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Number System, Problems on Number

মোট প্রশ্ন১,৭৩৬এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Number System, Problems on Number

PrepBank · পাতা ১ / ১৮ · ১–১০০ / ১,৭৩৬

১.
The difference between the two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are -
  1. ক
    ক) 9 and 25
  2. খ
    খ) 12 and 28
  3. গ
    গ) 33 and 49
  4. ঘ
    ঘ) 56 and 72
ব্যাখ্যা

Let the numbers be x and (x + 16)
Then, x/3 - (x + 16)/7 = 4
⇔ 7x - 3(x +16) = 84
⇔ 4x = 84 + 48 = 132
⇔ x = 132/4 = 33
hence, the numbers are 33 and 49
Answer: 33 and 49

২.
Four fifth of a number is 12 more than the two third of the number. The number is -
  1. ক
    55
  2. খ
    65
  3. গ
    80
  4. ঘ
    90
ব্যাখ্যা
Question: Four fifth of a number is 12 more than the two third of the number. The number is -

Solution:
Let,
The number is x.
ATQ
(4/5) × x = {(2/3) × x} + 12
⇒ (4x)/5 = (2x)/3 + 12
⇒ 3 × 4x = 5 × 2x + 180 [multiply with 15]
⇒ 12x = 10x + 180
⇒ 2x = 180
∴ x = 90
৩.
After dividing a number successively by 3, 4, and 7, the remainder obtained is 2, 1, and 4 respectively. What will be the remainder if 84 is divided by the same number? 
  1. ক
    49
  2. খ
    51
  3. গ
    53
  4. ঘ
    57
ব্যাখ্যা
Question: After dividing a number successively by 3, 4, and 7, the remainder obtained is 2, 1, and 4 respectively. What will be the remainder if 84 is divided by the same number? 

Solution:
As the Number gives a remainder of 4 when it is divided by 7, then the number = (7x + 4)
The same gives remainder 1 when it is divided 4, then the number = {4 × (7x + 4) + 1}
Also, the number when divided by 3 gives the remainder 2, thus number = [3 × {4 × (7x + 4) + 1} + 2]
= 3 × (28x + 16 + 1) + 2
= 84x + 51 + 2
= 84x + 53

We get the final number 53 more than a multiple of 84 Hence, if the number is divided by 84,
The remainder will be 53 
৪.
If the sum of a number and 10 exceeds 1/21 of 5/7 of the number by 294, then what will be the number?
  1. ক
    21
  2. খ
    127
  3. গ
    294
  4. ঘ
    81
ব্যাখ্যা

Let the number be X.
Then the sum of the number and 10 = X + 10.
and 1/21 of 5/7 of the number = 1/21 × 5/7 × X
From given,
We have X + 10 - 294 = 1/21 × 5/7 × X
X + 10 - 294 = 1/21 × 5/7 × X
⇒ X - 284 = 5X/147
⇒ 5X/147 - X = -284
⇒ 5X - 147X/147 = -284
⇒ -142X = 147 × (-284)
⇒ X = 147 × 2
⇒ X = 294
Hence the required number is 294.

৫.
If you count from 1 to 100, how often will you encounter the number 5?
  1. ক
    25
  2. খ
    19
  3. গ
    12
  4. ঘ
    20
ব্যাখ্যা

Question: If you count from 1 to 100, how often will you encounter the number 5?

Solution:
১ - ১০ পর্যন্ত ৫ আছে = ১ টি
১১ - ২০ পর্যন্ত ৫ আছে = ১ টি
২১ - ৩০ পর্যন্ত ৫ আছে = ১ টি
৩১ - ৪০ পর্যন্ত ৫ আছে = ১ টি
৪১ - ৫০ পর্যন্ত ৫ আছে = ২ টি
৫১ - ৬০ পর্যন্ত ৫ আছে = ১০ টি
৬১ - ৭০ পর্যন্ত ৫ আছে = ১ টি
৭১ - ৮০ পর্যন্ত ৫ আছে = ১ টি
৮১ - ৯০ পর্যন্ত ৫ আছে = ১ টি
৯১ - ১০০ পর্যন্ত ৫ আছে = ১ টি

∴ মোট ৫ রয়েছে = ২০ টি

৬.
If x2 + 4x + 3 is odd, then which one of the following could be the value of x?
  1. ক
    ক) 3
  2. খ
    খ) 5
  3. গ
    গ) 9
  4. ঘ
    ঘ) 16
ব্যাখ্যা
x = 3 হলে x2 + 4x + 3 = 32 + 4 × 3 + 3 = 9 + 12 + 3 = 24 
x = 5 হলে x2 + 4x + 3 = 52 + 4 × 5 + 3 = 25 + 20 + 3 = 48
x = 9 হলে x2 + 4x + 3 = 92 + 4 × 9 + 3 = 81 + 36 + 3 = 120
x = 16 হলে x2 + 4x + 3 = 162 + 4 × 16 + 3 = 256 + 64 + 3 = 323 
৭.
0.014 × 0.014 = ?
  1. ক
    0.000196
  2. খ
    0.00196
  3. গ
    19.6
  4. ঘ
    196
ব্যাখ্যা
Question: 0.014 × 0.014 = ?

Solution:
0.014 × 0.014 = 0.000196
৮.
If the ratio of two numbers is 3:4 and their L.C.M is 84, then find the greater number.
  1. ক
    ক) 28
  2. খ
    খ) 34
  3. গ
    গ) 82
  4. ঘ
    ঘ) 54
ব্যাখ্যা

Let, these two numbers be 3x and 4x and their LCM = 12x
ATQ,
12x = 84
Or, x = 7
So, the greater number is = 4 × 7 = 28

৯.
What is the greatest number of four digits which is divisible by 15, 25, 40 and 75?
  1. ক
    ক) 9200
  2. খ
    খ) 9400
  3. গ
    গ) 9600
  4. ঘ
    ঘ) 9800
ব্যাখ্যা

Greatest number of four digits = 9999
LCM of 15, 25, 40 and 75 = 600
9999/600 = 16,
remainder = 399
Hence, greatest number of four digits which is divisible by 15, 25, 40 and 75
= 9999 - 399
= 9600

১০.
The HCF and LCM of the two numbers are 165 and 33 respectively. If the ratio of the two numbers 1 : 5, then what is the largest number?
  1. ক
    ক) 105
  2. খ
    খ) 115
  3. গ
    গ) 135
  4. ঘ
    ঘ) 165
ব্যাখ্যা
Question: The HCF and LCM of the two numbers are 165 and 33 respectively. If the ratio of the two numbers 1 : 5, then what is the largest number?

Solution:
Let the numbers be x and 5x respectively.

ATQ,
x × 5x = 165 × 33
⇒ x2 = (165 × 33)/5
⇒ x2 = 33 × 33
⇒ x = 33

The largest number is = 5 × 33 = 165
১১.
Find the square of a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.
  1. ক
    9
  2. খ
    16
  3. গ
    25
  4. ঘ
    18
ব্যাখ্যা
Question: Find the square of a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

Solution: 
Let the number be x

Then, x + 17 = 60/x
⇒ x2 + 17x − 60 = 0
⇒ x2 + 20x − 3x − 60 = 0
⇒ (x + 20)(x − 3) = 0
∴ x = - 20, 3 

The positive number = 3
Hence, the square of the positive number = 9
১২.
If both 112  and 33  are factors of the number a* 43 * 62 * 1311 , then what is the smallest possible value of ‘a’?
  1. ক
    ক) 363
  2. খ
    খ) 121
  3. গ
    গ) 3267
  4. ঘ
    ঘ) 33
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:
প্রদত্ত সংখ্যাটির একটি উৎপাদক হলো 112 
 a * 43 * 62 * 1311 সংখ্যাটি 11 এর গুণিতক নয়। 
a * 43 * 62 * 1311 সংখ্যাটি 11 দ্বারা বিভাজ্য হলে a এর মান 112 হবে।  

আবার,
33 হলো a * 43 * 62 * 1311 এর আরও একটি উৎপাদক 
a * 43 * 62 * 1311 এ  62 হলো সংখ্যাটির একটি অংশ। 
এখানে,
62 = 32 * 22
a * 43 * 62 * 1311 সংখ্যাটিতে  32 রয়েছে। 
a সংখ্যাটিতে আরো একটি 3 লাগবে। 

a এর কমপক্ষে মান হবে = 112 × 3 = 363
১৩.
How many 5 will you pass on the way when you count from 1 to 100?
  1. ক
    ক) 18
  2. খ
    খ) 19
  3. গ
    গ) 20
  4. ঘ
    ঘ) 21
ব্যাখ্যা

Numbers in which digit 5 is used between 1 & 100 are the following = 5 , 15 , 25 , 35 , 45 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 65 , 75 , 85 , 95 = 20 times

১৪.
How many numbers from 11 to 50 are there which are exactly divisible by 7 not by 3?
  1. ক
    ক) 6
  2. খ
    খ) 5
  3. গ
    গ) 4
  4. ঘ
    ঘ) 2
ব্যাখ্যা
Question: How many numbers from 11 to 50 are there which are exactly divisible by 7 not by 3?

Solution:
11 থেকে 50 পর্যন্ত 7 দ্বারা বিভাজ্য সংখ্যাগুলো হলো 14, 21, 28, 35, 42, এবং  49
14, 28, 35, এবং 49 সংখ্যাগুলো 3 দ্বারা বিভাজ্য নয়।
 7 দ্বারা বিভাজ্য এবং 3 দ্বারা বিভাজ্য নয় এমন সংখ্যা = 4
১৫.
Find the LCM of 2.5, 0.5 and 0.175.
  1. ক
    ক) 0.5
  2. খ
    খ) 0.05
  3. গ
    গ) 17.5
  4. ঘ
    ঘ) 1.75
ব্যাখ্যা
Question: Find the LCM of 2.5, 0.5 and 0.175.

Solution: 
2.5 = 25/10
0.5 = 5/10
0.175 = 175/1000

দুই বা ততোধিক ভগ্নাংশের ল.সা.গু = (লব গুলোর ল.সা.গু)/(হর গুলোর গ.সা.গু)
= (২৫, ৫, ১৭৫ এর ল.সা.গু)/(১০, ১০, ১০০০ এর গ.সা.গু)
= ১৭৫/১০
= ১৭.৫
১৬.
  1. ক
    2.11
  2. খ
    2.13
  3. গ
    2.58
  4. ঘ
    2.03
ব্যাখ্যা
Question:

Solution: 
১৭.
A gardener planted trees in rows and columns such that the number of rows is seven more than the number of columns. If the total number of rows and columns is 87, find the number of trees.
  1. ক
    1880
  2. খ
    2160
  3. গ
    1680
  4. ঘ
    2450
ব্যাখ্যা

Question: A gardener planted trees in rows and columns such that the number of rows is seven more than the number of columns. If the total number of rows and columns is 87, find the number of trees.

Solution:
Let the number of columns = x.
Then, number of rows = x + 7

According to the question,
x + (x + 7) = 87
⇒ 2x + 7 = 87
⇒ 2x = 80
⇒ x = 40

Number of rows = x + 7 = 40 + 7 = 47

Total number of trees = rows × columns
= 47 × 40 = 1880

১৮.
If (89)2 is added to the square of a number, the answer so obtained is 16202. What is the 1/26 of that number?
  1. ক
    ক) 5.65
  2. খ
    খ) 2.70
  3. গ
    গ) 3.50
  4. ঘ
    ঘ) 6.66
ব্যাখ্যা
Solution: 
let the number is = x 

ATQ,
x2 + (89)2 = 16202
x2 = 16202 - 7921
x2 = (91)2
x = 91

1/26 of x = 91/26 = 3.5
১৯.
a2 is divisible by both 40 and 75 if a has exactly three distinct prime factors, which of the following could be the value of 'a'?
  1. ক
    ক) 30
  2. খ
    খ) 60
  3. গ
    গ) 200
  4. ঘ
    ঘ) 420
  5. ঙ
    ঙ) None
ব্যাখ্যা
LCM of 40 and 75 = 600.
600 = 2² × 5² × 3 × 2.
As a² is a perfect square number,
the least value of a² is = 2²×5² × 3²×2² = 3600.
So, the value of a = √3600 = 60.
২০.
If the sum of two numbers is 36 and their H.C.F and L.C.M are 3 and 105, respectively, the sum of the reciprocals of the two number is - 
  1. ক
    4/35
  2. খ
    4/39
  3. গ
    2/17
  4. ঘ
    4/33
ব্যাখ্যা
Question: If the sum of two numbers is 36 and their H.C.F and L.C.M are 3 and 105, respectively, the sum of the reciprocals of the two number is - 

Solution: 
let,
the numbers are x and y 
∴ x + y = 36 
and xy = (3 × 105) = 315

sum of their reciprocals = 1/x + 1/y 
= (x + y)/xy
= 36/315
= 4/35
২১.
In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?
  1. ক
    45
  2. খ
    72
  3. গ
    60
  4. ঘ
    54
ব্যাখ্যা

Question: In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?

​​Solution:
Number of students who play basketball, n(B) = 30
Number of students who play volleyball, n(V) = 20
Number of students who play both basketball and volleyball, n(B ∩ V) = 8
Number of students who play neither = 12

n(B ∪ V) = n(B) + n(V) - n(B ∩ V)
= 30 + 20 - 8 = 42

Total students in the class = students who play basketball or volleyball + students who play neither
n(U) = n(B ∪ V) + neither = 42 + 12 = 54

∴ Total 54 students in the class.

২২.
The next term of the series: 36 + 81 + 144 + 225 + ____ is
  1. ক
    ক) 256
  2. খ
    খ) 289
  3. গ
    গ) 324
  4. ঘ
    ঘ) 369
ব্যাখ্যা

Given, 36 + 81 + 144
The series is = 62 + 92 + 122 + 152 + 182
So, next term is 182 = 324

২৩.
Which of the following is a factor of x3 - x - 24?
  1. ক
    ক) x - 5
  2. খ
    খ) x - 3
  3. গ
    গ) x - 2
  4. ঘ
    ঘ) x - 1
ব্যাখ্যা
Question: Which of the following is a factor of x3 - x - 24?

Solution: 
f(x) = x3 - x - 24
f(3) = 33 - 3 - 24
f(3) =27 - 27
      = 0
(x - 3) is a factor of x3 - x - 24.
২৪.
When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?
  1. ক
    23x + 17y = 19
  2. খ
    17x - 23y = 9
  3. গ
    17x + 23y = 19
  4. ঘ
    14x + 5y = 6
  5. ঙ
    5x - 14y = - 6
ব্যাখ্যা
Question: When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?

Solution:
From the problem it follows:
n = 17x + 5
n = 23y + 14

So,
17x + 5 = 23y + 14
⇒ 17x - 23y = 9
২৫.
Out of six consecutive natural numbers if the sum of first three is 27, what is the sum of the other three?
  1. ক
    ক) 24
  2. খ
    খ) 25
  3. গ
    গ) 35
  4. ঘ
    ঘ) 36
ব্যাখ্যা

Let the six numbers be, x, (x + 1), (x + 2), (x + 3), (x + 4) and (x + 5)
Then,
⇔ x + (x + 1) + (x + 2) = 27
⇔ 3x + 3 = 27
Required sum :
= (x + 3) + (x + 4) + (x + 5)
= 3x + 12 
= (3x + 3) + 9
= 27 + 9
= 36

২৬.
A whole number is added to 100 and the same number is subtracted from 100. The sum of the resulting numbers is-
  1. ক
    0
  2. খ
    50
  3. গ
    100
  4. ঘ
    200
ব্যাখ্যা
Question: A whole number is added to 100 and the same number is subtracted from 100. The sum of the resulting numbers is-

Solution:
Let the whole number be x.

∴ (x + 100) + (100 - x)
= x + 100 + 100 - x
= 200
২৭.
A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?
  1. ক
    ক) 12
  2. খ
    খ) 11
  3. গ
    গ) 10
  4. ঘ
    ঘ) 9
  5. ঙ
    ঙ) 8
ব্যাখ্যা
Question: A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?

Solution: 
খাবার বাকি থাকে = 91 - 39 = 52 পাউন্ড 

52 পাউন্ড খাবার খায় = 6 দিনে 
1 পাউন্ড খাবার খায় = 6 × 52 দিনে 
1 পাউন্ড খাবার খায় = (6 × 52)/39 দিনে 
= 8 দিনে 
২৮.
If A, B, and C are three numbers, such that the L.C.M of A and B is B and L.C.M. of B and C is B then the L.C.M of A, B and C is-
  1. ক
    ক) A
  2. খ
    খ) B
  3. গ
    গ) C
  4. ঘ
    ঘ) (A + B + C)/3
ব্যাখ্যা

L.C.M of A and B is ''B'';
L.C.M of B and C is ''B';
⇒ L.C.M of A, B, C is ''B''
Answer : B

২৯.
  1. ক
    ক) 16
  2. খ
    খ) 8
  3. গ
    গ) 4
  4. ঘ
    ঘ) 1
ব্যাখ্যা
Question:

Solution: 
৩০.
In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?
  1. ক
    ক) 10
  2. খ
    খ) 12
  3. গ
    গ) 14
  4. ঘ
    ঘ) 16
ব্যাখ্যা

ধরি, সংখ্যাগুলো x - 3, x - 2, x - 1, x, x + 1, x + 2, x + 3, x + 4
শেষ চারটি সংখ্যার যোগফল = 4x + 10
প্রথম চারটি সংখ্যার যোগফল = 4x - 6
∴ ( - ) পার্থক্য = 16

৩১.
  1. ক
    ক) 1
  2. খ
    খ) 2
  3. গ
    গ) 4
  4. ঘ
    ঘ) 643
ব্যাখ্যা
প্রশ্ন:

সমাধান: 
৩২.
If a = m - 1, then which of the following is true when m = 1/2
  1. ক
    a0 > a2 > a3 > a
  2. খ
    a0 > a2 > a > a3
  3. গ
    a0 > a > a2 > a3
  4. ঘ
    a2 > a > a0 > a3
  5. ঙ
    None of these
ব্যাখ্যা
Question: If a = m - 1, then which of the following is true when m = 1/2

Solution:
a = m - 1
⇒ a = 1/2 - 1
∴ a = - (1/2)

a0 = 1
a = - (1/2)
a2 = 1/4
a3 = - (1/8)

∴ a0 > a2> a3 > a
৩৩.
Mamun bought 50 shares at Tk. 60, and 2 months later, he purchased 25 shares at Tk 56, at what price should he purchase 25 additional shares in order to have an average price of Tk 58 per share?
  1. ক
    ক) 53
  2. খ
    খ) 54
  3. গ
    গ) 55
  4. ঘ
    ঘ) 56
ব্যাখ্যা
Question: Mamun bought 50 shares at Tk. 60, and 2 months later, he purchased 25 shares at Tk 56, at what price should he purchase 25 additional shares in order to have an average price of Tk 58 per share?

Solution: 
ধরি 
x  টাকা দরে আরো ২৫ টি শেয়ার কিনেছিল 

প্রশ্নমতে 
(50 × 60) + (25 × 56) + (25 × x)/(50 + 25 + 25) = 58 
(3000 + 1400 + 25x)/100 = 58
25x + 4400 = 5800
25x = 5800 - 4400
25x = 1400
x = 1400/25
x = 56 
৩৪.
If 738A6A is divisible by 11, then the value of A is:
  1. ক
    ক) 7
  2. খ
    খ) 12
  3. গ
    গ) 9
  4. ঘ
    ঘ) 10
ব্যাখ্যা
A number is divisible by 11 when the difference between the total number of odd places and the total number of even places is equal to zero or multiple of 11

Therefore, A + A + 3 = 7 + 8 + 6
⇒ 2A = 18
⇒ A = 9
৩৫.
The product of 0.09 and 0.007 is :
  1. ক
    ক) 0.00630
  2. খ
    খ) 0.000663
  3. গ
    গ) 0.06300
  4. ঘ
    ঘ) 0.00063
ব্যাখ্যা

9 × 7 = 63
Sum of decimal places = 5
∴ 0.09 × 0.007 = 0.00063

৩৬.
The numbers of prime factors in the expression 68 × 717 ×1127 is equal to-
  1. ক
    ক) 60
  2. খ
    খ) 64
  3. গ
    গ) 66
  4. ঘ
    ঘ) 68
ব্যাখ্যা
Question: The numbers of prime factors in the expression 68 × 717 ×1127 is equal to- 

Solution:
   68 × 717 × 1127
= (2 × 3)8 × 717 × 1127
= 28 × 38× 717 × 1127
Number of prime factors in the given expression
= (8 + 8 + 17 + 27)
= 60
৩৭.
85% of a number is added to 24, the result is the same number. Find the number?
  1. ক
    ক) 150
  2. খ
    খ) 140
  3. গ
    গ) 130
  4. ঘ
    ঘ) 160
ব্যাখ্যা
Question: 85% of a number is added to 24, the result is the same number. Find the number?

Solution: 
ধরি,
সংখ্যাটি x 

প্রশ্নমতে,
x এর 85% + 24 = x
(85x/100) + 24 = x
(17x /20) + 24 = x
x - (17x /20)  = 24
(20x - 17x)/20 = 24
3x/20 = 24
x/20 = 8
x = 160
৩৮.
One-third of Ratan's investment in National Savings Certificate is equal to one-half of his investment in FDR. If he has Tk. 1,50,000 as total investment. how much he invested in savings certificate?
  1. ক
    ক) Tk. 30,000
  2. খ
    খ) Tk. 50,000
  3. গ
    গ) Tk. 60,000
  4. ঘ
    ঘ) Tk. 90,000
ব্যাখ্যা
Question: One-third of Ratan's investment in National Savings Certificate is equal to one-half of his investment in FDR. If he has Tk. 1,50,000 as total investment. how much he invested in savings certificate?

Solution: 
Let saving in N.S.C and P.P.F. be Tk. x and Tk.(150000 - x) respectively

Now 
x/3 = (1/2)(150000 - x)
x/3 = 75000 - x/2
(x/3) + (x/2) = 75000
(2x + 3x)/6= 75000
5x/6 = 75000
x = (75000 × 6)/5
x = 90000
৩৯.
A is 3 years older than B and 3 years younger than C. B and D are twins. How older is C than D?
  1. ক
    ক) 12 years
  2. খ
    খ) 6 years
  3. গ
    গ) 3 years
  4. ঘ
    ঘ) Cannot be Determined
ব্যাখ্যা
Question: A is 3 years older than B and 3 years younger than C. B and D are twins. How older is C than D?

Solution: 
A এর বয়স = B  +  3 বছর 
A এর বয়স = C - 3 বছর 

যেহেতু 
 B এবং D যমজ
B = D

B  +  3 = C - 3
D  +  3 = C - 3
D = C - 6
C = D + 6

C, D এর চেয়ে 6 বছরের বড়। 
৪০.
The difference between two integers is 5. Their product is 500. Find the numbers.
  1. ক
    ক) 15, 20
  2. খ
    খ) 20, 25
  3. গ
    গ) 30, 25
  4. ঘ
    ঘ) 21, 26
ব্যাখ্যা

Let the integers be x and (x + 5)
Then,
⇔x(x+5)=500
⇔x2+5x−500=0
⇔(x+25)(x−20)=0
⇔x=20 
So, the numbers are 20 and 25

৪১.
One-third of one-fourth of a number is 12. Then the number is:
  1. ক
    72
  2. খ
    96
  3. গ
    144
  4. ঘ
    166
ব্যাখ্যা
Question: One-third of one-fourth of a number is 12. Then the number is:

Solution:
Let the number be p.
We know that the term "of" indicates the sign of multiplication.

Now, it is given that one-third of one-fourth of p is 12, therefore, we have:

1/3 × 1/4 × p = 12
⇒ p/12 = 12
⇒ p = 12 × 12
⇒ p = 144

Hence, the number is 144.
৪২.
The L.C.M. of (6x2 - x - 1), (3x2 + 7x + 2) and (2x2 + 3x - 2) is
  1. ক
    ক) (2x - 1) (3x + 1) (3x + 2)
  2. খ
    খ) (2x - 1) (3x + 1) (x - 2)
  3. গ
    গ) (2x + 1) (3x - 1) (x + 2)
  4. ঘ
    ঘ) (2x - 1) (3x + 1) (x + 2)
ব্যাখ্যা
6x2 - x - 1 = 6x2 - 3x + 2x - 1
                = 3x(2x - 1) + 1(2x - 1)
                 = (2x - 1) (3x + 1)


(3x2 + 7x + 2) = 3x2 + 6x + x + 2
                      = 3x(x + 2) + 1(x + 2)
                       = (x + 2)(3x + 1)

(2x2 + 3x - 2) = 2x2 + 4x - x - 2
                      = 2x(x + 2) - 1(x + 2)
                      = (x + 2) (2x - 1)

∴ Required L.C.M. is = (2x - 1) (3x + 1) (x + 2)
৪৩.
The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows except-
  1. ক
    4
  2. খ
    20
  3. গ
    30
  4. ঘ
    40
ব্যাখ্যা
Question: The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows except-

Solution:
Obviously the number of rows must be a factor of 180. The only option which is not a factor of 180 is D (40).
180/4=45
180/20=9
180/30=6
180/40=4.50
৪৪.
Find the least number which will leave the remainder 7 when divided by 8, 12, 16, and 20.
  1. ক
    237
  2. খ
    243
  3. গ
    267
  4. ঘ
    247
ব্যাখ্যা
Question: Find the least number which will leave the remainder 7 when divided by 8, 12, 16, and 20.

Solution: 
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 that is exactly divisible by 8, 12, 16, and 20
Thus,
The required number which leaves the remaining 7 is,
240 + 7 = 247
৪৫.
The average of 6 numbers is 25. If 3 more numbers, with an average of 22 are added to these numbers, what will be the average of the combined 9 numbers?
  1. ক
    24
  2. খ
    25
  3. গ
    26
  4. ঘ
    28
ব্যাখ্যা
Question: The average of 6 numbers is 25. If 3 more numbers, with an average of 22 are added to these numbers, what will be the average of the combined 9 numbers?

Solution:
The average of 6 numbers is 25
∴ The total of 6 numbers is 25 × 6 = 150

The average of 3 numbers is 22
∴ The total of 3 numbers is 22 × 3 = 66

The sum of 9 numbers is = 150 + 66 = 216
∴ The average of 9 numbers is 216/9 = 24
৪৬.
If you count from 1 to 100, how many 5s will you pass on the way?
  1. ক
    ক) 22
  2. খ
    খ) 18
  3. গ
    গ) 20
  4. ঘ
    ঘ) 21
ব্যাখ্যা
প্রশ্ন : If you count 1 to 100, how many 5s will you pass on the way?
সমাধান : 
১ থেকে 100 পর্যন্ত গণনায় ৫, ১৫, ২৫, ৩৫, ৪৫, ৫০, ৫১, ৫২, ৫৩, ৫৪, ৫৫, ৫৬, ৫৭, ৫৮, ৫৯, ৬৫, ৭৫, ৮৫, ৯৫ এই সংখ্যাগুলোতে ৫ পাওয়া যায় মোট ২০ বার (বিঃ দ্রঃ ৫৫ তে দুটি ৫ গণনা করতে হবে)
৪৭.
Which one of the following numbers is divisible by 3 ?
  1. ক
    ক) 4006020
  2. খ
    খ) 2345678
  3. গ
    গ) 9566003
  4. ঘ
    ঘ) 2876423
ব্যাখ্যা
Sum of the digits in 4006020 = 4 + 0 + 0 + 6 + 0 + 2 + 0 =12, which is divisible by 3
Hence, 4006020 is divisible by 3
৪৮.
Which of the following is closest to √3 = ?
  1. ক
    1.70
  2. খ
    1.45
  3. গ
    9/4
  4. ঘ
    1.73
  5. ঙ
    1.78
ব্যাখ্যা
Square root of 3 : 

     = 1.73
৪৯.
In a raw of trees, a tree is 7th from the left and 14th from the right end. How many trees are there in the row?
  1. ক
    18
  2. খ
    19
  3. গ
    20
  4. ঘ
    21
ব্যাখ্যা
Total number of tree = 7 + 14 - 1 = 20
1 - 2 - 3 - 4 - 5 - 6 - 7 (a tree from the left as well as 14th from the right end) - 8 (13th) - 9 (12th) - 10 (11th) - 11 (10th) - 12 (9th) - 13 (8th) - 14 (7th) - 15(6th) - 16 (5th) - 17 (4th) - 18 (3rd) - 19 (2nd) - 20 (1st)
So total number of tree = 20
৫০.
If the difference of the squares of two natural numbers 25. Find the sum of the square of these numbers?
  1. ক
    351
  2. খ
    420
  3. গ
    313
  4. ঘ
    245
  5. ঙ
    321
ব্যাখ্যা
Question: If the difference of the squares of two natural numbers 25. Find the sum of the square of these numbers?

Solution:
Bigger number = (difference of squares + 1)/2 = (25 + 1)/2 = 13
Smaller number = (difference of squares - 1)/2 = (25 - 1)/2 = 12

So sum of the square of these numbers = 132 + 122 = 169 + 144 = 313
৫১.
The product of two whole number is 37. The square root of the difference of the numbers is 
  1. ক
    ক) 8
  2. খ
    খ) 4
  3. গ
    গ) 6
  4. ঘ
    ঘ) 3
ব্যাখ্যা
Since 37 is a prime number 37 = 1 × 37
∴ Required square root =√(37 - 1) = √36 = 6
৫২.
The sum of first n odd natural numbers is be given by-
  1. ক
    (n + 1)2
  2. খ
    n2 + 1
  3. গ
    n2
  4. ঘ
    n2 - 1
ব্যাখ্যা
Question: The sum of first n odd natural numbers is be given by-

Solution:
প্রথম n সংখ্যক স্বাভাবিক বিজোড় সংখ্যার সমষ্টি = 1 + 3 + 5 + 7 +....... + n
প্রথম পদ, a = 1
সাধারন অন্তর, d = 3 - 1 = 2
পদ সংখ্যা = n

আমরা জানি,
সমষ্টি = (n/2){2a + (n - 1)d}
= (n/2){2.1 + (n - 1).2}
= (n/2)(2 + 2n - 2)
= (n/2).2n
= n2
৫৩.
The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number?
  1. ক
    240
  2. খ
    270
  3. গ
    295
  4. ঘ
    360
ব্যাখ্যা
Question: The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number?

Solution:
Let the smaller number be x.
Then larger number = (x + 1365).

ATQ,
x + 1365 = 6x + 15
⇒ 5x = 1350
∴ x = 270

∴ Smaller number = 270.
৫৪.
0.009/p = 0.01, what is the value of p?
  1. ক
    .09
  2. খ
    .0003
  3. গ
    .9
  4. ঘ
    .0009
ব্যাখ্যা
Question: 0.009/p = 0.01, what is the value of p?

Solution:
Here,
.009/p = .01 

Then p = .009/.01
= .9/1
= .9
৫৫.
Three numbers are in the ratio 1 : 2 : 3, and the sum of their cubes is 7776. The smallest number will be -
  1. ক
    8
  2. খ
    6
  3. গ
    12
  4. ঘ
    18
ব্যাখ্যা

Question: Three numbers are in the ratio 1 : 2 : 3, and the sum of their cubes is 7776. The smallest number will be -

Solution:
Given,
The numbers be in the ratio 1 : 2 : 3
So, let:
Smallest number = x
Middle number = 2x
Largest number = 3x

ATQ,
x3 + 8x3 + 27x3 = 7776
⇒ 36x3 = 7776
⇒ x3 = 7776/36 = 216
∴ x = 6

So the smallest number is 6

৫৬.
x, y are positive integers. When x is divided by y, the remainder is 5. If x/y=5.20, what is the value of x?
  1. ক
    ক) 130
  2. খ
    খ) 155
  3. গ
    গ) 330
  4. ঘ
    ঘ) 425
ব্যাখ্যা
Question: x, y are positive integers. When x is divided by y, the remainder is 5. If x/y = 5.20, what is the value of x
Solution: 
দেয়া আছে 
x/y = 5.20
x/y = 520/100
x/y = 26/5

এখানে 5 দিয়ে 26  ভাগ করলে ভাগশেষ 1 থাকে 
কিন্তু বলা আছে ভাগশেষ 5 থাকবে 
তাই 
26/5 এর লব ও হরের সাথে 5 গুণ করতে হবে। 
x/y = 26/5 = (26 × 5)/(5 × 5) = 130/25
130 কে 25 দ্বারা ভাগ করলে 5 ভাগশেষ থাকে। 
x এর মান = 130
৫৭.
In a Mathematics examination, the number scored by 5 candidates is 5 successive odd integers. If their total marks are 185, the highest score is:
  1. ক
    33
  2. খ
    37
  3. গ
    39
  4. ঘ
    41
ব্যাখ্যা
Question: In a Mathematics examination, the number scored by 5 candidates is 5 successive odd integers. If their total marks are 185, the highest score is:

Solution:
Let the five successive odd numbers be,
x, x + 2, x + 4, x + 6, x + 8

Then, according to given information,
x + x + 2 + x + 4 + x + 6 + x + 8 = 185 
⇒ 5x + 20 = 185 
⇒ 5x = 165 
⇒ x = 33

∴ Highest number = 33 + 8 = 41
৫৮.
Two-fifths of one-fourth of three-seventh of a number is 15. What is the one-seventh of the number?
  1. ক
    30
  2. খ
    40
  3. গ
    50
  4. ঘ
    75
ব্যাখ্যা
Question: Two-fifths of one-fourth of three-seventh of a number is 15. What is the one-seventh of the number?

Solution:
Let the number be x

According to the question
(2/5) . (1/4) . (3/7) . x = 15
⇒ (1/10) × (3/7) × x = 15
⇒ 3/70 × x = 15
⇒ x = (15 × 70)/3
∴ x = 350

The one-seventh of the number = 350/7
= 50
৫৯.
A number when multiplied by 3 and increased by 5 gives 20. Find the number.
  1. ক
    4
  2. খ
    5
  3. গ
    6
  4. ঘ
    8
ব্যাখ্যা
Question:  A number when multiplied by 3 and increased by 5 gives 20. Find the number.

Solution:
Let, the number be = x

According to the question,
3x + 5 = 20
⇒ 3x = 20 - 5 
⇒ 3x = 15 
⇒ x =15/3
⇒ x = 5

Therefore, the number is 5
৬০.
Rahim ranks seventh from the top and twenty-sixth from the bottom in class. How many students are there in the class?
  1. ক
    ক) 32
  2. খ
    খ) 33
  3. গ
    গ) 34
  4. ঘ
    ঘ) 31
ব্যাখ্যা

Total number of students = (26 + 7) - 1 = 32

৬১.
The capacity of two pots is 12 liters and 48 liters respectively. Find the capacity of a which can exactly measure the content of the two pots.
  1. ক
    ক) 10000 cc
  2. খ
    খ) 12000 cc
  3. গ
    গ) 8000 cc
  4. ঘ
    ঘ) 16000 cc
ব্যাখ্যা
Question: The capacity of two pots is 12 liters and 48 liters respectively. Find the capacity of a which can exactly measure the content of the two pots.

Solution: 
Required capacity = H.C.F. of 12 liters and 48 liters = 12 liters

1 liter = 1000 cc
12 liters = 12000 cc
৬২.
To make 5686 exactly divisible by 7 what number should be subtracted from it??
  1. ক
    3
  2. খ
    2
  3. গ
    1
  4. ঘ
    0
ব্যাখ্যা
Question: To make 5686 exactly divisible by 7 what number should be subtracted from it??

Solution: 
if we divide 5686 by 7 the remainder is 2
to make it divisible by 7 we need to subtract 2 from it.

so the number is = 5686 - 2 = 5684
dividing 5684 by 7 we get the result = 812
৬৩.
The sum of four consecutive multiples of 4 is 200. What is the largest number?
  1. ক
    48
  2. খ
    52
  3. গ
    56
  4. ঘ
    60
ব্যাখ্যা
Question: The sum of four consecutive multiples of 4 is 200. What is the largest number? 

Solution:
ধরি,
চারটি ক্রমিক 4 এর গুণিতক যথাক্রমে x, (x + 4), (x + 8) এবং (x + 12)

প্রশ্নমতে,
x + (x + 4) + (x + 8) + (x + 12) = 200
⇒ 4x + 24 = 200
⇒ 4x = 200 - 24
⇒ 4x = 176
⇒ x = 176/4
⇒ x = 44

∴ বৃহত্তম সংখ্যা = x + 12
= 44 + 12
= 56
৬৪.
The difference between 4/5 of a number and 45% of the number is 56. What is 55% of the number?
  1. ক
    82
  2. খ
    88
  3. গ
    98
  4. ঘ
    96
ব্যাখ্যা
Question: The difference between 4/5 of a number and 45% of the number is 56. What is 55% of the number?
 
Solution:
Let the number be x.

According to the question,
4/5 of x - 45% of x = 56
⇒ (4x/5) − (45x/100) = 56
⇒ (4x/5) − (9x/20) = 56
⇒ (16x - 9x)/20 = 56
⇒ 7x/20 = 56
⇒ x = (56 × 20)/7
⇒ x = 8 × 20
∴ x = 160

∴ 55% of x = (55/100) × 160 = 88
৬৫.
Let N be the smallest positive integer that is divisible by both 6 and 9. How many distinct prime factors does N have?
  1. ক
    6
  2. খ
    5
  3. গ
    2
  4. ঘ
    3
  5. ঙ
    1
ব্যাখ্যা
Question: Let N be the smallest positive integer that is divisible by both 6 and 9. How many distinct prime factors does N have?

Solution:
We are told that N is divisible by both 6 and 9. So, the smallest such number is their LCM.

Now, LCM(6, 9) = 2 × 3 × 3 = 18
Therefore, N = 18
The prime factorization of 18 is 2 × 32. The distinct prime factors are 2 and 3.

∴ There are exactly 2 distinct prime factors.
∴ N has 2 distinct prime factors.
৬৬.
The difference between two integers is 6. If their product is 352, then the sum of two numbers is?
  1. ক
    48
  2. খ
    38
  3. গ
    32
  4. ঘ
    42
ব্যাখ্যা
Question: The difference between two integers is 6. If their product is 352, then the sum of two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 6,
∴ x - y = 6 and their product is, xy = 352

Now, We know
⇒ (x + y)2 = (x - y)2 + 4xy
⇒ (x + y)2 = 62 + 4 × 352
⇒ (x + y)2 = 36 + 1408
⇒ (x + y)2 = 1444
⇒ x + y = √1444
∴ x + y = 38

∴ The sum of the two numbers is 38.
৬৭.
A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each. If the total cost does not exceed Tk. 97, what is the maximum number of notebooks he has bought?
  1. ক
    ক) 4
  2. খ
    খ) 5
  3. গ
    গ) 7
  4. ঘ
    ঘ) 9
ব্যাখ্যা
Question: A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each. If the total cost does not exceed Tk. 97, what is the maximum number of notebooks he has bought?

Solution:
A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each.
Total cost = 5x + 8(x + 4)
= 5x + 8x + 32
= 13x + 32

So, 13x + 32 ≤ 97
⇒ 13x ≤ 97 - 32
⇒ 13x ≤ 65
⇒ x ≤ 5
⇒ x + 4 ≤ 5 + 4
∴ x + 4 ≤ 9

So, the maximum number of notebooks he has bought is 9
৬৮.
How many prime numbers are there between 50 and 60? 
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    4
ব্যাখ্যা

Question: How many prime numbers are there between 50 and 60?

Solution:
A prime number is a number that is divisible only by 1 and itself.

Numbers between 50 and 60 are:
51, 52, 53, 54, 55, 56, 57, 58, 59

Among these, the prime numbers are:
53, 59.

∴ Total number of prime numbers = 2.

৬৯.
In a class of 72 children are seated in rows and columns in such a way that the number of children in each row is 12.5% more than the number of children in each column. How many children are there in each row?
  1. ক
    10
  2. খ
    9
  3. গ
    7
  4. ঘ
    8
ব্যাখ্যা
Let the number of children in each column be x.
Thus, the number of children in each row is 1.125x

ATQ,
72 = x × (1.125x)
=> x2 = 72/1.125
=> x2 = 64
∴x=8

Thus, number of children in a row = 1.125x = 1.125 × 8 = 9
৭০.
One number is 3 more than 2 times of another number. Their sum is 21. Find the numbers.
  1. ক
    ক) 7, 14
  2. খ
    খ) 2, 19
  3. গ
    গ) 6, 15
  4. ঘ
    ঘ) 10, 11
ব্যাখ্যা
প্রশ্ন: একটি সংখ্যা অপর একটি সংখ্যার দ্বিগুণের থেকে 3 বেশি। সংখ্যাদ্বয়ের সমষ্টি 21 হলে, সংখ্যাদ্বয়-

সমাধান: 
ধরি,
একটি সংখ্যা = Y
∴ অপর সংখ্যাটি = 2Y + 3

প্রশ্নমতে,
Y + 2Y + 3 = 21
3Y = 18
Y = 6

অন্য সংখ্যাটি = 15
৭১.
Three numbers are in the ratio 2 : 5 : 8 and their average is 75. What is the difference between larger and smaller numbers?
  1. ক
    90
  2. খ
    80
  3. গ
    95
  4. ঘ
    55
ব্যাখ্যা
Question: Three numbers are in the ratio 2 : 5 : 8 and their average is 75. What is the difference between larger and smaller numbers?

Solution:
Let the three numbers are 2x, 5x, and 8x, where x is a constant multiplier.
The average of the three numbers is given as 75.
The average of the three numbers is,
⇒ (2x + 5x + 8x)/3 = 75
⇒ 15x/3 = 75
⇒ 5x = 75
⇒ x = 75/5
∴ x = 15
The first number is = 2x = 2 × 15 =30
The third number is = 8x = 8 × 15 = 120

∴ The difference between the larger and smaller numbers is, 120 − 30 = 90
৭২.
A rectangular block 4 cm by 12 cm by 8 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.
  1. ক
    ক) 6
  2. খ
    খ) 12
  3. গ
    গ) 16
  4. ঘ
    ঘ) 40
ব্যাখ্যা
Volume of the block
= (4 × 12 × 8) cm3
= 384 cm3

Side of the largest cube
= H.C.F of 4 cm, 12 cm, 8 cm
= 4 cm.

Volume of this cube = (4 × 4 × 4) cm3
= 64 cm3

Number of cubes
= 384/64
= 6
৭৩.
Chose the pair of numbers which comes next: 75, 65, 85, 55, 45, 85, 35, _______
  1. ক
    ক) 25, 15
  2. খ
    খ) 35, 25
  3. গ
    গ) 35, 85
  4. ঘ
    ঘ) 25, 85
ব্যাখ্যা
প্রশ্ন: Chose the pair of numbers which comes next: 75, 65, 85, 55, 45, 85, 35, _______

সমাধান: 
এই সিরিজে ১০ বিয়োগ করে পরবর্তী সংখ্যা নেওয়া হয়েছে। 
প্রতি দুই ঘর পরপর ৮৫ সংখ্যাটি পুনরাবৃত্তি হয়েছে। 

পরবর্তী সংখ্যাটি 35 - 10 = 25
যেহেতু, দুই সংখ্যা পরপর 85 সংখ্যাটি পুনরাবৃত্তি হচ্ছে। 
অতএব, 25 এর পরবর্তী সংখ্যাটি হবে 85 

∴পরবর্তী জোড়াটি হবে 25, 85
৭৪.
The three rational numbers between 3 and 4 are :
  1. ক
    ক) 5/2, 6/2, 7/2
  2. খ
    খ) 13/4, 14/4, 15/4
  3. গ
    গ) 12/7, 13/7, 14/7
  4. ঘ
    ঘ) 11/4,12/4, 13/4
ব্যাখ্যা

Rational number is any number that can express in the form of qp of two integers, where 'q' cannot be zero.

i) First rational number between 3 and 4 can be calculated by finding average between them, which is
(3 + 4)/2 = 7/2 or 14/4
Now, we have three numbers i.e. 3, 7/2 and 4, so other remaining rational numbers can be calculated by taking average between 3 and 7/2, and between 7/2 and 4.

ii) Second rational number between 3 and 7/2 can be calculated by finding average between them.
(7/2 + 3) / 2 = {(7 + 6)/2} / 2 = 13/4

iii) The third rational number between 4 and 7/2 can be calculated by finding the average between them.
(7/2 + 4)/2 = {(7 + 8)/2}/2 = 15/4

iv) Similarly fourth rational number between 3 and 13/4 can be calculated by finding the average between them.
(13/4 + 3) / 2 = {(13 + 12)/4}/2 = 25/8

v) Similarly, fifth rational number between 4 and 13/4 can be calculated by finding average between them.
{(13/4) + 4}/2 = {(13 + 16)/4} / 2 = 29/8


Then the rational numbers between 3 and 4 are : 7/2 or 14/4, 13/4, 15/4, 25/8, 29/8

 

৭৫.
A boy has 75 litres oil in one cane and 45 litres in another. The maximum capacity of container which can measure oil of either container in exact number?
  1. ক
    ক) 1 litre
  2. খ
    খ) 5 litres
  3. গ
    গ) 25 litres
  4. ঘ
    ঘ) 15 litres
ব্যাখ্যা
প্রশ্ন: একজন বালকের দুইটি পাত্রে যথাক্রমে 75 লিটার ও 45 লিটার তেল আছে। সর্বোচ্চ কত লিটারের পাত্র দিয়ে সঠিক ভাবে মাপা যাবে?

সমাধান: 
75 = 5 × 5 × 3
45 = 5 × 3 × 3

∴ 75 এবং 45 এর গ.সা.গু = 15

অর্থাৎ 15 লিটার এর পাত্র দিয়ে সঠিক ভাবে মাপা যাবে।
৭৬.
What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in the form of a solid square?
  1. ক
    900
  2. খ
    400
  3. গ
    1600
  4. ঘ
    2500
ব্যাখ্যা

In this type of question, We need to find out the LCM of the given numbers.
LCM of 12, 15, 18 and 20;
12=2 × 2 × 3;
15=3 × 5;
18=2 × 3 × 3;
20=2 × 2 × 5

Hence, LCM = 2 × 2 × 3 × 5 × 3

Since, the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5,
hence,
The required number of soldiers = 2 × 2 × 3 × 3 × 5 × 5
= 900

৭৭.
The average of the two numbers is 6.5 and square root of their product is 6. What are the numbers?
  1. ক
    ক) 11 and 2
  2. খ
    খ) 8 and 5
  3. গ
    গ) 10 and 3
  4. ঘ
    ঘ) 9 and 4
ব্যাখ্যা
Question: The average of two numbers is 6.5 and square root of their product is 6. What are the numbers?

Solution:
Let the number be x and y

According to the question,
(x + y)/2 = 6.5,
⇒ x + y = 13

and,
√xy = 6
⇒ xy = 36
⇒ x = 36/y

Now,
(36/y) + y = 13
⇒ (36 + y2)/y = 13
⇒ 36 + y2 = 13y
⇒ y2 - 13y + 36 = 0
⇒ (y - 9)(y - 4) = 0
So, y = 4 or, 9

If y = 4 then, x = 9
If y = 9 then, x = 4

So, the numbers are 4 and 9
৭৮.
The difference between a number and its square is 72. What is the number?
  1. ক
    ক) 6
  2. খ
    খ) 7
  3. গ
    গ) 9
  4. ঘ
    ঘ) 8
ব্যাখ্যা
Question: The difference between a number and its square is 72. What is the number?

Solution:
Let, the number be x.

ATQ,
x2 - x = 72
⇒ x2 - x - 72= 0
⇒ x2 - 9x + 8x - 72 = 0
⇒ x(x - 9) + 8(x - 9) = 0
⇒ (x - 9) (x + 8) = 0
⇒ x - 9 = 0

Here,
x - 9 = 0
∴ x = 9
৭৯.
Which of the following is divisible by 3?
  1. ক
    34616
  2. খ
    46904
  3. গ
    10984
  4. ঘ
    14589
ব্যাখ্যা
Question: Which of the following is divisible by 3?

Solution:
We have to simply find the sum of the digits for each option and check if the sum is divisible by 3....

34616 = 3 + 4 + 6 + 1 + 6 = 20
46904 = 4 + 6 + 9 + 0 + 3 = 23
10984 = 1 + 0 + 9 + 8 + 4 = 22
14589 = 1 + 4 + 5 + 8 + 9 = 27

Hence we can see that only 27 is divisible by 3. Thus, 14589 is divisible by 3.
৮০.
Six consecutive whole numbers are given. The sum of the last three numbers is 36. What is the sum of the first three numbers?
  1. ক
    22
  2. খ
    25
  3. গ
    27
  4. ঘ
    31
ব্যাখ্যা
Question: Six consecutive whole numbers are given. The sum of the last three numbers is 36. What is the sum of the first three numbers?

Solution: 
let,
The numbers be x - 2, x - 1, x, x + 1, x + 2, x + 3 

ATQ,
x + 1 + x + 2 + x + 3 = 36
⇒ 3x + 6 = 36
⇒ 3x = 30 
⇒ x = 30/3
∴ x = 10 

The sum of the first three numbers is = x - 2 + x - 1 + x
= 10 - 2 + 10 - 1 + 10 
= 30 - 3
= 27
৮১.
Simplify the expression using BODMAS rule:
(3/2) of (4/7){(10 × 3) - (8 × 2)}
  1. ক
    6
  2. খ
    12
  3. গ
    14
  4. ঘ
    18
  5. ঙ
    20
ব্যাখ্যা
Question: Simplify the expression using BODMAS rule:
(3/2) of (4/7){(10 × 3) - (8 × 2)}

Solution:
(3/2) of (4/7){(10 × 3) - (8 × 2)}
= (6/7){30 - 16}
= (6/7)(14)
= 6 × 2
= 12
৮২.
Which of the following numbers should be added to 8567 to make it exactly divisible by 4?
  1. ক
    ক) 4
  2. খ
    খ) 1
  3. গ
    গ) 2
  4. ঘ
    ঘ) 3
ব্যাখ্যা
On dividing 8567 by 4, the remainder is 3.

To make it divisible by 4, we must add (4 - 3) =1 
৮৩.
If the sum of three consecutive numbers is multiplied by 4, the result is 132. What is the largest number?
  1. ক
    10
  2. খ
    12
  3. গ
    14
  4. ঘ
    16
ব্যাখ্যা

Question: If the sum of three consecutive numbers is multiplied by 4, the result is 132. What is the largest number?

Solution:
ধরি,
তিনটি ধারাবাহিক সংখ্যা যথাক্রমে x, x + 1 এবং x + 2

প্রশ্নমতে,
⇒ {x + (x + 1) + (x + 2)} × 4 = 132
⇒ (3x + 3) × 4 = 132
⇒ 3x + 3 = 132 / 4
⇒ 3x + 3 = 33
⇒ 3x = 33 - 3
⇒ x = 30/3
∴ x = 10

বৃহত্তম সংখ্যাটি হলো = x + 2
= 10 + 2
= 12

৮৪.
3640 ÷ 14 × 16 + 340 = ?
  1. ক
    0.70
  2. খ
    3525
  3. গ
    4480
  4. ঘ
    None of these
ব্যাখ্যা
Question: 3640 ÷ 14 × 16 + 340 = ?

Solution:
3640 ÷ 14 × 16 + 340 
= 260 × 16 + 340 
= 4160 + 340 
= 4500
৮৫.
The sum of four consecutive even numbers A, B, C and D is 180. What is the sum of the set of next four consecutive even numbers?
  1. ক
    210
  2. খ
    204
  3. গ
    202
  4. ঘ
    212
ব্যাখ্যা
Question: The sum of four consecutive even numbers A, B, C and D is 180. What is the sum of the set of next four consecutive even numbers?

Solution: 
let, the consecutive even numbers are a, a + 2, a + 4, a + 6
∴ a + a + 2 + a + 4 + a + 6 = 180
or, 4a = 180 - 12
or, 4a = 168
∴ a = 42

the numbers are 42, 44, 46, 48.

next 4 consecutive numbers are 50, 52, 54, 56
sum = 50 + 52 + 54 + 56
= 212
৮৬.
The difference between the greatest and least prime numbers which are less than 100 is = ?
  1. ক
    94
  2. খ
    91
  3. গ
    95
  4. ঘ
    97
  5. ঙ
    89
ব্যাখ্যা
Greatest prime number = 97
Least prime number = 2
So, their difference 97 - 2 = 95
৮৭.
If 1/5 : 1/N = 1/N : 1/1.25, then the value of N is? 
  1. ক
    ক) 1.25
  2. খ
    খ) 1.5
  3. গ
    গ) 2.25
  4. ঘ
    ঘ) 2.5
ব্যাখ্যা
1/5 : 1/N = 1/N : 1/1.25
⇒ 5 : N = N : 1.25
⇒ 5/N =N/1.25
⇒ N2 = 6.25
⇒ N = 2.5
৮৮.
Find the H.C.F, if the numbers are in the ratio of 4 : 5 : 6 and their L.C.M. is 2400.
  1. ক
    35
  2. খ
    20
  3. গ
    40
  4. ঘ
    67
ব্যাখ্যা
Question: Find the H.C.F, if the numbers are in the ratio of 4 : 5 : 6 and their L.C.M. is 2400.

Solution:
Let the numbers be 4x, 5x and 6x.

Then,their L.C.M. = 60x.
So, 60x = 2400
∴ x = 40

∴ The numbers are (4 × 40) = 160 , (5 × 40) = 200 and (6 × 40) = 240.

Hence, required H.C.F = 40.
৮৯.
What is the number of divisor of 540?
  1. ক
    ক) 20
  2. খ
    খ) 21
  3. গ
    গ) 23
  4. ঘ
    ঘ) 24
ব্যাখ্যা
540 = 2 × 2 × 3 × 3 × 3 × 5
         = 22 × 33 × 5
নির্ণেয় ভাজক সংখ্যা = (2 + 1)(3 + 1)(1 + 1) 
                                  = 3 × 4 × 2 = 24
৯০.
The sum of the squares of three numbers is 532 and the ratio of the first and the second as also of the second and the third is 3:2. Then the first number is -
  1. ক
    ক) 18
  2. খ
    খ) 6
  3. গ
    গ) 12
  4. ঘ
    ঘ) 20
ব্যাখ্যা

Let the numbers are x, y, z
so, x2+ y2 + z2 = 532
Given,
x : y = 3 : 2 = 9 : 6
y : z = 3 : 2 = 6 : 4
Then x : y : z = 9 : 6 : 4
Now, Let the numbers are 9a, 6a, 4a
So, (9a)2 + (6a)2 + (4a)2 = 532
Or, 81a2 + 36a2 + 16a2 = 532
Or, 133a2 = 532
Or, a2 = 532/133 = 4
Or, a = 2
∴ the first number = 9×2 = 18

৯১.
Find the missing numeral: (? - 2763)/86 × 13 = 208
  1. ক
    ক) 3031
  2. খ
    খ) 4037
  3. গ
    গ) 4421
  4. ঘ
    ঘ) 4139
ব্যাখ্যা

ধরি, শূণ্য স্থানের সংখ্যাটি x
প্রশ্নমতে, (x - 2763) ÷ 86 × 13 = 208
⇒ (x - 2763) = 208/13 × 86
⇒ x - 2763 = 1376
⇒ x = 1376 + 2763
⇒ x = 4139

৯২.
If 6.4 is 80 percent of 20 percent of a certain number, what is the number?
  1. ক
    ক) 9
  2. খ
    খ) 36
  3. গ
    গ) 40
  4. ঘ
    ঘ) 81
ব্যাখ্যা
ধরি 
সংখ্যাটি x 

প্রশ্নমতে,
x × 80% × 20% = 6.4
x × 80/100 × 20/100 = 6.4
(x × 80× 20)/(100 × 100) = 6.4 
16x/100 = 6.4
16x = 640
x = 640 /16 
x = 40
৯৩.
P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?
  1. ক
    16
  2. খ
    20
  3. গ
    35
  4. ঘ
    65
ব্যাখ্যা
Question: P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

Solution:
Given,
P and Q are two positive integers and PQ = 64

64 = 1 × 64
= 2 × 32
= 4 × 16
= 8 × 8
The possible combinations of (P, Q)
So that, the product is 64 are (1, 64), (2, 32), (4, 16) and (8, 8)
∴ P + Q cannot be 35.
৯৪.
What will be the next number of the series?
4, 7, 13, 25, 49,..........
  1. ক
    ক) 96
  2. খ
    খ) 98
  3. গ
    গ) 97
  4. ঘ
    ঘ) 99
ব্যাখ্যা
Question: What will be the next number of the series?
4, 7, 13, 25, 49,..........

Solution: 
4 + 3 = 7
7 + 6 = 13
13 + 12 = 25
25 + 24 = 49
49 + 48 = 97
৯৫.
If a + b = 10 and ab = 21, then what is the value of a2 + b2 = ?
  1. ক
    142
  2. খ
    82
  3. গ
    79
  4. ঘ
    58
ব্যাখ্যা
Question: If a + b = 10 and ab = 21, then what is the value of a2 + b2 = ?

Solution:
Given that,
a + b = 10
ab = 21

We know,
a2 + b2 =(a + b)2 - 2ab
= (10)2 - (2 × 21)
= 100 - 42
= 58
৯৬.
Which of the following is equivalent to the pair of inequalities x + 6 > 10 and x - 3 ≤ 5?
  1. ক
    2 < x < 16
  2. খ
    2 ≤ x < 4
  3. গ
    2 < x ≤ 8
  4. ঘ
    4 < x ≤ 8
ব্যাখ্যা
Question: Which of the following is equivalent to the pair of inequalities x + 6 > 10 and x - 3 ≤ 5?

Solution:
x + 6 > 10
⇒ x > 4

x - 3 ≤ 5
⇒ x ≤ 8

Bringing these two inequalities together we get:
4 < x ≤ 8
৯৭.
If n is even, which of the following cannot be odd?
  1. ক
    ক) n + 1
  2. খ
    খ) 5(n + 1)
  3. গ
    গ) 2(n + 1)
  4. ঘ
    ঘ) n2 + 1
ব্যাখ্যা
Question: If n is even, which of the following cannot be odd?

Solution:
এখানে,
n জোড় সংখ্যা 
তাহলে, n + 1 একটি বিজোড় সংখ্যা 

এখন, (n + 1) কে 2 দ্বারা গুণ করলে জোড় সংখ্যা হবে। 
∴ 2(n + 1) একটি জোড় সংখ্যা
৯৮.
The difference between the local value and the face value of 7 in the numeral 32675149 is-
  1. ক
    75142
  2. খ
    64851
  3. গ
    5149
  4. ঘ
    69993
ব্যাখ্যা
Question: The difference between the local value and the face value of 7 in the numeral 32675149 is-

Solution:
32675149

(Local value of 7) - (Face value of 7)
= 70000 - 7
= 69993
৯৯.
The LCM of three different numbers is 120. Which of the following cannot be their HCF?
  1. ক
    ক) 8
  2. খ
    খ) 12
  3. গ
    গ) 24
  4. ঘ
    ঘ) 35
  5. ঙ
    ঙ) Cannot be determined
ব্যাখ্যা
Since HCF is always a factor of LCM, we cannot have three numbers with HCF 35 and LCM 120.
১০০.
  1. ক
    ক) 100
  2. খ
    খ) 100/61
  3. গ
    গ) 19/53
  4. ঘ
    ঘ) 61
ব্যাখ্যা
Question:


Solution: