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Tricky Shifts in Average

Introduction

Aptitude exams में कई बार ऐसे questions आते हैं जहाँ किसी group का average बदल जाता है जब एक नया value जोड़ा जाता है या कोई value हटाई जाती है। इसे shift in average कहते हैं।

देखने में tricky लगता है, लेकिन pattern समझ में आ जाए तो calculation बहुत आसान हो जाती है।

Pattern: Tricky Shifts in Average

Pattern

सबसे ज़रूरी idea: Average × Number of Terms = Total Sum

जब कोई value add/remove होती है, तो नए average में जो परिवर्तन होता है, वह सीधे उस value और पुराने average के अंतर को दर्शाता है।

Average बढ़े → न्यू value old average से बड़ी है।
Average घटे → न्यू value old average से छोटी है।

Step-by-Step Example

Question

10 students की average age 20 years है। एक नया student join होता है और average 2 years बढ़ जाती है। नए student की age निकालें।

Options:

  • A) 40
  • B) 38
  • C) 42
  • D) 44

Solution

  1. Step 1: पुराना total निकालें

    Old total = 10 × 20 = 200.

  2. Step 2: नया average निकालें

    New average = 20 + 2 = 22.

  3. Step 3: नया total निकालें

    New total = 11 × 22 = 242.

  4. Step 4: नए student का योगदान (value) निकालें

    Age = New total - Old total = 242 - 200 = 42.

  5. Final Answer:

    42 years → Option C

  6. Quick Check:

    (200 + 42) ÷ 11 = 22 → बिल्कुल सही average आया ✅

Quick Variations

Case 1: One number added → देखें नया number पुरानी average से बड़ा/छोटा है।

Case 2: One number removed → Average ऊपर-नीचे इस पर निर्भर करता है कि हटाया गया number average से कैसा था।

Case 3: Average किसी fixed value से बढ़ जाए/घट जाए → total difference method उपयोग करें।

Trick to Always Use

  • Step 1: Total = Average × Number of terms
  • Step 2: Replacement/addition/removal के बाद नया total निकालें
  • Step 3: कुल अंतर = जोड़ा/हटाया गया item
  • Step 4: Average का नया मान verify करने के लिए जल्दी से divide करें

Summary

Summary

  • पहले और बाद के totals हमेशा Average × Count से निकालें।
  • Totals का अंतर ही added/removed item का value देता है।
  • Average बढ़े → नया item बड़ा था।
  • Average घटे → नया item छोटा था।
  • Quick check: (Old total ± item) ÷ New count से average verify करें।

Remember Example:
Old total = 200 (10 × 20)
New total = 242 (11 × 22)
Difference = 42 → New student’s age = 42

Practice

(1/5)
1. The average age of 6 students is 12 years. A new student joins, and the average increases by 1 year. Find the age of the new student.
easy
A. 18
B. 19
C. 20
D. 21

Solution

  1. Step 1: Compute the old total age

    Total age of 6 students = 6 × 12 = 72.

  2. Step 2: Compute the new total with increased average

    New average = 13 → New total = 7 × 13 = 91.

  3. Step 3: Find the age of the new student

    Age of new student = 91 - 72 = 19.

  4. Final Answer:

    19 years → Option B

  5. Quick Check:

    (72 + 19)/7 = 91/7 = 13 ✅
Hint: Multiply average by number of students to get total sum quickly.
Common Mistakes: Using difference of averages directly without adjusting totals.
2. The average of 10 numbers is 25. If one number is removed, the average becomes 24. Find the removed number.
easy
A. 34
B. 35
C. 36
D. 37

Solution

  1. Step 1: Compute the original total

    Total of 10 numbers = 10 × 25 = 250.

  2. Step 2: Compute the new total after removal

    Total of remaining 9 numbers = 9 × 24 = 216.

  3. Step 3: Subtract to find the removed number

    Removed number = 250 - 216 = 34.

  4. Final Answer:

    34 → Option A

  5. Quick Check:

    (250 - 34)/9 = 216/9 = 24 ✅
Hint: Subtract new total from original total to find the removed number.
Common Mistakes: Dividing by wrong count (forgetting one number removed).
3. The average marks of 5 students is 60. A new student joins and the average increases to 62. Find the marks of the new student.
easy
A. 70
B. 72
C. 74
D. 76

Solution

  1. Step 1: Compute original total marks

    Total of 5 students = 5 × 60 = 300.

  2. Step 2: Compute new total marks with increased average

    Total of 6 students = 6 × 62 = 372.

  3. Step 3: Subtract to find new student's marks

    Marks of new student = 372 - 300 = 72.

  4. Final Answer:

    72 → Option B

  5. Quick Check:

    (300 + 72)/6 = 372/6 = 62 ✅
Hint: Difference in average × new total students = extra marks added.
Common Mistakes: Multiplying difference by old count instead of new count.
4. The average of 8 students is 15. One student leaves, and the average rises to 16. Find the age of the student who left.
medium
A. 6
B. 7
C. 8
D. 9

Solution

  1. Step 1: Compute original total age

    Total of 8 students = 8 × 15 = 120.

  2. Step 2: Compute new total age

    Total of remaining 7 students = 7 × 16 = 112.

  3. Step 3: Find age of student who left

    Age of student who left = 120 - 112 = 8.

  4. Final Answer:

    8 years → Option C

  5. Quick Check:

    (120 - 8)/7 = 112/7 = 16 ✅
Hint: Leaving a low value increases the average.
Common Mistakes: Using 8 instead of 7 when recalculating new total.
5. The average of 20 students is 25. If one new student joins, the average decreases by 1. Find the marks of the new student.
medium
A. 2
B. 4
C. 6
D. 8

Solution

  1. Step 1: Compute the original total marks

    Total marks of 20 students = 20 × 25 = 500.

  2. Step 2: Compute new total marks with decreased average

    New average = 24 → Total for 21 students = 21 × 24 = 504.

  3. Step 3: Find marks of new student

    Marks of new student = 504 - 500 = 4.

  4. Final Answer:

    4 marks → Option B

  5. Quick Check:

    (500 + 4)/21 = 504/21 = 24 ✅
Hint: When average decreases, new member is below the old average.
Common Mistakes: Using wrong total count (20 instead of 21).

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