Bank Math Master

Question: Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 12 seconds respectively and they cross each other in 24 seconds. The ratio of their speeds is:

Solution:
Let,
the speeds of the two trains be x m/sec and y m/sec respectively.

Then,
length of the first train = 27x metres,
and length of the second train = 12y metres.

ATQ,
(27x + 12y)/(x + y) = 24
⇒ 27x + 12y = 24x + 24y
⇒ 27x - 24x = 24y - 12y
⇒ 3x = 12y
⇒ x/y = 12/3
∴ x : y = 4 : 1

Question: The side of an equilateral triangle is 4m. What is the height of the triangle?

Solution:
Given,
The side of an equilateral triangle = 4m

We know,
Area of an equilateral triangle = (√3/4) × 4²
= (√3/4) × 16
= 4√3

Let,
the height of the triangle = h

We also know,
(1/2) × base × height = area
⇒ (1/2) × 4 × h = 4√3
⇒ h = (4√3 × 2)/4
∴ h = 2√3

So, the height of the triangle = 2√3 m

Question: You visit a fruit market that has 10 different types of fruits. But the Pineapple and Durian are too spiky and smelly for your taste, so you decide to never choose them. You want to make a fruit basket with 6 different fruits from the ones you like. How many ways can you make your basket?

Solution:
Number of different types of fruits =10
Number of fruits available to pick from = 10 - 2 = 8
You are choosing 6 different fruits out of 8.

∴ Number of ways = ⁸C₆
=8!/6!(8 - 6)!
= (8 × 7 × 6!)/(6! × 2!)
= 56/2
= 28 ways

Question: The difference between the ages of two friends is 25% of the older one’s age. If the younger one is 24 years old, what is the age of the older friend?

Solution:
Let
the age of the older friend be x.

Then,
x - 24 = 25% of x
⇒ x - 24 = 25x/100
⇒ x - 24 = x/4
⇒ x - x/4 = 24
⇒ 4x - x = 96
⇒ 3x = 96
∴ x = 32

So the older friend is 32 years old.

Question: If cotA = 3/4 then, sinA = ?

Solution:
We know,
cosec²A = 1 + cot²A
⇒ cosec²A = 1 + (3/4)²
⇒ cosec²A = 1 + 9/16
⇒ cosec²A = (16 + 9)/16
⇒ cosec²A = 25/16
⇒ cosecA = √(25/16)
⇒ 1/sinA = 5/4
∴ sinA = 4/5

Question: A bag costs 20% more than a purse. A wallet costs 30% less than the bag. If the price of the purse is 200 Tk, then by what percentage is the wallet cheaper than the purse?

Solution:
Given,
the price of the purse = 200 tk

∴ Price of the bag = 200 + (200 × 20%)
= 240 tk

Price of the wallet = 240 − (240 × 30%)
= 240 − 72
= 168 Tk

Difference is = (200 - 168) = 32 tk

∴ Percentage = (32 × 100)/200 = 16%

Question: Find the value of k if (x + 2) is a factor of 3x{x + (k​/3x) + (2​/3)}

Solution:
Here,
3x{x + (k​/3x) + (2​/3)}
= 3x² + k + 2x
= 3x² + 2x + k

Given,
(x + 2) is a factor,
So, x + 2 = 0
∴ x = - 2

ATQ,
3 × (- 2)² + 2(- 2) + k = 0
⇒ 12 - 4 + k = 0
⇒ 8 + k = 0
∴ k = - 8

Question: If x and y are positive integers such that x + y = 6, what is the probability that x ≥ 3?

Solution:
Since both x and y must be positive integers,
total possible ways = (1, 5), (2 ,4), (3, 3), (4, 2), (5,1) = 5
Number of favorable outcomes = (3, 3), (4, 2), (5, 1) = 3

∴ So the probability that (x ≥ 3) = 3/5

Question: A boat moves downstream at the rate of 1 km in 15/2 minutes and upstream at the rate of 6 km an hour. What is the speed of the boat in the still water?

Solution:
দেওয়া আছে,
স্রোতের অনুকূলে ১৫/২ মিনিটে যায় ১ কিমি
∴ স্রোতের অনুকূলে ১ মিনিটে যায় ১/(১৫/২) কিমি
∴ স্রোতের অনুকূলে ৬০ মিনিট বা (১ ঘণ্টায়) যায় (১ × ৬০)/(১৫/২) কিমি
= (৬০ × ২)/১৫ কিমি
= ৮ কিমি

স্রোতের প্রতিকূলে ১ ঘণ্টায় যায় ৬ কিমি

∴ স্থির পানিতে নৌকার বেগ = (৮ + ৬)/২ কিমি/ঘণ্টা
= ১৪/২ কিমি/ঘণ্টা
= ৭ কিমি/ঘণ্টা

Question: Find the smallest 5-digit number that is exactly divisible by 24, 36, and 60

Solution:
Required smallest number must be divisible by the L.C.M. of 24, 36, and 60
L.C.M. of 24, 36, and 60
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
60 = 2 × 2 × 3 × 5

L.C.M. = 2 × 2 × 2 × 3 × 3 × 5
= 360

Now divide 10000 by 360 we get,
10000 ÷ 360 = 27.78

Take the ceiling of 27.78, which is 28
we get,
28 × 360 = 10080
10080 is the smallest 5-digit number which is divisible by 24, 36, and 60.

Question: Two pipes A and B can fill a tank in 15 and 30 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
Part filled by A in 1 hour = 1/15

Part filled by B in 1 hour = 1/30

Part filled by (A + B) in 1 hour
= (1/15) + (1/30)
= (2 + 1)/30
= 3/30
= 1/10

∴ Both pipes can fill the tank in 10 hours

Question: 30 men can complete a road construction project in 20 days, working 8 hours a day. How many men need to be appointed extra to complete the same work in 12 days, working 10 hours a day?

Solution:
দৈনিক ৮ ঘণ্টা করে কাজ করে ২০ দিনে শেষ করতে লোক লাগে ৩০ জন
∴ দৈনিক ১ ঘণ্টা করে কাজ করে ১ দিনে শেষ করতে লোক লাগে (৩০ × ৮ × ২০) জন
∴ দৈনিক ১০ ঘণ্টা করে কাজ করে ১২ দিনে শেষ করতে লোক লাগে (৩০ × ৮ × ২০)/(১০ × ১২) জন
= ৪০ জন

∴ অতিরিক্ত লোক লাগবে = (৪০ - ৩০) = ১০ জন

Question: In a code FLOWER is written as EMPXER, GARDEN is written as FBSEEN, then how can PLANET be written in the same code?

Solution:

Given,
F         L          O          W          E        R
↓(- 1)  ↓(+ 1)   ↓(+ 1)   ↓(+ 1)    ↓(0)    ↓(0)
E        M         P           X           E        R

G         A          R          D         E        N
↓(- 1)   ↓(+ 1)   ↓(+ 1)   ↓(+ 1)  ↓(0)    ↓(0)
F         B           S          E          E        N

Similarly,
P          L          A          N         E        T
↓(- 1)   ↓(+ 1)   ↓(+ 1)   ↓(+ 1)  ↓(0)    ↓(0)
O         M         B          O         E        T

Question: If 7^{(x + 2)} = 49^{(3x - 4)} then the value of x = ?

Solution:
7^{(x + 2)} = 49^{(3x - 4)}
⇒ 7^{(x + 2)} = (7²)^{(3x - 4)}
⇒ 7^{(x + 2)} = 7^{(6x - 8)}
⇒ x + 2 = 6x - 8
⇒ 5x = 10
∴ x = 2

Question: If α, β are the roots of the equation x² - 15x + 36 = 0, then αβ equals to:

Solution:
x² - 15x + 36 = 0
⇒ x² - 12x - 3x + 36 = 0
⇒ x(x - 12) - 3(x - 12) = 0
⇒ (x - 12)(x - 3) = 0
∴ x = 12, 3

Hence, α = 12, β = 3
Hence, The value of αβ = 12 × 3 = 36

Question: The area of a trapezium is 96 square cm. The length of one of the parallel sides is 12 cm, and the distance between the parallel sides is 8 cm. Find the length of the other parallel side.

Solution:
Given,
Area of the trapezium = 96 cm²
One parallel side a = 12 cm
Distance between the parallel sides h = 8 cm

Let
the other parallel side = b cm

We know,
The area of a trapezium = (1/2) × (a + b) × h
⇒ 96 = (1/2) × (12 + b) × 8
⇒ 96 = (12 + b) × 4
⇒ (12 + b) = 96/4
⇒ 12 + b = 24
⇒ b = 24 - 12
∴ b = 12

∴ The other parallel side is 12 cm.

Question: If log₃{log₂(x² - 4x - 24)} = 1, where ‘x’ is a natural number, find the value of x.

Solution:
Given,
log₃{log₂(x² - 4x - 24)} = 1
⇒ log₂(x² - 4x - 24) = 3
⇒ x² - 4x - 24 = 8
⇒ x² - 4x - 32 = 0
⇒ (x - 8) (x + 4) = 0
⇒ x = 8, - 4
Since x is a natural number, so x = 8.

Question: Which number will complete the series:
155, 153, 149, 141, 125, ........?

Solution:
The pattern is followed by
155 - 2 = 153
153 - 4 = 149
149 - 8 = 141
141 - 16 = 125
125 - 32 = 93

Hence the number= 93.

Question: The first number is 5% greater and the second number is 25% greater than a third number. What is the ratio of first two numbers?

Solution:
Let
the third number be x

Then, first number = 105% of x
= 105x/100
= 21x/20

Second number = 125% of x
= 125x/100
= 5x/4

∴ Ratio of first two numbers = 21x/20 : 5x/4
= (21x × 4) : (5x × 20)
= 21x : 25x
= 21 : 25

Question: How many triangles are there in the given figure?

Solution:

The hidden triangles in the given figure are: 1, 2, 3, 4, 5, 6, 12, 13, 26, 45, 134, 256, 123456.

Total number of triangles = 13.