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Finding Missing Number (Average)

Introduction

In many average problems, some numbers are given and the average is known. Your task is to find the missing number that keeps the average correct.

This pattern is important because it teaches how to reverse the average formula.

Pattern: Finding Missing Number (Average)

Pattern

The key idea is: Total = Average × Number of terms.

If one number is missing, use:
Missing number = Total - Sum of given numbers

Step-by-Step Example

Question

The average of 5 numbers is 18. Four of them are 12, 15, 20, and 25. Find the missing number.

Options:

  • A) 15
  • B) 16
  • C) 18
  • D) 20

Solution

  1. Step 1: Calculate the total required sum

    Total of all 5 numbers = 18 × 5 = 90.

  2. Step 2: Add the known numbers

    Sum of given 4 numbers = 12 + 15 + 20 + 25 = 72.

  3. Step 3: Subtract to find the missing value

    Missing number = 90 - 72 = 18.

  4. Final Answer:

    18 → Option C

  5. Quick Check:

    (12 + 15 + 20 + 25 + 18) ÷ 5 = 90 ÷ 5 = 18 ✅

Quick Variations

1. Missing one value when average and total count are given.

2. Finding multiple missing values if one more condition is provided.

3. Common in marks, ages, and scorecard questions.

Trick to Always Use

  • Total = Average × Number of terms
  • Subtract the sum of known values

Summary

Summary

  • Always compute the total using the formula Total = Average × Count.
  • Subtract the sum of all given values to find the missing number.
  • Use a quick verification by recalculating the average.
  • This technique is useful in marks, runs, ages, and data table problems.

Example to remember:
If average = 18 for 5 terms → total = 90; if known sum = 72 → missing number = 18.

Practice

(1/5)
1. The average of 4 numbers is 10. Three of them are 8, 12, and 9. Find the missing number.
easy
A. 11
B. 12
C. 13
D. 10

Solution

  1. Step 1: Calculate the total required sum

    Total of 4 numbers = 10 × 4 = 40.

  2. Step 2: Add the known numbers

    Sum of given numbers = 8 + 12 + 9 = 29.

  3. Step 3: Subtract to find the missing value

    Missing number = 40 - 29 = 11.

  4. Final Answer:

    11 → Option A

  5. Quick Check:

    (8 + 12 + 9 + 11) ÷ 4 = 40 ÷ 4 = 10 ✅
Hint: Total = Average × Count, then subtract known sum.
Common Mistakes: Dividing sum of given numbers directly by total count instead of subtracting.
2. The average of 5 numbers is 20. Four numbers are 18, 22, 25, and 15. Find the missing number.
easy
A. 20
B. 18
C. 22
D. 25

Solution

  1. Step 1: Determine the total sum required

    Total of 5 numbers = 20 × 5 = 100.

  2. Step 2: Add the known four numbers

    Sum of given numbers = 18 + 22 + 25 + 15 = 80.

  3. Step 3: Subtract to get the missing number

    Missing number = 100 - 80 = 20.

  4. Final Answer:

    20 → Option A

  5. Quick Check:

    (18 + 22 + 25 + 15 + 20) ÷ 5 = 100 ÷ 5 = 20 ✅
Hint: Subtract known numbers’ sum from the total to get the missing value.
Common Mistakes: Forgetting to multiply average by total terms first.
3. The average of 6 numbers is 14. If five of them are 12, 15, 18, 10, and 16, find the missing number.
easy
A. 12
B. 13
C. 14
D. 15

Solution

  1. Step 1: Compute the required total

    Total of 6 numbers = 14 × 6 = 84.

  2. Step 2: Sum the known five numbers

    12 + 15 + 18 + 10 + 16 = 71.

  3. Step 3: Subtract to find the missing number

    Missing number = 84 - 71 = 13.

  4. Final Answer:

    13 → Option B

  5. Quick Check:

    (12 + 15 + 18 + 10 + 16 + 13) ÷ 6 = 84 ÷ 6 = 14 ✅
Hint: Always verify total - given sum matches an option.
Common Mistakes: Mis-summing the given numbers.
4. The average of 8 numbers is 25. If 7 numbers are 20 each, find the missing number.
medium
A. 60
B. 50
C. 30
D. 40

Solution

  1. Step 1: Calculate the total sum needed

    Total of 8 numbers = 25 × 8 = 200.

  2. Step 2: Compute the sum of the repeated values

    Sum of 7 numbers = 7 × 20 = 140.

  3. Step 3: Subtract to find the missing value

    Missing number = 200 - 140 = 60.

  4. Final Answer:

    60 → Option A

  5. Quick Check:

    (140 + 60) ÷ 8 = 200 ÷ 8 = 25 ✅
Hint: If several numbers are equal, compute quickly with multiplication.
Common Mistakes: Mistaking average for one of the numbers.
5. The average of 10 numbers is 12. If 9 numbers add up to 100, find the missing number.
medium
A. 22
B. 20
C. 18
D. 15

Solution

  1. Step 1: Find the total of all numbers

    Total of 10 numbers = 12 × 10 = 120.

  2. Step 2: Identify the known sum

    Sum of 9 numbers = 100.

  3. Step 3: Subtract to find the missing number

    Missing number = 120 - 100 = 20.

  4. Final Answer:

    20 → Option B

  5. Quick Check:

    (100 + 20) ÷ 10 = 120 ÷ 10 = 12 ✅
Hint: Compute total first and subtract the known sum.
Common Mistakes: Multiplying given sum by average instead of subtracting.

Mock Test

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