Fraction and Simplification, Average and Mean

Question: In a class, if 5 students are seated on each bench, 5 benches remain vacant. But if 4 students are seated on each bench, 8 students have to stand. How many students are there in the class?

Solution:
Let the number of benches = b

1^st condition,  
5 students are seated per bench then 5 benches remain vacant  
∴ Students = 5 × (b - 5)

2^nd condition, 
4 students are seated per bench then 8 students are left standing  
∴ Students = 4 × b + 8

ATQ,
5(b - 5) = 4b + 8
⇒ 5b - 25 = 4b + 8  
⇒ 5b - 4b = 8 + 25  
∴ b = 33

Now, total students = 5 × (33 - 5)  
= 5 × 28  
= 140

∴ There are 140 students in the class.

Total weight = 45 × 50 = 2250 kg
New average = 45 - 0.1 = 44.9
The total weight of 49 = 49 × 44.9 = 2200.1
The weight of the student who left the class = (2250 - 2200.1) kg
= 49.9 kg

As per question,
Total cost of 10 books = Tk. 120
Total cost of 8 books = Tk. 94
∴ The cost of 2 books = Tk. 26

Let the price of one book is x TK.
∴ The other book must be 160% of x or 1.6x TK.

∴ x + 1.6x = 26
Or, x = 10.

∴ Cost of a book is = 10 TK
and cost of the other book is = 26 - 10 = 16 TK.

Let 1st no = x
2nd no = y
and, 3rd no = z
ATQ,
(x + y)/2 = 15 + (y + z)/2
Or, x + y = 30 + y + z
Or, x - z = 30
So, Difference of 1st and 3rd no is 30

Adding the given equations:
12a + 3b + 7b - 2a = 10
Or, 10a + 10b = 10
Or, 10(a + b) = 10
Or, a + b = 1
So, average of a and b is 0.5

Question: Find the value of x, if (x/7) - (x/9) = 2

Solution:
Given that,
(x/7) - (x/9) = 2
⇒ (9x - 7x)/63 = 2
⇒ 2x = 2 × 63
∴ x = 63

Question: - 5x - [4y - {9x - (3y - 7x)}] simplifies to 

Solution:
- 5x - [4y - {9x - (3y - 7x)}]
= - 5x - [4y - {9x - 3y + 7x}]
= - 5x - [4y - 9x + 3y - 7x]
=  - 5x - [7y - 16x]
= - 5x - 7y + 16x
= 11x - 7y

Question: The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41° C. If temperature on Saturday was 44° C, what was the temperature on Wednesday?

Solution:
Average temperature for Wednesday, Thursday and Friday = 40° C
∴ Total temperature = 3 × 40 = 120° C

Average temperature for Thursday, Friday and Saturday = 41° C
∴ Total temperature = 41 × 3 = 123° C

And,
Temperature on Saturday = 44° C

Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 123 - 120
⇒ Saturday - Wednesday = 3
∴ Wednesday = 44 - 3 = 41° C

Question: The average of x and y is 45, and the average of y and z is 50. If y = 42, then find the value of (x + z).

Solution:
Given y = 42

Average of x and y = 45
∴ x + y = 45 × 2
⇒ x + y = 90
⇒ x + 42 = 90
⇒ x = 90 - 42
⇒ x = 48

Average of y and z = 50
∴ y + z = 50 × 2
⇒ y + z = 100
⇒ 42 + z = 100
⇒ z = 100 - 42
⇒ z = 58

Now, x + z = 48 + 58 = 106

Question: If the average of x and y is 70 and the average of y and z is 90. What is the value of z - x?

Solution: 
(x + y)/2 = 70
∴ x + y = 140 .........(1)

Again,
(y + z)/2 = 90
∴ y + z = 180 .........(2)

From (2) - (1) we get,
y + z - x - y = 180 - 140
∴ z - x = 40

Let original average be x

then (12x+ 96) / 13 = x + 5
⇒ 12x + 96 = 13x + 65
⇒ x = 96 - 65 = 31
∴ His average after 13 innings = 31 + 5 = 36

Question: If 2 is added to the numerator of a fraction, the fraction becomes 1. If 9 is added to the denominator, the fraction becomes 1/2. Find the fraction.

Solution:
ধরি,
লব x, হর y
শর্তমতে,
(x +2)/y = 1
⇒ x + 2 = y   ................(1) 

আবার,
x/(y + 9) = 1/2 
⇒ 2x = y + 9
⇒ 2x - 9 = y ..................(2) 

(1) ও (2) হতে পাই,
2x - 9 = x + 2
⇒ x = 11

x  এর মান (1) নং এ বসিয়ে পাই,
11 + 2 = y
⇒ y = 13 

∴ ভগ্নাংশটি 11/13

Given, Captain's age 26 years and Wicketkeeper's age 29 years.
Let, Average age be X years

ATQ,
9(x - 1) + 26 + 29 = 11x
⇒ 9x - 9 + 26 + 29 = 11x
⇒ 2x = 46
⇒ x = 23

Question: If the average of 'p' numbers is 3q² and the average of 'q' numbers is 3p², what is the average of the combined (p + q) numbers?

Solution:
দেওয়া আছে,
'p' সংখ্যার গড় = 3q²
∴ p সংখ্যার সমষ্টি = p × 3q²

'q' সংখ্যার গড় = 3p²
∴ 'q' সংখ্যার সমষ্টি = q × 3p²

∴ মোট সমষ্টি = (p × 3q²) + (q × 3p²)
= 3pq² + 3qp²
= 3pq(q + p)

∴ তাদের গড় = মোট সমষ্টি/(p + q)
= 3pq(p + q)/(p + q)
= 3pq

Question: On simplification √{(0.65)² - (0.16)²} reduces to-

Solution:
Given that,
√{(0.65)² - (0.16)²}
Since, a² - b² = (a - b)( a + b)
= √{(0.65 + 0.16)(0.65 - 0.16)}
= √{(0.81)(0.49)}
= √{(0.9)(0.9)×(0.7)(0.7)}
= 0.9 × 0.7
= 0.63

Question: A batsman scored 45, 68, 52, 71, and 59 runs in five innings. How many runs must he score in the sixth innings to have an average of exactly 60 runs?

Solution:
ধরি, 6-তম ইনিংসে রান = x
প্রথম 5টি ইনিংসে মোট রান = (45 + 68 + 52 + 71 + 59) = 295

প্রশ্নমতে,
(295 + x)/6 = 60
⇒ 295 + x = 60 × 6
⇒ 295 + x = 360
⇒ x = 360 - 295
⇒ x = 65

∴ 6-তম ইনিংসে 65 রান করতে হবে।

Question: A man has Tk. 480 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. What is the total number of notes that he has ?

Solution: 
Let number of notes of each denomination be x.
Then x + 5x + 10x = 480
⇒ 16x = 480
∴ x = 30.

Hence, total number of notes = 3x = 90

Question: The average score of a class of 90 students in an exam was 40. The average score of the students who had passed is 50, and the average score of the students who had failed is 30. How many students failed in the exam?

Solution:
Let
Total number of students who had failed = x
So, the total number of students who had passed = 90 - x

ATQ,
50(90 - x) + 30x = 90 × 40
⇒ 4500 - 50x + 30x = 3600
⇒ 20x = 4500 - 3600
⇒ 20x = 900
∴ x = 45

Average weight of 15 girls = 24 kg
Total weight of 15 girls = 24 x 15 = 360 kg
Average after including a new girl = 24 + 3 = 27 kg
Total weight of 16 girls = 27 x 16 = 432 kg
Weight of the new girl = Total weight of 16 girls - Total weight of 15 girls = 432 - 360 = 72 kg
Hence the required answer is 72 kg.

10 টি সংখ্যার সমষ্টি = 10 × 7 = 70
8 দ্বারা ওই দশটি সংখ্যার প্রত্যেককে গুণ করার পর সমষ্টি হবে = 70 × 8 = 560
∴ নতুন গড় = 560/10 = 56 

Question: A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is -

Solution:
Let, the fraction is a/b

Now, (a - 1)/b = 1/3
⇒ b = 3a - 3

Again, a/(b + 8) = 1/4
⇒ b + 8 = 4a
⇒ b = 4a - 8

∴ 3a - 3 = 4a - 8
⇒ 4a - 3a = 8 - 3
⇒ a = 5

And, b = (3 × 5) - 3
= 15 - 3
= 12

∴ The fraction = a/b = 5/12 

Trepidation (noun)
- English Meaning: An uncomfortable feeling of nervousness or worry about something that is happening or might happen in the future.
-  Bangla Meaning: সচকিত উত্তেজিত মনোভাব।  

Synonyms:
• Anxiety - ভবিষ্যৎ বিষয়ে ভয় ও অনিশ্চয়তাবোধ; উদ্বেগ; দুশ্চিন্তা।

• Apprehension - আশঙ্কা; ভবিষ্যৎ বিষয়ে উৎকণ্ঠার অনুভূতি; উপলব্ধি; চেতনা; বোধ।

• Disquietude - মানসিক অস্থিরতা বা উদ্বেগ।

Antonyms:
• Calmness - শান্ততা, বিশ্রান্ততা।

• Equanimity - মনমেজাজের প্রশান্তি।

• Composure - শান্তি; স্থৈর্য; আত্মসংবরণ।

Other options:
খ) Very comfortable situation.
- Translations: খুবই আরামদায়ক অবস্থা। 

গ) Find a solution.
- Translations:  সমাধান বের করা। 

ঘ) Always being confident.
- Translations: সবসময় আত্মবিশ্বাসী।

Source: Live MCQ Lecture.

Question:  = ?

Solution:

Members in A = 71, B = 18, C = 53

Average spent per member: A = Tk. 397, B = Tk. 421 and C = Tk. 137

Now,
Total amount by A = 71 × 397 = 28187
Total amount by B = 18 × 421 = 7578
Total amount by C = 53 × 137 = 7261

∴ Total amount = 28187 + 7578 + 7261 = 43026

∴ Total members = 71 + 18 + 53 = 142

∴ Average = 43026/142 = 303

Total average amount spent per member = Tk. 303

Question: The average weight of 12 students in a class is 45.5 kg. What should be the weight of a 13th student so that the average weight of all 13 students becomes 47.2 kg?

Solution:
Average weight of 12 students = 45.5 kg
∴ Total weight of 12 students = (45.5 × 12) kg
= 546 kg

Again,
Average weight of 13 students = 47.2 kg
Total weight of 13 students = (47.2 × 13) kg
= 613.6 kg

∴ Weight of the 13th student = (613.6 - 546) kg
= 67.6 kg

Therefore, the weight of the 13th student should be 67.6 kg.

Question: The average of 60 numbers is 25. If two numbers, namely 30 and 40, are discarded, what is the average of the remaining numbers?

Solution:
ATQ,
The average of 60 numbers is = 25
The sum of 60 numbers is = 25 × 60 = 1500
The two numbers discarded = 30 + 40 = 70
The sum of the remaining 58 numbers = 1500 - 70 = 1430

∴ The new average = 1430/58 = 24.66 (approximately)

Let, there was x exams (excluding the makeup exam)
ATQ, 80x + 92 = 84(x + 1)
Or, 80x + 92 = 84x + 84
Or, x = 2
So, total number of exams including the makeup exam = 2 + 1 = 3

Question: A student find the average of ten 2 digit numbers, and while copying numbers by mistake, he writes one number with its digit interchanged, as a result of that his average is 3.6 less that the correct answer, then find the difference of the digits of the number in which he made the mistake.

Solution:
The number = 10x + y
After interchanging the digit, the number becomes 10y + x
The difference in average after interchanging the digits = 3.6
As there are 10 numbers,
The difference between the numbers will be = 3.6 × 10 = 36

Now
10x + y - (10y + x) = 36
9x - 9y ⇒ 36
9(x - y) = 36
x - y = 4

Difference in digits, (x - y) = 4

∴ The difference of the digits of the number in which he made the mistake is 4.

Question: The average of six consecutive numbers A, B, C, D, E and F is 62. What is the sum of B and F?

Solution:
Let the Numbers A, B, C, D, E, F be x, x + 1, x + 2, x + 3, x + 4, x + 5.
According to question, 
x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 = 62 × 6
⇒ 6x + 15 = 372
⇒ 6x = 372 - 15 = 357
⇒ 6x = 357
⇒ x = 59.5

∴ B = x + 1 = 60.5 
∴ F = x + 5 = 64.5

∴ B + F = 60.5 + 64.5 = 125

7 টি পরপর ধনাত্মক বিজোড় সংখ্যার গড় p
ধরি, সংখ্যাগুলোঃ p-6, p-4, p-2, p, p+2, p+4, p+6

এবং এর পরবর্তি 7 টি ধনাত্মক বিজোড় সংখ্যা হবেঃ p+8, p+10, p+12, p+14, p+16, p+18, p+20
সুতরাং পরবর্তি 7 টি ধনাত্মক বিজোড় সংখ্যার গড় হবে = (7p+98) / 7 = p+14

Let the number of female students be x
Let the weight of female students = 57x
Number of male students = 66
Total weights of male students = 65.9 × 66
The average weight of all the students = 60.3 kg
Total weights of all the students = 60.3 (66 + x)

According to the given information,
Then,
⇒ 60.3 (66 + x) = 66 × 65.9 + 57x
⇒ 60.3 × 66 + 60.3x = 66 × 65.9 + 57x
⇒ 60.3x - 57x = 66 (65.9 - 60.3)
⇒ 3.3x = 66 × 5.6

∴ x = (66 × 5.6)/3.3
⇒ x = 2 × 56
⇒ x = 112