Bank Math

Question: What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

Solution:
সরল রেখার সাধারণ সমীকরণ,
y = mx + c ......(1) (এখানেm = ঢাল)

যদি কোনো রেখার ঢাল হয় m, তবে তার লম্ব (perpendicular) রেখার ঢাল হবে,
m' = - (1/m)

এখন,
20x - 2y = 6
2y = 20x - 6
y = 10x - 3
(1) নং এর সাথে তুলনা করে পাই,
m = 10

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - (1/10)

Let the speed of the stream be x km/hr
Upstream Speed = 15 + x
Downstream Speed = 15 - x

So,
{30/(15+x)} + {30/(15-x)} = 4(1/2)(4 hours 30 minutes)
⇒ {900/(225-x2)} = 9/2
⇒ 9x² = 225
⇒ x² = 25
⇒ x = 5.
Hence, the speed of the stream is 5 km/hr.

Question: A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done is 23 days?

Solution: 
Ratio of times taken by A and B = 100 : 130 = 10 : 13

Suppose B takes x days to do the work.
Then,
10 : 13 :: 23 : x
⇒ x = (23 × 13)/10
⇒ x = 299/10
A's 1 day's work = 1/23
B's 1 day's work = 10/299
(A + B)'s 1day's work = 1/23 + 10/299
= 23/299
= 1/13

∴ A and B together can complete the work in 13 days.

Question: If x + (1/x) = 4 then, x - (1/x) = ?

Solution:
দেওয়া আছে, x + 1/x = 4

আমরা জানি,
(x - 1/x)² = (x + 1/x)² - 4 . x . 1/x
⇒ (x - 1/x)² = 4² - 4 [মান বসিয়ে]
⇒ (x - 1/x)² = 16 - 4
⇒ (x - 1/x)² = 12
⇒ x - (1/x) = ± √12
∴ x - (1/x) = ± 2√3

Assume that the reservoir is filled by the first pipe in 'x' hours.
So, the reservoir is filled by a second pipe in 'x + 20' hours.

Now, from these above conditions,
we can form the equations as,
1/x + 1/(x + 20) = 1/24
[x + 20 + x]/[x(x + 20)] = 1/24
x²– 28x – 480 = 0

By solving this quadratic equation , we get the factors (x – 40) (x+12) = 0
Hence, we get two values :
(x – 40) = 0 and (x+12) = 0
⇒ x = 40 and x = -6

Since the filling of the reservoir is positive work, we can neglect the negative value of 'x'.
Thus, x = 40

This means that the second pipe will take (x+ 20) hrs = 40 + 20 = 60 hrs to fill the reservoir.

Question: There are 7 non-collinear points. How many triangles can be drawn by joining these points?

Solution: 
A triangle is formed by joining any three non-collinear points in pairs.
There are 7 non-collinear points.

∴ The number of triangles formed = ⁷C₃ = 35

Question: A sum of money triples itself in 25 years at simple interest. What is the annual interest rate?

Solution:
ধরি, আসল = P টাকা
∴ 25 বছরে সুদে-আসলে হয় (A) = 3P টাকা
সুতরাং, সুদ (I) = A - P = 3P - P = 2P টাকা
সময় (n) = 25 বছর
সুদের হার (r) = ?

আমরা জানি,
I = (Pnr)/100
⇒ r = (I × 100)/(P × n)
⇒ r = (2P × 100)/(P × 25)
⇒ r = (2 × 100)/25
⇒ r = 2 × 4
⇒ r = 8%
সুতরাং, বার্ষিক সুদের হার 8%।

Let, ABCD be a || gm in which AB = 30 m, BC = 14 m & AC = 40 m.
Clearly, area of || gm ABCD = 2 (area of ∆ABC).
Let, a = 30, b = 14 & c = 40.
Then, s = (1/2)(a+b+c) = 42
Therefore, area of ∆ABC = √s(s-a)(s-b)(s-c)
= √42×12×28×2 = 168 m²
Therefore area of || gm = (2×168) m² = 336 m²

One person can select one house out of 3 = ³C₁ ways =3 ways
Hence, three persons can select one  house in = 3 × 3 × 3 = 27 ways
So, All three apply for the same house, its probability = 3/27 = 1/9

Question: A train takes 50 seconds to cross a platform 300 meters long and 35 seconds to cross another platform 150 meters long. Find the length of the train.

Solution:
Let the length of the train = x meters

Then,
For the first platform, the distance covered by the train = (x + 300) meters
And,
For the second platform, the distance covered by the train = (x + 150) meters

According to the question,
(x + 300)/50 = (x + 150)/35
⇒ 50(x + 150) = 35(x + 300)
⇒ 50x + 7500 = 35x + 10500
⇒ 50x - 35x = 10500 - 7500
⇒ 15x = 3000
⇒ x = 3000/15 
⇒ x = 200

∴ The length of the train is 200 meters

Question: Two pipes can fill a tank in 12 minutes working together. After working together for 8 minutes, the first pipe is closed. It then takes 10 more minutes for the second pipe to fill the tank completely. How long would the second pipe take alone to fill the tank?

Solution: 
Let the first pipe fill the tank at rate A tanks per minute.
Let the second pipe fill the tank at rate B tanks per minute.
Both pipes together fill the tank in 12 minutes. so,
A + B = 1/12  ....... (1)

They work together for 8 minutes.
Work done in 8 minutes = 8 × (1/12) = 8/12 = 2/3 of the tank

∴ Remaining work = 1 - (2/3) = 1/3 of the tank
This remaining 1/3 is filled by the second pipe alone in 10 minutes.
⇒ B × 10 = 1/3
∴ B = (1/3)/10 = 1/30 tank per minute

Therefore, time taken by the second pipe alone to fill the full tank = 1/1/30 = 30 minutes.

Selling price = 9
Loss = 20%
Cost price = (9 × 100)/80
= 45/4
To make a profit of 5% selling price
= (45/4) × (105/100)
= (9 × 21)/(4 × 4)
= 11.81

Question: 30 workers can manufacture 30 machines working 5 hours a day. How many workers need to be appointed extra to triple the production if they work 10 hours a day?

Solution:
5 hours to manufacture 30 machines by 30 workers
∴ 1 hour to manufacture 1 machine by = (30 × 5)/30 workers
∴ 10 hours to manufacture 90 machines by = (5 × 90)/10 workers
= 45 workers

∴ Extra workers required
= (45 - 30)
= 15 workers

Let 'B' alone can do the work in 'x' days
6/30 + 18/x = 1
=> x = 22.5
1/30 + 1/22.5 = 7/90
=> 90/7 = 12 (6/7) days

Question: The sum of digits of a 3-digit number is divisible by 7. Which of these numbers satisfies it?

Solution:
A number is divisible by 7 in terms of digit sum if the sum of its digits is divisible by 7.
Check each number:
234 → 2 + 3 + 4 = 9 → 9/7 = 1 remainder 2 
351 → 3 + 5 + 1 = 9 → 9/7 = 1 remainder 2 
142 → 1 + 4 + 2 = 7 → 7/7 = 1 
429 → 4 + 2 + 9 = 15 → 15/7 = 2 remainder 1

142 satisfies it.

Question: A college has 10 basketball players. A 5 member's team and a captain will be selected out of these 10 players. How many different selections can be made?

Solution:
We can select the 5 member team out of the 10 in =  ¹⁰C₅ ways
= 252 ways

The captain can be selected from amongst the remaining 5 players in 5 ways.

∴ The total ways the selection of 5 players and a captain can be made = 252 × 5 ways
= 1260 ways

Question: Numbers from 10 to 20 are written on cards. One card is drawn at random. What is the probability that the number is prime or divisible by 3?

Solution:
The numbers are, 
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
∴ Total numbers = 11

∴ Prime number are, 11, 13, 17, 19 = 4

And, List the numbers divisible by 3 = 3 numbers (12, 15, 18)

∴ Total favorable outcomes = Primes + Divisible by 3 - Overlap
= 4 + 3 - 0 = 7

∴ Probability = Favorable outcomes/Total outcomes
= 7/11

So the probability is 7/11.

Question: Tk. 10000 becomes Tk. 12000 in 4 years at a certain rate of simple interest. If the rate becomes 2.5 times of itself, the amount of same principal in 6 years will be -

সমাধান:
10000 টাকা 4 বছরে 12000 টাকা হয়।
∴ 4 বছরে সুদ = 12000 - 10000 টাকা = 2000 টাকা
1 বছরে সুদ = 2000/4 = 500 টাকা

2.5 গুণ বৃদ্ধিতে নতুন 1 বছরের সুদ = 500 × 2.5 টাকা
= 1250 টাকা

6 বছরে মোট সুদ = 1250 × 6 টাকা = 7500 টাকা
সুতরাং, 6 বছর পর সুদাসলে হবে = 10000 + 7500 টাকা
= 17500 টাকা

Question: Find out the missing number in the following series. 256, 16, 4, ____?

Solution:
√256 = 16
√16 = 4
√4 = 2

So, missing number is = √4 = 2

4, 9, 6,11, 8, 13
Here in both in even and in odd position numbers increased by 2
4, 6, 8, 10
and, 9, 11, 13, 15

So, the next number in the series is 10

Speed upstream = 3/(20/60)
= 9 km/hr.
Speed downstream = 3/(18/60)
= 10 km/hr.
Rate of current = (10 - 9)/2
= 1/2 km/hr.

Question: What is the square root of 0.16?

Solution:
 The square root of 0.16 = √0.16
= 0.4

ধরি, জামালের আয় = y টাকা
সুতরাং তপুর আয় = y + 2000 টাকা
প্রশ্নমতে,
y + (y + 2000) = x
বা, 2y + 2000 = x
বা, 2y = x - 2000
বা, y = x/2 - 1000.

Question: A man rowed 4 miles upstream in 2 hours. If the river current is 1 mile per hour, how long will it take him to return downstream?

Solution:
Let,
the velocity of the boat in still water = x mph
and the velocity of the stream = y mph = 1 mph

In upstream, he covers 4 miles in 2 hours.
∴ Upstream speed = 4 ÷ 2 = 2 mph

ATQ,
x - y = 2
⇒ x - 1 = 2
⇒ x = 3

So, downstream speed = x + y = 3 + 1 = 4 mph

Time to row 4 miles downstream = 4 ÷ 4 = 1 hour

∴ The return trip will take 1 hour.

Question: Given 16^x = 4⁴, what is x?

Solution:
16^x = 4⁴
⇒ (4²)^x = 4⁴
⇒ 4^2x = 4⁴
⇒ 2x = 4
⇒ x = 4/2
∴ x = 2

Given, p = 4q
if q = 1 then p = 4
if q = 2 then p = 8 not acceptable as p have to be less than 8
So, the correct answer is 1

Question: The least number by which 294 must be multiplied to make it a perfect square is :

Solution:
294 = 7 × 7 × 2 × 3
Here, 2 and 3 have odd exponents.
Multiplying by 2 × 3 = 6 will make 294 a perfect square.

∴ Multiplying by 6 will make 294 a perfect square.

Let, K = Kabir’s current weight, in pounds
S = Sister’s current weight, in pounds

We are told that “If Kabir loses 8 pounds, he will weigh twice as much as his sister.'' We put this into an equation:
K – 8 = 2S
∴ K = 2S + 8...... (i)

Next, we are told that “Together they now weigh 278 pounds.” We can also put this into an equation.
K + S = 278........ (ii)
To solve this equation, we can substitute 2S + 8 from Equation (i) for the variable K in Equation 2:
2S + 8 + S = 278
3S = 270
S = 90

From equation (ii) we can find,
K = 278 - 90 = 188

Question: The slope of a line perpendicular to one with slope (- 3/4) is:

Solution:
আমরা জানি, 
দুটি সরলরেখা পরস্পর লম্ব হলে তাদের ঢালদ্বয়ের গুণফল - 1 হয়।
অর্থাৎ, যদি কোনো সরলরেখার ঢাল (m) হয়, তাহলে তার উপর লম্ব রেখার ঢাল হবে - 1/m.

এখানে, মূল রেখার ঢাল = - 3/4 । 
তাই লম্ব রেখার ঢাল হবে = -1/(- 3/4) = 4/3

অতএব, লম্ব রেখার ঢাল = 4/3.

Total age of 10 students = 150 years
Total age of 15 students = 240 years
Total age of 5 new students = 240 - 150 = 90 years
Therefore,
Average age of 5 new students = 90/5 = 18 years

Question: A cyclist completes a journey in 8 hours. He travels the first half of the journey at the rate of 15 km/hr and the second half at the rate of 25 km/hr. Find the total journey in km.

Solution:
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)/15 ঘন্টা
= D/30 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়, সময় = দূরত্ব/গতিবেগ
= (D/2)/25 ঘন্টা
= D/50 ঘন্টা

প্রশ্নমতে,
 D/30 + D/50 = 8
⇒ (5D + 3D)/150 = 8
⇒ 5D + 3D = 8 × 150
⇒ 8D = 1200
⇒ D = 1200/8
⇒ D = 150 কিমি

সুতরাং, মোট যাত্রার দূরত্ব হলো 150 কিমি।

The man starts with 7 kmph
Distance covered in first 1 hour = 7 km
He increases his speed every hour by 1 km.
Speed in 2nd hour = 8 km/hr
Distance covered in 2nd hour = 8 km
Remaining distance = 19.5 - (7 + 8)
19.5 - 15
= 4.5 km
Speed in the third hour = 9 km/hr
Time taken to cover 4.5 km at 9 km/hr
= 4.5/9
= 1/2 hour.
Therefore,
Total time = 1 + 1 + (1/2) hours = 2 (1/2) hours.
Hence, the answer is 2(1/2) hours.