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Fraction and Simplification, Average and Mean

মোট প্রশ্ন৯৪৮এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Fraction and Simplification, Average and Mean

PrepBank · পাতা ৩ / ১০ · ২০১–৩০০ / ৯৪৮

২০১.
The average of five numbers is 16. If one number is excluded, the average becomes 14. The excluded number is -
  1. ক
    24
  2. খ
    22
  3. গ
    23
  4. ঘ
    26
ব্যাখ্যা
Question: The average of five numbers is 16. If one number is excluded, the average becomes 14. The excluded number is -

Solution:
The average of five numbers is 16
The sum of five numbers is (16 × 5) = 80

let, the excluded number is x

so,
(80 - x)/4 = 14
or, 80 - x = 56
or x = 80 - 56
∴ x = 24
২০২.
The average of a, b, c is 6 and a - b = 4, ab = 21, what is the value of c? 
  1. ক
    ক) 6
  2. খ
    খ) 7
  3. গ
    গ) 8
  4. ঘ
    ঘ) 9
ব্যাখ্যা
দেয়া আছে, 
(a + b + c)/3 = 6 
a + b + c = 18 ............ (1)

a - b = 4,
ab = 21

আমরা জানি 
(a + b)2 = (a - b)2 + 4ab 
(a + b)2 = (4)2 + 4 × 21
(a + b)2 = 16 + 84
(a + b)2 = 100
a + b = 10 

(1) নং এ a + b এর মান বসিয়ে পাই, 
a + b + c = 18
10 + c = 18 
c = 18 - 10 
c = 8
২০৩.
When is converted into fraction, the result will be?
  1. ক
    2/15
  2. খ
    4/20
  3. গ
    3/15
  4. ঘ
    1/4
ব্যাখ্যা
Question: When is converted into fraction, the result will be?

Solution:
২০৪.
If the average of x numbers is y2 and that of y number is x2, then the average of (x + y) numbers is-
  1. ক
    ক) x/y
  2. খ
    খ) 2xy
  3. গ
    গ) x + y
  4. ঘ
    ঘ) xy
ব্যাখ্যা
Sum of x numbers = x × y2 = xy2
Similarly,
sum of y numbers = y × x2 = x2y
Total number of numbers = x + y
∴ Average of (x + y) numbers = (xy2 + x2y)/(x + y)
= [xy × (y + x)]/(x + y)
= xy
২০৫.
If the average of 'm' numbers is 2n2 and the average of 'n' numbers is 2m2, what is the average of the combined (m + n) numbers?
  1. ক
    6mn/(m + n)
  2. খ
    2mn
  3. গ
    mn
  4. ঘ
    4mn/(m + n)
ব্যাখ্যা

Question: If the average of 'm' numbers is 2n2 and the average of 'n' numbers is 2m2, what is the average of the combined (m + n) numbers?

Solution:
দেওয়া আছে,
'm' সংখ্যার গড় = 2n2
∴ m সংখ্যার সমষ্টি = m × 2n2

'n' সংখ্যার গড় = 2m2
∴ 'n' সংখ্যার সমষ্টি = n × 2m2

∴ মোট সমষ্টি = (m × 2n2) + (n × 2m2)
= 2mn(n + m)

∴ তাদের গড় = মোট সমষ্টি/(m + n)
= 2mn(m + n)/(m + n)
= 2mn

২০৬.
The average weight of 40 students in a class is 45 kg. If 10 new students are admitted, the average weight increases by 1 kg. What is the average weight of the new students?
  1. ক
    46 kg
  2. খ
    47.8 kg
  3. গ
    48.5 kg
  4. ঘ
    50 kg
  5. ঙ
    51 kg
ব্যাখ্যা

Question: The average weight of 40 students in a class is 45 kg. If 10 new students are admitted, the average weight increases by 1 kg. What is the average weight of the new students?

Solution:
40 জন শিক্ষার্থীর মোট ওজন = 40 × 45 = 1800 kg
10 জন নতুন শিক্ষার্থী ভর্তি হওয়ায় মোট শিক্ষার্থীর সংখ্যা = 40 + 10 = 50 জন
নতুন গড় ওজন = 45 + 1 = 46 kg
সুতরাং, 50 জন শিক্ষার্থীর মোট ওজন = 50 × 46 = 2300 kg

নতুন 10 জন শিক্ষার্থীর মোট ওজন = 2300 - 1800 = 500 kg
∴ নতুন 10 জন শিক্ষার্থীর গড় ওজন = 500/10 = 50 kg

সুতরাং, নতুন শিক্ষার্থীদের গড় ওজন 50 kg

২০৭.
Average cost of 5 apples and 4 mangoes is Tk. 36. The average cost of 7 apples and 8 mangoes is Tk. 48. Find the total cost of 24 apples and 24 mangoes.
  1. ক
    1044
  2. খ
    2088
  3. গ
    720
  4. ঘ
    3240
ব্যাখ্যা

Average cost of 5 apples and 4 mangoes = Tk. 36
Total cost = 36 × 9 = 324

Average cost of 7 apples and 8 mangoes = Tk. 48
Total cost = 48 × 15 = 720

Total cost of 12 apples and 12 mangoes = 324 + 720 = 1044
Therefore, cost of 24 apples and 24 mangoes = 1044 × 2 = 2088

২০৮.
  1. ক
    7/25
  2. খ
    7/16
  3. গ
    9/25
  4. ঘ
    7/13
ব্যাখ্যা
Question: 


Solution: 
২০৯.
The average of a and b is 30, and the average of b and c is 40. If b = 36 than find the value of (a + c).
  1. ক
    60
  2. খ
    64
  3. গ
    68
  4. ঘ
    72
ব্যাখ্যা

Question: The average of a and b is 30, and the average of b and c is 40. If b = 36 than find the value of (a + c).

Solution: 
Given b = 36

Average of a and b = 30
a + b = 30 × 2
⇒ a + b = 60
⇒ a + 36 = 60
⇒ a = 60 - 36
∴ a = 24

Average of b and c = 40
b + c = 40 × 2
⇒ b + c = 80
⇒ c = 80 - 36
∴ c = 44

Now,
a + c = 24 + 44 = 68

২১০.
Having scored 96 runs in the 19th innings a cricketer increase his average score by 4.what will be his average score after the 19th innings?
  1. ক
    24
  2. খ
    26
  3. গ
    28
  4. ঘ
    32
ব্যাখ্যা
Question: Having scored 96 runs in the 19th innings a cricketer increase his average score by 4.what will be his average score after the 19th innings?

Solution:
Let, his average score is = x
His total score after 19th innings = 19x

ATQ,
(19x - 96)/18 = x - 4
⇒ 19x - 96 = 18x - 72
⇒ 19x - 18x = 96 - 72
∴ x = 24

So, His average score is 24.
২১১.
When a student weighing 45 kg left a class, the average weight of the remaining 59 students increased by 200g. What is the average weight of the remaining 59 students?
  1. ক
    62 kg
  2. খ
    57 kg
  3. গ
    45 kg
  4. ঘ
    None of these
ব্যাখ্যা
Question: When a student weighing 45 kg left a class, the average weight of the remaining 59 students increased by 200g. What is the average weight of the remaining 59 students?

Solution:
let, the average weight of 59 students is x kg
Total age of 59 students = 59x
Total age of 60 students = 59x + 45 

ATQ,
x = {(59x + 45)/60} + 0.2
⇒ x - 0.2 = (59x + 45)/60
⇒ 60x - 12 = 59x + 45
⇒ 60x - 59x = 45 + 12 
∴ x = 57 kg
২১২.
If the average marks of three classes of 45, 60 and 55 students are 60, 55, 50 respectively, find the average marks of all the students.
  1. ক
    52.85
  2. খ
    45.75
  3. গ
    64.68
  4. ঘ
    54.68
ব্যাখ্যা
Question: If the average marks of three classes of 45, 60 and 55 students are 60, 55, 50 respectively, find the average marks of all the students.

Solution:
Average = (45 × 60 + 60 × 55 + 55 × 50)/(45 + 60 + 55)
= (2700 + 3300 + 2750)/160
= 8750/160
= 54.68
২১৩.
In a class of 120 students, the average score in science is 75. If the 70 girls scored an average of 78, what is the average score of the remaining boys?
  1. ক
    68.5
  2. খ
    74
  3. গ
    73.4
  4. ঘ
    70.8
ব্যাখ্যা

Question: In a class of 120 students, the average score in science is 75. If the 70 girls scored an average of 78, what is the average score of the remaining boys?

Solution:
ধরি, ছাত্রদের গড় নম্বর = x

120 জন শিক্ষার্থীর মোট নম্বর = 120 × 75 = 9000
70 জন ছাত্রীর মোট নম্বর = 70 × 78 = 5460

প্রশ্নমতে,
5460 + (120 - 70) × x = 9000
⇒ 5460 + 50x = 9000
⇒ 50x = 9000 - 5460
⇒ 50x = 3540
⇒ x = 3540/50
⇒ x = 70.8

∴ 50 জন ছাত্রের গড় নম্বর = 70.8

২১৪.
'Ambidexter' means-
  1. ক
    ক) সুবোধ
  2. খ
    খ) পন্ডিত
  3. গ
    গ) সব্যসাচী
  4. ঘ
    ঘ) কঠোর
ব্যাখ্যা
Ambidexter (adjective)
- one who can draw a bow with both hands; ambidextrous.
- সব্যসাচী
উৎস: বাংলা একাডেমী অভিধান।
২১৫.
A passenger travels from Dhaka to Cumilla at a speed of 30 kmph and return with a speed of 60 kmph. What is the average speed?
  1. ক
    17 kmph
  2. খ
    20 kmph
  3. গ
    30 kmph
  4. ঘ
    40 kmph
ব্যাখ্যা

Question: A passenger travels from Dhaka to Cumilla at a speed of 30 kmph and return with a speed of 60 kmph. What is the average speed?

Solution:
Let the total distance be 60 km [L.C.M. of 30 kmph and 60 kmph]

Time to cover 60 km distance at the speed of 30 km/hr = 60/30 = 2 hrs
Time to cover 60 km distance at the speed of 60 km/hr = 60/60 = 1 hr

Total distance = 60 + 60 = 120 km

Total time = 2 hrs + 1 hrs = 3 hrs

∴ Average speed = 120/3 = 40 km/hr

২১৬.
  1. ক
    8
  2. খ
    10
  3. গ
    12
  4. ঘ
    14
ব্যাখ্যা
Question:

Solution:
২১৭.
The average of the first five multiples of 13 is-
  1. ক
    ক) 33
  2. খ
    খ) 35
  3. গ
    গ) 37
  4. ঘ
    ঘ) 39
ব্যাখ্যা
Question: The average of the first five multiples of 13 is-

Solution: 
The  first five multiples of 13: (13 × 1), (13 × 2), (13× 3), (13 × 4), (13 × 5)
their sum = (13 × 1) + (13 × 2) + (13 × 3) + (13 × 4) + (13 × 5)
= 13 (1 + 2 + 3 + 4 + 5)
= 13 × 15

∴ average = (13 × 15)/5
= 39
২১৮.
The average age of a group of 15 employees is 24 years. If 5 more employees join the group, the average age increases by 2 years. Find the average age of the new employees.
  1. ক
    ক) 35
  2. খ
    খ) 30
  3. গ
    গ) 24
  4. ঘ
    ঘ) 32
ব্যাখ্যা

The average age of a group of 15 employees is 24 years.
Therefore, the sum of the ages of all 15 of them is, 15 × 24 = 360
When 5 more employees join the group, the average age increases by 2 years.
So, New average = 26.
Now, there are 20 employees.
The sum of the ages of everyone = 20×26 = 520
Therefore, the sum of the ages of the 5 new employees = 520 - 360 = 160
∴ The average age of the 5 new employees = 160/5 = 32 years.

২১৯.
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
    ক) 24
  2. খ
    খ) 25
  3. গ
    গ) 26
  4. ঘ
    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:
৬টি সংখ্যার গড় = ২৫
সমষ্টি = (২৫ × ৬) = ১৫০

পরবর্তী ৩টি সংখ্যার গড় ২২ হলে সমষ্টি = (২২ × ৩) = ৬৬
৯টি সংখ্যার মোট সমষ্টি = (১৫০ + ৬৬) = ২১৬

∴ গড় = ২১৬/৯ = ২৪
২২০.
The average of 4 positive integers 60. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers?
  1. ক
    52
  2. খ
    65
  3. গ
    73
  4. ঘ
    78
ব্যাখ্যা
Question: The average of 4 positive integers 60. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers?

Solution:
The average of 4 positive integers 60.
The sum of 4 positive integers is = 60 × 4 = 240
Given the two numbers are 83 and 29.
The difference between the remaining two integers is 28

Now,
Let higher integer number is x and other is (x - 28)

According to the question,
⇒ 83 + 29 + x + x - 28 = 240
⇒ 2x = 240 + 28 - 83 - 29
⇒ 2x = 156
⇒ x = 156/2
∴ x = 78
Thus, the higher of the remaining two integers is 78.
২২১.
Mr. Nur was hired for a job for 7 days. Each day, he was paid Tk. 10 more than what he was paid for their previous day of work. The total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his starting pay?
  1. ক
    Tk. 160
  2. খ
    Tk. 148
  3. গ
    Tk. 90
  4. ঘ
    Tk. 80
  5. ঙ
    None
ব্যাখ্যা
Question: Mr. Nur was hired for a job for 7 days. Each day, he was paid Tk. 10 more than what he was paid for their previous day of work. The total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his starting pay?

Solution:
Let,
The pays of seven days respectively: x, x + 10 x + 20 x + 30, x + 40, x + 50, x + 60

According to the question,
x + x + 10 + x + 20 + x + 30 = x + 40 + x + 50 + x + 60
⇒ 4x + 60 = 3x + 150
∴ x = 90

∴ His starting pay is Tk. 90
২২২.
Q (33-56): Read the following questions carefully and choose the right answer.
৩৩) Which one of the following numbers can be removed from the set S = (0, 2, 4, 5, 9) without changing the average of set S?
  1. ক
    ক) 0
  2. খ
    খ) 2
  3. গ
    গ) 4
  4. ঘ
    ঘ) 5
ব্যাখ্যা
The average of the elements in the original set S is:( 0+2+4+5+9)/5
                                                                                 =20/5
                                                                                 =4
If we remove an element that equals the average, then the average of the new set will remain unchanged.
The new set after removing 4 is {0, 2, 5, 9}.
The average of the elements is, (0+2+5+9)/4
                                                 =16/4
                                                 =4
২২৩.
The average age of X, Y and Z was 25 years and that of Y and Z was 25 years. X’s present age is-
  1. ক
    27 years
  2. খ
    35 years
  3. গ
    25 years
  4. ঘ
    22 years
  5. ঙ
    None of the above
ব্যাখ্যা
Question: The average age of X, Y and Z was 25 years and that of Y and Z was 25 years. X’s present age is-

Solution:
The average of X, Y and Z is 25

So, the sum of their ages = 75
Now, the sum of Y and Z will be 50 (because their average is 25)

So age of X =75 - 50 = 25 years
২২৪.
  1. ক
    45
  2. খ
    81
  3. গ
    405
  4. ঘ
    25
ব্যাখ্যা
Question:


Solution:

২২৫.
When a person weighing 60 kg leaves a group of 50 people, the average weight of the remaining people increases by 0.3 kg. What is the new average weight of the remaining 49 people?
  1. ক
    60 kg
  2. খ
    65 kg
  3. গ
    68 kg
  4. ঘ
    75 kg
  5. ঙ
    None
ব্যাখ্যা
Question: When a person weighing 60 kg leaves a group of 50 people, the average weight of the remaining people increases by 0.3 kg. What is the new average weight of the remaining 49 people?

Solution:
let,
The average weight of 49 people is x kg
Total weight of 49 people = 49x
Total weight of 50 people = 49x + 60

ATQ,
50(x - 0.3) = (49x + 60)
⇒ 50x - 15 = 49x + 60
⇒ 50x - 49x = 60 + 15
∴ x = 75

∴ the new average weight of the remaining 49 people is 75 kg.
২২৬.
The average age of three men is 30 years. If their ages are in ratio 3 : 5 : 7, the age of the youngest man is-
  1. ক
    12 years
  2. খ
    18 years
  3. গ
    30 years
  4. ঘ
    42 years
ব্যাখ্যা

Question: The average age of three men is 30 years. If their ages are in ratio 3 : 5 : 7, the age of the youngest man is-

Solution:
The sum of the ages of three men = (30 × 3) years
= 90 years

Let the ages be 3x, 5x and 7x

Now,
3x + 5x + 7x = 90
⇒ 15x = 90
⇒ x = 6

So, age of the youngest man = 3x
= 3 × 6
= 18 years

২২৭.
Find the value of 'x' if the mean of the set of the numbers 8, 5, x, 10, 15, 21 is given as 11.
  1. ক
    3
  2. খ
    7
  3. গ
    9
  4. ঘ
    14
ব্যাখ্যা
প্রশ্ন: Find the value of 'x' if the mean of the set of the numbers 8, 5, x, 10, 15, 21 is given as 11.

সমাধান:
ATQ,
(8 + 5 + x + 10 + 15 + 21)/6 = 11
⇒ (59 + x)/6 = 11
⇒ 59 + x  = 66
⇒ x = 66 - 59
∴ x = 7
২২৮.
The average earnings of Rohan for the first three months of the calendar year 2023 is Tk. 1200. If his average earnings for the second and third months is Tk. 1300 then find his earnings in the first month.
  1. ক
    Tk. 900 
  2. খ
    Tk. 1000 
  3. গ
    Tk. 1200 
  4. ঘ
    Tk. 1500 
ব্যাখ্যা
Question: The average earnings of Rohan for the first three months of the calendar year 2023 is Tk. 1200. If his average earnings for the second and third months is Tk. 1300 then find his earnings in the first month.

Solution: 
earnings of Rohan for the first three months = 3 × 1200 = 3600 
earnings of Rohan for the second and third months = 2 × 1300 = 2600 

 his earnings in the first month = 3600 - 2600 
= Tk. 1000 
২২৯.
A student's marks were wrongly entered as 89 instead of 62. Due to that the average marks for the class got increased by 3/2. The number of students in the class is-
  1. ক
    20
  2. খ
    10
  3. গ
    18
  4. ঘ
    24
ব্যাখ্যা
Question: A student's marks were wrongly entered as 89 instead of 62. Due to that the average marks for the class got increased by 3/2. The number of students in the class is-

Solution:
Let the number of students in the class be x.
Total increase in marks = x × 3/2 = 3x/2

ATQ,
3x/2 = (89 - 62)
⇒ 3x/2 = 27
⇒ 3x = 54
∴ x = 18

∴ The total number of students in the class is 18.
২৩০.
The average of 25 results is 18. The average of the first 12 of those is 14 and the average of the last 12 is 17. What is the 13th result?
  1. ক
    68
  2. খ
    72
  3. গ
    76
  4. ঘ
    78
ব্যাখ্যা
Question: The average of 25 results is 18. The average of the first 12 of those is 14 and the average of the last 12 is 17. What is the 13th result?

Solution:
Sum of 1st 12 results = 12 × 14 = 168
Sum of last 12 results = 12 × 17 = 204
Let,
13th result = x 

ATQ,
168 + 204 + x = (25 × 18)
⇒ 372 + x = 450
⇒ x = 450 - 372
∴ x = 78
২৩১.
A group of 10 boxes has their average weight increased by 4 kg after replacing a 60 kg box with a new one. What is the weight of the new box?
  1. ক
    75
  2. খ
    85
  3. গ
    92
  4. ঘ
    100
ব্যাখ্যা

Question: A group of 10 boxes has their average weight increased by 4 kg after replacing a 60 kg box with a new one. What is the weight of the new box?

Solution:
Let the weight of the new box be x kg.
Let the total weight of the original 10 boxes = W
Original average = W/10

After replacing the 60 kg box with x,
The total weight becomes: W - 60 + x
The new average = (W - 60 + x) / 10

Accordingly:
New average = Old average + 4
(W - 60 + x) / 10 = W/10 + 4
⇒ W - 60 + x = W + 40
⇒ x = 40 + 60
⇒ x = 100

২৩২.
(5√5)3 =?
  1. ক
    125√5
  2. খ
    625√5
  3. গ
    25√5
  4. ঘ
    625
ব্যাখ্যা
Question: (5√5)3 =?

Solution: 
Given that,
= (5√5)3
= 5√5 × 5√5 × 5√5
= (5 × 5 × 5)(√5 × √5 × √5)
= 125 × 5√5
= 625√5
২৩৩.
The average of 10 numbers is 23. If each number is increased by 4, what will the new average be?
  1. ক
    29
  2. খ
    27
  3. গ
    30
  4. ঘ
    33
ব্যাখ্যা

Question: The average of 10 numbers is 23. If each number is increased by 4, what will the new average be?

Solution:
Given,
Average of 10 numbers = 23
⇒ Sum/Total numbers = 23
⇒ Sum/10 = 23
∴ Sum of the 10 numbers = 230

If each number is increased by 4, the total increase = 4 × 10 = 40
New sum = 230 + 40 = 270

Therefore, the new average = 270/10 = 27

২৩৪.
The captain of a football team of 11 members is 28 years old, and the goalkeeper is 4 years older than him. If the ages of these two are excluded, the average age of the remaining players is 2 years less than the average age of the whole team. What is the average age of the team?
  1. ক
    20 years
  2. খ
    21 years
  3. গ
    22 years
  4. ঘ
    none of the above
ব্যাখ্যা
Question: The captain of a football team of 11 members is 28 years old, and the goalkeeper is 4 years older than him. If the ages of these two are excluded, the average age of the remaining players is 2 years less than the average age of the whole team. What is the average age of the team?

Solution:
Let the average age of the whole team be x years.

ATQ,
11x - (28 + 32) = 9(x - 2)
⇒ 11x - 60 = 9(x - 2)
⇒ 11x - 60 = 9x - 18
⇒ 11x - 9x = 60 - 18
⇒  2x = 42
∴ x = 21
২৩৫.
The price of 6 notebooks is equal to the price of 2 calculators. The total price of 9 notebooks and 4 calculators is Tk. 2688. Find the price of one calculator?
  1. ক
    Tk. 384
  2. খ
    Tk. 330
  3. গ
    Tk. 426
  4. ঘ
    Tk. 530
ব্যাখ্যা
Question: The price of 6 notebooks is equal to the price of 2 calculators. The total price of 9 notebooks and 4 calculators is Tk. 2688. Find the price of
one calculator?

Solution:
Let the price of one notebook is n
and the price of one calculator = c

ATQ,
6n = 2c
c = 3n ........(1)
And,
⇒ 9n + 4c = 2688
⇒ 9n + 4(3n) = 2688
⇒ 21n = 2688
⇒ n = 2688/21
∴ n = 128

From (1),
∴ c = 3n = 3 × 128 = 384

∴ Price of one calculator c = Tk. 384
২৩৬.
180 mangoes are distributed among 70 men and women such that each men gets 2 and each woman gets 3 mangoes. The number of men is - 
  1. ক
    20
  2. খ
    25
  3. গ
    30
  4. ঘ
    40
ব্যাখ্যা
Question: 180 mangoes are distributed among 70 men and women such that each men gets 2 and each woman gets 3 mangoes. The number of men is - 

Solution:
Let, the number of men be x.
The number of women = 70 - x

ATQ,
2x + 3(70 - x) = 180
⇒ 2x + 210 - 3x = 180
⇒ x = 210 - 180
∴ x = 30

∴ The number of men is 30.
২৩৭.
The average daily wage of 10 workers is Tk. 500. If the lowest wage is Tk. 400, then what is the possible maximum wage?
  1. ক
    Tk. 1600
  2. খ
    Tk. 1300
  3. গ
    Tk. 1250
  4. ঘ
    Tk. 1400
ব্যাখ্যা
Question: The average daily wage of 10 workers is Tk. 500. If the lowest wage is Tk. 400, then what is the possible maximum wage?

Solution:
The average wage of 10 workers is Tk. 500.
So total wages = Average × Number of Workers = 500 × 10 = 5000
To find the maximum possible wage of one worker, we must minimize the wages of the other 9 workers.
Let the 10th worker earn the maximum possible wage (M).

Now,
Total Wages = Wages of 9 Workers + Maximum Wage (M)
⇒ 5000 = (9 × 400) + M
⇒ 5000 = 3600 + M
⇒ M = 5000 - 3600
∴ M = 1400

So the maximum possible wage among the workers is Tk. 1400.
২৩৮.
While on a holiday, X persons have decided to rent a van. The rent of the van is Tk D and each person is to pay an equal share. If Y persons cancel their trip, which of the following represents the additional amount of Tk per person that each remaining person must pay in order to still rent the van?
  1. ক
    YD/(X(X - Y))
  2. খ
    D/(X - Y)
  3. গ
    YD/(X - Y)
  4. ঘ
    YD
  5. ঙ
    None
ব্যাখ্যা

Question: While on a holiday, X persons have decided to rent a van. The rent of the van is Tk D and each person is to pay an equal share. If Y persons cancel their trip, which of the following represents the additional amount of Tk per person that each remaining person must pay in order to still rent the van?

Solution: 
Total rent = Tk. D
Original number of persons = X
∴ Original share per person = D/X

And, 
If Y persons cancel then, 
∴ Remaining persons = X - Y
∴ New share per person = D/(X - Y)

∴ Additional amount each remaining person must pay = {D/(X - Y)} - (D/X)
= D{1/(X - Y)} - (1/X)}
= D{(X - X + Y)/X(X - Y)}
= YD/(X(X - Y))

∴ The additional amount per remaining person is YD/(X(X - Y)).

২৩৯.
If the average of two numbers is T and the larger number is x, what is the other number?
  1. ক
    2T - x
  2. খ
    2T/x
  3. গ
    x - 2T
  4. ঘ
    2T + x
ব্যাখ্যা
Question: If the average of two numbers is T and the larger number is x, what is the other number?

Solution:
The average of two numbers is T
∴ The total of two numbers is 2T

The larger number is x
∴ The other number is 2T - x
২৪০.
A waiter's income consists of his salary and tips. During one week his tips were 5/4 of his salary. What fraction of his income came from tips?
  1. ক
    4/9
  2. খ
    5/4
  3. গ
    5/8
  4. ঘ
    5/9
ব্যাখ্যা
Question: A waiter's income consists of his salary and tips. During one week his tips were 5/4 of his salary. What fraction of his income came from tips?

Solution:
Let, salary = Tk. x
Then, tips = Tk. 5x/4

Total income = x + (5x/4)
= 9x/4

∴ Required fraction = (5x/4) × (4/9x) = 5/9
২৪১.
If the average number of 8 terms is given to be 40 and the average of first 6 terms is given to be 35. What is the average of the remaining 2 terms?
  1. ক
    30
  2. খ
    55
  3. গ
    40
  4. ঘ
    42
  5. ঙ
    None of these
ব্যাখ্যা
Question: If the average number of 8 terms is given to be 40 and the average of first 6 terms is given to be 35. What is the average of the remaining 2 terms?

Solution:
Sum of all the 8 terms = 320
The sum of first 6 terms = 210
Now , the sum of remaining terms = 320 - 210 = 110
So , the average of 2 terms would be = 110/2 = 55
২৪২.
If 1 is added to the numerator of a fraction, the fraction becomes 1. If 1 is added to the denominator, the fraction becomes 1/2. Find the fraction.
  1. ক
    1/4
  2. খ
    1/2
  3. গ
    2/3
  4. ঘ
    3/8
ব্যাখ্যা

Question: If 1 is added to the numerator of a fraction, the fraction becomes 1. If 1 is added to the denominator, the fraction becomes 1/2. Find the fraction.

Solution:
ধরি, ভগ্নাংশটি = x/y

শর্তমতে,
(x + 1) / y = 1
⇒ x + 1 = y ............(1)

আবার,
x / (y + 1) = 1/2
⇒ 2x = y + 1
⇒ 2x - 1 = y ............(2)

(1) ও (2) থেকে,
2x - 1 = x + 1
⇒ x = 2

x = 2 হলে,
y = x + 1 = 2 + 1 = 3

∴ ভগ্নাংশটি = 2/3

২৪৩.
If the average of p numbers is q2 and the average of q numbers is p2, find the average of all (p + q) numbers.
  1. ক
    p + q
  2. খ
    pq 
  3. গ
    p2 + q2
  4. ঘ
    p2q2
ব্যাখ্যা

Question: If the average of p numbers is q2 and the average of q numbers is p2, find the average of all (p + q) numbers.

Solution:
Sum of p numbers = p × q2
Sum of q numbers = q × p2
Total sum = pq2 + qp2 = pq(p + q)

Total numbers = p + q

∴ Average of all (p + q) numbers = Total sum/Total numbers 
= [pq(p + q)]/(p + q) 
= pq

২৪৪.
Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?
  1. ক
    30
  2. খ
    26
  3. গ
    24
  4. ঘ
    22
ব্যাখ্যা
Question: Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

Solution: 
Let,
these numbers are x, y and z respectively.

ATQ,
{(x + y)/2} - {(y + z)/2} = 12
⇒ {(x + y) - (y + z)}/2 = 12
⇒ (x + y - y - z)/2 = 12
∴ x - z = 24

∴ the difference between the first and the third number is = 24.
২৪৫.
The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly:
  1. ক
    19.68
  2. খ
    20
  3. গ
    29.68
  4. ঘ
    23
ব্যাখ্যা
Question: The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly:

Solution:
The total sum of 48 numbers,
= (50 × 30) - (35 +40)
= 1500 - 75
= 1425

Average = 1425/48 = 29.68
২৪৬.
If the average (arithmetic mean) of x and y is 60 and the average (arithmetic mean) of y and z is 80, what is the value of (z-x)?
  1. ক
    ক) 70
  2. খ
    খ) 40
  3. গ
    গ) 20
  4. ঘ
    ঘ) None
ব্যাখ্যা

এখানে (x + y)/2 = 60 বা, x + y = 120…..(1)
এবং (y + z)/2 = 80 বা, y + z = 160…..(2)
(2) নং সমীকরণ থেকে (1) নং বিয়োগ করে পাই,
z - x = 40

২৪৭.
If A : B = 1 : 2, B : C = 4 : 3 and A + B + C = 630, what is the value of C?
  1. ক
    210
  2. খ
    70
  3. গ
    140
  4. ঘ
    270
ব্যাখ্যা
Question: If A : B = 1 : 2, B : C = 4 : 3 and A + B + C = 630, what is the value of C?

Solution: 
Given,
A : B = 1 : 2 = 2 : 4
B : C = 4 : 3

∴ A : B : C = 2 : 4 : 3

∴ Value of C = 630 × 3/9
= 210
২৪৮.
Hamed's average on 4 tests is 80. Assuming he can earn no more than 100 on any test, what is the least he can earn on his 5th test and still have a chance for an 85 average after seven tests?
  1. ক
    ক) 60
  2. খ
    খ) 70
  3. গ
    গ) 75
  4. ঘ
    ঘ) 85
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
Question: Hamed's average on 4 tests is 80. Assuming he can earn no more than 100 on any test, what is the least he can earn on his 5th test and still have a chance for an 85 average after seven tests?

Solution: 
হামিদ 4টি পরীক্ষায় মোট পায় = 4 × 80 = 320 নম্বর 
হামিদ 7টি পরীক্ষায় মোট পায় = 7 × 85 = 595 নম্বর 

হামিদ 3টি পরীক্ষায় মোট পায় = 595 - 320 = 275 নম্বর 

হামিদ ৬ষ্ঠ ও ৭ম পরীক্ষায় সর্বোচ্চ নম্বর পায় = 100 + 100 = 200
হামিদ ৫ম পরীক্ষায় সর্বনিম্ন নম্বর পায় = 275 - 200 = 75
২৪৯.
What is the average from 1 to 59?
  1. ক
    28.5
  2. খ
    29
  3. গ
    29.5
  4. ঘ
    30
ব্যাখ্যা
Question: What is the average from 1 to 59?

Solution:
We know,
Average of n natural numbers = (n + 1)/2
Here, n = 59

∴ Average = (59 + 1)/2
= 60/2
= 30
২৫০.
The average weight of 3 friends is 66 kg. None of the friends weights less than 62 kg. What can be the maximum weight of any three friends?
  1. ক
    70 kg
  2. খ
    65 kg
  3. গ
    72 kg
  4. ঘ
    74 kg
ব্যাখ্যা
Question: The average weight of 3 friends is 66 kg. None of the friends weights less than 62 kg. What can be the maximum weight of any three friends?

Solution: 
তিনজনের গড় ওজন ৬৬ কেজি 
মোট ওজন ৬৬ × ৩ কেজি 
= ১৯৮ কেজি 

প্রতিজনের ওজন সর্বনিম্ন ৬২ কেজি 
দুজনের সর্বনিম্ন ওজন ৬২ × ২ কেজি 
= ১২৪ কেজি 

একজনের সর্বোচ্চ ওজন হতে পারে = ১৯৮ - ১২৪ কেজি 
= ৭৪ কেজি 
২৫১.
Average of 60 numbers are 42. When 5 more numbers are included, the average of 65 numbers become 45. Find the average of 5 numbers.
  1. ক
    70
  2. খ
    78
  3. গ
    81
  4. ঘ
    90
ব্যাখ্যা
Question: Average of 60 numbers are 42. When 5 more numbers are included, the average of 65 numbers become 45. Find the average of 5 numbers.

Solution:
Total of 60 numbers = 60 × 42 = 2520
Now, total of 65 numbers = 65 × 45 = 2925

Hence, sum of 5 numbers = 2925 - 2520 = 405

∴ Average of five numbers = 405/5
 = 81
২৫২.
The average age of 50 students in a class is 18 years. When 10 new students are admitted, the average is increased by 0.5 years. The average age of new students is?
  1. ক
    15 years
  2. খ
    18 years
  3. গ
    20 years
  4. ঘ
    21 years
  5. ঙ
    22.5 years
ব্যাখ্যা

Question: The average age of 50 students in a class is 18 years. When 10 new students are admitted, the average is increased by 0.5 years. The average age of new students is?

Solution:
50 জন শিক্ষার্থীর মোট বয়স = 50 × 18 = 900 বছর
10 জন নতুন শিক্ষার্থী ভর্তি হওয়ায় মোট শিক্ষার্থীর সংখ্যা = 50 + 10 = 60 জন

এখন, গড় বয়স বৃদ্ধি পাওয়ায় নতুন গড় বয়স = 18 + 0.5 = 18.5 বছর
সুতরাং, 60 জন শিক্ষার্থীর মোট বয়স = 60 × 18.5 = 1110 বছর

নতুন 10 জন শিক্ষার্থীর মোট বয়স = 1110 - 900 = 210 বছর
সুতরাং, নতুন 10 জন শিক্ষার্থীর গড় বয়স = 210/10 = 21 বছর।

২৫৩.
5 ÷ √5 = ?
  1. ক
    √5
  2. খ
    5
  3. গ
    1/√5
  4. ঘ
    0.05
ব্যাখ্যা
Question: 5 ÷ √5 = ?

Solution:
5 ÷ √5
= 5/√5
= √5
২৫৪.
The average weight of A, B and C is 45kg. If the average weight of A and B is 40 kg and that of B and C is 43 kg, then the weight of B is-
  1. ক
    31 kg
  2. খ
    20 kg
  3. গ
    17 kg
  4. ঘ
    None of these
ব্যাখ্যা
Question: The average weight of A, B and C is 45kg. If the average weight of A and B is 40 kg and that of B and C is 43 kg, then the weight of B is-

Solution: 
Let A, B, C represent their respective weights.

Then, we have:
A + B + C =(45 × 3) = 135..............(i)
A + B = (40 × 2) = 80.................(ii)
B + C=(43 × 2) = 86.................(iii)

Adding (ii) and (iii),
we get: A + 2B + C =80 + 86
A + 2B + C =166 .....(iv)

Subtracting (i) from (iv),
we get:
A + 2B + C - (A + B + C) = 166 - 135 
B = 31

∴ B's weight =31 kg.
২৫৫.
What is the average of the sum of the first 10 logical terms of the Fibonacci series if the series starts with zero?
  1. ক
    ক) 6
  2. খ
    খ) 7.5
  3. গ
    গ) 8
  4. ঘ
    ঘ) 8.8
ব্যাখ্যা
Question: What is the average of the sum of the first 10 logical terms of the Fibonacci series if the series starts with zero?

Solution:
The first 10 logical terms of the Fibonacci series if the series starts with zero = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
So, the average = (0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34)/10
= 88/10
= 8.8
২৫৬.
If the average of 'm' numbers is √2n2 and the average of 'n' numbers is √2m2, what is the average of the combined (m + n) numbers?
  1. ক
    2√2mn
  2. খ
    √2mn
  3. গ
    m2n2
  4. ঘ
    4mn
ব্যাখ্যা

Question: If the average of 'm' numbers is √2n2 and the average of 'n' numbers is √2m2, what is the average of the combined (m + n) numbers?

Solution:
দেওয়া আছে,
 m সংখ্যার গড় = √2n2
∴ m সংখ্যার সমষ্টি = m × √2n2

n সংখ্যার গড় = √2m2
∴ n সংখ্যার সমষ্টি = n × √2m2

∴ মোট সমষ্টি = m + n = (m × √2n2) + (n × √2m2)
= √2mn2 + √2m2n
= √2mn(m + n)

∴ তাদের গড় = মোট সমষ্টি/(m + n)
= √2mn(m + n)/(m + n)
= √2mn

২৫৭.
Average age of 15 students of a class is 15 years. Out of these the average age of 5 students is 14 years and that of the other 9 students is 16 years. The age of the 15th student is-
  1. ক
    15 years
  2. খ
    11 years
  3. গ
    9 years
  4. ঘ
    18 years
ব্যাখ্যা
Question: Average age of 15 students of a class is 15 years. Out of these the average age of 5 students is 14 years and that of the other 9 students is 16 years. The age of the 15th student is-

Solution:
Given that
Average age of 15 students = 15 years
Average age of 5 students = 14 years
Average age of 9 students = 16 years

Now,
Total age of all 15 students = 15 × 15 = 225 years
Total age of 5 students = 5 × 14 = 70 years
Total age of 9 students = 9 × 16 = 144 years 

∴ Age of the 15th student = 225 - (70 + 144) = 225 - 214 = 11 years
২৫৮.
The average weight of A, B and C is 50 kg. If the average weight of A and B be 46 kg and that of B and C be 52 kg, then the weight of B is-
  1. ক
    36 kg
  2. খ
    40 kg
  3. গ
    48 kg
  4. ঘ
    56 kg
  5. ঙ
    46 kg
ব্যাখ্যা
Question: The average weight of A, B and C is 50 kg. If the average weight of A and B be 46 kg and that of B and C be 52 kg, then the weight of B is-

Solution:
Let A, B, C represent their respective weights.
Then, we have,
A + B + C = (50 × 3) = 150 ........(i)
A + B = (46 × 2) = 92 ........(ii)
B + C = (52 × 2) = 104 .........(iii)

Adding (ii) and (iii), we get:
A + 2B + C = 92 + 104 = 196 ........(iv)
And,
Subtracting (iv) from (i),
A + 2B + C - (A + B + C) = 196 - 150
B = 46

∴ B's weight = 46 kg
২৫৯.
10 years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born. Still, the average age of the family is the same today. If the two children differ in age by 2 years, find the present age of the younger child.
  1. ক
    2 years
  2. খ
    3 years
  3. গ
    4 years
  4. ঘ
    5 years
ব্যাখ্যা
Question: 10 years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born. Still, the average age of the family is the same today. If the two children differ in age by 2 years, find the present age of the younger child.

Solution:
Total age of 4 members, 10 years ago = (24 × 4) years = 96 years.
Total age of 4 members now = 4 × (24 + 10) = 136 years
Total age of 4 members now = (24 × 6) years = 144 years.

The sum of the ages of 2 children = (144 - 136) years
= 8 years

Let, the age of the younger child be x years
Then, age of elder child = (x + 2) years
So, x + x + 2 = 8
⇒ 2x = 6
∴ x = 3

So, the age of the younger child 3 years.
২৬০.
The sum of the three consecutive even numbers is 48 more then the average of these numbers. Which of the following is the third largest of these numbers?
  1. ক
    26
  2. খ
    18
  3. গ
    22
  4. ঘ
    24
ব্যাখ্যা
Question: The sum of the three consecutive even numbers is 48 more then the average of these numbers. Which of the following is the third largest of these numbers?

Solution: 
Let, the numbers be = x, (x + 2) and (x + 4)

Then,
(x + x + 2 + x + 4) - (x + x + 2 + x + 4)/3 = 48
⇒ (3x + 6) - (3x + 6)/3 = 48
⇒ 9x + 18 - 3x - 6 = 144
⇒ 6x + 12 = 144
⇒ 6x = 144 - 12
⇒ 6x = 132
∴ x = 22

The third largest of these numbers is = x + 4 = 22 + 4 = 26
২৬১.
The average weight of 20 students is 50 kg, and the average weight of another 30 students is 60 kg. What is the average weight of all 50 students combined?
  1. ক
    54.5 kg
  2. খ
    55 kg
  3. গ
    56 kg
  4. ঘ
    57.5 kg
ব্যাখ্যা

Question: The average weight of 20 students is 50 kg, and the average weight of another 30 students is 60 kg. What is the average weight of all 50 students combined?

Solution:
20 জন ছাত্রের গড় ওজন = 50 কেজি
∴ 20 জন ছাত্রের ওজনের সমষ্টি = 20 × 50 = 1000 কেজি

30 জন ছাত্রের গড় ওজন = 60 কেজি
∴ 30 জন ছাত্রের ওজনের সমষ্টি = 30 × 60 = 1800 কেজি

মোট ওজনের সমষ্টি = 1000 + 1800 = 2800 কেজি
মোট ছাত্র সংখ্যা = 20 + 30 = 50 জন

সুতরাং, সম্মিলিত গড় ওজন = 2800/50 = 56 কেজি

২৬২.
The average expenditure of a man for the first five months is TK.1200 and for the next seven months is TK.1300. If he saves TK.2900 in that year, his monthly average income is 
  1. ক
    TK. 1400
  2. খ
    TK. 1500
  3. গ
    TK. 1600
  4. ঘ
    TK. 1800
ব্যাখ্যা
প্রশ্ন: The average expenditure of a man for the first five months is TK.1200 and for the next seven months is TK.1300. If he saves TK.2900 in that year, his monthly average income is 

Solution:
Total annual expenditure = TK.(5 × 1200) + (7 × 1300)
= TK.(6000 + 9100)
= TK. 15100
His total annual income = Total expenditure + Total savings
= (15100 + 2900)
= TK. 18000

∴ Average monthly income = 18000/12 = TK. 1500
২৬৩.
After replacing an old member with a new member it was found that the average age of five members of a club is the same as it was 3 years ago. What is the difference between the ages of the replaced and the new member?
  1. ক
    ক) 15
  2. খ
    খ) 20
  3. গ
    গ) 25
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
Question: After replacing an old member with a new member it was found that the average age of five members of a club is the same as it was 3 years ago. What is the difference between the ages of the replaced and the new member?

Solution:
(পাঁচ সদস্যের একটি গ্রুপে পুরাতন একজন সদস্যের পরিবর্তে নতুন একজন সদস্য নেওয়া হলে তাদের গড় বয়স ৩ বছর আগে যত ছিল তত হয়।  পরিবর্তিত সদস্য এবং নতুন সদস্যের বয়সের পার্থক্য কত?)


এখানে
⇒ ৫ জন সদস্যের মোট বয়স ৩ বছর আগে যতই থাকুক ৩ বছর পর (৫ × ৩ ) = ১৫ বেড়ে যাওয়ার কথা।
⇒ কিন্তু নতুন সদস্য আসার কারণে ১৫ বছর কমে গিয়েছে, অর্থাৎ ৩ বছর আগের গড় বয়সের সমান হয়ে যায়।
⇒ অর্থাৎ নতুন সদস্যের বয়স পরিবর্তিত সদস্যের বয়স থেকে ১৫ বছর বেশি।
২৬৪.
If the sum of 12 numbers is 756, the average of the first 6 is 54 and the average of the last 5 is 72, what is the 7th number?
  1. ক
    ক) 72
  2. খ
    খ) 70
  3. গ
    গ) 62
  4. ঘ
    ঘ) 74
ব্যাখ্যা
Question: If the sum of 12 numbers is 756, the average of the first 6 is 54 and the average of the last 5 is 72, what is the 7th number?

Solution:
প্রথম ১২টি সংখ্যার সমষ্টি = ৭৫৬
প্রথম ৬টি সংখ্যার সমষ্টি = (৫৪ × ৬) = ৩২৪

আবার,
শেষ ৫টি সংখ্যার গড় = ৭২
শেষ ৫টি সংখ্যার সমষ্টি = (৭২ × ৫) = ৩৬০

∴ ৭ম সংখ্যাটি = {৭৫৬ - (৩২৪ + ৩৬০)}
= ৭২
২৬৫.
What is the square root of (8 + 2√15)?
  1. ক
    √5 + √3
  2. খ
    2√2 + 2√6
  3. গ
    2√5 + 2√3
  4. ঘ
    √2 + √6
ব্যাখ্যা
Question: What is the square root of (8 + 2√15)?

Solution:
২৬৬.
In a class average age of 15 boys is 11. If 5 boys each of age 9 years are added, what would be the new average?
  1. ক
    20 years
  2. খ
    10 years
  3. গ
    10.5 years
  4. ঘ
    23 years
ব্যাখ্যা
Question: In a class average age of 15 boys is 11. If 5 boys each of age 9 years are added, what would be the new average?

Solution:
Sum of ages of 15 boys = 15 × 11= 165
Sum of ages of 5 boys = 5 × 9 = 45
Total age of 20 boys = 165 + 45 = 210
∴ Average of ages of 20 boys = 210/20 = 10.5 years
২৬৭.
Three math classes, X, Y, and Z, take an algebra test. The average score in class X is 83. The average score in class Y is 76. The average score in class Z is 85. The average score of all students in classes X and Y together is 79. The average score of all student classes Y and Z together is 81. What is the average for all the three classes?
  1. ক
    ক) 81
  2. খ
    খ) 81.5
  3. গ
    গ) 82
  4. ঘ
    ঘ) 84.5
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
Question: Three math classes, X, Y, and Z, take an algebra test. The average score in class X is 83. The average score in class Y is 76. The average score in class Z is 85. The average score of all students in classes X and Y together is 79. The average score of all student classes Y and Z together is 81. What is the average for all the three classes?

Solution: 
ধরি, X ক্লাসে ছাত্র আছে x জন 
Y ক্লাসে ছাত্র আছে y জন 
Z ক্লাসে ছাত্র আছে z জন 

X ক্লাসে মোট নম্বর 83x 
Y ক্লাসে মোট নম্বর 76y
Z ক্লাসে মোট নম্বর 85z

X ক্লাসে ও Y ক্লাসে মোট নম্বর = 79(x + y)
∴ 83x + 76y = 79(x + y)
⇒ 83x + 76y = 79x + 79y
⇒ 83x - 79x = 79y - 76y
⇒ 4x = 3y
⇒ y = 4x/3

Y, Z ক্লাসে মোট নম্বর = 81(y + z)

76y + 85z = 81(y + z)
⇒ 76y + 85z = 81y + 81z
⇒ 85z - 81z = 81y - 76y
⇒ 4z = 5y 
⇒ z = 5y/4
= 5x/3

মোট গড় = (83x + 76y + 85z)/(x + y + z)
= (83x + 76× 4x/3 + 85 × 5x/3)/(x +4x/3 + 5x/3)
= 978/12
= 81.5 
২৬৮.
The ratio of Pens ant Pencils in a shop is 3 : 2 respectively. The average number of Pens and Pencils is 170. What is the number of Pens in the shop? 
  1. ক
    ক) 144
  2. খ
    খ) 154
  3. গ
    গ) 184
  4. ঘ
    ঘ) 204
ব্যাখ্যা
Question: The ratio of Pens ant Pencils in a shop is 3 : 2 respectively. The average number of Pens and Pencils is 170. What is the number of Pens in the shop? 

Solution: 
ধরি,
দোকানে কলম আছে = 3x 
দোকানে পেন্সিল আছে = 2x 

প্রশ্নমতে,
(3x + 2x)/2 = 170 
5x = 340 
x = 340/5
x = 68

দোকানে কলম আছে = 3 × 68 = 204টি
২৬৯.
8 + 4 ÷ 2 × 5 =?
  1. ক
    18
  2. খ
    30
  3. গ
    22
  4. ঘ
    50
ব্যাখ্যা
Question: 8 + 4 ÷ 2 × 5 =? 

Solution:
8 + 4 ÷ 2 × 5 
= 8 + 2 × 5
= 8 + 10
= 18.
২৭০.
The mean weight of three club members is 42 kg. If none of them weighs less than 40 kg, what is the maximum possible weight of one of the members?
  1. ক
    41 kg
  2. খ
    42 kg
  3. গ
    43 kg
  4. ঘ
    45 kg
  5. ঙ
    46 kg
ব্যাখ্যা

Question: The mean weight of three club members is 42 kg. If none of them weighs less than 40 kg, what is the maximum possible weight of one of the members?

Solution:
Given,
the mean weight of three members is 42 kg
Total weight of three members = (42 × 3) kg = 126 kg

According to the question,
Minimum weight of any member = 40 kg
 So, Minimum weight of 2 members = (40 × 2) = 80 kg

∴ Maximum weight of any of three members = (126 - 80) kg = 46 kg 

২৭১.
The average of 9 observations was found to be 35. Later on, it was detected that observation 81 was misread as 18. The correct average of the observation is -
  1. ক
    40
  2. খ
    43
  3. গ
    45
  4. ঘ
    47
  5. ঙ
    None of the above
ব্যাখ্যা
Question: The average of 9 observations was found to be 35. Later on, it was detected that observation 81 was misread as 18. The correct average of the observation is -

Solution:
Here, the incorrect mean of 9 observations = 35

So, incorrect sum of observations = 35 × 9 = 315

Since 81 is misread as 18.

So, the correct sum of observations = 315 -18 + 81 = 378

Hence, the correct average of the observation is = 378/9 = 42
২৭২.
The average of 11 numbers is 60. If the average of the first six numbers is 58 and that of the last six numbers is 63, then the middle number is:
  1. ক
    ক) 58
  2. খ
    খ) 60
  3. গ
    গ) 62
  4. ঘ
    ঘ) 64
  5. ঙ
    ঙ) 66
ব্যাখ্যা
Middle numbers = [(58 × 6 + 63 × 6) - 60 × 11]
= 66
২৭৩.
A cake is divided into 27 pieces. If Saiful takes 1/3 of the cake and Mahin takes 1/3 of the rest that are left, how many pieces are still left?
  1. ক
    15 pieces
  2. খ
    12 pieces
  3. গ
    10 pieces
  4. ঘ
    9 pieces
ব্যাখ্যা
Question: A cake is divided into 27 pieces. If Saiful takes 1/3 of the cake and Mahin takes 1/3 of the rest that are left, how many pieces are still left?

Solution:
Saiful takes 1/3 of cake
Left after Saiful takes = (1 - 1/3) of cake
= 2/3 of cake

Mahin takes = (2/3) × (1/3)
= 2/9 of cake
Saiful and Mahin takes = (1/3 + 2/9) = (3 + 2) / 9 = 5/9 of cake

Left after both take = (1 - 5/9)
= 4/9 of cake

Full cake divided into 27 pieces
∴ 4/9 of cake divided into = (27 × 4)/9 pieces
= 12 pieces
২৭৪.
What is the median of the modes in the dataset {-5, 4, 3, 7, 2, 1, 3, 4, 5,-1, 7, 8, -4, 2, 6}?
  1. ক
    3.5
  2. খ
    3
  3. গ
    4
  4. ঘ
    5.5
ব্যাখ্যা
Question: What is the median of the modes in the dataset {- 5, 4, 3, 7, 2, 1, 3, 4, 5,- 1, 7, 8, - 4, 2, 6}?

Solution: 
The given dataset, when arranged in order, is:
- 5, - 4, -1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8

The numbers 2, 3, 4, and 7 each appear twice.
The frequencies are: 2, 3, 4, 7

Here,
n = 4
The median = {(4/2)th term and ((4/2) + 1)th term} / 2
= {2nd and 3rd terms added together}/2
= (3 + 4)/2
= 3.5
২৭৫.
Which of the following is closest to the value of 999/{100 + (1/999)}?
  1. ক
    0.001
  2. খ
    0.01
  3. গ
    0.1
  4. ঘ
    1
  5. ঙ
    10
ব্যাখ্যা
Question: Which of the following is closest to the value of 999/{100 + (1/999)}?

Solution:
Since it is an approximation, we can say that 1/999 is so small (almost 0.001) that we can neglect the same.

What remains is the simple fraction of 999/100 = 9.99 ≈ 10
The closest answer is 10.
২৭৬.
A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half. The number of pupils in the class is-
  1. ক
    40
  2. খ
    45
  3. গ
    39
  4. ঘ
    37
ব্যাখ্যা
Question: A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half. The number of pupils in the class is-

Solution:
Let there be x pupils in the class.
Total increase in marks = x × (1/2) = x/2

Here,
x/2 = (83 - 63)
⇒ x/2 = 20 
⇒ x = 40
২৭৭.
Among 100 students, the average marks in English is 78. If the 60 girls scored an average of 84, determine the average score of the remaining boys.
  1. ক
    72.5
  2. খ
    67
  3. গ
    68
  4. ঘ
    69
ব্যাখ্যা

Question: Among 100 students, the average marks in English is 78. If the 60 girls scored an average of 84, determine the average score of the remaining boys.

Solution:
ধরি,
ছাত্রদের গড় নম্বর = x 

100 জন শিক্ষার্থীর মোট নম্বর = 100 × 78 = 7800 
এবং 60 জন ছাত্রীর মোট নম্বর = 60 × 84 = 5040

প্রশ্নমতে,
5040 + (100 - 60)x = 7800
⇒ 5040 + 40x = 7800
⇒ 40x= 7800 - 5040
⇒ 40x = 2760
⇒ ক = 2760/40 
⇒ ক = 69

∴ 40 জন ছাত্রের গড় নম্বর = 69

২৭৮.
If the sum of A and B is 40, and if C = 32, what is average value of A, B and C?
  1. ক
    ক) 24
  2. খ
    খ) 26
  3. গ
    গ) 28
  4. ঘ
    ঘ) 30
ব্যাখ্যা
Question: If the sum of A and B is 40, and if C = 32, what is average value of A, B and C?

Solution: 
average value of A, B and C is = (A + B + C)/3
= (40 + 32)/3
= 72/3
= 24
২৭৯.
When a person weighing 60 kg leaves a group of 50 people, the average weight of the remaining people increases by 0.3 kg. What is the new average weight of the remaining 49 people?
  1. ক
    69 kg
  2. খ
    74 kg
  3. গ
    65 kg
  4. ঘ
    77 kg
  5. ঙ
    75 kg
ব্যাখ্যা

Question: When a person weighing 60 kg leaves a group of 50 people, the average weight of the remaining people increases by 0.3 kg. What is the new average weight of the remaining 49 people?

Solution:
let,
The average weight of 49 people is x kg
Total weight of 49 people = 49x
Total weight of 50 people = 49x + 60

ATQ,
50(x - 0.3) = (49x + 60)
⇒ 50x - 15 = 49x + 60
⇒ 50x - 49x = 60 + 15
∴ x = 75

∴ the new average weight of the remaining 49 people is 75 kg.

২৮০.
২৪ জন ছাত্রের গড় ওজন ৩৫ কেজি। যদি একজন শিক্ষকের ওজন যোগ করা হয়, তবে গড় ওজন ৪০০ গ্রাম বেড়ে যায়। শিক্ষকের ওজন কত?
  1. ক
    ক) ৩৫ কেজি
  2. খ
    খ) ৪০ কেজি
  3. গ
    গ) ৪৫ কেজি
  4. ঘ
    ঘ) ৬০ কেজি
ব্যাখ্যা
প্রশ্ন- ২৪ জন ছাত্রের গড় ওজন ৩৫ কেজি। যদি একজন শিক্ষকের ওজন যোগ করা হয়, তবে গড় ওজন ৪০০ গ্রাম বেড়ে যায়। শিক্ষকের ওজন কত?

সমাধান-
২৪ জন ছাত্রের মোট ওজন = (৩৫ × ২৪) = ৮৪০ কেজি
শিক্ষকসহ ২৫ জনের মোট ওজন = (৩৫.৪ × ২৫) = ৮৮৫ কেজি  [৪০০ গ্রাম = ০.৪ কেজি]

শিক্ষকের ওজন = ৮৮৫ - ৮৪০ = ৪৫ কেজি

২৮১.
A farmer has cows and ducks. There are 40 animals and a total of 120 legs. How many ducks are there?
  1. ক
    22
  2. খ
    25
  3. গ
    20
  4. ঘ
    16
  5. ঙ
    None of these
ব্যাখ্যা
Question: A farmer has cows and ducks. There are 40 animals and a total of 120 legs. How many ducks are there?

Solution:
Let, cows = x 
ducks = 40 - x
Now,
Legs = 4x + 2(40 - x) 
= 4x + 80 - 2x
= 2x + 80

ATQ,
⇒ 2x + 80 = 120
⇒ 2x = 120 - 80
⇒ 2x = 40
∴ x = 20

∴ Number of ducks = 40 - 20 = 20 
২৮২.
Which fraction is greater than 2/5 and less than 8/9?
  1. ক
    4/9
  2. খ
    3/4
  3. গ
    5/12
  4. ঘ
    All of them
ব্যাখ্যা

Question: Which fraction is greater than 2/5 and less than 8/9?

Solution: 
2/5 = 0.40
8/9 = 0.88

Now,
4/9 = 0.44;  greater than 0.40, less than 0.88
3/4 = 0.75 → greater than 0.40, less than 0.88
5/12 = 0.41;  greater than 0.40, less than 0.88

All the given fractions lie between 2/5 and 8/9.

So correct answer: d) All of them 

২৮৩.
The average weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
  1. ক
    56 kg
  2. খ
    60 kg
  3. গ
    58 kg
  4. ঘ
    52 kg
  5. ঙ
    54 kg
ব্যাখ্যা
Question: The average weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
 
Solution:
Average weight of 7 boys = 56 kg.
Total weight of 7 boys = (56 × 7) kg = 392 kg.
 
Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg
= 338 kg.
 
Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)
= (392 - 338) kg
= 54 kg.
 
Therefore, the weight of the seventh boy is 54 kg.
২৮৪.
The average run of a cricket player of 10 innings was 35. How many runs must be made in his next innings so as to increase his average of runs by 5?
  1. ক
    80
  2. খ
    90
  3. গ
    100
  4. ঘ
    110
ব্যাখ্যা
Question: The average run of a cricket player of 10 innings was 35. How many runs must be made in his next innings so as to increase his average of runs by 5?

Solution: 
Average after 11 innings = 35 + 5 = 40
Required number of runs,
= (40 × 11) - (35 × 10)
= 440 - 350
= 90
২৮৫.
What is the average of the sum of the first 20 natural numbers?
  1. ক
    ক) 10
  2. খ
    খ) 10.5
  3. গ
    গ) 20
  4. ঘ
    ঘ) 21
ব্যাখ্যা
Question: What is the average of the sum of the first 20 natural numbers?

Solution:
The average is = {20(20 + 1)} / (2 × 20)
= 420/40
= 10.5
২৮৬.
  1. ক
    15
  2. খ
    23/15
  3. গ
    6/17
  4. ঘ
    17/6
ব্যাখ্যা
Question:


Solution:
 
২৮৭.
The average of runs of a cricket player of 10 innings was 35. How many runs must he make in his next innings so as to increase his average of runs by 5?
  1. ক
    ক) 90
  2. খ
    খ) 135
  3. গ
    গ) 100
  4. ঘ
    ঘ) 145
ব্যাখ্যা
প্রশ্ন: The average of runs of a cricket player of 10 innings was 35. How many runs must he make in his next innings so as to increase his average of runs by 5?

Solution: 
Total runs =35 × 10 = 350
Now increase in average is 5 runs
so, 
New average =35 + 5 = 40 runs
Total runs = 40 × 11= 440
Runs made in the 11th inning =440 - 950 = 90
২৮৮.
Find the average of all the numbers between 7 and 46 which are divisible by 5 -
  1. ক
    25.5
  2. খ
    32.7
  3. গ
    19.2
  4. ঘ
    27.5
ব্যাখ্যা
Question: Find the average of all the numbers between 7 and 46 which are divisible by 5 -

Solution:
First, let's list all the numbers between 7 and 46 that are divisible by 5.
So, the numbers divisible by 5 in this range are,
10, 15, 20, 25, 30, 35, 40, 45 

∴ The sum of these numbers is,
10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 = 220

∴ Average = 220/​8 = 27.5
Thus, the average of all numbers between 7 and 46 that are divisible by 5 is 27.5.
২৮৯.
If the mean of numbers 28, x, 42, 78 and 104 is 62, what is the mean of 48, 62, 98, 124 and x?
  1. ক
    ক) 58
  2. খ
    খ) 390
  3. গ
    গ) 78
  4. ঘ
    ঘ) 310
ব্যাখ্যা
Question: If the mean of numbers 28, x, 42, 78 and 104 is 62, what is the mean of 48, 62, 98, 124 and x?

Solution: 
এখানে,
(28 + x + 42 + 78 + 104)/5 = 62
252 + x = 62 × 5
252 + x = 310
x = 310 - 252 
x = 58

48, 62, 98, 124 এবং 58 এর গড় = (48 + 62 + 98 + 124 + 58)/5
                                                  = 390/5
                                                  = 78
২৯০.
What is the value of the expression?
(√3 + √12)2 = ?
  1. ক
    24
  2. খ
    27
  3. গ
    30
  4. ঘ
    21
ব্যাখ্যা

Question: What is the value of the expression?
(√3 + √12)2 = ?

Solution:
(√3 + √12)2
= {√3 + √(3 × 4)}2
= (√3 + 2√3)2
= (3√3)2
= 32 × (√3)2
= 9 × 3
= 27

২৯১.
The average age of the 20 aspirants of a class is 19.2 years. After some time two more aspirants join them and then average is increased by 0.3 years. Find the difference between the age of new aspirants.
  1. ক
    ক) 12
  2. খ
    খ) 15
  3. গ
    গ) 18
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
২০ জন ছাত্রের গড় বয়স = ১৯.২ বছর 
২০ জন ছাত্রের মোট বয়স = (১৯.২ × ২০) বছর =৩৮৪ বছর 

২২ জন ছাত্রের গড় বয়স = ১৯.৫ বছর
 ২২ জন ছাত্রের মোট বয়স = (১৯.৫× ২২) বছর =৪২৯ বছর
 
২ জন ছাত্রের মোট বয়স = (৪২৯ - ৩৮৪) বছর  
                                      = ৪৫ বছর

এখান থেকে তাদের বয়সের পার্থক্য বের করা সম্ভব নয়।
২৯২.
The average of a group of men is increased by 6 years when a person aged of 16 years is replaced by a new person of aged 40 years. How many men are there in the group?
  1. ক
    ক) 5
  2. খ
    খ) 6
  3. গ
    গ) 3
  4. ঘ
    ঘ) 4
ব্যাখ্যা
Question: The average of a group of men is increased by 6 years when a person aged of 16 years is replaced by a new person of aged 40 years. How many men are there in the group?

Let
The no. of persons in the group be X 
Now 
Member in group × aged increased = difference of replacement
X × 6 = 40 - 16
Or, 6X = 24
Or, X = 4
২৯৩.
In a set of 3 numbers, the average of first two numbers is 2, the average of the last two numbers is 3, and the average of the first and the last numbers is 4. What is the average of three numbers?
  1. ক
    3
  2. খ
    2
  3. গ
    2.5
  4. ঘ
    3.5
ব্যাখ্যা

Question: In a set of 3 numbers, the average of first two numbers is 2, the average of the last two numbers is 3, and the average of the first and the last numbers is 4. What is the average of three numbers?

Solution: 
let, the numbers are x, y, z 

x + y = 2 × 2 = 4
y + z = 2 × 3 = 6 
z + x = 2 × 4 = 8

2 (x + y + z) = 4 + 6 + 8 = 18 
⇒ (x + y + z) = 9 

∴ the average of three numbers is = 9/3 = 3 

২৯৪.
A group of 8 people has their average weight increased by 2.5 kg after replacing a 65 kg person with someone new. What is the possible weight of the replacement?
  1. ক
    70 Kg
  2. খ
    78 Kg
  3. গ
    82 Kg
  4. ঘ
    85 Kg
ব্যাখ্যা
Question: A group of 8 people has their average weight increased by 2.5 kg after replacing a 65 kg person with someone new. What is the possible weight of the replacement?

Solution:
ধরি,
 8 জন ব্যাক্তির গড় ওজন = x কেজি
তাহলে, মোট ওজন = 8x
65 কেজি ওজনের ব্যক্তি চলে যাওয়ার পর মোট ওজন= (8x - 65) কেজি

আবার,
মনে করি, নতুন ব্যাক্তির ওজন= y kg

তাহলে নতুন ব্যক্তি প্রবেশ করার পর নতুন মোট ওজন হবে = (8x - 65 + y) কেজি
এবং নতুন গড় = x + 2.5

প্রশ্নমতে,
(8x - 65 + y)/8 = x + 2.5
⇒ 8x - 65 + y = 8x + 20
⇒ 8x + y - 8x = 20 + 65
⇒ y = 85

সুতরাং, নতুন ব্যাক্তির ওজন =  85 কেজি।
২৯৫.
The average temperature for the first 4 days of a week is 38.2°C and that of the last 4 days is 39.3°C. If the average temperature for the whole week is 38.6°C, then the temperature on the fourth day is-
  1. ক
    36.5°C
  2. খ
    38.8°C
  3. গ
    39.8°C
  4. ঘ
    39.3°C
ব্যাখ্যা
Question: The average temperature for the first 4 days of a week is 38.2°C and that of the last 4 days is 39.3°C. If the average temperature for the whole week is 38.6°C, then the temperature on the fourth day is-

Solution:
Temperature on the fourth day
= [(38.2 × 4 + 39.3 × 4) - (38.6 × 7)]° C
= 39.8° C
২৯৬.
If 
  1. ক
    8
  2. খ
    12
  3. গ
    16
  4. ঘ
    14
ব্যাখ্যা

Question: If 

Solution:

২৯৭.
The average of 8 numbers is 8. If 4 is subtracted from each of 6 of the numbers, what is the new average?
  1. ক
    3.5
  2. খ
    5
  3. গ
    4
  4. ঘ
    6.5
ব্যাখ্যা

Question: The average of 8 numbers is 8. If 4 is subtracted from each of 6 of the numbers, what is the new average?

Solution:
Given that, 
The average of 8 numbers = 8
∴ Sum of the 8 numbers = (8 × 8) = 64

If 4 is subtracted from each of 6 of these numbers, then the new sum becomes,
= 64 - (6 × 4)
= 64 - 24
= 40

Therefore, the new average of the 8 numbers will be,
= 40/8
= 5

So the new average is 5.

২৯৮.
The average marks of a student in 5 subjects is 75. Later, marks in one subject was increased by 12 and marks in another subject was decreased by 17. Find the corrected average of marks.
  1. ক
    ক) 74
  2. খ
    খ) 72
  3. গ
    গ) 71
  4. ঘ
    ঘ) 69
ব্যাখ্যা
Sum of all numbers = 75 × 5 = 375
Later, marks in one subject was increased by 12 and marks in another subject was decreased by 17.
Corrected marks = 375 + 12 – 17
                       = 370
Corrected Average = Correct sum / number of subjects
                           = 370/5
                           = 74
∴ Corrected average is 74
২৯৯.
The average of 6 numbers in 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
    ক) 24
  2. খ
    খ) 25
  3. গ
    গ) 26
  4. ঘ
    ঘ) 28
ব্যাখ্যা
6টি সংখ্যার গড় = 25 
6টি সংখ্যার সমষ্টি = 25× 6= 150

3টি সংখ্যার গড় = 22
3টি সংখ্যার সমষ্টি = 22 × 3= 66

9টি সংখ্যার সমষ্টি = 150 + 66 = 216
9টি সংখ্যার গড় =216/9 
                          = 24
৩০০.
The average temperature on Monday, Tuesday, and Wednesday was 26°C. The average temperature on Tuesday, Wednesday, and Thursday was 25°C. If the temperature on Monday was 28°C, what was the temperature on Thursday?
  1. ক
    23°
  2. খ
    26°
  3. গ
    25°
  4. ঘ
    29°
ব্যাখ্যা

Question: The average temperature on Monday, Tuesday, and Wednesday was 26°C. The average temperature on Tuesday, Wednesday, and Thursday was 25°C. If the temperature on Monday was 28°C, what was the temperature on Thursday?

Solution:
The total temperature on Monday, Tuesday, and Wednesday = 26° × 3 = 78°
The total temperature on Tuesday, Wednesday, and Thursday = 25° × 3 = 75°
 
ATQ,
(Mon + Tue + Wed) - (Tue + Wed + Thu) = 78° - 75°
⇒ Mon - Thu = 3°
⇒ Thu = Mon - 3°
⇒ Thu = 28° - 3°
∴ Thu = 25°

∴ The temperature on Thursday = 25°