3
⇒ y = 1.5 km/hr

Question: Which number is to be added to the numerator and denominator of 7/17 to form 3/5?

Solution: 
ধরি, 
সংখ্যাটি x

প্রশ্নমতে,
(7 + x)/(17 + x) = 3/5
⇒ 5(7 + x) = 3(17 + x)
⇒ 35 + 5x = 51 + 3x
⇒ 5x - 3x = 51 - 35
⇒ 2x = 16
∴ x = 8

Question: A merchant mixes two varieties of tea costing Taka 250/kg and Taka 350/kg in the ratio 3:2. If he sells the mixture at Taka 360/kg, his profit percentage is - 

Solution: 
Cost of first variety = 250 Taka/kg
Cost of second variety = 350 Taka/kg
Ratio = 3:2

Cost price of mixture = {(3×250) +(2×350)}/(3+2)
= 1450/5
= 290 Taka/kg

Profit = Selling price - Cost price
= 360 - 290 
= 70 Taka/kg

Profit percentage = (70/290) × 100%
= 24.13%
= 25%

Let,
The number of boys and girls in the class are 3x and x respectively.
Let the average score of the girls be y.
Then, 3x(A + 1) + xy = (3x +x)A
⇒ 3(A + 1) + y = 4A
⇒ 3A + 3 + y = 4A
⇒ y = A - 3

Question: 7.2 is 60 percent of 30 percent of a certain number. What is the number?

Solution:
ধরি, সংখ্যাটি x

প্রশ্নমতে,
60% of 30% of x = 7.2
⇒ 60% of 30x/100 = 7.2
⇒ 60% of 3x/10 = 7.2
⇒ (60 × 3x)/(100 × 10) = 7.2
⇒ (6 × 3x)/(10 × 10) = 7.2
⇒ 18x/100 = 7.2
⇒ 18x = 7.2 × 100
⇒ 18x = 720
∴ x = 40

Question: If p is odd and q is even, which expression is even?

Solution: 
We know, 
Odd + Even = Odd
And Odd × Even = Even

Now, let p = 3 and q = 4
ক) p + q = 3 + 4 = 7 ; Odd

খ) pq = 3 × 4 = 12 ; Even

গ) p + 2q + 1 = 3 + 2 × 4 + 1 = 4 + 8 = 12 ; Even

So, correct answer is Both b and c

Product A’s percentage of profit = {(250 - 200)/200} × 100 = 25%
Product B’s percentage of profit = {(180 - 150)/150} × 100 = 20%
Product C’s percentage of profit = {(110 - 100)/100} × 100 = 10%

So, only product A has the highest profit margin.