61. Find the average of all prime numbers between 30 and 50?

30 और 50 के बीच सभी अभाज्य संख्याओं का औसत ज्ञात कीजिए।

Option “B” is correct.

Formula:

Average = Sum of all observations/total number of all observations

Calculation:

Prime number between 30 and 50 = 31, 37, 41, 43, 47

⇒ Sum of prime number between 30 and 50 = 31 + 37 + 41 + 43 + 47 = 199

∴ Average of all prime number between 30 and 50 = 199/5 = 39.8

62. If the average of five observations a1, a2, a3, a4 and a5 is N, then what is the average of five new observations a1 − 100, a2 − 100, a3 − 100, a4 − 100, and a5 − 100

यदि पांच अवलोकनों a1, a2, a3, a4 और a5 का औसत N है, फिर पाँच नए अवलोकनों a1 − 100, a2 − 100, a3 − 100, a4 − 100, और a5 − 100 का औसत क्या है

Option “B” is correct.

Using the above formulae, we have

Average of a1, a2, a3, a4 and a5 is N

Sum of a1, a2, a3, a4 and a5 is 5N

Sum of (a1 – 100 + a2 – 100 + a3 – 100 + a4 – 100 + a5 – 100) = 5N – 500

Average = {5N – (100 + 100 + 100 + 100 + 100)}/5

Average = {(N × 5) – (100 × 5)}/5 = N – 100

∴ Required average = N – 100

63. The average of six numbers N1, N2, N3, N4, N5, and N6 is A. 5 is subtracted from each number. What is the new average?

छह संख्या N1, N2, N3, N4, N5, और N6 का औसत A है। प्रत्येक संख्या से 5 घटाया जाता है। नया औसत क्या है?

Option “D” is correct.

Concept:

If the average of n numbers is N and x is subtracted from each numbers then the new average will

be (N – x)

And if the average of n numbers is N and x is added to each numbers then the new average will be

(N + x)

Formula used:

Average = Sum of all numbers/Total numbers

Calculation:

According to question,

A = (N1 + N2 + N3 + N4 + N5 + N6)/6

⇒ 6A = N1 + N2 + N3 + N4 + N5 + N6

If 5 is subtracted from each number then

New average = (N1 – 5 + N2 – 5 + N3 – 5 + N4 – 5 + N5 – 5 + N6 – 5)/6

⇒ New average = (N1 + N2 + N3 + N4 + N5 + N6 – 30)/6

∴ New average = (6A – 30)/6 = A – 5 

64.If a 12 years old boy is replaced by a new boy, then the average age of 18 boys increases by 1.5 years.
What is the age of the new boy?     

12 वर्ष के लड़के को एक नए लड़के से प्रतिस्थापित कर दिया जाता है, तो 18 लड़कों की औसत आयु 1.5 वर्ष से बढ़ जाती है।
नए लड़के की आयु क्या है?

Option “D” is correct.

Formula used:

Average = Sum of all quantities / Total number of quantities

Calculation:

Let the initial average be x and the 12 years old boy is replaced by a new boy whose age is y

According to question,

18x – 12 + y = 18(x + 1.5)

⇒ 18x – 12 + y = 18x + 27

⇒ y = 39

∴ The age of new boy = 39 years[/bg_collapse]

65. There are 60 students in a class. A boy whose weight is 80 kg leaves the class and a girl joined the class, as a result, there is a decrease of 0.5 kg as average weight. Find the weight of the girl.

एक कक्षा में 60 छात्र हैं। एक लड़का जिसका वजन 80 किग्रा है वह कक्षा छोड़ देता है और एक लड़की कक्षा में शामिल हो जाती है, जिसके फलस्वरूप औसत वजन 0.5 किलो ग्राम कम हो जाता है, लड़की का वजन ज्ञात कीजिये।

Option “C” is correct.

Given: 

Total number of students in class = 60

Formula used:

Average = Sum of observations/Total number of observation

Calculation:

Let the weight of the new girl joined the class be ‘y’ kg

Let the average weight of the class be ‘x’ kg

Reduced average weight of class = (x – 0.5) kg

⇒ 60x – 80 + y = 60(x – 0.5)

⇒ 60x – 80 + y = 60x – 30

∴ The weight of the girl is 50 kg

SHORTCUT TRICK

Weight of new person = weight of left person ± change in average × total number

” + ” sign is used when the average is increasing and ” – ” when the average is decreasing

Average weight reduced by 0.5 kg then, weight reduced = 0.5 × 60 = 30 kg

Weight of girl = 80 – 30 = 50 kg

∴ The weight of the girl is 50 kg

66. The average weight of a family of 18 persons decreases by 1 kg when the weight of a child is added to it. If the average weight of the family after adding the weight of the child is 19 kg, what will be the weight of the child?

एक बच्चे का वजन जोड़ने पर 18 व्यक्तियों के एक परिवार का औसत वजन 1 किलो कम हो जाता है। यदि बच्चे का वजन शामिल करने के बाद परिवार का औसत वजन 19 किलो है, तो बच्चे का वजन क्या होगा?

Option “B” is correct.

Given:

The average weight of the family of 18 persons decreases by 1 kg when the weight of the child is added to it. The average weight of the family after adding the weight of the child is 19 kg.

Concept used:

Total weight = Average weight × Total number of person

Calculation:

The average weight of the family after adding the weight of the child is decreased by 1 kg and now it is 19 kg

That means before the addition of the child average weight of 18 person ware (19 + 1) = 20 kg

Total weight was (20 × 18) = 360 kg

The present number of members in the family is 19

Present total weight = (19 × 19) = 361 kg

Weight of the child = (361 – 360) = 1 kg

∴ Weight of the child is 1 kg

Shortcut Trick

Weight of the child = Average weight after adding the weight of the child – number of members excluding the child × decrease in average weight

⇒ Weight of the child = 19 – 1 × 18 = 19 – 18 = 1 kg

67.Average of 9 consecutive numbers is 26. Which is the largest number out of these 9?

9 क्रमागत संख्याओं का औसत 26 है। इन 9 में से सबसे बड़ी संख्या कौन सी है?

Option “B” is correct.

Let, 9 consecutive numbers,

⇒ (x – 4), (x – 3), (x – 2), (x – 1), x, (x + 1), (x + 2), (x + 3), (x + 4)

∴ Sum of 9 consecutive numbers,

⇒ (x – 4) + (x – 3) + (x – 2) + (x – 1) + x + (x + 1) + (x + 2) + (x + 3) + (x + 4)

⇒ 9x

∴ average of 9 consecutive numbers,

⇒ 9x/9

⇒ x

According to problem,

⇒ x = 26

∴ largest number among these,

⇒ x + 4 = 26 + 4 = 30

68. The average weight of some people in the bus is 62.5 kg. A person weighted 68 kg de – boarded at the next stop and 2 persons whose average weight is 54.5 kg boarded the bus. If the average weight got decreased by 0.5 kg, then find the number of people in the bus previously.

बस में कुछ व्यक्तियों का औसत वजन 62.5 किग्रा है। अगले स्टॉप पर 68 किग्रा वजन का एक व्यक्ति उतर जाता है और 54.5 किग्रा वजन के दो व्यक्ति बस में चढ़ते हैं। यदि औसत वजन में 0.5 किग्रा की कमी आती है, तो आरंभ में बस में लोगो की संख्या ज्ञात कीजिए

Option “C” is correct.

Suppose total people are ‘x’

∴ Total weight = 62.5x

A person weighted 68 kg de – boarded in next stop and 2 persons whose average weight is 54.5 kg boarded the bus;

∴ Total weight = 62.5x – 68 + 54.5 × 2 = (62.5x + 41)

Since total number of people is now increased by 1;

∴ (62.5x + 41) / (x + 1) = 62

⇒ 62.5x + 41 = 62x + 62

⇒ x = 42

∴ Number of people in the bus = 42

69. The average of first 11 multiples of a number is 150. Find the number.

एक संख्या के पहले 11 गुणजों का औसत 150 है। संख्या ज्ञात कीजिए।

Option “A” is correct.

Let the number be ‘x’

Sum of first 11 multiples of x = x(1 + 2 + 3 + … + 11) = x × (11 × 12)/2 = 66x

⇒ Average of first 11 multiples of x = 66x/11 = 6x

But, Average = 150

⇒ 6x = 150

⇒ x = 150/6 = 25

70. A student calculated the average of 10 three digit number, but by mistake he reversed the digit of a number and thus his average increased by 29.7. Find the difference between the unit digit and hundredth digit of a number.

एक छात्र तीन अंकों वाली 10 संख्याओं का औसत ज्ञात करता है, लेकिन गलती से वह एक संख्या को उल्टा कर देता है और इसलिए उसके औसत में 29.7 की वृद्धि हो जाती है। संख्या के इकाई अंक और सैकड़ा अंक के बीच अंतर ज्ञात कीजिये।

Option “C” is correct.

Given:

While calculationg, the student reversed a digit of the number.

Concept used:

A number xyz = 100x + 10y + z

Calculations:

Let the number be xyz,

⇒ xyz = 100x + 10y + z         —-(1)

When the number is reversed, the number becomes zyx,

⇒ zyx = 100z + 10y + x         —-(2)

Subtraction equation 1 from equation 2,

⇒ 99z – 99x,

Since the average is of 10 numbers,

⇒ Total increase in number = 10 × 29.7 = 297

⇒ 99z – 99x = 297

⇒ z – x = 3

∴ Difference in unit and hundredth digit = z – x = 3

 

IMPORTANT POINT
xyz is a number like 123 which is read as one hundred and twenty three

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