Bank Math

PrepBank · পাতা ৬৫ / ১৬১ · ৬,৪০১–৬,৫০০ / ১৬,১২৪

৬,৪০১.

7 years ago, son’s age was 1/5th of the father’s age. If the ratio of their present ages is 1:3, what will be the ratio of their ages after 7 years?

ক
ক) 2:5
খ
খ) 3:5
গ
গ) 5:11
ঘ
ঘ) 3:7
ঙ
ঙ) None of above

ব্যাখ্যা

Let,
7 years ago son’s age was x years old and father’s age was 5x year.
Now,
(x+7)/(5x+7) = 1/3,
so x = 7
Son’s age = 7 + 7 = 14
Father’s age= 5 × 7 + 7 = 42
After 7 years,
Son’s age : Father’s age = 21:49 = 3:7

৬,৪০২.

One morning after sunrise, Sakib was standing facing a pole. The shadow of the pole fell exactly to his left. Which direction was Sakib facing?

ক
East
খ
West
গ
North
ঘ
South

ব্যাখ্যা

Question: One morning after sunrise, Sakib was standing facing a pole. The shadow of the pole fell exactly to his left. Which direction was Sakib facing?

Solution:
1. In the morning, the sun is in the east, so all shadows fall toward the west.
2. If the shadow (west) is to Sakib's left, he must be facing north.If he were facing south, the shadow would be to his right

৬,৪০৩.

A man can walk a certain distance at a uniform speed in 100 days. How long will it take him to cover twice the distance at half the normal speed?

ক
400 days
খ
200 days
গ
50 days
ঘ
25 days

ব্যাখ্যা

Question: A man can walk a certain distance at a uniform speed in 100 days. How long will it take him to cover twice the distance at half the normal speed?

Solution:
Earlier time = 100 days.
Distance is doubled and speed is reduced to half.
∴ time will become 2 × 2 = 4 times.
Hence now it will take 100 × 4 = 400 days.

৬,৪০৪.

In a school the ratio of boys and girls is 3 : 4 respectively. When 100 girls leave the ratio becomes 5 : 6 respectively. How many boys are there in the school?

ক
ক) 750
খ
খ) 825
গ
গ) 850
ঘ
ঘ) 900

ব্যাখ্যা

Question: In a school the ratio of boys and girls is 3 : 4 respectively. When 100 girls leave the ratio becomes 5 : 6 respectively. How many boys are there in the school?

Solution:
Let the number of boys and girls be 3x and 4x respectively
Then,
3x/(4x -100) = 5/6
⇒ 18x = 20x - 500
⇒ 2x = 500
⇒ x = 250
Total number of boys = 3 × 250 = 750

৬,৪০৫.

then x = ?

ক
ক) 14
খ
খ) 16
গ
গ) 12
ঘ
ঘ) 144

ব্যাখ্যা

Question: then x = ?

Solution:
x/√128 = √162/x
⇒ x² = √(128 × 162)
⇒ x² = √(2 × 64 × 2 × 81)
⇒ x² = 2 × 8 × 9
⇒ x² = 144
⇒ x = √144
∴ x = 12

৬,৪০৬.

If the chance that a plane arrives safely at an airport is 9/10, then what is the chance that out of 5 planes expected at least 4 will arrive safely?

ক
9⁴/10⁴
খ
15 × 9⁵/10⁴
গ
14 × (9⁴/10⁵)
ঘ
None of these

ব্যাখ্যা

Question: If the chance that a plane arrives safely at an airport is 9/10 then what is the chance that out of 5 planes expected at least 4 will arrive safely?

Solution:
chance that a plane arrives safely 9/10
chance that a plane not arriving safely (1 - 9/10)
= 1/10

the chance that out of 5 planes expected at least 4 will arrive safely = ⁵C₄ (9/10)(9/10) (9/10)(9/10)(1/10) + (9/10)(9/10) (9/10)(9/10)(9/10)
= 5 (9⁴/10⁵) + (9⁵/10⁵)
= 9⁴(5 + 9) /10⁵
= 14 × (9⁴/10⁵)

৬,৪০৭.

The first, second and third terms of the proportion are 42, 36, 35. Find the fourth term.

ক
50
খ
40
গ
37
ঘ
32
ঙ
30

ব্যাখ্যা

Question: The first, second and third terms of the proportion are 42, 36, 35. Find the fourth term.

Solution:
Let the fourth term be x.

Thus 42, 36, 35, x are in proportion.
Product of extreme terms = 42 × x
Product of mean terms = 36 × 35

Since, the numbers make up a proportion
Therefore,
42 × x = 36 × 35
or, x = (36 × 35)/42
or, x = 30

Therefore, the fourth term of the proportion is 30.

৬,৪০৮.

If the difference between the squares of two consecutive natural numbers is 21, what is the sum of the squares of these two numbers?

ক
223
খ
421
গ
300
ঘ
121
ঙ
221

ব্যাখ্যা

Question: If the difference between the squares of two consecutive natural numbers is 21, what is the sum of the squares of these two numbers?

Solution:
Bigger number = (difference of squares + 1)/2 = (21 + 1)/2 = 11
Smaller number = (difference of squares - 1)/2 = (21 - 1)/2 = 10

So, sum of the squares of these numbers = 11² + 10²
= 121 + 100
= 221

৬,৪০৯.

A number is as much greater than 31 as less than 55. Which of the following is that numbers?

ক
ক) 47
খ
খ) 52
গ
গ) 39
ঘ
ঘ) 43

ব্যাখ্যা

Question: A number is as much greater than 31 as less than 55. Which of the following is that numbers?

Solution:
Let, the number be x
ATQ,
x - 31 = 55 - x
⇒ 2x = 86
⇒ x = 43

৬,৪১০.

Person 1 to 4 receive equal shares of an income, while Person 5 receives half of what each of Persons 1 to 4 receives. If the total income is 18000 taka, how much does Person 5 get?

ক
1500
খ
2000
গ
2500
ঘ
3000

ব্যাখ্যা

Question: Person 1 to 4 receive equal shares of an income, while Person 5 receives half of what each of Persons 1 to 4 receives. If the total income is 18000 taka, how much does Person 5 get?

Solution:
Let the amount each of Persons 1 to 4 receives = x taka.
Then Person 5 receives = x/2 taka.

ATQ,
Total income = amount received by all 5 persons
⇒ x + x + x + x + (x/2) = 18000
⇒ 4x + x/2 = 18000
⇒ (8x + x)/2 = 18000
⇒ 9x/2 = 18000
⇒ 9x = 36000
⇒ x = 36000/9
∴ x = 4000

Therefore, Person 5 receives = x/2 = 4000/2 = Tk. 2000

৬,৪১১.

If '+' means '÷', 'x' means '-' , '- ' means x' and '÷' means +' then find the value of 36 +18 - 17 × 16 ÷ 3

ক
ক) 12
খ
খ) 21
গ
গ) 16
ঘ
ঘ) 30

ব্যাখ্যা

36 + 18 - 17 × 16 ÷ 3
= 36 ÷ 18 × 17 - 16 + 3
= 2 × 17 - 16 + 3
= 34 - 16 + 3
= 37 - 16
= 21

৬,৪১২.

An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?

ক
ক) 120°
খ
খ) 150°
গ
গ) 180°
ঘ
ঘ) 140°

ব্যাখ্যা

Question: An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?

Solution:
From 8 o'clock to 20'clock total time is 6 hours.

Hours hand makes,
in 12 hours = 360° angle
in 6 hours = (360° × 6)/12
= 180°

৬,৪১৩.

One-third of Farid's capital was invested at 7%, one-fourth at 8%, and the rest at 10%. He earned 561 taka annually. How much is his total capital?

ক
6,600 taka
খ
7,600 taka
গ
9,000 taka
ঘ
None of the above

ব্যাখ্যা

Question: One-third of Farid's capital was invested at 7%, one-fourth at 8%, and the rest at 10%. He earned 561 taka annually. How much is his total capital?
(ফরিদের মূলধনের এক-তৃতীয়াংশ ৭% সুদে, এক-চতুর্থাংশ ৮% সুদে এবং বাকি অংশ ১০% সুদে বিনিয়োগ করা হয়েছিল। তার বার্ষিক আয় ছিল ৫৬১ টাকা। তার মোট মূলধন কত?)

Solution:
ধরি,
আসল x টাকা
আমরা জানি, I = Pnr

561 = {(x × 7)/(100 × 3)} + {(x × 8)/(100 × 4)} + {(5x × 10)/(12 × 100)}
⇒ 561 = (7x/300) + (x/50) + (x/24)
⇒ 561 = 102x/1200
⇒ 561 = 51x/600
∴x = 6,600 taka

৬,৪১৪.

Three unbiased coins are tossed. What is the probability of getting at most two heads?

ক
3/8
খ
7/8
গ
5/8
ঘ
1/8

ব্যাখ্যা

Question: Three unbiased coins are tossed. What is the probability of getting at most two heads?

Solution:
Getting at most Two heads means 0 to 2 but not more than 2.

Here S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}

Let E = event of getting at most two heads
Then E = {TTT, TTH, THT, HTT, THH, HTH, HHT}

∴P(E) = n(E)/n(S) = 7/8

৬,৪১৫.

40 is subtracted from 60% of a number, the result is 50. Find the number.

ক
130
খ
50
গ
70
ঘ
150

ব্যাখ্যা

Question: 40 is subtracted from 60% of a number, the result is 50. Find the number.

Solution:
Let the number be x.

According to the question,
60% of x - 40 = 50

⇒ (60/100)x - 40 = 50
⇒ (3/5)x = 90
⇒ x = (5 × 90)/3
⇒ x = 150

∴ The required number is 150.

৬,৪১৬.

The present ages of three persons are in the ratio of 4 : 7 : 9. Six years ago, the sum of their ages was 62. What was the age of the oldest person six years ago?

ক
36 years
খ
30 years
গ
28 years
ঘ
21 years
ঙ
16 years

ব্যাখ্যা

Question: The present ages of three persons are in the ratio of 4 : 7 : 9. Six years ago, the sum of their ages was 62. What was the age of the oldest person six years ago?

Solution:
Let their present ages are 4x, 7x and 9x years respectively.
∴ 6 years ago, their present ages are (4x - 6), (7x - 6) and (9x - 6) respectively.

ATQ,
(4x - 6) + (7x - 6) + (9x - 6) = 62
⇒ 20x - 18 = 62
⇒ 20x = 62 + 18
⇒ x = 80/20
⇒ x = 4

∴ Their present ages are (4 × 4) =16, (7 × 4) = 28 and (9 × 4) = 36 years respectively.
∴ Age of the oldest person six years ago is (36 - 6) = 30 years.

৬,৪১৭.

In how many different ways can the letters of the word PUNCTUAL be arranged?

ক
ক) 16020
খ
খ) 12060
গ
গ) 20160
ঘ
ঘ) 12600

ব্যাখ্যা

Given word Punctual

The word PUNCTUAL consists of 8 letters in total.
The word has five consonants - p, n, c, t, l
The word has three vowels -a, u and u - where the letter ‘U’ comes twice.

The number of arrangements for the said word will be
= 8! / 2!
= (8 × 7 × 5 × 4 × 3 × 2 × 1)/(1 × 2)
= 20160

৬,৪১৮.

A company makes a profit of 5% on its first TK. 1000 of sales each day, and 4% on all sales in excess of 1000 for that day. What would be the profit of the company in a day when sales are TK. 6000?

ক
Tk. 200
খ
Tk. 220
গ
Tk. 250
ঘ
Tk. 300

ব্যাখ্যা

Question: A company makes a profit of 5% on its first TK. 1000 of sales each day, and 4% on all sales in excess of 1000 for that day. What would be the profit of the company in a day when sales are TK. 6000?

Solution:
Profit for first TK. 1000 = 1000 × (5/100) = TK. 50

Total sales excess of Tk. 1000 = (6000 - 1000) = Tk. 5000

∴ Profit for excess of Tk. 1000 = 5000 × (4/100) = Tk. 200

∴ Total Profit (50+ 200) = Tk. 250

৬,৪১৯.

A cistern has two taps that fill it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the there are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern?

ক
20 minutes
খ
15 minutes
গ
10 minutes
ঘ
8 minutes
ঙ
None

ব্যাখ্যা

Question: A cistern has two taps that fill it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the there are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern?

Solution:
Let, the waste pipe can empty the cistern in x min.

According to the question,
1/12 + 1/15 - 1/x = 1/20
⇒ 1/12 + 1/15 - 1/20 = 1/x
⇒ 1/x = (5 + 4 + 3)/60
⇒ 1/x = 6/60
⇒ 1/x = 1/10
∴ x = 10

∴ The waste pipe will empty the full cistern in 10 minutes.

৬,৪২০.

How many 4-letter words can be formed using the letters A, B, C with repetition allowed?

ক
64
খ
81
গ
12
ঘ
4³ - 4

ব্যাখ্যা

Question: How many 4-letter words can be formed using the letters A, B, C with repetition allowed?

Solution:

Number of choices for each letter = 3 (A, B, or C)

Length of word = 4

Total number of 4-letter words = 3⁴ = 81

৬,৪২১.

The quadratic equation whose one rational root is 4 + √3 is-

ক
x² - 12x + 21 = 0
খ
x² + 8x + 12 = 0
গ
x² + 4x - 3 = 0
ঘ
x² - 8x + 13 = 0

ব্যাখ্যা

Question: The quadratic equation whose one rational root is 4 + √3 is-

Solution:
ধরি, একটি মূল হলো 4 + √3
যেহেতু এটি একটি দ্বিঘাত সমীকরণ এবং একটি মূলে মূলদ আছে, তাই অপর মূলটি হবে এর অনুবন্ধী (conjugate)।
অতএব, অপর মূলটি হলো 4 - √3

মূলদ্বয়ের যোগফল = (4 + √3) + (4 - √3) = 8

মূলদ্বয়ের গুণফল = (4 + √3)(4 - √3)
= 4² - (√3)²
= 16 - 3
= 13

আমরা জানি,
মূলদ্বয় α এবং β হলে দ্বিঘাত সমীকরণটি হয়:
x² - (α + β)x + α × β = 0

সুতরাং, নির্ণেয় দ্বিঘাত সমীকরণটি হলো,
x² - 8x + 13 = 0

৬,৪২২.

In how many different ways can the letters of the word 'RUMOUR' be arranged?

ক
160
খ
180
গ
200
ঘ
220

ব্যাখ্যা

Question: In how many different ways can the letters of the word 'RUMOUR' be arranged?

Solution:

The word 'RUMOUR' consists of 6 letters in which each of 'R' and 'U' comes twice.

Here,
Number of letters, n = 6
Number of letter 'R', p = 2
Number of letter 'U', q = 2

∴ Number of arrangements
= n!/(p! × q!)
= 6!/(2! × 2!)
= (6 × 5 × 4 ×3 × 2 × 1)/(2 × 2)
= 180

Hence, The letters of the word 'RUMOUR' be arranged in 180 ways.

৬,৪২৩.

P and Q are 27 km away. Two trains with speed of 24 km/hr and 18 km/hr respectively start simultaneously from P and Q and travel in the same direction. They meet at a point R beyond Q. Distance QR is -

ক
ক) 126 km
খ
খ) 81 km
গ
গ) 48 km
ঘ
ঘ) 36 km

ব্যাখ্যা

Relative speed= 24-18= 6 km/hr
Time required by faster train to overtake slower train
= 27/6 hr
= 4(1/2) hr
∴ Distance between Q and R:
= 18×4(1/2)
= 81 km

৬,৪২৪.

An individual is cycling at a speed of 25 km per hour. He catches his predecessor who had started earlier in two hours. What is the speed of his predecessor who had started 3 hours earlier?

ক
16 km
খ
14 km
গ
10 km
ঘ
8 km
ঙ
None of the above

ব্যাখ্যা

Question: An individual is cycling at a speed of 25 km per hour. He catches his predecessor who had started earlier in two hours. What is the speed of his predecessor who had started 3 hours earlier?

Solution:
The distance covered in two hours,
= 2 × 25 = 50 km

Time taken by Predecessor = (3 + 2) = 5 hours

Then, the speed of the predecessor,
= 50/5
= 10 kmph

উত্তর None of the above হবে, কারণ speed এর একক kmph.

৬,৪২৫.

Find the average of first 20 consecutive natural numbers.

ক
12.20
খ
10.78
গ
16.45
ঘ
10.5

ব্যাখ্যা

Question: Find the average of first 20 consecutive natural numbers.

Solution:
The average of first n consecutive natural numbers is
=(n + 1)/2

Here,
n=20

So, average = (20 + 1)/2
= 21/2
= 10.5

৬,৪২৬.

If (m/n)^y-1=(n/m)^{y - 3}, the value of y is -

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Given that,
(m/n)^{y - 1} = (n/m)^{y - 3}
⇒ (m/n)^{y - 1} = (m/n)^{-(y - 3)}
⇒ y -1 = - (y - 3)
⇒ y -1 =- y +3
⇒ y + y = 3 + 1
⇒ 2y = 4
⇒ y = 4/2
∴ y = 2

৬,৪২৭.

A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work they can finish in a day?

ক
1/2
খ
1/3
গ
1/4
ঘ
1/5
ঙ
None of the above

ব্যাখ্যা

Question: A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work they can finish in a day?

Solution:
A's 1 day's work = 1/18

B can do the same work in half the time taken by A, that is 9 days.
B's 1 day's work = 1/9

Then, working together, part of the same work they can finish in a day
= (1/9) + (1/18)
= (2 + 1)/18
= 3/18
= 1/6

৬,৪২৮.

(0.003)² = ?

ক
ক) 0.009
খ
খ) 0.0009
গ
গ) 0.00009
ঘ
ঘ) 0.000009

ব্যাখ্যা

Question: (0.003)² = ?

Solution:
(0.003)² = 0.003 × 0.003
= 0.000009

৬,৪২৯.

The average weight of 17 students is 90 kg. If the weight of teacher is also included, then the average weight is increased by 200 grams. Find the weight of the teacher?

ক
ক) 93.6 kg
খ
খ) 94 kg
গ
গ) 93.4 kg
ঘ
ঘ) 94.6 kg

ব্যাখ্যা

Total weight including teacher = 18 × 90.2 = 1623.6 kg
Total weight of 17 students = 17 × 90 = 1530 kg
So, weight of the teacher = 1623.6 – 1530 = 93.6 kg

৬,৪৩০.

There are 45 students in a certain class 2/3 of the students are girls and 1/2 of the girls are blue-eyed. How many blue- eyed girls are there in the class?

ক
30
খ
15
গ
28
ঘ
None

ব্যাখ্যা

Question: There are 45 students in a certain class 2/3 of the students are girls and 1/2 of the girls are blue-eyed. How many blue- eyed girls are there in the class?

Solution:
মোট ছাত্র-ছাত্রীর সংখ্যা = 45 জন
বালিকার সংখ্যা = (45 × 2)/3 জন
= 30 জন

blue- eyed বালিকার সংখ্যা = (30 × 1)/2 জন
= 15 জন

৬,৪৩১.

The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is:

ক
2520 m²
খ
2480 m²
গ
2420 m²
ঘ
1520 m²

ব্যাখ্যা

Question: The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is:

Solution:
We have,
(l - b) = 23 and
2(l + b) = 206
⇒ (l + b) = 103

Solving the two equations, we get:
l = 63 and b = 40

∴ Area = (l x b) = (63 x 40) m² = 2520 m²

৬,৪৩২.

Two numbers are in the ratio 5 : 2. If the difference of their squares is 189, then find the largest number-

ক
ক) 6
খ
খ) 9
গ
গ) 12
ঘ
ঘ) 15

ব্যাখ্যা

Question: Two numbers are in the ratio 5 : 2. If the difference of their squares is 189, then find the largest number-

Solution:
দুটি সংখ্যার অনুপাত ৫ : ২
সংখ্যা দুটি ৫x, ২x

প্রশ্নমতে,
(৫x)^২ - (২x)^২ = ১৮৯
⇒ ২৫x^২ - ৪x^২ = ১৮৯
⇒ ২১x^২ = ১৮৯
⇒ x^২ = ৯
∴ x = ৩

সংখ্যা দুটি ১৫, ৬
বড় সংখ্যাটি ১৫

৬,৪৩৩.

If ⁶P_r = 360 and If ⁶C_r = 15, find r?

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

ⁿP_r = ⁿC_r × r!
⁶P_r = 15 × r!
360 = 15 × r!
r! = 360/15
= 24
r! = 4 × 3 × 2 × 1
⇒ r! = 4!
Therefore, r = 4
----------------------------
Alternative way:
⁶P_r/⁶C_r = 360/15
or, 6!/(6 - r)! ÷ 6!/{r!(6 - r)!} = 24
or, r! = 4!
∴ r = 4

৬,৪৩৪.

What should come in place of the question mark (?) in the following number series?
588, 587, 583, 574, 558, ?, 497

ক
538
খ
527
গ
533
ঘ
541
ঙ
None of these

ব্যাখ্যা

Question: What should come in place of the question mark (?) in the following number series?
588, 587, 583, 574, 558, ?, 497

Solution:
588 - 1² = 587
587 - 2² = 583
583 - 3² = 574
574 - 4² = 558
558 - 5² = 533
533 - 6² = 497

৬,৪৩৫.

P, Q, and R started a business by investing Tk. 120,000, Tk. 180,000 and Tk. 200,000 respectively. Find the share of Q out of an annual profit of Tk. 125,000.

ক
Tk. 45,000
খ
Tk. 50,000
গ
Tk. 55,000
ঘ
Tk. 35,000

ব্যাখ্যা

Question: P, Q, and R started a business by investing Tk. 120,000, Tk. 180,000 and Tk. 200,000 respectively. Find the share of Q out of an annual profit of Tk. 125,000.

Solution:
The ratio of investments of P, Q, and R is:
P : Q : R = 120,000 : 180,000 : 200,000
= 12 : 18 : 20
= 6 : 9 : 10

Sum of the ratios = 6 + 9 + 10 = 25
Total annual profit = Tk. 125,000

Q's share of profit = (9/25) × 125,000
= 9 × 5,000
= 45,000 Tk.

৬,৪৩৬.

If x : y = 5 : 3, then (11x + 2y) : (11x - 2y) = ?

ক
ক) 67 : 49
খ
খ) 57 : 43
গ
গ) 57 : 49
ঘ
ঘ) 61 : 49

ব্যাখ্যা

x : y = 5 : 3
⇒ x/y = 5/3
⇒ 11x/2y = (11 × 5)/(2 × 3)
⇒ 11x/2y = 55/6
⇒ (11x + 2y)/(11x - 2y) = (55 + 6)/(55 - 6)
⇒ (11x + 2y)/(11x - 2y) = 61/49
⇒ (11x + 2y) : (11x - 2y) = 61 : 49

৬,৪৩৭.

The breadth of a rectangular field is 60% of its length. If the perimeter of the field is 800 metre, what is the area of the field in square metres?

ক
30500
খ
32500
গ
40000
ঘ
37500

ব্যাখ্যা

Let, the length of the rectangle is l and breadth is b.
Given that the breadth of the rectangular field is 60% of its length.
b = 60l/100
=3l/5
Perimeter of the field 800 m
⇒ 2(l + b) = 800
⇒2{l + (3l/5)} = 800
⇒ l + (3l/5) = 400
⇒ 8l/5 = 400
⇒ l = 250 m.
b = 3l/5
= (3 × 250)/5
= 150 m
Area = lb
= (250 × 150)
= 37500 m².

৬,৪৩৮.

What percentage of the whole week does Sami spend in school except lunch time, if his school times are 8 am to 4 pm from Saturday to Thursday where as there is 1 hour to his lunch time?

ক
18%
খ
20%
গ
25%
ঘ
30%

ব্যাখ্যা

Question: What percentage of the whole week does Sami spend in school except lunch time, if his school times are 8 am to 4 pm from Saturday to Thursday where as there is 1 hour to his lunch time?

Solution:
Time spend by Sami in a day = {(12 + 4) - 8} hours
= 8 hours

Except lunch time Sami spend in a day = (8 - 1) hours
= 7 hours

Percentage time spend in a week = {(7 × 6)/(7 × 24)} ×100%
= 25%

৬,৪৩৯.

If ¹⁸C_r= ¹⁸C_{r + 2} ,find ^rC₅ = ?

ক
ক) 28
খ
খ) 32
গ
গ) 36
ঘ
ঘ) 56

ব্যাখ্যা

Question: If ¹⁸C_r= ¹⁸C_{r + 2} ,find ^rC₅ = ?

Solution:
¹⁸C_r= ¹⁸C_{r + 2}
So, r + r + 2 = 18
2r + 2 = 18
2r = 18 - 2
2r = 16
r = 8

⁸C₅ = 56

৬,৪৪০.

A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

ক
8 Years
খ
2 Years
গ
5 Years
ঘ
10 Years

ব্যাখ্যা

Question: A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

Solution:
Let,
'x' years ago the ratio of their ages was 3 : 2

ATQ,
(50 - x) : (35 - x) = 3 : 2
⇒ (50 - x)/(35 - x) = 3/2
⇒ 105 - 3x = 100 - 2x
⇒ 3x - 2x = 105 - 100
∴ x = 5

৬,৪৪১.

A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

ক
20^th day
খ
30^th day
গ
31^th day
ঘ
40^th day

ব্যাখ্যা

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে ২৫০ টাকা।
২য় দিনে ব্যয় করে ২০০ টাকা।

∴ ২ দিনে তার জমা থাকে (২৫০ - ২০০) = ৫০ টাকা।

এখন, (১০০০ - ২৫০) = ৭৫০ টাকা।

৫০ টাকা জমা থাকে ২ দিনে
১ টাকা জমা থাকে ২/৫০ দিনে
৭৫০ টাকা জমা থাকে (২ × ৭৫০)/৫০ দিনে
= ৩০ দিন।

৩০ দিন পর তার হাতে থাকে ৭৫০ টাকা
এবং ৩১ তম দিনে সে আয় করে ২৫০ টাকা।
তাহলে মোট টাকা হয় (৭৫০ + ২৫০) = ১০০০ টাকা,

সুতরাং ৩১ দিনে তার কাছে ১০০০ টাকা ছিল।

৬,৪৪২.

The sum of the digits of two - digit number is 10, while when the digits are reversed, the number decrease by 36. Find the changed number.

ক
28
খ
73
গ
37
ঘ
82
ঙ
None of these

ব্যাখ্যা

Question: The sum of the digits of two - digit number is 10, while when the digits are reversed, the number decrease by 36. Find the changed number.

Solution:
Let, number be = (10a + b)

ATQ
(10a + b) - (10a + b) = 36
⇒ 10a - 10b + b - a = 36
⇒ 9a - 9b = 36
⇒ a - b = 4 ............. (1)

Sum of digits,
a + b = 10 ............... (2)

(1) + (2)⇒
a - b + a + b = 4 + 10
⇒ 2a = 14
∴ a = 7
Put the value of a in (2) We get,
a + b = 10
⇒ b = 10 - 7
∴ b = 3

∴ The required number is = (10a + b)
= (10 × 7) + 3
= 73

So, Changed number = 37

৬,৪৪৩.

A mother is 20 years older than her daughter. In 10 years, her age will be twice the age of her daughter. What is the present age of the daughter?

ক
10 years
খ
12 years
গ
5 years
ঘ
6 years

ব্যাখ্যা

Question: A mother is 20 years older than her daughter. In 10 years, her age will be twice the age of her daughter. What is the present age of the daughter?

Solution:
Let the daughter’s present age be x years.
Then, the mother’s present age is = x + 20 years.

ATQ
In 10 years, the mother’s age will be twice the daughter’s age.
So, (x + 20) + 10 = 2 (x + 10)
⇒ 2x + 20 = x + 30
⇒ 2x - x = 30 - 20
∴ x = 10

Therefore, the present age of the daughter is 10 years.

৬,৪৪৪.

Which of the following describes all values of x for which 1 - x² ≥ 0?

ক
x ≤ 1
খ
0 ≤ x ≤ 1
গ
1 ≤ x ≤ - 1
ঘ
- 1 ≤ x ≤ 1

ব্যাখ্যা

Question: Which of the following describes all values of x for which 1 - x² ≥ 0?

Solution:
1 - x² ≥ 0
⇒ - x² ≥ - 1
⇒ x² ≤ 1
⇒ x² ≤ 1²
∴ - 1 ≤ x ≤ 1

৬,৪৪৫.

The midpoints of the sides of a square are connected to form a new inscribed square. How many times greater than the area of the inscribed square is the area of the original square?

ক
1/2
খ
4
গ
2
ঘ
√2

ব্যাখ্যা

Question: The midpoints of the sides of a square are connected to form a new inscribed square. How many times greater than the area of the inscribed square is the area of the original square?

Solution:

ধরি, বর্গক্ষেত্রটির বাহুর দৈর্ঘ্য n মিটার
ক্ষেত্রফল = n² বর্গমিটার

অন্ত:স্থ বর্গক্ষেত্রের বাহুর দৈর্ঘ্য n/√2 মিটার
ক্ষেত্রফল = (n/√2)²
= n²/2

বর্গক্ষেত্রটি অন্ত:স্থ বর্গক্ষেত্রের = n²/n²/2
= 2 গুণ

৬,৪৪৬.

The marked price of a t-shirt was Tk. 800. A man bought the same for Tk. 420 after getting two successive discounts. the first being 25%. then the second discount rate is-

ক
25%
খ
20%
গ
30%
ঘ
40%
ঙ
None

ব্যাখ্যা

Question: The marked price of a t-shirt was Tk. 800. A man bought the same for Tk. 420 after getting two successive discounts. the first being 25%. then the second discount rate is-

Solution:
Marked price = 800
Actual price = 420
First discount = 25%
Let the second discount be x%

Then, we can write

after 25% discount,
discounted amount = 800 × (25/100)
= 200

New price = 800 - 200
= 600 Tk

Again, second discount,
discounted amount = 600 × (x/100)
= 6x

New price = 600 - 6x

ATQ,
600 - 6x = 420
⇒ 600 - 420 = 6x
⇒ 6x = 180
⇒ x = 30

∴ Second discount = 30%

৬,৪৪৭.

From 6 men and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done, if the committee is to include at least one lady?

ক
ক) 328
খ
খ) 246
গ
গ) 252
ঘ
ঘ) 356

ব্যাখ্যা

To committee can be formed in the following ways,
(1 lady + 4 gents) or (2 ladies + 3 gents) or (3 ladies + 2 gents) or (4 ladies + 1 gents) or (5 ladies + 0 gents).
Total number of possible arrangements,
= (⁴C₁ × ⁶C₄) + (⁴C₂ × ⁶C₃) + (⁴C₃ × ⁶C₂) + (⁴C₄ × ⁶C₁)
= 60 + 120 + 60 + 6
= 246

[ From 6 men and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done, if there is no restriction about its formation?
Solution:
From 6 men and 4 ladies, a committee of 5 is to be formed.
Total number of possible arrangements, if there is no restriction about its formation
= ^{(6 + 4)}C₅
= ¹⁰C₅
= 252 ]

৬,৪৪৮.

There is a provision of food in fort for 800 soldiers for 40 days. After 10 days, 200 soldiers leave the fort. Remaining food will last for how many days?

ক
ক) 25
খ
খ) 30
গ
গ) 35
ঘ
ঘ) 40

ব্যাখ্যা

Question: There is a provision of food in fort for 800 soldiers for 40 days. After 10 days, 200 soldiers leave the fort. Remaining food will last for how many days?

Solution:
After 10 days 200 soldiers leave the fort.
So, remainig days (40-10)= 30 days
& remaing soldiers (800-200)= 600 soldiers

Now, 800 soldiers can have the food for 30 days
1 soldier can have the food for 800×30 days
600 soldiers can have the food for (800×30)/600 days = 40 days
Hence the remaining food will last for 40 days.

৬,৪৪৯.

The age of a father 10 years ago was thrice the age of his son. 10 years hence, the father's age will be twice that of his son. Father's present age is :

ক
70 years
খ
60 years
গ
80 years
ঘ
90 years

ব্যাখ্যা

Question: The age of a father 10 years ago was thrice the age of his son. 10 years hence, the father's age will be twice that of his son. Father's present age is :

Solution:
Let, son's age 10 years ago be X years
Then, father age 10 years ago = 3X years

Son's age now = (X + 10) years,
Father age now = (3X + 10) years

( 3X + 10 ) + 10 = 2[( X + 10 )+10]
⇒ 3X + 20 = 2( X + 20 )
⇒ 3X + 20 = 2X + 40
⇒ X = 20

∴ fathers age is = 60 + 10 = 70 years

৬,৪৫০.

A student is provided with three iron pieces of different lengths—44 cm, 22 cm, and 55 cm—and must form rods of the greatest possible length with no leftover iron. What is the maximum length of such rods?

ক
9 cm
খ
17 cm
গ
11 cm
ঘ
15 cm

ব্যাখ্যা

Question: A student is provided with three iron pieces of different lengths—44 cm, 22 cm, and 55 cm—and must form rods of the greatest possible length with no leftover iron. What is the maximum length of such rods?

Solution:
Maximum possible length of such rod = (H.C.F. of 44, 22, 55) cm = 11 cm.

৬,৪৫১.

Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?

ক
22
খ
25
গ
28
ঘ
32
ঙ
56

ব্যাখ্যা

Question: Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?

Solution:
Machine A produces 120/40 = 3 bolts per second
Machine B produces 100/20 = 5 bolts per second.
Running simultaneously, they produce 3 + 5 = 8 bolts per second

Thus it will take them 200/8 = 25 seconds to produce 200 bolts.

৬,৪৫২.

If n is a negative number, then which of the following is the least number?

ক
ক) 0
খ
খ) -n
গ
গ) 2n
ঘ
ঘ) n2

ব্যাখ্যা

n < 0 ⇒ 2n < 0,
again, - n > 0 and n² = (-n)2 > 0.
Thus, out of the numbers 0, -n, 2n and n²
We find that 2n is the least number here.
Answer: 2n

৬,৪৫৩.

If x + 2y = 6 and xy = 4 what is (2/x) + (1/y)?

ক
3/2
খ
1/3
গ
1/2
ঘ
1

ব্যাখ্যা

Question: If x + 2y = 6 and xy = 4 what is (2/x) + (1/y)?

Solution:
Given, x + 2y = 6
xy = 4

Now,
2/x + 1/y
= (2y + x)/xy
= (x + 2y)/xy
= 6/4
= 3/2

৬,৪৫৪.

Out of the 30000 tickets for the cricket tournament Dhaka, 1/4 were sold at Tk. 300, 1/3 were sold at Tk. 250 and the rest were sold for Tk. 125. How many tickets were sold at Tk. 125?

ক
5000
খ
7500
গ
12500
ঘ
None

ব্যাখ্যা

Question: Out of the 30000 tickets for the cricket tournament Dhaka, 1/4 were sold at Tk. 300, 1/3 were sold at Tk. 250 and the rest were sold for Tk. 125. How many tickets were sold at Tk. 125?

Solution:
300 টাকার টিকেট বিক্রি করে = 30000 × 1/4 টি
= 7500 টি

250 টাকার টিকেট বিক্রি করে = 30000 × 1/3 টি
= 10,000 টি

125 টাকার টিকেট বিক্রি করে = 30000 - (7500 + 10000) টি
= 12,500 টি

৬,৪৫৫.

A hall is 20m long and 10m broad. If the sum of the areas of the floor and the ceiling is equal to of the areas of four walls, the volume of the hall is:

ক
ক) 1333.33 m³
খ
খ) 2633.33 m³
গ
গ) 1233.33 m³
ঘ
ঘ) 2733.33 m³

ব্যাখ্যা

ধরি
হল ঘরের উচ্চতা = h
দেওয়া আছে,
ঘরের দৈর্ঘ্য 20 মি.
ঘরের প্রস্থ 10 মি.

প্রশ্নমতে,
2 ( 20 × 10 ) = 2 × ( 20 + 10 ) × h
বা, 2 × 30 × h = 2 ( 20 × 10 )
বা, h × 30 = 200
বা, h = 20/3
আমরা জানি,
আয়তন= 20 × 10 × 20/3
= 1333.33 ঘন মিটার

৬,৪৫৬.

How much time will it take for an amount of Tk. 450 to yield Tk. 99 as interest at 5.5% per annum of simple interest?

ক
3 years
খ
4 years
গ
5 years
ঘ
6 years

ব্যাখ্যা

Question: How much time will it take for an amount of Tk. 450 to yield Tk. 99 as interest at 5.5% per annum of simple interest?

Solution:
Here
Principal = Tk. 450
S.I = Tk. 99
R = 5.5%

Time = (100 × 99)/(450 × 5.5)
= 4 years

৬,৪৫৭.

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 2 : 1?

ক
8
খ
6
গ
9
ঘ
4
ঙ
5

ব্যাখ্যা

Question: The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 2 : 1?

Solution:
Let the two digit number be 10x + y
And the number obtained after interchanging be 10y + x
Difference = 9(x - y) = 36
⇒ x - y = 4
Possible combinations are (5, 1) (6, 2) (7, 3) (8, 4) (9, 5)
Also, given that the ratio of the digits is 2 : 1
Only combination possible is (8, 4)
Sum of the digits = 8 + 4 = 12
Difference of the digits = 8 - 4 = 4
Difference between these two is 12 - 4 = 8

৬,৪৫৮.

A man invested tk.1552 in a stock at 97 to obtain an income of tk. 128. The dividend from the stock is:

ক
ক) 7.5%
খ
খ) 8%
গ
গ) 9.7%
ঘ
ঘ) None of these

ব্যাখ্যা

By investing tk. 1552, income = tk. 128.
By investing tk. 97, income = tk.128/1552 x 97= tk. 8.
Dividend = 8%

৬,৪৫৯.

A student's marks were wrongly entered as 83 instead of 63. Due to the error the average marks for the class got increased by 0.5. Find the number of students in the class.

ক
ক) 10
খ
খ) 20
গ
গ) 40
ঘ
ঘ) 73

ব্যাখ্যা

নাম্বার বেশি পায় = 83 - 63 = 20
প্রশ্নমতে,
0.5 অংশ বৃদ্ধি = 20
∴ 1 অংশ বৃদ্ধি = 20/0.5 = 40
∴ ছাত্র-ছাত্রী সংখ্যা = 40

৬,৪৬০.

Find the difference between the compound profit and the simple profit on Tk. 12,000 for 2 years at 25% per annum.

ক
Tk 550
খ
Tk 350
গ
Tk 750
ঘ
Tk 480

ব্যাখ্যা

Question: Find the difference between the compound profit and the simple profit on Tk. 12,000 for 2 years at 25% per annum.

Solution:
Given that,
P = 12000 Tk
r = 25%
= 25/100
= 1/4

t = 2 years

We know,
The compound profit = P (1 + r)ⁿ- P
= 12000 (1 + 1/4)²- 12000
= (12000 × 25/16)- 12000
= 18750 - 12000
= 6750 Tk

and
Simple profit I = Prn
= 12000 × 1/4 × 2
= 6000

∴ The different between compound profit and simple profit = (6750 - 6000) Tk
= Tk 750

৬,৪৬১.

Five litres of wine is removed from a cask full of wine and is replaced with water. Five litres of this mixture is then removed and replaced with water. If the ratio of wine to water in the cask is now 16 : 9, how much wine did the cask hold?

ক
25 litres
খ
50 litres
গ
100 litres
ঘ
150 litres

ব্যাখ্যা

Question: Five litres of wine is removed from a cask full of wine and is replaced with water. Five litres of this mixture is then removed and replaced with water. If the ratio of wine to water in the cask is now 16 : 9, how much wine did the cask hold?

Solution:
Let the cask holds x liters of wine.
5 liters of wine is replaced with water. This operation is done 2 times.
∴ [(x - 5)/x]² = 16/(16 + 9)
⇒ [(x - 5) / x]² = 16/25
⇒ [(x - 5)/x] = 4/5
⇒ 5x - 25 = 4x
⇒ x = 25 liters

৬,৪৬২.

If 12 men can finish a task in 15 days, how many days will 10 men take to finish the same task?

ক
20
খ
18
গ
16
ঘ
12

ব্যাখ্যা

Question: If 12 men can finish a task in 15 days, how many days will 10 men take to finish the same task?

Solution:
12 men can finish a task in 15 days
∴ 1 men can finish a task in 15 × 12 days
∴ 10 men can finish a task in (15 × 12)/10 days
= 18 days

৬,৪৬৩.

Two trains A and B start simultaneously in the opposite direction from two points P and Q and arrive at their destinations 16 and 9 hours respectively after their meeting each other. At what speed does the second train B travel if the first train travels at 120 km/h?

ক
90 km/h
খ
160 km/h
গ
67.5 km/h
ঘ
None of these

ব্যাখ্যা

Question: Two trains A and B start simultaneously in the opposite direction from two points P and Q and arrive at their destinations 16 and 9 hours respectively after their meeting each other. At what speed does the second train B travel if the first train travels at 120 km/h?

Solution:
After meeting with each other the 1st train travels 120 × 16 km=1920km.
If the speed of the 2nd train be X km/h, it travels 1920/X hour before meeting.
After meeting, the 2nd train travels 9X km.
Before meeting, the first train covered this distance in 9X/120 hour.
As the two trains started simultaneously, before meeting their journey time was same.
So, 1920/X = 9X/120
⇒ 9X²= 1920 × 120
⇒ X² = (1920 × 120)/9
⇒ X = √[640 × 40]
∴ X=160.

∴ The train B travels at 160km/h.

৬,৪৬৪.

A shirt is sold for Tk. 1500 at a profit of 20%. What would have been the actual profit or loss percentage if it had been sold for Tk. 1200?

ক
4% loss
খ
5% profit
গ
5% loss
ঘ
4.5% profit

ব্যাখ্যা

Question: A shirt is sold for Tk. 1500 at a profit of 20%. What would have been the actual profit or loss percentage if it had been sold for Tk. 1200?

সমাধান:
ধরি, শার্টের ক্রয়মূল্য = x টাকা

20% লাভে বিক্রয়মূল্য = x + x এর 20%
= x + (x × 20/100)
= x + x/5 = 6x/5

প্রশ্নমতে,
6x/5 = 1500
⇒ 6x = 1500 × 5
⇒ 6x = 7500
⇒ x = 7500/6
⇒ x = 1250

∴ শার্টের ক্রয়মূল্য = Tk. 1250

বিক্রয়মূল্য = Tk. 1200 ক্রয়মূল্য = Tk. 1250

যেহেতু বিক্রয়মূল্য < ক্রয়মূল্য, তাই ক্ষতি হবে।

ক্ষতি = ক্রয়মূল্য - বিক্রয়মূল্য
= 1250 - 1200 = 50 টাকা

শতকরা ক্ষতি = (ক্ষতি/ক্রয়মূল্য) × 100%
= (50/1250) × 100%
= 5000/1250
= 4%

∴ 4% ক্ষতি হবে

৬,৪৬৫.

The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is-

ক
15
খ
16
গ
18
ঘ
25

ব্যাখ্যা

Question: The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is-

Solution:

Let
C.P. of each article be Tk. 1
C.P. of x articles = Tk. x.

S.P. of x articles = Tk. 20.

Profit = Tk. (20 - x).

∴ {(20 - x)/x} × 100 = 25

⇒ 2000 - 100x = 25x
⇒ 125x = 2000
∴ x = 16.

৬,৪৬৬.

A salesman travels a distance of 45 km in 2 hours and 30 minutes. How much faster, in kilometers per hour, on an average, must he travel to make such a trip in 5/6 hour less time?

ক
16 km/h
খ
15 km/h
গ
12 km/h
ঘ
9 km/h

ব্যাখ্যা

Question: A salesman travels a distance of 45 km in 2 hours and 30 minutes. How much faster, in kilometers per hour, on an average, must he travel to make such a trip in 5/6 hour less time?

Solution:
Given,
Time = 2 hours and 30 minutes
= (2 + 30/60) hours
= (2 + 1/2) hours
= 5/2 hours

∴ Initial speed = (45 ÷ 5/2) km/h
= (45 × 2/5) km/h
= 18 km/h

New time to travel 45 km = (5/2 - 5/6) = 10/6 = 5/3 hours

∴ New speed = (45 ÷ 5/3) km/h
= (45 × 3/5) km/h
= 27 km/h

∴ Speed should increased = (27 - 18) = 9 km/h

৬,৪৬৭.

দুটি ট্রেনের গতিবেগের অনুপাত ৭ : ৮. যদি দ্বিতীয় ট্রেনটি ৪ ঘণ্টায় ৩৮৪ কি,মি. যায়। তাহলে, প্রথম ট্রেনের গতিবেগ কত?

ক
ক) ৭২ কি.মি./ঘণ্টা
খ
খ) ৭৮ কি.মি./ঘণ্টা
গ
গ) ৮২ কি.মি./ঘণ্টা
ঘ
ঘ) ৮৪ কি.মি./ঘণ্টা

ব্যাখ্যা

দেয়া আছে
দুটি ট্রেনের গতিবেগের = ৭ : ৮

ধরি
১ম ট্রেনের গতিবেগ = ৭ক
২য় ট্রেনের গতিবেগ = ৮ক

আবার
২য় ট্রেনের গতিবেগ = ৩৮৪/৪ কি.মি./ঘণ্টা
= ৯৬ কি.মি./ঘণ্টা
প্রশ্নমতে
৮ক = ৯৬
ক = ১২

প্রথম ট্রেনের গতিবেগ = (৭ × ১২) কি.মি./ঘণ্টা = ৮৪ কি.মি./ঘণ্টা

৬,৪৬৮.

After a 20% discount, the price of a television is Tk. 12000. What was the price before the discount?

ক
Tk. 18000
খ
Tk. 13000
গ
Tk. 14000
ঘ
Tk. 15000
ঙ
None of the above

ব্যাখ্যা

Question: After a 20% discount, the price of a television is Tk. 12000. What was the price before the discount?

Solution:
In 20% discount,
Discount price 80 taka when original price 100 taka

∴ Discount price 1 taka when original price 100/80 taka
∴ Discount price 12000 taka when original price
(12000 × 100)/80 taka
= 15000 taka

৬,৪৬৯.

If a : b is the ratio of two whole numbers and c is their HCF, then the LCM of those two numbers is -

ক
ক) ab/c
খ
খ) bc/a
গ
গ) ac/b
ঘ
ঘ) abc

ব্যাখ্যা

Question: If a : b is the ratio of two whole numbers and c is their HCF, then the LCM of those two numbers is -

Solution:
The ratio of the numbers = a : b
HCF of the numbers = c

So, c is the common factor of the numbers

Then, First number = ac
Second Number = bc

Now,
First Number × Second Number = HCF and LCM of the numbers
⇒ ac × bc = c × LCM
⇒ LCM = abc

৬,৪৭০.

The difference of the two numbers is 20% of the large number, if the smaller number is 20, then the larger number is-

ক
15
খ
20
গ
25
ঘ
30

ব্যাখ্যা

Question: The difference of the two numbers is 20% of the large number, if the smaller number is 20, then the larger number is-

Solution:
Let the large number be x.
Then,
x - 20 = 20% of x = 20x/100 = x/5
⇒ x - x/5 = 20
⇒ 5x - x = 100
⇒ 4x = 100
∴ x = 25

৬,৪৭১.

If Machine X can produce 1,000 bolts in 4 hours and Machine Y can produce 1,000 bolts in 5 hours, in how many hours can Machines X and Y, working together at these constant rates, produce 1,000 bolts?

ক
10/9 hours
খ
20/3 hours
গ
20/9 hours
ঘ
None of these

ব্যাখ্যা

Question: If Machine X can produce 1,000 bolts in 4 hours and Machine Y can produce 1,000 bolts in 5 hours, in how many hours can Machines X and Y, working together at these constant rates, produce 1,000 bolts?

Solution:
Machine X can produce 100 bolts in 4 hrs.
In an hour X can produce 1000/4 = 250 bolts

Machine Y can produce 1000 bolts in 5 hrs.
In an hour Y can produce 1000/5 = 200 bolts

Together they can produce 250 + 200 = 450 bolts in an hour.
They will produce 1000 bolts in 1000/450 hours.
= 20/9 hours.

৬,৪৭২.

A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

ক
100 m
খ
150 m
গ
190 m
ঘ
200 m

ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman = (11 - 10) km/hr = 1 km/hr

Distance covered in 6 minutes = (1/60) × 6 km = 1/10 km = 100 m

Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

৬,৪৭৩.

The average of 17 numbers is 45. The average of first 9 of these numbers is 51 and the last 9 of these numbers is 36. What is the ninth number?

ক
18
খ
22
গ
20
ঘ
24
ঙ
None

ব্যাখ্যা

Question: The average of 17 numbers is 45. The average of first 9 of these numbers is 51 and the last 9 of these numbers is 36. What is the ninth number?

Solution:
Total of 17 numbers = 17 × 45 = 765
Total of first 9 numbers = 9 × 51 = 459
Total of last 9 numbers = 9 × 36 = 324

∴ Tenth number = ( 459 + 324) - 765
= 783 - 765
= 18

৬,৪৭৪.

A team has 5 students named A, B, C, D, and E. A is taller than B; C is shorter than E; A is shorter than D; If B is taller than E, which student is the shortest?

ক
E
খ
B
গ
A
ঘ
C

ব্যাখ্যা

Question: A team has 5 students named A, B, C, D, and E. A is taller than B; C is shorter than E; A is shorter than D; If B is taller than E, which student is the shortest?

Solution:
প্রশ্নমতে,
A, B- এর চেয়ে লম্বা; A > B,
C, E- এর চেয়ে খাটো; E > C,
A, D- এর চেয়ে খাটো; D > A,
এবং B, E- এর চেয়ে লম্বা; B > E,

এই সবগুলোকে একসাথে লিখলে সবার অবস্থান দাঁড়াবে D > A > B > E > C
সুতরাং সব থেকে ছোট হলো C

৬,৪৭৫.

Find the value of .

ক
ক) 2
খ
খ) 3
গ
গ) 1
ঘ
ঘ) 4

ব্যাখ্যা

Question: Find the value of .

Solution:
log₂√6 + log₂√(2/3)
= log₂√(6 × 2/3)
= log₂√2²
= log₂2
= 1

৬,৪৭৬.

The length of a rectangular plot is 20 meters more than its breadth. If the cost of fencing the plot at Tk. 26.50 per meter is Tk. 5,300. What is the length of the plot in meters?

ক
ক) 60
খ
খ) 100
গ
গ) 75
ঘ
ঘ) 50

ব্যাখ্যা

বাগানের পরিসীমা = 5300/26.5
= 200 মিটার
এখন ধরি, বাগানের প্রস্থ x মিটার
∴ বাগানের দৈর্ঘ্য (x + 20) মিটার
প্রশ্নমতে, 2(x + x + 20) = 200
⇒ 2(2x + 20) = 200
⇒ 2x + 20 = 100
⇒ 2x = 100 - 20
⇒ 2x = 80
⇒ x = 40
∴ বাগানের দৈর্ঘ্য = 40 + 20 = 60 মিটার

৬,৪৭৭.

The number of subsets of a set with 5 elements is:

ক
10
খ
25
গ
30
ঘ
32

ব্যাখ্যা

Question: The number of subsets of a set with 5 elements is:

Solution:
- কোনো সেট থেকে যতগুলো সেট গঠন করা যায়, এদের প্রত্যেকটি সেটকে ঐ সেটের উপসেট (subset) বলা হয়।
কোনো সেটের উপাদানের সংখ্যা, n = 5
ঐ সেটের উপসেট (subset) সংখ্যা = 2ⁿ
=2⁵
=32

৬,৪৭৮.

A cistern is filled by Pipe A and Pipe B together in 2 hours. Pipe A alone can fill the cistern at the rate of 100 litres per hour. Pipe B alone can fill the cistern in 4 hours. What is the capacity of the cistern?

ক
400 litres
খ
380 litres
গ
450 litres
ঘ
520 litres

ব্যাখ্যা

Question: A cistern is filled by Pipe A and Pipe B together in 2 hours. Pipe A alone can fill the cistern at the rate of 100 litres per hour. Pipe B alone can fill the cistern in 4 hours. What is the capacity of the cistern?

Solution:
Let the capacity of the cistern = x litres.

Pipe A fills at 100 litres per hour
∴ Time taken by A alone = x/100 hours
Pipe B alone fills the cistern in 4 hours
∴ Pipe B's rate = x/4 litres per hour
And Combined rate of A + B = 100 + (x/4) litres per hour

They together fill the cistern in 2 hours, so:
Combined rate = x/2 litres per hour

Therefore, 100 + (x/4) = x/2
⇒ (x/2) - (x/4) = 100
⇒ (2x - x)/4 = 100
⇒ x = 4 × 100
∴ x = 400

So the capacity of the cistern is 400 litres.

৬,৪৭৯.

What word can be formed by arranging the letters of 'AEEHLPTN'?

ক
A name of a disease
খ
A name of an animal
গ
A name of a game
ঘ
A name of a sea

ব্যাখ্যা

Question: What word can be formed by arranging the letters of 'AEEHLPTN'?

Solution:
By rearranging the letters 'AEEHLPNT', the name of an animal (ELEPHANT ⇒ Elephant) can be formed.

৬,৪৮০.

Practicing for a competition, a swimmer saw that he could swim 20 km downstream in just 1 hr while it took 2 hrs to swim upstream. Find the speed of the river and that of the swimmer respectively.

ক
ক) 4 km/hr ; 16 km/hr
খ
খ) 5 km/hr ; 15 km/hr
গ
গ) 6 km/hr ; 14 km/hr
ঘ
ঘ) 8 km/hr ; 12 km/hr

ব্যাখ্যা

Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

Downstream Speed = Distance covered/Time taken
= 20/1
= 20 km/hr
Upstream Speed = 20/2 = 10 km/hr

X + Y = 20 km/hr
and, X - Y = 10 km/hr

Adding them we get,
X + Y + X - Y = 30 km/hr
∴ X=15 km/hr = Speed of swimmer in still water

∴ Y = 20 - 15 = 5 km/hr = Speed of river.

৬,৪৮১.

a + b + c + d = ?

ক
160°
খ
360°
গ
320°
ঘ
540°

ব্যাখ্যা

Question: a + b + c + d = ?

Solution:

We know that,
Total amount of internal angles of pentagon is 540°
∴ 180° - a + 140° + 180° - b + 180° - c + 180° - d = 540°
⇒ 860° - a - b - c - d = 540°
⇒ a + b + c + d = 860° - 540° = 320°

৬,৪৮২.

A train can travel 50% faster than a car. Both start from point A at the same time and reach point B, 150 km away from A at the same time. On the way, however, the train lost about 25 minutes while stopping at the stations. The speed of the car is:

ক
80 kmph
খ
120 kmph
গ
132 kmph
ঘ
140 kmph

ব্যাখ্যা

Question: A train can travel 50% faster than a car. Both start from point A at the same time and reach point B, 150 km away from A at the same time. On the way, however, the train lost about 25 minutes while stopping at the stations. The speed of the car is:

Solution:
Let speed of the car be x kmph.
Then, speed of the train = x + (50x)/100 = x + x/2
= (3x)/2

Now
(150/x) - [150/{(3x)/2)}] = 25/60
⇒ (150/x) - {300/(3x)} = 5/12
⇒ (150/x) - (100/x) = 5/12
⇒ (150 - 100)/x = 5/12
⇒ 50/x = 5/12
⇒ 5x = 12 × 50
∴ x = 120 kmph

৬,৪৮৩.

Which of the followings is the smallest value?

ক
0.000064/0.00016
খ
0.00003/0.000006
গ
0.00025/0.0005
ঘ
0.00049/0.007

ব্যাখ্যা

Question: Which of the followings is the smallest value?

Solution:
0.00025/0.0005 = 0.5
0.00003/0.000006 = 5
0.00049/0.007 = 0.07
0.000064/0.00016 = 0.4

৬,৪৮৪.

A man travels equal distances of his journey at 40, 30 and 15 km/h respectively. Find his average speed for whole journey.

ক
ক) 24 km/hour
খ
খ) 14 km/hour
গ
গ) 32 km/hour
ঘ
ঘ) 36 km/hour

ব্যাখ্যা

Required average speed
= (3 ×15 ×30 × 40) / {(40 × 30) + (40 × 15) + (30 × 15)}
= 54000/(1200 + 600 + 450)
= 54000/2250
= 24 km/hour

৬,৪৮৫.

Length of the train is 170 meters and the speed of the train is 63 km/hour. This train can pass a bridge in 30 seconds, then find the length of the bridge.

ক
ক) 355 m
খ
খ) 325 m
গ
গ) 365 m
ঘ
ঘ) 312 m

ব্যাখ্যা

Given speed = 63 km/hr = 63 × 5/18 = 35/2 m/s
Let the length of the bridge = x mts
Given time taken to cover the distance of (170 + x) mts is 30 sec.
We know speed = distance / time
⇒ 35/2 = (170+x)/30
⇒ 340 + 2x = 1050
⇒ 2x = 710
∴ x = 355

৬,৪৮৬.

Jamil can run 1 km in 4 minutes. He has a car that has a top speed of 75 km/hour. Jamil's speed is what percentage of his car's top speed?

ক
10%
খ
20%
গ
25%
ঘ
30%
ঙ
None of these

ব্যাখ্যা

Question: Jamil can run 1 km in 4 minutes. He has a car that has a top speed of 75 km/hour. Jamil's speed is what percentage of his car's top speed?

Solution:
4 মিনিট= 4/60 ঘণ্টা
= 1/15 ঘণ্টা

জামিলের গতিবেগ = 1/(1/15) কি.মি./ঘণ্টা
= 15 কি.মি./ঘণ্টা

গাড়ির গতিবেগ = 75 কি.মি./ঘণ্টা
∴ জামিলের গতিবেগ তার গাড়ির গতিবেগের = {(15/75) × 100}%
= 20%

৬,৪৮৭.

If 5 students run a mile in 5 minutes, how much time will 50 students take to run a mile?

ক
ক) 5 minutes
খ
খ) 10 minutes
গ
গ) 50 minutes
ঘ
ঘ) None of these

ব্যাখ্যা

Question: If 5 students run a mile in 5 minutes, how much time will 50 students take to run a mile?

Solution:
5 students run a mile in 5 minutes
1 students run a mile in 5 minutes
50 students run a mile in 5 minutes

৬,৪৮৮.

If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?

ক
720 Tk
খ
680 Tk
গ
600 Tk
ঘ
580 Tk
ঙ
620 Tk

ব্যাখ্যা

Question: If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?

Solution:
Whole work is done by A in (3 × 4) = 12 days
∴ A's 1 day's work = 1/12 part
Whole work is done by B in (4 × 9) = 36 days
∴ B's 1 day's work = 1/36 part

A's 1 day's work : B's 1 day's work
= A's wages : B's wages
= 1/12 : 1/36
= 3 : 1

∴ A's share = (800 × 3/4) Tk
= 600 Tk

৬,৪৮৯.

Rakib studies with the help of flash cards. He has a set of 30 flash cards out of which 17 cards are white and rest are grey. 4 white and 5 grey cards are marked ENGLISH. Find the possibility of choosing a grey card or an ''ENGLISH'' Card randomly from the set.

ক
ক) 9/13
খ
খ) 13/30
গ
গ) 17/30
ঘ
ঘ) 22/30

ব্যাখ্যা

We know,
Probability = what we want/Total
Or = add; AND = multiply

We want a grey card OR ENGLISH card
There are 30 - 17 = 13 grey cards
There are 4 + 5 = 9 ENGLISH cards
Total cards = 30
Also, 5 grey cards are ENGLISH cards.

So Probability = 13/30 + 9/30 - 5/30 = (13 + 9 - 5)/30
= 17/30 [This subtraction is needed a grey card gets counted twice - once in 13 grey cards and once again in 9 ENGLISH cards.]

৬,৪৯০.

A boat can travel with a speed of 15 km/hr in still water. If the speed of the stream is 10 km/hr, find the time taken by the boat to go 75 km downstream.

ক
3 hours
খ
4 hours
গ
5 hours
ঘ
6 hours
ঙ
7 hours

ব্যাখ্যা

Speed downstream = (15 + 10) km/hr or 25 km/hr
Time taken to travel 75 km downstream = 75/25 hours or 3 hours

৬,৪৯১.

The difference between simple interest for 4 years and 6 years at 5.5% per annum is BDT 220. Find the principal amount.

ক
Tk. 2000
খ
Tk. 3600
গ
Tk. 1800
ঘ
Tk. 2400

ব্যাখ্যা

Question: The difference between simple interest for 4 years and 6 years at 5.5% per annum is BDT 220. Find the principal amount.

Solution:
Given that,
Rate of interest, r = 5.5%
Difference in simple interest for 6 years and 4 years = 220
Time difference, n = 6 - 4 = 2 years

We know that,
I = (P × r × n)/100
⇒ 220 = (P × 5.5 × 2)/100
⇒ 11 × P = 22000
⇒ P = 22000/11
∴ P = 2000
∴ The sum is Tk. 2000.

৬,৪৯২.

How much time will it take for an amount of Tk. 900 to yield Tk. 81 as interest at 9% per annum of simple interest?

ক
2 years
খ
1.5 years
গ
1 years
ঘ
3 years

ব্যাখ্যা

Question: How much time will it take for an amount of Tk. 900 to yield Tk. 81 as interest at 9% per annum of simple interest?

Solution:
I = Pnr
⇒ 81 = 900 × n × (9/100)
⇒ n = (81 × 100)/(900 × 9)
∴ n = 1 years.

৬,৪৯৩.

A gym class can be divided into 8 teams with an equal number of players on each team or into 12 teams with an equal number of players on each team. What is the lowest possible number of students in the class?

ক
ক) 20
খ
খ) 24
গ
গ) 36
ঘ
ঘ) 48

ব্যাখ্যা

The L.C.M. of 8 and 12 is the lowest possible number of students in the class.
The L.C.M. of 8 and 12 is 24.

৬,৪৯৪.

What is the least number which when doubled is exactly divisible by 8, 14, 18, and 24?

ক
240
খ
252
গ
264
ঘ
270

ব্যাখ্যা

Question: What is the least number which when doubled is exactly divisible by 8, 14, 18, and 24?

Solution:
Let the number be x.
Double the number is 2x.

8 = 2 × 2 × 2
14 = 2 × 7
18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3

∴ LCM = 2 × 2 × 2 × 3 × 3 ×7
= 504

∴ x = 504/2 = 252

৬,৪৯৫.

In Figure, we have BX = (1/2)AB, BY = (1/2)AB and AB = BC, then-

ক
BX = BY
খ
BX ≠ BY
গ
AX = AC
ঘ
More than one of the above
ঙ
None of the above

ব্যাখ্যা

Question: In Figure, we have BX = (1/2)AB, BY = (1/2)AB and AB = BC, then-

Solution:
4th axiom of Euclid which state that, “the things which coincide with one another will be equal to one another.

In the given Figure, we have
BX = (1/2)AB
BY = (1/2)AB
AB = BC

Here, both BX and BY are equal to half of the line segment AB. Thus, from Euclid's axiom, all these three parts are equal to each other.
BX = (1/2)AB = BY
∴ BX = BY

৬,৪৯৬.

If the price of sugar increases by 10%, by what percentage a housewife should reduce the consumption of sugar so that expense is not increased?

ক
5.5%
খ
7.1%
গ
9.1%
ঘ
10%

ব্যাখ্যা

Question: If the price of sugar increases by 10%, by what percentage a housewife should reduce the consumption of sugar so that expense is not increased?

Solution:
Let the price be Tk. x per kg
Consumption be y kg

Expenditure = price × consumption
⇒ Expenditure = x × y

Price of sugar is increased by 10%
New price of sugar = Tk. 1.1x per kg

Let new consumption be z kg
New Expenditure = (1.1x) × z

New Expenditure = old Expenditure
⇒ (1.1x) × z = x × y
⇒ z = y/1.1

Reduction in consumption = (y - z) = y - y/1.1 = y/11

Percentage reduction in consumption = (reduction in consumption/original consumption)× 100
Percentage reduction in consumption = [(y/11)/y] × 100 = 100/11 = 9.09% ≈ 9.1%

৬,৪৯৭.

The value of 1 + {(tan 30° - tan 45°)/(cot 45° - cot 60°)} is -

ক
-1
খ
0
গ
1
ঘ
2

ব্যাখ্যা

Question: The value of 1 + {(tan 30° - tan 45°)/(cot 45° - cot 60°)} is -

Solution:
1 + (tan 30° - tan 45°)/(cot 45° - cot 60°)
= 1 + (tan 30° - tan 45°)/{cot (90° - 45°) - cot (90° - 60°)}
= 1 + (tan 30° - tan 45°)/(tan 45° - tan 30°)
= 1 + (tan 30° - tan 45°)/(-1)(tan 30° - tan 45°)
= 1 - 1
= 0

৬,৪৯৮.

A pipe can fill a tank in a hours and another pipe can empty it in b (b > a) hours. If both pipes are open, in how many hours will the tank is filled?

ক
(b - a)/ab hours
খ
(a + b) hours
গ
ab/(b - a) hours
ঘ
None of these

ব্যাখ্যা

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

Solution:
Net part filled in 1 hour

∴ Net rate = Filling rate - Emptying rate
= (1/a) - (1/b)
= (b - a)/ab

∴ The tank will be filled in = ab/(b - a) hours.

৬,৪৯৯.

If 4y - 3x = 5, what is the smallest integer value of x for which y > 100?

ক
395
খ
134
গ
132
ঘ
131

ব্যাখ্যা

Question: If 4y - 3x = 5, what is the smallest integer value of x for which y>100?

Solution:
4y - 3x = 5
⇒ 4y = 3x + 5
⇒ y = (3x + 5)/4

(3x + 5)/4 > 100
⇒ 3x + 5 > 400
⇒ 3x > 395
⇒ x > 395/3
⇒ x > 131.67

The smallest integer value for x 132

৬,৫০০.

If both 11² and 3³ are factors of the number a × 4³ × 6² × 13¹¹, then what is the smallest possible value of 'a'?

ক
33
খ
121
গ
363
ঘ
3267

ব্যাখ্যা

Question: If both 11² and 3³ are factors of the number a × 4³ × 6² × 13¹¹, then what is the smallest possible value of 'a'?

Solution:
a × 4³ × 6² × 13¹¹ can be expressed in terms of its prime factors as a × 2⁸ × 3² × 13¹¹

11² is a factor of the given number.
If we do not include 'a', 11 is not a prime factor of the given number.
If 11² is a factor of the number, 11² should be a part of 'a'

3³ is a factor of the given number.
If we do not include 'a', the number has only 3² in it.
Therefore, if 3³ has to be a factor of the given number 'a' has to contain 31 in it.

Therefore, 'a' should be at least 11² × 3 = 363 if the given number has 11² and 3³ as its factors.

The question is "what is the smallest possible value of 'a'?"
The smallest value that 'a' can take is 363